Paulo Lima-Filho and Jay R. Walton 2004Topics in Mathematical Ecology1 Paulo Lima-Filho and Jay R. Walton 2004 1Copyrigh tc 2004 by P. Lima-Filho and J. R. Walton.All righ reserved.

Topics in Mathematical Ecology1

Paulo Lima-Filho and Jay R. Walton

2004

1Copyright c© 2004 by P. Lima-Filho and J. R. Walton. All right reserved.

Contents

1 Background and History 1

2 Modelling in Ecology 3

3 Initial approach 4

4 Linear Equations: The Harmonic Oscillator 9

5 Non-Linear Equations 145.1 Important Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 The Phase Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.2.1 Nodal Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.2.2 Saddle Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.2.3 Center Points or Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2.4 Spiral Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6 Lotka–Volterra Models 266.1 Predator-Prey Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.2 Competition Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7 The Diffusion Model 377.1 Basic Model for a Single Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377.2 Tracking Number and Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397.3 Multiple Species Diffusion Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

8 Interacting Populations on Non-Flat Landscapes 448.1 Differential Geometry of Curves and Surfaces . . . . . . . . . . . . . . . . . . . . . . 44

8.1.1 Change of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468.1.2 Calculus on Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

8.2 Population Movement on Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

9 The Topology of Landscapes 559.1 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599.4 Parametric Layout of Landscapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649.5 Hastings Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

iii

Chapter 1

Background and History

Ecology is the study of interactions between organisms and their surroundings including otherorganisms. Organisms grow, multiply, occupy different regions, compete with other species, preyupon others, migrate, evolve over a long period and overcome the hostility of nature by adjustingto the surroundings and devising ingenious adaptations Many interesting questions arise. How doesthe growth of one species affect the other? Why are some species common and others rare? Whydo some species become extinct? How do some species self-regulate themselves? How much we canhumans share environments with animals without destroying nature’s balance? Why are equatorialregions home to a greater diversity of plant and animal life compared to temperate areas? Why doanimals behave the way they do?

Nature appears to balance itself. The principle of natural selection is a prominent force gov-erning evolution. The human factor has emerged as a force dramatically accelerating the pace ofevolution and natural selection. The most important single human factor affecting all animal lifehas been the expansion of “civilization” driven by the accelerating pace of technological develop-ment. The destruction of forests and the irresponsible misuse of natural resources to meet theneeds of growing and spreading human populations have disturbed natural balances over much ofthe earth. This has prompted some researchers to find ways to recover the natural balance whilekeeping human interests as central. Researchers began gradually to quantify forces that drive natu-ral selection, such as resource availability, population pressure etc. to achieve better understandingof ecosystems. It was not enough just to believe that when the environment is favorable, organismsmultiply and their chances of long term success improve. It has proved necessary to understandmyriad other factors influencing environmental health.

Experimental biology, as a laboratory based science, has yielded important insights on ecologicalquestions. By their very nature, laboratories provide controlled environments for testing hypothesesand isolating variables. However this is not what animals encounter in nature. So, critics ofthese techniques advocate methods based upon field observations. But such observations not onlyneed quantitative studies, they also require qualitative studies. For example, if a population ofmosquitoes in a certain rural area suddenly appears to grow substantially, it might be due todeterioration in health administration or possibly to mosquitoes developing a strong resistance tothe insecticide used, or many other factors. Thus, there can be a bewilderingly large number ofpossible causative factors most of which are uncontrollable. Modeling has emerged as an importantnew tool to be used in coordination with laboratory and field studies. Field studies might be closeto reality, but lab studies are more controllable if artificial and modeling offers generality, flexibilityand controllability.

Thus iterations from field to modelling to laboratory and back has become a productive way to

1

2 CHAPTER 1. BACKGROUND AND HISTORY

coax nature into revealing her secrets. This interactive process has resulted into many rewardingconcepts. For example Darwin expounded the importance of competition in evolution throughfield observations. The Russian biologist Gausse demonstrated competitive exclusion in test tubesby using two species of a single cell organisms called paramoecium. Lotka–Volterra developedmodels to explain coexistence and exclusion mathematically followed by McArthur’s field studieson coexistence of several species of warblers which revealed flaws in Gausse’s theories. EventuallyHutchinson attempted to synthesize all these through his concept of niche all of which has givenus a much better picture of animal behavior.

The beginning of mathematical ecology dates back at least as far as Lotka’s and Volterra’sseminal work on competition and prey-predator interactions. It was believed that self-regulatingforces limiting populations were an integral part of the process of growth. But with the adventof mathematical ecology such ideas got transformed into the concept of dynamic equilibrium ofmulti-species interactions. Although field and laboratory work were the initial modalities in thedevelopment of ecology as a scientific enterprise, modeling and quantitative methods have assumeda prominent role in the ecologists scientific tool box.

Chapter 2

Modelling in Ecology

The purpose of these notes is to give a brief introduction to various models of proven or potential usein mathematical ecology. Some of the models have now become standard tools for the ecologist, butsome of them have only recently been formulated and are yet to be tested against field or laboratorydata. All of the models discussed herein are necessarily overly simplified. Many assumptions aremade which limit a model’s ability to accurately reflect or capture all of the features and subtletiesat work in real habitats. These caveats notwithstanding, such models often do help in furtheringour understanding of a few of the most important and dominant processes influencing the viabilityand evolution of complex ecosystems.

We take as a basic assumption or axiom that all attempts at quantification and modelling in theecological context are provisional and subject to improvement. A pioneer in the field of mathemat-ical ecology, Vito Volterra, wrote, “In order to approach any ecological question mathematically, itis convenient to start with hypotheses, which, although departing from reality, give an approximateimage of it. The representation will be a gross one, at least to begin with, but at the same timeit will be simple. One can deal with it analytically, and verify quantitatively or even just quali-tatively whether the results agree with observation. One can thereby test the correctness of theinitial hypothesis and lay the ground for new results. To facilitate the analysis, it is convenient torepresent the phenomenon schematically, by isolating the factors one wishes to examine, assumingthey act alone, and by neglecting others.”

Once a model is considered at least provisionally satisfactory in its ability to explain real phe-nomena, numerical parameters assume importance. These have to be estimated using suitablestatistical methods. If a model gives a poor fit to observations for any choice of parameter values,one must conclude that the logical framework of the model is inadequate and must be reconsid-ered. However, even if a model gives good fit to observational data, caution must be exercised inextrapolation and interpretation of the results; the good fit might only reflect serendipity in themodel selection or an abundance of parameters. Indeed, there may be an infinite number of com-peting models that do equally well in fitting the same observational data. Thus modeling involvesa dichotomy of approaches. One approach focuses upon simulations in close agreement with obser-vational data. The other focuses upon a qualitative understanding of the fundamental physical orbiological processes behind macroscopic ecological systems with accurate quantitative predictionbeing a secondary goal. Both approaches have there successes and contribute to deepening ourunderstanding of nature.

3

Chapter 3

Initial approach

We shall be modeling populations of one or more species, where by a species we mean a groupof functionally similar organisms. Populations are assemblages of individuals of the same speciesthat may interact. Our interest will be mainly in the sizes of selected species populations and theirfluctuations. Our tilting towards the study of selected populations is guided by the fact that thestudy of individuals is often too chaotic and unpredictable, while the study of entire ecosystemsseems too difficult (and sometimes even unproductive, in the sense that it offers no special advantagein answering specific questions about individual species viability). It should be mentioned thatindividuals based modeling is gaining popularity largely do to enhanced computational capabilities.However, conventional wisdom still asserts that the behavior of individuals seems to be totallyrandom and defy prediction, whereas the behavior of large populations of individuals can often bevery predictable and obey relatively simple mathematical laws.

Populations, whether of mice or men, whether in labs or in zoos or in nature, fluctuate. Thereare births and deaths. There is immigration and migration. In case of microorganisms like bacteriaor small insects like mosquitoes, the population size is measured by their biomass or biodensity (i.e.biomass per unit cubic volume). In larger animals we often go by their numbers, if the populationis small or by their number densities, if the population is large. Other factors may be monitoredsuch as proportion of male-female, or as in the case of animals with lengthy lifespans, proportionsin various age groups. Sometimes we study various stages, for instance in case of insects theproportions of eggs, larvae, pupas and adults.

Variation in the size of populations is frequently a response to the fluctuations in the surroundingenvironment. If some essential resource such as food, nesting sites etc. are on the decline, populationgrowth may be reversed or retarded. Competition may cause a similar effect. Competition may beamong the same species, such as for example, water-hyacinths growing in a lake— when they coverthe entire water’s surface, growth must cease. Examples of competitions between different speciesare weeds and crops competing for growth in the same space, different carnivores competing forsame species of hearbivores and so on.

In the investigations that we intend to pursue our tools might seem primitive and goals modest.However, they will still provide a modeling environment sufficiently rich to yield useful and insightfulpredictions and simulations. Initially, we shall review popular models of interacting populationsin which only aggregate population densities across entire habitats are considered. Such modelsyield systems of ordinary differential equations in which the only independent variable is time.Subsequently, we shall consider models incorporating spatial variability and mobile populationsunder the assumption of flat or planar landscapes. Recognizing that many species one might wishto consider do not necessarily live in planar habitats, we shall then develop models accounting for

4

5

a non-flat topography. That is we shall consider habitats which include hills and valleys. Hence, apopulation’s size will be a function of both, space and time.

Typically, we shall be attempting to model the evolution in space and time of a density describingthe size of the a population. More specifically, denote by the set X a habitat and by

ρ(x, t) the mass (or number) density at location x ∈ X and time t. The concept of mass densityis usually employed when we deal with large populations (insects, grasses, small fish, etc). If weare dealing with species which consist of relatively small numbers of individuals of relatively largesize such as whales, elephants or even sheep, we might prefer the use of number density, by whichwe mean number of animals per unit region (volume or surface area as the case may be). In thefollowing diagram X denotes a habitat imbeded in a Euclidean space of dimension 1, 2 or 3 (ormore generally in a Riemannian manifold of dimension 1, 2, or 3). Let x denote the position vector(relative to some suitably chosen origin) of an aggregation of a species at a location P in the habitat,and let P(x) denote a small open neighborhood of P in X. We call P(x) a part of X. The totalmass (or number) of the species occupying the part is given by

(x)P

X

x

0

P

in P(x)(t) =∫

P

ρ(x, t)dx. Early attempts to model the evolution of human and animal population

sizes considered only aggregate data over entire habitats or geographic areas. The first such modelto be expoited is the so called Malthusian model named after the famous British economist ThomasRobert Malthus. Specifically, considering the average density over the entire habitat

ρ(t) :=1

|X|

∫

X

ρ(x, t)dx,

(where |X| denotes the volume (area) of the entire habitat X) Malthus conjectured that the humanpopulation of the earth would evolve according to

dρ(t)

dt= aρ(t).

Solving this equation gives

ρ(t) = ρ0eat, where ρ0 is the

6 CHAPTER 3. INITIAL APPROACH

initial density where the evolutionary clock has been artitrarily set to zero at some convenientstarting time.

If a > 0, the population grows at the exponential rate. Malthus used this model to argue thatthe human population would quickly outgrow the earth’s capacity to sustain it. If a < 0, theopposite conclusion holds, i.e. the species decays with time and ultimately becomes extinct.

Similar considerations applied to number density leads to the differential equation

dN(t)

dt= aN(t)

The constant ‘a’ is called the intrinsic growth rate for the population. Inspired by such consid-erations Charles Darwin calculated that a single pair of elephants would have at least 15 milliondescendants after five centuries. That this does not happen clearly points at the inevitability ofcompetition among individuals of the same species for the limited resources that the earth has tooffer. In fact this competition is the key to Darwin’s theory of evolution through natural selection.Exponential growths have been observed on many occasions for limite periods of time. For ex-ample, agricultural insect pests often increase at incredible rates and are known to cause famines.Similarly, when New Zealand was colonized, rabbits were introduced and their colonies grew at anexponential rate wrecking havoc on the environment.

Exponential growth however, is not a common feature over extended period of time in mosthabitats. Sooner or later this type of growth is mitigated by shortages of resources. There are factorsother than limited resources that show up occasionally such as sudden change in temperatures,floods or volcanic eruptions etc. A natural modification to the exponential growth model is tointroduce a non-constant growth factor. If such variability is due simply to seasonal influences, onemight want to assume a = a(t). The resulting equation

dρ(t)

dt= a(t)ρ(t)

is readily integrated to produce

ρ(t) = ρ0eA(t)

where A(t) :=∫ t0 a(r) dr.

However, a population’s growth rate can vary due to factors intrinsic to species interactions evenwithout seasonal influences. Thus, for example, one might postulate a growth factor of the forma(t) = f(ρ(t)) which reflects the belief that growth is regulated by population size. The simplestform for f(ρ) is linear which, since the growth rate should decrease as the population increases,leads to the famous logistic differential equation

1

ρ

dρ

dt= r − cρ

or

1

N

dN

dt= r − cN

whose solutions are readily shown to be

ρ

r − cρ=

ρ0

r − cρ0ert and

N

r − cN=

N0

r − cN0ert.

7

Solving the first of these for ρ, we get

ρ =rρ0e

rt

r − cρ0(1 − ert)

=r/c

rcρ0

e−rt − e−rt + 1

=K

1 + e−rt(

Kρ0

− 1) where K = r/c.

Thus ρ(t) = K1+qe−rt where q = K

ρ0− 1. As t → ∞, ρ(t) → K, which is called the carrying capacity

of the species. Thus, the population does not increase indefinitely but is limited to a maximumdensity equal to K. The graph of the above function resembles S and is, hence, called the sigmoidalgraph

densityρ

time t

K = carrying capacity

The precise evolution of the logistic growth law depends upon the initial condition. Specifically,there are three distinct cases to consider.Case 1): Suppose initially ρ(0) is at the carrying capacity, i.e. ρ0 = K. Then q = 0 and ρ(t) = K∀t.

Thus the population (or number) density stays constant and there is neither growth nor decayfor the species.

Case 2): If ρ(0) < K =, then Kρ0

> 1 and q > 0. Thus ρ(t) > 0, and since

ρ′(t) = ρ(r − cρ)

= cρ(K − ρ) > 0,

it follows that ρ increases till it reaches its maximum = K.

Case 3): If ρ(0) > K, then ρ′(t) < 0 and hence the population declines until its density reachesthe carrying capacity K.

This discussion leads to the concept of equilibrium. More specifically, for the logistic model ofρ′(t) = ρ(t)(r− cρ), the population ceases to grow and becomes stationary when ρ ′(t) = 0 i.e. when

8 CHAPTER 3. INITIAL APPROACH

ρ(t) = 0 or ρ(t) = r/c = K. We saw above that initially if ρ0 > K or ρ0 < K, ρ(t) → K as t → ∞.Thus the stationary value ρ(t) = K gives a promise of stability. We call this equilibrium, a stableequilibrium. By equilibrium here we mean the invariability of mass density, i.e. a zero growth rate.Thus a small fluctuation around K ultimately returns to the equilibrium value K.

On the other hand, if ρ(t) = 0 at some time, say t0, then the equilibrium is unstable. Indeed,a slight perturbation from the equilibrium point ρ(t) = 0, causes a large and permanent deflection.Specifically, if for some t0 > 0 it ever happens that 0 < ρ(t0) < K, then ρ′(t) > 0 and the populationincreases for all t > t0 getting farther and farther away from the zero equilbrium point.

Suppose ρ(t) > 0 before reaching equilibrium (which is generally the case, since ρ(t) ≥ 0 ∀t),i.e. at some time t1 < t0, suppose ρ(t1) > 0. Then ρ′(r1) = cρ(t1)(K − ρ(t1)) > 0.

Hence ρ increases with time and since it stays positive ρ′(t) > 0 ∀t and equilibrium cannot bereached. Thus at t0 we have unstable equilibrium.

The above discussion of equilibrium for the logistic model motivates the more general discussionof equilibria for dynamical systems given below. We shall start with one dimensional equations ofthe form

u′(t) = u(t)F (u(t))

and look for equilibria, i.e. constant solutions u′(t) = 0, which clearly must be such that eitheru(t) = 0 or F (u(t)) = 0. In each case, we investigate what it means for an equilibrium state to beeither stable or unstable.

Chapter 4

Linear Equations: The Harmonic

Oscillator

A convenient starting point for describing the theory of equilibria for dynamical systems is theclassical harmonic oscillator equation. Consider the 1-dimensional motion of a particle of mass m.Then by Newton’s second law,

mu′′(t) = F (t) (mass × acceleration=Force)

where u(t) denotes the deflection of the particle from some reference position and F denotes theresultant force acting on the particle. (Here were are implicitly assuming that the mass is constantin time.) For example, if the mass is acted upon only by a (linear) spring attached to a fixed, rigidsupport, then F (t) = −Ku(t) and the equation of motion for the particle becomes

u′′(t) = −k2u(t) (1)

in which k2 := K/m. This equation has the great virtue of readily admitting a closed form generalsolution. In particular, its general solution is

u(t) = a cos kt + b sin kt. (2)

Before investigating the notion of equilibrium for this equation, it proves useful to make somegeneral observations based upon energy considerations.

To that end, we multiply (1) by u′(t) and integrate over the time interval [0, t] obtaining

1

2u′(t)2 +

1

2k2u(t)2 =

1

2u′(0)2 +

1

2k2u(0)2. (3)

It follows that the expression

1

2u′(t)2 +

k2

2u(t)2

is a constant (first integral) of the system.

9

10 CHAPTER 4. LINEAR EQUATIONS: THE HARMONIC OSCILLATOR

m=1

m=1x

T = k x2

The various terms in (3) have interesting physical interpretations. The amount of work done instretching the spring through a distance u(t) − u(0) is

W (t) − W (0) :=

∫ t

0Ku(r)u′(r)dr =

1

2Ku(t)2 − 1

2Ku(0)2.

This is stored in the spring-mass system as potential energy. The instantaneous kinetic energy ofthe mass is defined as

K(t) :=1

2mv2 =

1

2mu′(t)2.

Thus, the constancy of 12u′(t)2 + 1

2K2u(t)2 implies that the total energy of the system is conserved.That is , the total energy of the system at any time equals that present initially; the energy of thesystem merely switches back and forth between its potential and kinetic forms.

Equilibrium states of the harmonic oscillator equation (1) must be constant in time. From (2)one sees that

u′(t) = k(b cos(t) − a sin(t))

which vanishes for all time if and only if a = b = 0. Thus, the only equilibrium state is that forwhich the spring is perpetually at its equilibrium length. Is this a stable equilibrium? From ourenergy considerations, we note that any departure from equilibrium, i.e. any initial conditions forwhich |u(0)| + |u′(0)| > 0, leads to a solution with W (t) + K(t) = W (0) + K(0) > 0 for all t > 0.Thus, the equilibrium state is never achieved. However, the solution stays close to the equilibriumstate for all time if it starts close to it, since the total energy of the system is a measure of departurefrom the zero equilibrium state and it is a constant of the system.

The stability of equilibrium states of a second order differential equation such as (1) can beinvestigated directly from the equation without needing to find its general solution, which is usuallyimpossible to find in closed form. To illustrate, notice that (1) can be re-cast as a 2 × 2 first ordersystem of ODEs. To that end, one defines

v(t) = u′(t).

Then

v′(t) = u′′(t) = −k2u(t).

11

Thus we arrive at a system of two equations:

du

dt= v(t)

dv

dt= −k2u(t)

.

Let u(t) =(u(t)v(t)

)

. Then the system can be rewritten as

u′(t) =

(

u(t)v(t)

)′=

(

u′(t)v′(t)

)

=

(

v(t)−k2u(t)

)

= F (u, v)

where F is a vector valued function (F : R2 → R2). Note that( v(t)−k2u(t)

)

can be written as

(

0 1−k2 0

)(

u(t)v(t)

)

= A

(

u(t)v(t)

)

.

Thus the equation finally takes up the form

u′(t) = Au(t).

Comparing this equation withu′(t) = au(t)

whose solution is u(t) = u0(t)eat motivates us to seek a solution to the vector equation

u′(t) = Au(t)

in the form

u(t) = eAtu0.

where the matrix valued exponential function eAt is defined through

eAt := I + At +A2t2

2!+

A3t3

3!. . .

with I denoting the identity matrix. This power-series representation for eAt is readily shown toconverge for all matrices A and for all t.

The eigenvalues of A are the roots of det[A − λI] = 0 or

det

[

0 − λ 1−k2 0 − λ

]

= 0.

It follows immediately that λ = ±ki. Since the eigenvalues of A are distinct, A is diagonalizable

and ∃ an invertible matrix P s.t. A = P(

ki 00 −ki

)

P−1 = PΛP−1 where Λ =(

λ1 00 λ2

)

, λ1 = ki,

λ2 = −ki.

∴ Λn =

(

λn1 00 λn

2

)

=⇒ eΛ =

∞∑

0

(

λn1

n! 0

0λn2

n!

)

=

(

∑ λn1

n! 0

0∑ λn

2

n!

)

=

(

eλ1 00 eλ2

)

.

12 CHAPTER 4. LINEAR EQUATIONS: THE HARMONIC OSCILLATOR

Since A = PΛP−1 we have

An = PΛnP−1

∴

∑ An

n!= P

(

∑ Λn

n!

)

P−1

=⇒ eA = PeΛP−1

or eAt = PeΛtP−1.

Thus the differential equation

u′(t) = Au(t) has solution

u(t) = PeΛtP−1u0.

For equilibrium, we need u′(t) = 0. This is possible only if either u(t) =(00

)

∀t or A = 0. Since

A =(

0 1−k2 0

)

, A is never zero, we conclude that u(t) =(

00

)

∀t i.e. u(t) = 0 and u′(t) = 0 ∀t. Thusthe only equilibrium state for

u′′(t) = −k2u(t)

is u(t) = 0 ∀t, in agreement with our previous analysis from the closed form solution (2).Observe that since

u′(t)2 + k2u(t)2 = constant, we

can write v2 + k2u2 = constant ∀t. Hence if we plot u against v, the graph of the above equationwill be an ellipse

u

v

whose dimension will depend upon the initial energy (and hence upon initial conditions).Now consider the damped harmonic oscillator whose equation is

u′′(t) + au′(t) + k2u(t) = 0.

Its general solution is

u(t) = c1e−a+

√a2−4k2

2t + c2e

„

−a−√

a2−4k2

2

«

t.

Suppose a > 2k(k > 0).Then

√a2 − uk2 is a real number and a2 − uk2 < a2

∴ ±√

a2 − uk2 < a

=⇒ − a ±√

a2 − uk2 < 0

13

Hence, as t → ∞, u(t) → 0 and the solution converges to the unique stable equilibrium stateu(t) ≡ 0. We call such an equilibrium state asymptotically stable.

Now suppose 2k > a > 0. Then u(t) = e−at2

(

c1ei√

4k2−a2

2t + c2e

„

−i√

4k2−a2

2

«

t)

or equivalently

u(t) = e−at2

(

c1 cos(

i√

4k2−a2

2 t)

+ c2 sin((

−i√

4k2−a2

2

)

t))

. It easily follows that u(t) → 0 as t → ∞and again u(t) ≡ 0 is the unique equilibrium state which is also asymptotically stable. On the otherhand, if a < 0 but |a| > 2k, we have a2 − 4k2 < a2 and −a ±

√a2 − 4k2 > 0. Hence u(t) → ∞ and

the unique equilibrium state u(t) ≡ 0 is unstable. Finally a < 0 and |a| < 2k leads to oscillationsof large magnitude and hence instability.

Chapter 5

Non-Linear Equations

5.1 Important Examples

We know that there is a constant struggle for survival among different species of animals livingin the same habitat. One kind of animal survives by consuming the other. The other survives bydeveloping methods of evasion to avoid being eaten and so on.

As a simple example of this universal conflict between the predator and its prey, let us imaginean island inhibited by foxes and rabbits. The foxes eat rabbits, and the rabbits eat clover. Weassume that there is so much clover that the rabbits always have an ample supply of food. Whenthe rabbits are abundant, then the foxes flourish and their population grows. When the foxesbecome too numerous and eat too many rabbits, they enter a period of famine and their populationbegins to decline. As the foxes decrease, the rabbits become relatively safe and their populationstarts to increase again. This triggers a new increase in the fox population, and as time goes on wesee an endlessly repeated cycle of interrelated increases and decreases in the populations of the twospecies. These fluctuations are represented graphically below where we plot the population sizes ofthe two species against time.

foxes

rabbits

t

If x is the number of rabbits at time t, we should have

dx

dt= ax, a > 0

14

5.1. IMPORTANT EXAMPLES 15

as a consequence of the unlimited supply of clover, if the number y of foxes is zero. It is onlynatural to assume that the number of encounters per unit time between the two species is jointlyproportional to x and y. If we further assume that a certain proportion of these encounters resultin rabbits being eaten, then we have

dx

dt= ax − bxy; a, b > 0.

Similarly,dy

dt= −cy + dx y; c, d > 0;

for in the absence of rabbits (x = 0) the foxes will die and their increase depends on running intorabbits. Thus, we obtain a nonlinear system of equations describing the interaction of these twospecies. The equations

dx

dt= x(a − by)

dy

dt= −y(c − dx)

are known as Volterra’s prey-predator equation. The Italian mathematician Vito Volterra developedand studied them.

Unfortunately these nonlinear equations can not be solved in closed form but must be solvednumerically. However, much valuable qualitative information can be obtained about equilibria andtheir stability. Prior to presenting that analysis, it is instructive to introduce the relevant ideasthrough another simple, classical problem from elementary mechanics, the swinging pendulum.

Let x denote the angle of deviation of an undamped pendulum of length ` whose bob hasmass m. Then the equation of its motion (derived using Newton’s second law as was done for theharmonic oscillator) is given by

d2x

dt2+

g

`sinx = 0.

If a damping force proportional to the velocity of the bob is present, then its motion is given by

d2x

dt2+

c

m

dx

dt+

g

`sinx = 0.

m

lx

Another second order nonlinear equation with important application is found in the study ofvacuum tubes—the van der Pol equation

d2x

dt2+ µ(x2 − 1)

dx

dt+ x = 0.

16 CHAPTER 5. NON-LINEAR EQUATIONS

We shall discuss second order nonlinear equations and some very interesting properties of theirsolutions briefly here. The above equations can be written in the general form

d2x

dt2= f

(

x,dx

dt

)

. (1)

5.2 The Phase Plane

The values of x (the position of the particle) and dxdt (the velocity of the particle) constitute the

state of the dynamical system (1), called the phase states or phases and the plane of the variablesx and dx

dt is called the phase plane. If we introduce the variable y = dxdt , then (1) can be replaced

by the equivalent system

dx

dt= y

dy

dt= f(x, y).

(2)

In general, a solution of (2) is a pair of functions x(t) and y(t) defining a curve in the xy-plane (thephase plane).

More generally we study dynamical systems of the form

dx

dt= F (x, y)

dy

dt= G(x, y)

(3)

where F and G are continuous (with continuous first order partial derivatives throughout the plane).The independent variable t does not appear in the functions F and G explicitly. Such a systemis known as an autonomous system. A theorem from the theory of system of ordinary differentialequations guarantees that if t0 is any number and (x0, y0) is any point in the phase plane, thenthere is a unique solution

{

x = x(t)y = y(t)

(4)

of (3) such that x(t0) = x0 and y(t0) = y0. The functions (4) defines a curve in the phase planecalled a path (or trajectory) of the system.

An important property of autonomous dynamical systems such as (3) is that their solutionpaths exhibit translation invariance, that is if (4) solves (3), the so does

{

x = x(t + c)y = y(t + c)

for every choice of the parameter c. It should also be noted that at most one path passes througheach point of the phase plane (Why?) and the direction of increasing t along a given path is thesame for all solutions representing the path. A path is therefore a directed curve and we use arrowsto indicate the direction in which the path is traced out as t increases. In general paths of (3) coverthe entire plane and do not intersect one another. The only exceptions to this statement occur atpoints (x0, y0) where both F and G vanish. These points are called the critical points, and at sucha point the unique solution must be a constant solution x = x0 an y = y0. A constant solutioncorresponds to a degenerate path. These critical points constitute the equilibria of the dynamicalsystem, i.e. constant states satisfying the dynamical system. Thus, for the dynamical system (2),

5.2. THE PHASE PLANE 17

the equilibria are points (x0, 0) at which y = 0 and f(x0, 0) = 0, that is states for which both thevelocity and acceleration are identically zero.

Useful insight can be gained by giving the abstract dynamical system (3) the following inter-pretation from elementary mechanics. To that end, consider the of a vector field defined by

V (x, y) = F (x, y)ı + G(x, y),

which at a point (x, y) has horizontal component F (x, y) and vertical component G(x, y). Since

F (x, y) =dx

dtand G(x, y) =

dy

dt,

this vector is tangent to the path at P and points in the direction of increasing t. If we consider tas time, then V can be viewed as the velocity vector of a particle moving along the path(x(t), y(t))with prescribed velocity (F (x, y), G(x, y)).

S

C Q

PF

G

V

R

By way of analogy imagine that the entire phase plane is filled with particles and that each pathis the trail of a moving particle preceded and followed by many others on the same path andsurrounded by similar particles moving on nearby paths. This situation is reminiscent of twodimensional fluid motion. Since the system (3) is autonomous, V (x, y) does not change and thefluid motion is called steady. The paths are the trajectories of the moving particles and criticalpoints (equilibria) P,Q,R are stagnation points of the fluid motion.

The figure offers some typical features occuring in phase diagrams or portraits:

(a) the critical points,

(b) the arrangements of paths near critical points,

(c) the stability and instability of critical points, that is, whether a particle near such a pointremains near or wanders off to another part of the plane,

(d) closed paths corresponding to periodic motion (i.e. periodic solutions).

Since in generalnonlinear equations and systems cannot be solved explicitly, these phase portraits are instructive

tools in developing a qualitative theory of two dimentional autonomous dynamical systems.


5.2.1 Nodal Points

Let (x0, y0) be a critical (equilibrium) point for the dynamical system (3). A solution path(x(t), y(t)) is said to enter the critical point (x0, y0) as t → ±∞ provided

limt→±∞y(t) − y0

x(t) − x0exits or equals ±∞.

B

0

C

A

D

An isolated critical point (x0, y0) is called a nodal point provided it has a neighborhood suchthat every path in this neighborhood approaches and enters the critical point as t → ∞ (or → −∞).For example, the system

dx

dt= x;

dy

dt= −x + dy,

has (0, 0) as a nodal point. To see this, consider the general solution

{

x = c1et

y = c1et + c2e

2t.

Clearly, every non-trivial solution enters the critical point (0, 0) as t → −∞.

5.2.2 Saddle Points

A saddle point is an isolated critical point for which

(a) there are two half line paths AO and BO which approach and enter the critical point ast → ∞ and these two paths lie on a line AB,

5.2. THE PHASE PLANE 19

(b) there are also other two half line paths CO and DO which approach and enter the criticalpoint as t → −∞ and these two paths lie on a straight line CD; and

(c) between the four half line paths there are four regions, and each contains a family of pathswhich do not approach O as t → ∞ or −∞, but are asymptotic to one or another of the halfline paths as t → ∞ or −∞.

B

D

A

C

A saddle point

5.2.3 Center Points or Vortices

A critical point which is surround by a family of closed paths is called a center. It is notapproached by any path as t → ∞ or −∞.

For example, the system

dx

dt= −y,

dy

dt= x

has (0,0) as a critical point and has

x = −c1 sin t + c2 cos t

y = c1 cos t + c2 sin t

for the general solution. The solutions are periodic and correspond to paths lying along circles ofthe form x2 + y2 = r2 none of which approach the critical point.


Center (vortex)

5.2.4 Spiral Points

A critical point which is approached in a spiral-like manner by a family of paths that wind aroundit an infinite number of times as t → ∞ or → −∞ is called a spiral point. Paths approach thecritical point but they do not enter it. In other words, as t → ∞ or → −∞, a point P movingalong such a path approaches O but the line OP does not approach a definite direction.

Spiral point

5.3. STABILITY 21

For example, consider the linear system dxdt = ax − y; dy

dt = x + ay, which in matrix form is

z′(t) = Az(t)

where z(t) := (x(t), y(t))T and the coefficient matrix is

A =

[

a −11 a

]

.

The eigenvalues of A are {a ± i} which leads to the general solution

[

x(t)y(t)

]

= eAtz0 = eat

[

c1 cos(t) − c2 sin(t)c2 cos(t) + c1 sin(t)

]

.

The origin is obviously a spiral point with paths spiralling away from the origin when a < 0 andtowards the origin when a < 0.

5.3 Stability

a > 0 a < 0

One of the most important questions in the study of a dynamical system is that of stability ofits steady-states. For example in case of a pendulum, intuition suggests that there should be twosteady-states, one when the bob is at rest at highest point and the other when it is at its lowestpoint. The first one is clearly unstable, since a slight disturbance leads to are large deviation awayfrom the steady-state, where as the latter seems stable in that small disturbances lead to smalldeparture from the steady-state. We now formulate these intuitive ideas in a more precise way.Consider an isolated critical point of system (3), and assume that it is located at O = (0, 0) in thephase plane. This critical point is called stable if for each positive number ε, there is a positivenumber δ ≤ ε such that every path which is inside of the circle x2 +y2 = δ2 for some t = t0 remainsinside the circle x2 + y2 = ε2 for all t > t0. Loosely speaking, a critical point is stable if all pathsthat get sufficiently close to the point stay close to the point. The point is called asymptotically

stable if it is stable and there is a circle x2 + y2 = ε20 such that every path which is inside this circle

for some t = t0, approaches the critical point as t → ∞. If the critical point is not stable, it iscalled unstable.


δ

ε

C

t = t 0

An asymptotically stable critical point

As stated previously, one of our goals is to learn as much as we can about second order nonlineardifferential equations by studying their phase portraits. One aspect of this is classifying the criticalpoints of systems such as (3) according to their nature and stability. Under suitable conditionsthis problem can be solved for a given nonlinear system by studying a related linear system. Wetherefore first analyze the critical points of linear autonomous systems.

Consider the system

dx

dt= a1x + b1y

dy

dt= a2x + b2y

(5)

If det[

a1 b1a2 b2

]

6= 0, then the system has the origin as its unique critical point. We seek solutions of

the form x = Aeλt, y = Beλt which exist if and only if the system, we get

(λ − a)A − bB = 0

−cA + (λ − d)B = 0.

This system has non-trivial solutions for A,B provided

det

[

λ − a −b−c λ − b

]

= λ2 − (a + d)λ + (ac − bd) = 0.

Note that (ad − bc) 6= 0 implies that λ = 0 cannot be a root. Let λ1, λ2 be roots of the equation.The nature of these roots decides the classification of the critical points. We list below (withoutproof) these conditions.

(a) If the roots λ1 and λ2 of the characteristic equation are real, unequal and of the same sign,then the critical point is a node. If λ1, λ2 are both negative, the node is asymptotically stable.If λ1, λ2 are both positive, it is unstable.

(b) If λ1, λ2 are real, unequal and of opposite signs, then the critical point is a saddle point. Itis unstable.

5.3. STABILITY 23

(c) If λ1, λ2 are real and equal, then the critical point is a node. Let λ1 = λ2 = λ. If λ > 0, thenode is unstable. If λ < 0 the node is asymptotically stable.

(d) If λ1, λ2 are conjugate complex numbers with nonzero real part, then the critical point is aspiral point. If the real part is negative, the critical point is asymptotically stable. If it ispositive, the critical point is unstable.

(e) If λ1, λ2 are purely imaginary, then the critical point is a center. It is stable but not asymp-totically stable.

As an example, consider the simple harmonic oscillator equation discussed previously:

{

u′(t) = v(t)v′(t) = −k2u(t).

Its equilibrium point is (0,0) and its characteristic equation is λ2+k2 = 0 with eigenvalues λ = ±ki.The critical point, therefore is a center, which is stable but not asymptotically stable. The dampedharmonic oscillator equation

u + au + k2u = 0

can be written as as equivalent linear system by introducing u = v.

{

u = vv = −av − k2u.

Its characteristic equation is λ2 + aλ + k2 = 0, whose roots are λ = −a±√

a2−4k2

2 . If 0 < a < 2k,then the critical point u = 0 = v is a spiral point that is asymptotically stable. If a > 2k > 0(overdamped oscillator), the two roots are real, unequal and negaive. Hence the critical point is anode which is asymptotically stable.

Let us re-consider the pendulum equation. It proves instructive to keep the following diagramin mind when analyzing the equation and its equilibria.

θ

m

l θP

θ

From the diagram, it follows from Newton’s second law that

m(`θ) = −mg sin θ

or `θ = −g sin θ

∴ θ = −g/` sin θ.


Introducing dθdt = y, one has

∴

d2θ

dt2=

dy

dt= −g/` sin θ.

Thus, the governing second order pendulum equation can be replaced by the equivalent system

dθ

dt= y

dy

dt= −g

`sin θ

At point P the bob is moving at a speed of `θ = `y. Hence its kinetic energy is 12m`2y2. Also the

potential energy stored by the bob in its new position = mg `(1 − cos θ). Hence the total energyof the bob in its position P is 1

2m`2y2 + mg`(1 − cos θ)

=1

2m`2y2 + m`2 g

`(1 − cos θ)

=1

2m`2y2 + m`2k2(1 − cos θ)

= m`2

(

1

2y2 + k2(1 − cos θ)

)

. (*)

As demonstrated previously for the linear harmonic oscillator equation, one can show thatenergy is conserved for the pendulum equation as well. It then follows that phase plane trajectoriesfor the pendulum equation satisfy

dy

dθ=

−k2 sin(θ)

y

or y dy = −k2 sin(θ)dθ.

Integrating, we get 12y2 − k2 cos θ = c1 or

1

2y2 + k2(1 − cos θ) = k2 + c1

∴ m`2

(

1

2y2 + k2(1 − cos θ)

)

= c.

For convenience let us put m = 1, and ` = g (and hence k = 1). The above observationcan be rewritten as 1

2y2 + (1 − cos θ) = constant, where y = dθdt , of course. We shall write:

12y2 + (1 − cos θ) = h, where h is the total energy of the system. Let V (θ) = 1 − cos θ. Then

y = ±√

2(h − V (θ)).

To construct the (θ, y) phase plane diagram we first construct the curve Y = V (θ) and linesY = h for various values of h ≥ 0 in the (θ, Y ) plane (figure 1 below). From this (θ, Y ) plane weread off the values of h − V (θ) and plot the paths in (θ, y) plane directly below (figure 2).

5.3. STABILITY 25

Y

-4 π -3 π -2 π π 02π 3π

h = 1

h = 3

h = 2(θ)ν

π θ

y

h = 3

h = 2

h = 1h = 2

23- -2 - ππππ θ0 3π

From the formula V (θ) = 1 − cos θ, we find V ′(θ) = sin θ, V ′′(θ) = cos θ. Thus V ′(θ) = 0 atθ = nπ, V ′′(nθπ) > 0 if n is even and < 0 if n is even. If follows that the critical points (2nπ, 0) arecenters and hence stable. The critical points ((2n + 1)π, 0) are saddle points and hence unstable.

From the second figure we observe that if the total energy h is less than 2, then the correspondingpaths are closed. Each path surrounds one of the centers (2nπ, 0). Physically all of these stablecenters correspond exactly to one physical state, namely, the pendulum at rest with the bob in itslowest position, the stable equiposition.

Thus, each closed path about a center corresponds to a periodic back and forth oscillatorymotion. If h > 2 the corresponding paths are not closed. Clearly θ → ∞ as t → ∞ (if dθ

dt > 0)and → −∞ otherwise. Thus the motion corresponding to such a path does not define θ as aperiodic function of t. All the unstable centers ((2n + 1)π, 0) correspond to exactly one physicalstate, namely the pendulum at rest with bob in the highest position and each of the unclosed pathscorresponds to a physically periodic round-and-round motion.

Chapter 6

Lotka–Volterra Models

6.1 Predator-Prey Models

In the mid-1920s Umberto D’Ancona, an Italian marine biologist, performed a statistical analysisof the fish that were sold in the markets of Trieste, Fiume and Venice between 1910 and 1923.Fishing was largely suspended in the upper Adriatic during the First World War, from 1914 to1918, and D’Ancona showed that this coincided with increases in the relative abundance of somespecies and decreases in the relative adundance of other predatory species. He attempted to modelthe predator-prey interaction using the following system of ordinary differential equations:

dH

dt= H(r − cP ) = rH − cHP

dP

dt= P (bH − m) = bPH − mP. (1)

The first term on the right side of first equation implies that the prey will grow exponentiallyin the absence of the predator (the prey are limited by predators). The second term describesthe loss of prey due to predation, which is proportional to the number of both the prey and thepredator. The second term is called the mass action term. For the second equation, we observethat consumption of prey leads to production of new predators and that the predator populationdecreases exponentially in the absence of prey. The parameter r is the natural growth rate forthe prey species in the absence of predators and the parameter m is the natural death rate forthe predator species. The parameter c reflects the “harvesting” efficiency of the predator speciesand often measures the rate of biomass harvested from the prey population. Correspondingly, theparameter b reflects the efficiency with which the predator assimilates the biomass harvested fromthe prey population to increase the predator population’s biomass. Thus, usually one takes b < csince the predator species is usually not 100% efficient in utilizing the harvested prey species.

The system (1) can be simplified by introducing the substitution

x =b

mH

y =c

rP

resulting in

dx

dt= rx(1 − y)

dy

dt= my(x − 1). (2)

26

6.1. PREDATOR-PREY MODELS 27

System (2) has the two equilibrium solutions x(t) = 0 = y(t) and x(t) = 1 = y(t).The table below shows the relative abundance of selachians (sharks and shark-like fishes) which

are predators. The abundance of these predators increased during the war and decreased with anincrease in fishing. The relative abundance of prey, in turn, followed the opposite pattern.

Year % of selachians

1914 11.91915 21.41916 22.11917 21.21918 36.41919 27.31920 16.01921 15.91922 14.81923 10.7

D’Ancona wanted to understand how intensity of fishing affects the fish population. He reasonedthat the number of predators increased since more prey were available. However, it was not clearwhy in spite of increased number of prey the catch did not show it. What he could not understandwas why a reduced level of fishing benefited predators more then the prey. He approached thefamous mathematician Vito Volterra, the father of his fiance, with the problem. Volterra wrotedown a simple pair of equations (1) to explain this phenomenon.

Lets examine the two critical points (0, 0) and (1, 1) of (2). For the critical point (0, 0), it seemsreasonable to conjecture that in a small neighborhood of (0, 0), the bilinear term xy can be ignoredin determining the nature of the critical point. The corresponding linear system is

dx

dt= rx

dy

dt= −my. (3)

Its auxiliary equation is (λ−r)(λ+m) = 0 and the eigenvalues are λ = r,−m. Since the eigenvaluesare real, unequal and of opposite signs, the critical point (0,0) is an unstable saddle point. Of course,the system (3) is uncoupled and trivial to solve with x(t) = x0e

rt and y(t) = y0e−mt, from which

the above conclusion can easily be verified.The critical point (1,1) requires more careful analysis. The differential equation of the trajec-

tories of the system isdy

dx=

my(x − 1)

rx(1 − y).

Separation of variables givesr(1 − y)dy

y=

m(x − 1)dx

x.

Consequently, we have

r

(

1

y− 1

)

dy − m

(

1 − 1

x

)

dx = 0

∴ r ln y − ry − mx + m lnx = c

28 CHAPTER 6. LOTKA–VOLTERRA MODELS

or

ln yrxm = c + ry + mx

∴ yrxm = eceryemx

i.e. yr

ery · xm

emx = K, for some constant K. Thus the trajectories are the family of curves defined by

yr

ery· xm

emx= K.

We show that these are closed curves for x, y > 0. Let f(y) = yr/ery and g(x) = xm

emx for x, y > 0.

Observe that f(0) = 0 and f(∞) = 0, and f(y) > 0 for y > 0,

f ′(y) =yr−1(1 − y)

ery

and

f ′′(y) =ryr−1(r(y − 1)2 − 1)

ery.

Consequently f attains its maximum at y = 1. Similarly g attains its maximum at x = 1. Themaximum value of f is f(1) = 1/er and that of g is g(1) = 1/em. Let us denote these maximumvalues by My,Mx respectively. Since

f(y)g(x) = K,

if K > MxMy, there can not be (x, y) satisfying f(y)g(x) = K. Moreover, (x, y) = (1, 1) is theonly solution of

f(y)g(x) = MxMy.

Thus we need only to consider the case K = λMy where 0 < λ < Mx. Observe that xm

emx = λ hasone solution say xm < 1 and one solution say xM > 1, as is clear from the graph of g(x).


Mx

xM

1

g(x)

f(y)

1

λ

xxm

y

If x < xm or x > xm, then g(x) cannot be λ. In fact, in either case g(x) < λ or λg(x) > 1 and

hence

λ

g(x)· My > 1.

Since we are interested in solving f(y)g(x) = λMy, we observe that the equation

f(y) =λ

g(x)My > My

has no solution if xm > x or xM < x. If xm < x < xM , then we observe that λ < g(x) and hencef(y) = λ

g(x)My has two solutions y1(x) and y2(x) with y1(x) < 1, y2(x) > 1. (At x = xm or xM it

has only one solution namely y = 1.) As x → xm or xM , both y1(x), y2(x) → 1. Consequently thetrajectories defined by f(y)g(x) = K are closed. With above analysis we can sketch the trajectoriesand the following diagrams show these trajectories.


f(y) g(x)

=

1x x2m

1 x2M

xM

x1

f(y) g(x)

2λ=

λ1M

y

1 (1,1)

m

My

y

Observe that the directions of these trajectories are determined by the following considerations:

x > 1, y > 1dx

dt= rx(1 − y) < 0,

dy

dt= my(x − 1) > 0

i.e. x moves towards origin, y moves up x > 1, y < 1, dxdt > 0, dy

dt > 0 i.e. both x and y increase

x < 1, y < 1, dxdt > 0, dy

dt < 0 i.e. x moves right, y moves down. Finally x < 1, y > 1

dx

dt< 0,

dy

dt< 0 i.e.

x moves left, y moves down.Another important observation is that over a period T , the average value of x,

x =1

T

∫ T

0x(t)dt

= 1

and similarly y = 1. To see this we start with

dx

dt= rx(1 − y) and

dy

dt= my(x − 1).

∴

dx

dt

1

x= r(1 − y)

∴

1

T

T∫

0

1

x

dx

dtdt =

r

T

T∫

0

(1 − y(t))dt

i.e.lnx(T ) − lnx(0)

T=

r

T

T∫

0

dt − r

T

T∫

0

y(t)dt.

Since x(T ) = x(0) (our solutions are periodic with period T ), we have

0 =r

T

T −T∫

0

y(t)dt


or

1

T

T∫

0

y(t)dt = 1 i.e. y = 1.

Similarly x = 1.

Let us see how all this solves D’Ancona’s puzzle. When fishing starts the population of food fishdecreases at a rate say ε1H(t) where ε1 indicates the intensity of fishing. Let suppose that ε1 < r.(Would that humans always act so sensibly!) Suppose also that fishing decreases the selachians bythe rate ε2P (t). The system now changes to

dH

dt= rH − cHP − ε1H

dP

dt= bPH − mP − ε2P.

We replaced P and H by ryc and mx

b earlier. This gives

m

b

dx

dt= r

mx

b− c

r

c

m

bxy − ε1

mx

b

i.e.dx

dt= rx − rxy − ε1x and

dy

dt= −my + mxy − ε2y

i.e.dx

dt= (r − ε1)x − rxy

dy

dt= −(m + ε2)y + mxy.

Provided that ε1 < r, this latter system has periodic solutions similar to (2). (We’ll see why a bitlater.) Hence we can consider a period T and repeat the previous derivation of average populationsizes over one period. To that end, 1

xdxdt = r − ε1 − ry

∴

1

T

T∫

0

1

x

dx

dtdt =

1

T

T∫

0

((r − ε1) − ry)dt

∴ 0 = r − ε1 − r1

T

T∫

0

y dt

or y = 1 − ε1/r.

Similarly x = 1 + ε2/r. Does this suggest a plausible explanation of why the average yield offood fish increases and the predator decreases when both (or either) species are fished? It seemsthat a moderate amount of fishing actually increases the yield of food fish provided the predatoryspecies is also harvested at some constant rate.

Unfortunately, the Lotka–Volterra model suffers from a number of deficiencies. For exampleit is sensitive to even small perturbations in its coefficients which can destroy the existence ofa neutrally stable family of periodic orbits. Modelers have modified the classical Lotka-Volterramodel in a number of ways to correct its deficiencies.


For one such example, consider an autonomous prey-predator system of the form

dH

dt= f1(H,P )

dP

dt= f2(H,P ), (4)

where H is the number of prey and P is the number of predators. The equilibria are found bysolving

f1(H,P ) = 0 = f2(H,P ).

Let us denote solutions by (H∗, P ∗). Thus f1(H∗, P ∗) = 0 = f2(H

∗, P ∗). In order to study thestability (H∗, P ∗), we study solutions (H(t), P (t)) which at some time are near to the steady-state(H∗, P ∗)

x(t) = H(t) − H∗

y(t) = P (t) − P ∗.

Then using Taylor’s Theorem to approximate (f1(H,P ), f2(H,P )) in a neighborhood of (H∗, P ∗),one obtains

dx

dtdy

dt

=

dH

dtdP

dt

=

(

f1(H,P )f2(H,P )

)

=

(

f1(H∗, P ∗)

f2(H∗, P ∗)

)

+

∂f1

∂H

∣

∣

∣

(H∗,P ∗)

∂f1

∂P

∣

∣

∣

(H∗,P ∗)

∂f2

∂H

∣

∣

∣

(H∗,P ∗)

∂f2

∂P

∣

∣

∣

(H∗,P ∗)

(

H − H∗

P − P ∗

)

+ higher order terms,.

Hence, ignoring terms of order two and higher we have arrive at the linearized approximation ofthe non-linear system (4)

dx

dtdy

dt

=

(

00

)

+

(

a11 a12

a21 a22

)(

xy

)

i.e.

(

xy

)

=

(

a11 a12

a21 a22

)(

xy

)

where aij’s are various partial derivatives. The Jacobian matrix J = ( a11 a12a21 a22

) is called the “commu-nity” matrix in ecology. It captures the strength of the interactions in a community at equilibrium.The characteristic equation of the above system is

λ2 − (a11 + a22)λ + (a11a22 − a12a21) = 0

orλ2 − pλ + q = 0

where p = trace J and q = det J . And from here on, we can apply criteria discussed earlier tocharacterize the nature of equilibrium points.


In light of this, let us consider the modification:

dH

dt= aH

(

1 − H

K

)

− bHP = f1(H,P )

dN

dt= cHP − dP = f2(H,P ).

For convenience we introduce

u =H

Kand v =

b

aP

thereby changing these equations to

du

dt= au(1 − u − v)

dv

dt= cKuv − dv. (5)

Further introduce T = at after which (5)

du

dT = u(1 − u − v);dv

dT =cK

auv − d

av

=cK

av

(

u − d

cK

)

= αv(u − β)

where α = cKa , β = d

cK . Thus our model simplifies to

du

dT = u(1 − u − v)

dv

dT = αv(u − β). (6)

This system has three steady-states (u∗, v∗) = (0, 0), or (1,0), or (β, 1−β), and its Jacobean matrixis

J =

∂

∂uu(1 − u − v)

∂

∂vu(1 − u − v)

∂

∂uαv(u − β)

∂

∂vαv(u − β)

(u∗,v∗)

=

(

1 − 2u − v −uαv αu − αβ

)

(u∗,v∗)

.

At (u∗, v∗) = (0, 0)

J =

(

1 00 −αβ

)

.

This matrix has eigenvalues 1,−αβ which are real, distinct and of opposite sign. Hence (0,0) is asaddle point. At (u∗, v∗) = (1, 0)

J =

(

−1 −10 α(1 − β)

)

.

The eigenvalues are -1, and α(1 − β). If β > 1, both roots are real, unequal and negative. Hence(1,0) is an asymptotically stable node corresponding to the prey population being at its carryingcapacity in the absence of the predator population.


Also note that β > 1 means d > cK. Here d represents per capita death rate of the predatorand cK its growth rate. (Strictly speaking, cKu is the growth rate for v.) Thus, when the deathrate exceeds of the predator population exceeds its growth, there is a stable node. If the death rateis smaller, β < 1 and (1,0) is a saddle point.

Finally consider (u∗, v∗) = (β, 1 − β) for which

J =

(

−β −βα(1 − β) 0

)

.

The characteristic equation is

λ2 + βλ + αβ(1 − β) = 0

whose eigenvalues are

−β ±√

β2 − 4αβ(1 − β)

2.

The discriminant is

∆ = β2 − 4αβ(1 − β).

∆ > 0 iff β > 4α − 4αβ iff β(1 + 4α) > 4α i.e. iff β > 4α1+4α . Thus, when β > 4α

1+4α , we have real,unequal roots, both negative. Hence (β, 1 − β) is a node. Since both roots are negative this nodeis asymptotically stable.

When β < 4α1+4α , we have complex roots with negative real parts. Hence the critical point is

a spiral point which is asymptotically stable. This model does not have any periodic orbits (i.e.limit cycles), in marked contrast to the classical Lotka–Volterra system. The addition of a smallamount of prey-density dependence, which seems to be a more realistic assumption than exponentialgrowth, destroys the family of periodic orbits that we observed in the earlier model and which areobserved in nature.

This begs the question: What biological factors create limit cycles? The inclusion of a morerealistic “functional response” is such a factor. The functional response is defined to be the rate atwhich each predator captures prey. So far we considered the functional response that was a linearlyincreasing function of prey density. In practice things may go quite differently. For example, thepredator may become satiated. Clearly a fixed number of predators cannot continue to devour afixed proportion of the prey population irrespective of the prey population’s size. Such limits onthe predator’s ability to consume prey can have a profound effect on the dynamics of the model.

Holling described three different functional response curves. A type I functional response is alinear relationship between the number of prey eaten per predator per unit line and prey density.The resulting straight line may increase up to some fixed maximum

N

Type I

(H)φ

6.2. COMPETITION MODELS 35

A type II functional response is a hyperbolic function that saturates because of the time it takesto handle prey.

Let t denote time, th denote handling time for each prey, H be the number of prey and letV denote the victims (i.e. actual number of prey caught). It is only natural to assume that V isproportional to both the time available for searching and H.

V = α(t − V th)H

or V =αtH

1 + αthH.

Let us define φ(H) = V/t, which represents the rate of predation. Then we have φ(H) = bH1+hH

where b and h are constants.The graph of ϕ(H) against H is a hyperbola

(H)ϕ

Type II

H

A type III functional response is given by φ(H) = bH2

1+hH2 .This is a sigmoidal curve that has predators foraging inefficiently at low prey densities.

Type III

In light of this discussion, we can modify the Lotka–Volterra equations as:

dH

dt= H

(

1 − H

K

)

− ϕ(H)P

dP

dt= ϕ(H)P − dP.

These equations can lead to limit cycles in certain cases.

6.2 Competition Models

Thus far we have considered Lotka-Volterra models for two interacting species of the predator andprey type. A equally important type of two species interaction involves the two species competing


for the same resource. The prototype system has the form

{

N ′1(t) = r1N1(1 − N1/K1) − a1N1N2

N ′2(t) = r2N2(1 − N2/K2) − a2N1N2.

(7)

Thus, in the absence of competition, each species grows according to a logistic curve with differentintrinsic growth rates and carrying capacities. Moreover, they might have unequal competitiveadvantages for their common food source. One can apply the mathematical methods illustratedpreviously to analyize the stability of the various equilibrium states of (7).

These ideas extend naturally to models involving more than two species. Thus, for a homoge-neous mixture of several species, one can construct systems of the form

d

dtρj(t) = ρj(t)fj(ρ1, ρ2, . . . , ρn)

or

d

dtNj(t) = Nj(t)fj(N1, N2, . . . , Nn).

Naturally, the complexity of the analysis rises dramatically with system size. Even studying 3 × 3systems can prove to be a formidable task.

Chapter 7

The Diffusion Model

Thus far we have not considered spatially varying population models. We’ve assumed spatiallyhomogeneous populations on spatially homogeneous habitats. Another perspective argues thatwe’ve been considering spatially averaged population densities. However, there are many situationsin which the ecologist might want to model the spatial variation of a habitat and the movementof populations around it. The simplest such models treat the movement of species as a diffusiveprocess. This chapter presents a brief look at this modeling approach.

7.1 Basic Model for a Single Species

Suppose R is a fixed habitat independent of time. For example R might be a national park or apart of ocean and so on. Let P be a part of R (i.e. P ⊂ R). Then the total biomass of creaturest0f ype j is

∫

P

ρj(t, x)dx. where the biomass density ρj(t, x) of species j is a function of both the

position and time. Many factors can cause the biomass in the part P to change; during any timeinterval there might be births, deaths, growth, predation, entry or exit through the boundary ofP , etc. Conservation of biomass in the habitat is one of our basic balance laws. It asserts that allbiomass must be accounted for by physical, chemical and biological processes. It can be expressedmathematically as follows.

n

xv j

P

R

37

38 CHAPTER 7. THE DIFFUSION MODEL

Let vj(x, t) be the velocity of creatures of species j at point x in R at a time t. vj(x, t) describesthe movement of species j throughout the habitat. Then the rate of change of biomass in P isequal to the rate of flow of biomass across the boundary of P (i.e. flux across P ) plus the biomassproduction in P (which can be negative). Thus,

d

dt

∫

P

ρj(t, x)dx = −∫

∂P

ρj(t, x)vj(t, x) · n dS +

∫

P

f(t, x)dx (1)

where f(t, x) is a supply function that covers all possible interactions along with birth, death andgrowth. By the Divergence Theorem,

∫

∂P

ρj(t, x)vj(t, x) · n dS =

∫

P

div(ρj vj)dx,

and assuming the part P is independent of time,

d

dt

∫

P

ρj(t, x)dx =

∫

P

∂ρj

∂t(t, x)dx.

It follows that (1) becomes∫

P

(

∂ρj

∂t+ div(ρj vj) − f

)

dx = 0.

Since P was arbitrary to begin with, we must have

∂ρj

∂t+ div(ρj vj) − f = 0. (2)

A popular choice for vj is given by ρj vj = −µj grad ρj . Since grad ϕ is the direction ofmaximum increase of ϕ, this choice means that the creatures move in the direction of maximumdecrease of ρj . For this choice for vj , we have

∂ρj

∂t− µj∇ · ∇ρj = f

i.e.∂ρj

∂t− µj∆ρj = f

where ∆ is the Laplacian, ∂2

∂x21

+ · · · + ∂2

∂x2n. Note that in general both µj and f can be density de-

pendent and the above equation turns nonlinear. However, for simplicity of analysis and numericalsolution, µj is usually taken as a constant. For a single species we have

∂ρ

∂t+ div(ρv) = f.

This equation is called the “biomass balance equation”. If instead of mass density, number densityis used, we have

dN

dt+ div(Nv) = g,

the “number balance equation”.

7.2. TRACKING NUMBER AND SIZE 39

7.2 Tracking Number and Size

For many species, size is more or less a measure of health of the species—bigger means healthier.Creature health affects its fecundity, ability to move, feed and viability. The classical approachto incorporating size information into population modeling is through the notion of a structured

population. The basic idea is to divide the total population into subsets indexed by size (or age orother structure parameters). Thus, N(x, t, s) would denote the number density of creatures of sizes located at position x at time t. One must then make some assumptions about how size evolveswith time. Many such models have been introduced and studied.

Another approach to introducing a measure of size is to track both the number and biomassdensities for a species. One can then define a measure of size through

s(x, t) =ρ(x, t)

N(x, t),

which is the average size per creature at x at time t.Tracking both number and biomass density leads to two natural choices for vj , namely v =

−µρ∇ρ or v = − µ

N ∇N . One must also model f and g. One example of how one might model f is

dρ

dt+ div(ρv) + s(D − B) − NG = 0,

where D is the death rate, B is the birth rate and G is the growth rate. Correspondingly, oneshould model g through

dN

dt+ div(Nv) + (D − B) = 0

From the above two equations we can obtain

s′ + v · ∇s = G.

To see this we start with

ρ = Ns

∴ ρ′ = N ′s + Ns′.

The first equation then reads:

N ′s + Ns′ + ∇ · (ρv) + s(D − B) = NG

∴ N ′s + Ns′ + ∇ρ · v + ρ∇ · v + s(D − B) = NG.

∴ N ′s + Ns′ + (N∇s + s∇N) · v + sN(∇ · v) + s(D − B) = NG.

From the second equation, we have

N ′ = −(D − B) −∇ · (Nv)

= −(D − B) − (∇N) · v − N∇ · v.

∴ Eliminating N ′ between these two, we have

− s(D − B) − s(∇N) · v − sN(∇ · v)

+ Ns′ + N∇s · v + s(∇N · v) + sN(∇ · v) + s(D − B) = NG

i.e. s′ + ∇s · v = G. For convenience, we list together the four fundamental size-number-massrelations derived above.


(1) ρ′ + div(ρv) + s(D − B) = NG

(2) N ′ + div(Nv) + D − B = 0

(3) s′ + ∇s · v = G

(4) s = ρ/N .

Suppose we now model v through Nv = −µ∇N . Then we can rewrite (2) and (3) as

(5) N ′ − div(µ∇N) = B − D,

(6) s′ + ∇s ·(

−µ∇NN

)

= G.

Further, taking µ to be a constant results in

(7) N ′ − µ∆N = B − D, and (6) changed to

(8) s′ − µ∇N ·∇sN = G.

Equations (7) and (8), although obtained after making several simplifying assumptions, are stillvery complicated to analyze. Equation (7) is a parabolic partial differential equation, whereas (8) isa first order hyperbolic partial differential equation. Moreover, B, D, and G are nonlinear functionsof N and s. Parabolic and hyperbolic PDEs have dramatically different qualitative properties. Toget a feel for their differences, it is instructive to look at simplified, uncoupled examples.

Setting B − D = 0, (7) becomes the classical diffusion equation which occurs, for example, inthe usual model of heat conduction based upon Fourier’s law for heat flow. Regardless of the initialstate, solutions to the heat equation (assuming no source term and uniform boundary conditions)are infinitely smooth and converge to a spatially uniform as t → ∞.

Now lets consider first order hyperbolic equations. When G = 0, equation (3) becomes

s′ + ∇s · v = 0.

In addition, let’s suppose v is constant. There now exists a closed form solution. To understandhow this equation is solved, first consider its one dimensional counterpart, namely

∂s

∂t+ v

∂s

∂x= 0.

Let us try s = f(x − vt)

∂s

∂t= −vf ′(x − vt)

∂s

∂x= f ′(x − vt)

∴∂s

∂t+ v

∂s

∂x= 0.

Thus f(x − vt) is a solution of the above equation. In fact, it is the general solution. To see this,let us introduce characteristic coordinate

ξ = x − vt, η = x + vt,

7.2. TRACKING NUMBER AND SIZE 41

so that x = 12(ξ + η), t = η−ξ

2v . Also let us write s(x, t) = s(η, ξ)

∂s

∂t=

∂s

∂η

∂η

∂t+

∂s

∂ξ

∂ξ

∂t= v

∂s

∂η− v

∂s

∂ξ

and∂s

∂x=

∂s

∂η

∂η

∂x+

∂s

∂ξ

∂ξ

∂x=

∂s

∂η+

∂s

∂ξ.

Since s is assumed to be a solution, we have ∂s∂t + v ∂s

∂x = 0

∴ v∂s

∂η− v

∂s

∂ξ+ v

∂s

∂η+ v

∂s

∂ξ= 0

i.e. ∂s∂η = 0. Hence s is independent of η. Thus s = s(ξ) = s(x − vt). (Remark: If v = −v, the

equation reduces to ∂s∂t − v ∂s

∂x = 0 whose solution is f(x + vt).) At t = 0, s(x, 0) = f(x). This iscalled a wave profile. As t increases this wave profile travels to the right.

f(x) = s(x,v)

f(x-vt) = s(x,v)

Thus this 1-dimensional first order partial differential equation has solutions which are nec-essarily traveling waves. This property is shared by higher dimensional first order PDEs. Thus,when v is constant, the solutions of the equation ∂s

∂t + v · ∇s = 0 are traveling waves in higherdimensions. When v is nonconstant but a function v = v(x), a similar method of solution againleads to traveling waves. However, the assumption v = v(s) can lead to what are known as shocks.(This means a dramatic change in the size of the species over a small time period, i.e. a singularityin the gradient of s.)


For the system

N ′ − µ∆N = B − D

s′ − µ∇N · ∇s

N= G,

equilibria correspond to solutions with N ′ = 0, s′ = 0. This gives µ∆N = B−D and µ∇N ·∇sN = G.

Remark: Imagine the distribution of animals along a line, i.e. imagine the animals are in a 1dimensional region. If µ does not converge to zero the creatures are not spatially uniform and atone place they tend to reduce and at some other they tend to concentrate. Even if they are evenlydistributed numerically the mass density may not be uniform, for instance unhealthy animals tendto loose mass and healthy animals tend to gain. In a nonuniform distribution of sick and healthyanimals, the aggregation where there is a higher concentration of sick animals eventually loosesboth the mass and the number while the other collection thrives. Thus, if there is no motionbetween the groups there will be a jump discontinuity at the dividing point and subsequently aphenomenon similar to sonic boom. If µ is small, but nonzero however, some healthy specimensmay travel towards the left side and the steepness may reduce.

dividing point

discontinuity

healthy specimenssick specimens

S

When µ is small the steepness reduces as in the following diagram.

unhealthyanimals

healthyanimals

7.3 Multiple Species Diffusion Models

Diffusion models for multiple interacting species can be derived in a straight forward mannerfrom the above considerations for single species diffusion models and multiple species, spatiallyhomogeneous ODE models. For example, consider two species with densities (mass or number)u(t, x) and v(t, x). Then the basic diffusion model for their spatially dependent interactions wouldlook like

u′(t, x) −∇ · (µ1∇u(t, x)) = F1(u(t, x), v(t, x))

v′(t, x) −∇ · (µ2∇v(t, x)) = F2(u(t, x), v(t, x)). (3)

7.3. MULTIPLE SPECIES DIFFUSION MODELS 43

In general, µj can bepend upon both u(t, x) and v(t, x), but usually they are taken to be constant,often the same constant. The interaction terms F1(u, v) and F2(u, v) can be of the predator-preyor competition type, depending upon the nature of the interaction being studied. The system(3) is called a reaction-diffusion system since such systems have been studied extensively in themodeling of chemical reactions. Even though much effort has gone into developing a theory forreaction-diffusion systems, many open questions remain. They can be exceedingly complicated tounderstand and are capable of exhibiting a wide array of fascinating behavior including spontaneous,complex pattern formation and evolution.

Chapter 8

Interacting Populations on Non-Flat

Landscapes

We now address the problem of modeling interactions among different species on non-flat land-scapes. Recall that by habitat of a species we mean both the physical environment as well as thelowest species in the food chain (with their carrying capacities). Now suppose we have more than2 species. The relation between any two of them may be prey-predator type or they may competefor the same resource. An important example we wish to model involves a native species at equilib-rium in a habitat when an invading competitor species enters the scene. Often the invading speciesenters at one point of entry point and then fans out and spreads. The spreading may be facilitatedby breeze, rain, insects or some other natural phenomenon as in the case of spreading of plantseeds. Sometimes the native species may defeat the invading species or the invading species mayannihilate the native species , specially when there is a massive flux flowing in and at a higher rateof movement. Also in general the landscape is not flat and our efforts to model such situations arehighly affected by topography of the land (for example altitudes greatly influence the distributionof different species and inhibit or enhance their diffusion.

To develop equations modeling diffusion or motion on non-flat landscapes we need ideas andtechniques from the differential geometry of curves and surfaces. Thus, we begin this chapter bypresenting these necessary geometric concepts and formulae.

8.1 Differential Geometry of Curves and Surfaces

We begin by introducing the concept of a Regular Surface.

Definition: A subset S ⊂ R3 is a regular surface if ∀ p ∈ S,∃ an open neighborhood U ′ ⊂ R3

of p and an open set U ⊂ R2 together with the map X : U → U ′ ∩ S, taking (u, v) to (x, y, z)satisfying: (a) X is a smooth map from U to R3, i.e. If X(u, v) = (x(u, v), y(u, v), z(u, v)), thenx, y, z ∈ C1(U, R).

(b) X is a homeomorphism from U to U ′ ∩ S. (i.e.X is a bijective map with continuous inverse.)

(c) Regularity condition: For each q ∈ U the differential dX q : R2 → R3 is an injective map.

Recall that if F : U ⊂ R2 → R3 is a smooth map and q ∈ R2, then dFq : R2 → R3 is the mapv 7→ dFq(v) defined by

dFq(v) =d

dt(F(q + tv))|

t=0.

44

8.1. DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES 45

Let q = (u0, v0), and v = (a, b). Then

dFq(v) =d

dt(F(q + tv))|

t=0

=d

dt(x(u0 + ta, v0 + tb), y(u0 + ta, v0 + tb), z(u0 + ta, v0 + tb))|

t=0

= (∂x

∂u(u0, v0)a +

∂x

∂v(u0, v0)b,

∂y

∂u(u0, v0)a +

∂y

∂v(u0, v0)b,

∂z

∂u(u0, v0)a +

∂z

∂v(u0, v0)b))

=

∂x

∂u(u0, v0)

∂x

∂v(u0, v0)

∂y

∂u(u0, v0)

∂y

∂v(u0, v0)

∂z

∂u(u0, v0)

∂z

∂v(u0, v0)

(

ab

)

.

One should note that regularity of S at p means that X : U ⊂ R2 → U ′ ⊂ R3 has the propertythat dXq is one-to-one, where X(q) = p.

Moreover, one readily sees that

dXq(e1) =

∂x

∂u(u0, v0)

∂y

∂u(u0, v0)

∂z

∂u(u0, v0)

and dXq(e2) =

∂x

∂v(u0, v0)

∂y

∂v(u0, v0)

∂z

∂v(u0, v0)

where e1 = (1, 0) and e2 = (0, 1).We sometimes write

Xu = dX(e1) =

∂x∂u∂y∂u∂z∂u)

and Xv = dX(e2) =

∂x∂v∂y∂v∂z∂v

.

Also dXq is one-to-one if and only if Xu × Xv 6= 0. Given p ∈ S as above, the map X : U →U ′ ∩ S,X(q) = p is called a parametrisation of S in a neighborhood of p. S is then a parametrisedsurface and X(U) is called the trace of X .Example: Consider S2 = {(x, y, z) ∈ R3 : x2+y2+z2 = 1} Define X3+(u, v) = (u, v,

√

1 − (u2 + v2)),with U = {(u, v) : u2 + v2 < 1}, U ′

3+ = {(x, y, z) ∈ R3 : z > 0}. We can cover S2 with sixparametrised surfaces,called patches. For this, besides X 3+, we use X3− defined by X3− : U →U ′

3−∩ R3, X3−(u, v) = (u, v,−√

1 − (u2 + v2) and four other similar functions,X2+, X2−, X1+, X1−where, X2+(u, v) = (u,

√

1 − (u2 + v2), v), X2−(u, v) = (u,−√

1 − (u2 + v2), v) etc.We need to show that all these functions are regular. Let us consider X 3+(u, v) = (u, v,

√

1 − (u2 + v2)),

the others being similar. For convenience let us write X 3+ as X. We note that Xu = (1, 0,−u

√

1 − (u2 + v2)), Xv = (0, 1,

−v√

1 − (u2 + v2)).

Therefore Xu × Xv = (u

√

1 − (u2 + v2),

v√

1 − (u2 + v2), 1) 6= 0. As an exercise one can check the

regularity of other functions.

46 CHAPTER 8. INTERACTING POPULATIONS ON NON-FLAT LANDSCAPES

Definition: If f : U ⊂ R2 → R is a smooth function then the set Γ(f) = {(x, y, f(x, y)) : (x, y) ∈U} is called the graph of f. It can be easily seen that Γ(f) is a regular surface. As an illustrationlet U = R2, f(x, y) = x2 + y2. In this case Γ(f) turns out to be a paraboloid.

To parametrize this surface we let X : U = R2 → Γ(f) be defined by X(u, v) = (u, v, u2 + v2).Then we have Xu = (1, 0, 2u), X v = (0, 1, 2v). Hence Xu × Xv = (−2u,−2v, 1) 6= 0.

Definition: Let F : R3 → R be a smooth function. A point a in R is called a regular value for

F if ∀ p ∈ R3 withF (p) = a we have dFp 6= 0.( Equivalently ∇Fp 6= 0). If a is a regular value,then the level surface S = F−1(a) is a regular surface. ( The proof requires the Implicit Function

Theorem).

8.1.1 Change of Parameters

We note the following fact which can be proved easily. Suppose X : U → S ∩ U ′ andY : V → S ∩V ′ are parametrizations of S and W ′ = U ′ ∩ V ′ ∩ S, then the map Y

−1◦X : X−1

(W ′) → Y−1

(W ′)is smooth.

Definition: Suppose f : S → R is a continuous function. It is said to be smooth at p ∈ S iff ◦X : U ⊂ R2 → R is smooth. That is there is a parametrization X : U ⊂ R2 → U ′ ∩ S, X(q) =p such that f ◦ X is smooth on U .

Note that this definition is independent of parametrization. To see that, suppose that Y :V → V ′ ∩ S is another parametrization around p, then we can define W ′ = U ′ ∩ V ′ ∩ S =

X′(U) ∩ X

′(V ) ∩ S and we can write f ◦ Y = (f ◦ X) ◦ (X

−1 ◦ Y ), the first being smooth at

q = X−1

(p) and the second being smooth at q′ = Y−1

(p).

8.1.2 Calculus on Surfaces

Definition: Smooth maps between surfaces;

Suppose S,S ′ ⊂ R3 and F : S → S ′. We say that F is smooth if each coordinate function ofF is smooth for all points p ∈ S. In other words the functions x(p), y(p), z(p) must be smooth.Note that if πi : R3 → R, i = 1, 2, 3 denote the canonical projections then we require that each ofπi ◦ F is smooth.

Let S ⊂ R3 be a regular surface with parameterization X(u, v) = (x(u, v), y(u, v), z(u, v)). Thetangent space to S at p, TpS, is defined as follows.

Definition: Given p ∈ S, a vector v ∈ R3 is said to be tangent to S at p, if there exists a smoothcurve

α : (−ε, ε) → S

such that α(0) = p, α ′(0) = v.

Proposition: Let p ∈ S as above and let X : U ⊂ R2 → U ′ ∩ S be a parametrisation of S nearp with X(q) = p, (q = (u0, v0) ∈ U). Then the tangent spaceTp(S) is dXq(R

2) + p.

Sketch of the proof: If v = α′(0) where α : (−ε, ε) → S satisfies α(0) = p, we can choose ε smallenough so that α(t) ∈ U ′ ∩ S. Then

α(t) =(X ◦ X−1 ◦ α)(t)

=X(u(t), v(t))

=(x(u(t), v(t)), y(u(t), v(t)), z(u(t), v(t)).

The goal is to show that v ∈ dX q(R2).


v =α ′(0)

=d

dt(x(u(t), v(t)), y(u(t), v(t)), z(u(t), v(t))|t=0

=

(

dx

dt,dy

dt,dz

dt

)

|t=0

=

(

∂x

∂uu′(0) +

∂x

∂vv′(0),

∂y

∂uu′(0) +

∂y

∂vv′(0),

∂z

∂uu′(0) +

∂z

∂vv′(0)

)

=

∂x

∂u

∂x

∂v

∂y

∂u

∂y

∂v

∂z

∂u

∂z

∂v

(

u′(0)v′(0)

)

=dXq(w)

where w =

(

u′(0)v′(0)

)

∈ R2.

Corollary: Given X : U → U ′ ∩ S as above then {Xu, Xv} form a basis for Tp(S). (Xu, Xv) are

linearly independent and (Xu, Xv)

(

u′(0)v′(0)

)

= dXq(w).

We illustrate the above by describing the tangent plane to TpS2. Without loss of generalitypick p = (x0, y0.z0) ∈ S2 with z0 > 0. Take U = {(u, v) : u2 + v2 < 1} and let X(u, v) =(u, v,

√1 − u2 − v2). We know that X(x0, y0) = p. Now TpS2 = span {Xu, Xv}. Recall that

Xu = (1, 0,−u√

1 − u2 − v2), Xv = (0, 1,

−v√1 − u2 − v2

). Therefore any v ∈ TpS2 can be written as

v =aXu + bXv

=(a, 0,−ax0

√

1 − x20 − y2

0

) + (0, b,−by0

√

1 − x20 − y2

0

)

(a, b,−(ax0 + by0)√

1 − x20 − y2

0

).

Now

p ◦ Xu(x0, y0) =(x0, y0, z0) ◦ (1, 0,−x0

√

1 − x20 − y2

0

)

=x0 −z0x0

√

1 − x20 − y2

0

=0.

Similarly p ◦ Xv(x0, y0) = 0. Thus p ◦ v = 0∀ v ∈ TpS2 It follows therefore that TpS2 = {v ∈ R3 :v ⊥ p}.

We now introduce the First fundamental form.

Consider a surface S ⊂ R3 and let p ∈ S. Observe that TpS ⊂ TpR3 ≡ R3 and TpS is a planein R3. The inner product in R3 induces an inner product on TpS ∀ p ∈ S.

Definition: The First Fundamental Form


〈·, ·〉p : TpS ×TpS → R is defined as 〈v, w〉p = v ·w where v ·w denotes the usual inner producton R3.Description in local coordinates.

If X : U → U ′ ∩ S is a local parametrization, then two vectors v, w ∈ TpS can be written asv = aXu + bXv, w = cXu + dXv and hence

〈v, w〉p =〈aXu + bXv, cXu + dXv〉=ac〈Xu, Xu〉 + (ad + bc)〈Xu, Xv〉 + bd〈Xv, Xv〉=acE + (ad + bc)F + bdG.

where E = 〈Xu, Xu〉, F = 〈Xu, Xv〉, G = 〈Xv, Xv〉.Note that E,F,G are functions of u and v called the coefficients of first fundamental form in

local coordinates. The first fundamental form defines an inner product on the tangent plane andhence has an associated norm define by: ‖v‖p :=

√

〈v, v〉p.Note that the matrix H :=

(

E FF G

)

is positive definite and EG−F 2 6= 0. The first fundamental

form has a convenient representation in matrix form for a given local parameterization. Specifically,using the coordinate representations

[v] =

[

ab

]

, [w] =

[

cd

]

one has〈v, w〉p = [v] · H[w].

Definition: If α : [a, b] → S is a piecewise smooth curve on S with α(a) = p1 and α(b) = p2,both p1, p2 ∈ S, then we define the length of α from p1, to p2 by:

L(α) =

∫ b

a‖α′(t)‖p dt.

Further if β(t) = (u(t), v(t)) ∈ U, ∀ t, where X : U → U ′ ∩ S, then defining α(t) := X(β(t)), onehas a path in U ′∩S. Moreover, one readily shows that α ′(t) = u′(t)Xu(u(t), v(t))+v′(t)Xv(u(t), v(t)).Hence,

L(α) =

∫ b

a

√

u′(t)2E(u(t), v(t)) + 2u′(t)v′(t)F (u(t), v(t)) + v′(t)2G(u(t), v(t))dt.

We now define the distance d(p1, p2) between two points p1, p2 ∈ S as d(p1, p2) = inf{L(α) : αis a piecewise smooth path from p1 to p2}. It is not terribly difficult to show that d(p1, p2) as definedabove is a metric on S. However, the proof is omitted.

We now obtain a relation for the area of a portion of a surface. If W ⊂ S is describedparametrically, where X : U → W , then the area of W is given by A(W ) =

∫

U ‖Xu × Xv‖du dv.

Note that ‖Xu × Xv‖ =√

EG − F 2. Thus A(W ) =∫

U

√EG − F 2du dv.

Remark: Let θ be the angle between the vectors Xu and Xv. Then

‖Xu × Xv‖2 = ‖Xu‖2‖Xv‖2 sin2 θ

= ‖Xu‖2‖Xv‖2(1 − cos2 θ)

= ‖Xu‖2‖Xv‖2 − (‖Xu‖‖Xv‖ cos θ)2

= ‖Xu‖2‖Xv‖2 − (〈Xu, Xv〉)2

= EG − F 2


Definition: The Differential of φ at p ∈ S.

Let φ : S −→ R be a smooth map. Then the differential of φ at p ∈ S is defined to be theunique linear transformation, denoted dφp(·) : TpS −→ R, satisfying

dφp(Xu) =∂

∂uφ ◦ X =: φu, dφp(Xv) =

∂

∂vφ ◦ X =: φv

for all local coordinate representations of S. It follows that if v ∈ TpS has coordinate representationv = aXu + bXv, then

dφp(v) = aφu + bφv.

Definition: Gradient of φ at p ∈ S.

Let φ : S −→ R be a smooth map. Then the gradient of φ at p ∈ S is defined to be the unique

vector in Tp, denoted ∇φp, satisfying for all v ∈ TpS

dφp(v) = 〈v,∇φp〉p.

The existence of the unique vector dφp is guaranteed by the famous Riesz Representation Theoremfrom linear algebra (or more generally from functional analysis).

This global definition for the surface gradient can be given a local description as follows. Since∇φp ∈ TpS, it has a local representation ∇φp = αXu + βXv. It follows that

φu := dφp(Xu) = 〈Xu,∇φp〉p = αE + βF

and

φv := dφp(Xv) = 〈Xv,∇φp〉p = αF + βG.

Solving these two equations for α and β, one finds that

α =Gφu − Fφv

EG − F 2, β =

Eφv − Fφu

EG − F 2.

Definition: Smooth Vector Field on a Surface.

A smooth vector field, w : S −→ R3, is a smooth function satisfying w(p) ∈ TpS for all p ∈ S.

Definition: Covariant Derivative of a Smooth Vector Field.

Let w be a smooth vector field on S. Then the Covariant Derivative of w is defined to be theunique linear transformation Dw : TpS −→ TpS satisfying for all v ∈ TpS

Dw(v) := PTpS

[

d

dt(w ◦ α)|t=0

]

where

1. α(·) is any path defined on some interval −ε < t < ε taking values on S and with α ′(0) = v,and where

2. PTpS denotes the perpendicular projection operator onto the tangent plane TpS.

To compute the covariant derivative in local coordinates, we first write a given smooth surfacevector field w in component form

w ◦ X(u, v) =: w(u, v) = a(u, v)Xu + b(u, v)Xv.


Similarly, any vector v ∈ TpS can be written v = αXu + βXv. It follows that

Dw(v) = αDw(Xu) + βDw(Xv).

We must now compute

Dw(Xu)|q=(u0,v0) = PTpS

(

d

dtw(u0 + t, v0)|t=0

)

= PTpS

(

∂w

∂u

)

= PTpS

(

∂

∂u(aXu + bXv)

)

= PTpS(

a,uXu + aXu,u + b,uXv + bXv,u

)

= (a,uXu + b,uXv) + a(Γ111Xu + Γ2

11Xv) + b(Γ112Xu + Γ2

12Xv)

= (a,u + aΓ111 + bΓ1

12b)Xu + (bu + aΓ211 + bΓ2

12)Xv

where we’ve introduced the Christoffel symbols Γikl defined through the identities

PTpSXu,u = Γ111Xu + Γ2

11Xv

PTpSXv,u = Γ112Xu + Γ2

12Xv

PTpSXv,v = Γ122Xu + Γ2

22Xv.

Similarly, one has

Dw(Xv) = (a,v + aΓ112 + bΓ1

22)Xu + (b,v + aΓ212 + bΓ2

22)Xv.

Definition: Surface Divergence of a Smooth Vector Field.

Let w be a smooth vector field defined on S. Then the Divergence, Div(w), of w is defined tobe the trace of the linear map Dw(·) : TpS −→ TpS.

In local coordinates,

Div(w) = (a,u + aΓ111 + bΓ1

12) + (b,v + aΓ212 + bΓ2

22). (1)

An even more convenient form for the surface divergence of a vector field can be derived in termsof the coefficients of the first fundamental form. Specifically, we claim

Div(w) = a,u + b,v + aδu

δ+ b

δv

δ(2)

whereδ(u, v) := ‖Xu × Xv‖ =

√

EG − F 2. (3)

The derivation of (2) requires expressing the Christoffel symbols in terms of the coefficients of thefirst fundamental form. To that end, we write

Xu,u = Γ111Xu + Γ2

11 + L1n

Xu,v = Γ112Xu + Γ2

12 + L2n

Xv,v = Γ122Xu + Γ2

22 + L3n

in which n is the unit normal to S. Recalling that E := 〈Xu, Xu〉, one has

Eu = 2〈Xu,u, Xu〉, Ev = 2〈Xu,v, Xu〉, etc . . . .


It follows that

1

2Eu = Γ1

11E + Γ211F, Fu = (Γ1

11 + Γ112)F + Γ2

11G + Γ112E

1

2Ev = Γ1

12E + Γ212F, Fv = (Γ1

12 + Γ222)F + Γ2

12G + Γ122E

1

2Gu = Γ1

12F + Γ212G,

1

2Gv = Γ1

22F + Γ222G.

Next notice that

(

E FF G

)(

Γ111

Γ211

)

=

(

12Eu

Fu − 12Ev

)

(

E FF G

)(

Γ112

Γ212

)

=

(

12Ev12Gu

)

(

E FF G

)(

Γ122

Γ222

)

=

(

Fv − 12Gu

12Gv

)

.

Finally, invert these last relations using

(

E FF G

)−1

=1

δ2

(

G −F−F E

)

.

Substitution of the resulting expressions into (1) yields (2).

Divergence Theorem on a Surface.

Theorem: Let R ⊂ S be a region with oriented boundary Γ := ∂R. If w is a smooth vector fieldon S, then

∫

Γ〈w,m〉 ds =

∫ ∫

RDivSw dA (4)

where m is the outward pointing normal to Γ lying in the tangent plane to S.

Proof: For simplicity, we assume that R is contained in a single coordinate patch. Thus, R = X(U)for some U ⊂ R2, and Γ = X(C) for C = ∂U . Let α(t) = X(u(t), v(t)), for 0 ≤ t ≤ 1, parametrizeΓ where C is parametrized by β(t) = (u(t), v(t)). Then α′(t) = u′(t)Xu + v′(t)Xv is tangent to Γ

and the vector α′(t) × n is normal to Γ but lies in the tangent plane to S, where n := Xu×Xv

‖Xu×Xv‖is a unit normal to S. In local coordinates, one has (exercise)

α′(t) × n =

(

u′(t)F + Gv′(t)δ

)

Xu −(

Fv′(t) + Eu′(t)δ

)

Xv.


Moreover, ‖α′(t) × n ‖ = ‖α′(t)‖. Define γ(t) := α′(t) × n . Then,

∫

Γ〈w,m〉 ds =

∫ 1

0〈w(t),

γ(t)

‖γ(t)‖ 〉‖α′(t)‖ dt

=

∫ 1

0

{[

a

(

u′F + Gv′

δ

)

E + b

(

u′F + Gv′

δ

)

F

]

+

[

−a

(

v′F + Eu′

δ

)

F − b

(

v′F − Eu′

δ

)

G

]}

=

∫ 1

0

[

av′ − bu′] δ dt

=

∫

C

(

δaδb

)

· n ds

=

∫

UDiv

(

δaδb

)

dudv =

∫

U

(

∂

∂u(δa) +

∂

∂v(δb)

)

dudv

=

∫

RDivS(w) dA

thereby proving the theorem. The last step in this calculation makes use of the identity (2).Definition: The Laplace Operator on S. Let φ : S −→ R be a smooth function. Then theLaplace operator on φ is defined by

4Sφ := Div(∇φ).

Example: We now illlustrate these definitions by deriving an expression for the Laplacian of asmooth scalar field defined φ on a surface S expressed as the graph of a smooth function h(u, v) :U ⊂ R2 −→ R. We claim that

4Sφ =1

nDivU

(

1

nB ∇Uφ

)

(5)

where

n :=√

1 + h2u + h2

v, B :=

(

(1 + h2v) −huhv

−huhv (1 + h2u)

)

.

The operators DivU and ∇U in (5) denote the ordinary divergence and gradient operators on the“flat” space U ⊂ R2.

The derivation of (5) is somewhat lengthy but straight forward. We begin by cataloging a fewuseful identities. The surface coordinate map is X(u, v) := (u, v, h(u, v)), for (u, v) ∈ U ⊂ R2. Itfollows that the local base vectors for the tangent plane TpS are Xu = (1, 0, hu) and Xv = (0, 1, hv).The unit normal vector to the surface is then

n :=Xu × Xv

|Xu × Xv|=

(−hu,−hv , 1)√

1 + h2u + h2

v

and the coefficients of the first fundamental form are

E = 〈Xu, Xu〉p = 1 + h2u, F = 〈Xu, Xv〉p = huhv, F = 〈Xv, Xv〉p = 1 + h2

v.

It is useful to note that EG − F 2 = n2.Define the vector field w := ∇Sφ. Then one has

∇Sφ = aXu + bXv


with

a :=Gφu − Fφv

n2, b :=

Eφv − Fφu

n2. (6)

To compute the surface divergence of w, we first compute its covariant derivative. To that end, weneed the projections

PTpSXu,u = Xu,u − 〈Xu,u, n 〉p n =huuhu

n2Xu +

huuhv

n2Xv

PTpSXv,v =hvvhu

n2Xu +

hvvhv

n2Xv

PTpSXu,v =huvhu

n2Xu +

huvhv

n2Xv

The covariant derivative of w has the form

Dw(Xu) = PTpS

(

∂w

∂u

)

= a,uXu + b,uXv + aPTpSXu,u + bPTpSXv,u

Dw(Xv) = PTpS

(

∂w

∂v

)

= a,vXu + b,vXv + aPTpSXu,v + bPTpSXv,v

The surface divergence of w is the trace of its covariant derivative. Thus, we have

4Sφ = Div(w) = tr(Dw)

= au + bv +a

n2(huuhu + huvhv) +

b

n2(huvhu + hvvhv)

= a,u + b,v +a

n

∂n

∂u+

b

n

∂n

∂v

=1

n

[

∂

∂u(an) +

∂

∂v(bn)

]

=1

nDivU

(

n

[

ab

])

where a and b are given by (6). The desired expression (5) for the surface Laplace operator nowreadily follows.

Exercise: A One Dimensional Landscape. We consider a 1-dimensional example to illustrate theabove calculus on a non-flat world. Specifically, consider a curve S ∈ R2 defined by y = h(u) fora ≤ u ≤ b. As an exercise, the reader should verify (i.e. prove) the following assertions. Thecoordinate map is X(u) = (u, h(u)) and the tangent vector to S is given by X u = (1, h′(u)).Let f(·) : S −→ R be a smooth function. Then the differential of f at p ∈ S is the mapdfp(·) : TpS −→ R defined by fu := dfp[Xu] = d

duf ◦X = ∇f ·Xu. The gradient of f must satisfy∇f ∈ TpS, i.e. ∇f = αXu for some α. Hence, fu = ∇f ·Xu = α〈Xu, Xu〉 = α E = α(1+(h′)2) fromwhich one obtains α = fu/E. The first fundamental form becomes: for v, w ∈ TpS, v = αXu andw = βXu which implies 〈v, w〉p = αβ E = αβ(1 + (h′)2) = αβδ2. Now let w(·) : S −→ R2 denotea smooth vector field on S, which in component form is w(u) = a(u)X u. The covariant derivativeof w is is the linear map on the tangent space to S defined through Dw[X u] := PTpS

[

dduw(u)

]

=

TTpS [a′(u)Xu + a(u)Xu,u] = a′(u)Xu + a(u)PTpSXu,u = a′(u)Xu + a(u)h′′h′

E Xu. The divergence of

w is given by Divw = tr(Dw) = a′(u) + a(u)h′h′′

E . One now computes the Laplace operator applied

to a smooth function f(·) : S −→ R as 4f := Div∇f =(

fu

E

)′+(

fu

E

)

h′h′′

E = 1√E

ddu

[

1√E

dfdu

]

.


8.2 Population Movement on Surfaces

Modeling interaction populations on non-flat landscapes follows the same basic steps employedpreviously on flat landscapes. In particular, the basic principle is the balance of biomass density ornumber density for each species. The difference here is that these balance laws must be formulatedon non-flat surfaces embedded in three dimensional

Euclidean space or possibly curves embedded in the two dimensional Euclidean plane. In par-ticular, one uses the version of the divergence derived for non-flat surfaces (or curves) to formulatethe basic balance law for biomass or number density for arbitrary subsurfaces and then localize thebalance laws to points on the surface obtaining thereby partial differential equations on the surfacein terms of the surface gradient, divergence and Laplace operators. Of course, to actually solveexplicit problems one must then cast the equations into their local coordinate forms.

Chapter 9

The Topology of Landscapes

Thus far we have focused upon models of interacting species on a habitat without saying muchabout the physical structure of the habitat. However, the structure of a habitat can have profoundeffect upon species interactions. In particular, different species can perceive different structuresfor the same habitat. For example, birds might see a certain national park as a single connectedhabitat, whereas small mammals and some plant species might perceive it as consisting of discrete,disconnected patches. These ideas can be made precise through the language of topology which webriefly introduce below.

9.1 Metric Spaces

In practice, distance is a concept that depends upon the topography of a region and the species thattravels along it. These are constraints (and in some cases advantages) depending upon physicalabilities and sizes of animals. For example in a terrain full of rivers and mountains the distancebetween two points may vary according to the paths chosen by two different species. The distancebetween two mountain peeks is a straight Euclidean distance for birds but may turn out muchlarger for humans depending upon the position of bridges available (or foods, as the case may be)across rivers and scalable sides of the mountains. For some species this distance could be infinite,if an unnegotiable hurdle lies in between. In any case, the purely mathematical formulation of thenotion of distance is called a metric.

Definition. A metric d on a nonempty set X is a function d : X × X → R satisfying, for allx, y, z ∈ X:

(i) d(x, y) ≥ 0,

(ii) d(x, y) = 0 if and only if x = y,

(iii) d(x, y) = d(y, x), and

(iv) d(x, y) ≤ d(x, y) + d(z, y).

Note again that ability to migrate depends upon the landscape and the definition above is purelymathematical. In particular, the same set (region) can be given several distinct metrics.

Example. Given any nonempty set X, one can define the discrete metric on X by

d(x, y) = 0, iff x = y

d(x, y) = 1, otherwise.

55

56 CHAPTER 9. THE TOPOLOGY OF LANDSCAPES

Example. The Euclidean distance in Rn = {(x1, x2, . . . , xn) = x, xi ∈ R}

d(x, y) ={

∑

(xi − yi)2}1/2

.

More generally:

Example. Let p be a positive integer and x, y ∈ Rn. Define the “p norm” by ‖x‖p = {Σ|xi|p}1/p.

Then ρp(x, y) = {Σ|xi − yi|p}1/p is a metric on Rn. The case p = 2 gives the usual Euclideanmetric.

Example. Let I be an open interval in R and ϕ : I → R be a positive, differentiable real valuedfunction. Let

X = graph {ϕ}= {(t, ϕ(t)) ∈ R2, t ∈ I}

bϕ

ϕ (x)

x x1x0

a

The arclength of the path joining a = (x0, f(x0)) to b = (x1, f(x1)) along X, where x0 ≤ x1 isgiven by

` =

x1∫

x0

√

1 + ϕ(t)2 dt.

We shall see later that using the notion of arclength one can construct a metric in regions ofEuclidean space. Such a metric is essential when we want to discuss distance between points on asurface, for example. As we know most habitats are not planar and hence, to effectively compute thedistance between two points, on a hilly surface for example, we need to choose the least (infimum)of the lengths of curves joining the two points and lying entirely on the surface.

a

One of the main concepts in this section is the following.

9.1. METRIC SPACES 57

Definition. A set X, together with a metric on X is called a metric space. We use the notation(X, d) to emphasize the metrics being used on X.

Besides its explicit use to measure distance between points in a region, a metric on a set Xcan be used to introduce special subsets of X. These subsets will form the important concept of atopology on a set X.

Definition. Given a point x0 ∈ X and ε > 0, the open ball in X with radius ε and centered at x0

is the set

Bd(x0, ε) = {x ∈ X : d(x, x0) < ε}.A subset U ⊂ X is called open if it is either empty or for every x0 ∈ U , there is ε > 0 such thatBd(x0, ε) ⊂ U . It is obvious that U is open if and only if it is either empty or a union of open balls.A set F ⊂ X is said to be closed if its complement X\F is open. The collection Td consisting ofall open subsets

Example. Consider R2 with the usual Euclidean metric ρ2. Then Bρ2(a, ε) is the usual open disk.

Example. If ‖x‖1 means |x1| + |x2| for x = (x1, x2) and ρ1(x, y) = ‖x − y‖1 then Bρ1(0, ε) is a

square with diagonals along the axes.

x

x

x + x = ε

2

1 2

1

Example. Let ‖x‖∞ = max{|x1|, |x2|} and ρ∞(x, y) = ‖x− y‖∞. Then Bρ∞(0, ε) is a square withsides parallel to axes.

ε ε

Exercise. Let (X, d) be a metric space.

(i) If U1, U2 . . . Un are open sets, then show that U1 ∩ U2 . . . ∩ Un is also open,

(ii) if {Uλ}λ∈Λ is an arbitrary collection of open sets then show that⋃

λ∈Λ

Uλ is open.

Exercise. Show that if d is the discrete metric on a set X, then any subset of X is open in thecorresponding metric topology. Start by proving that Bd(x0, ε) = {x0} for 0 < ε < 1 and if ε > 1,B(x0, ε) = X.


Exercise. Prove that (R2, ρ2), (R2, ρ1) and (R2, ρ∞) have the same collection of open sets (i.e.same metric topology).(HINT: Prove first that, given any point x inside any ball for one metric, we can always find a smallenough radius such that the ball around x in another of these metrics is contained in the initiallygiven ball.

9.2 Continuity

One of the main uses of a topology lies in the notion of continuity. More precisely, consider metricspaces (X, d) and (Y, ρ). A function f : X → Y is called continuous if whenever U ⊂ Y is open inY then f−1(U) is open in X where f−1(U) = {x ∈ X : f(x) ∈ U}.Exercise. One says that f : X → Y is continuous at x0 ∈ X if given ε > 0, there is a δ = δ(x0, ε)such that for all x ∈ Bd(x0, δ), one has f(x) ∈ Bρ(f(x), ε). Show that f : X → Y is continuous ifand only if it is continuous at every x0 ∈ X.

Exercise. Let f : (X, d1) → (Y, d2) and g : (Y, d2) → (Z, d3) be continuous functions. Show thatg ◦ f : (X, d1) → (Z, d3) is a continuous function.

Let I = [0, 1] ⊂ R be a closed unit interval and let (X, d) be a metric space. A path betweenx, y ∈ X, parametrized by I, is a continuous function f : I → X such that f(0) = x and f(1) = y.

Definition. Let (X, d) be a metric space. Given two points x, y ∈ X, we say they lie in the same

path component if there is a path γ : I → X between x and y. If x and y lie in the same pathcomponent, we write x ∼

p.c.y.

This notion gives an equivalence relation on X. Indeed:

The relation is reflexive: Given x ∈ X, define τ : [0, 1] → X by τ(t) = x for all t. This pathconnects x to x; hence x ∼

p.c.x.

The relation is symmetric: Suppose x ∼p.c.

y, hence there is a path γ : [0, 1] → X such that γ(0) = x

and γ(1) = y. Define σ : [0, 1] → X by σ(t) = γ(1 − t). Then, since γ and t 7→ 1 − t are bothcontinuous, so is their composite σ. Also σ(0) = γ(1) = y and σ(1) = γ(0) = x, hence y ∼

p.c.x.

The relation is transitive: Suppose x ∼p.c.

y and y ∼p.c.

z. Then there are continuous functions

α, β → [0, 1] → X such that α(0) = x, α(1) = y, β(0) = y and β(1) = z. Define τ : [0, 1] → X asfollows.

τ(t) =

{

α(2t) 0 ≤ t ≤ 12

β(2t − 1) 12 ≤ t ≤ 1.

It is easy to see that the function τ : [0, 1] → R is continuous and satisfies τ(0) = x and τ(1) = z.This proves that x ∼

p.c.z.

Since ∼p.c.

is an equivalence relation, one can partition X into mutually disjoint equivalence

classes. These equivalence classes are called the path components of X. When X denotes a landscapeor habitat, we prefer to call these path components patches.

Observe that the connectedness of a domain X affects the solution of a differential equationdefined on X. For example, consider the rather trivial differential equation dy

dt = 0 with an initial

9.3. COMPACTNESS 59

condition y(1) = 2. If X = (1, 3)∪ (4, 5) is the metric space in which are seek its solution, then thesolution has the form

y(t) =

{

2 t ∈ (1, 3)c t ∈ (4, 5)

where c is any constant. If X = R, then the only possible solution is simply y(t) = 2 for all t.

9.3 Compactness

Let (X, d) be a metric space. A sequence {xn} in X is said to converge to a point x∞ ∈ X if forall ε > 0, one can find N(ε) > 0 such that for all n ≥ N one has d(xn, x∞) < ε, i.e. for all n ≥ N ,xn ∈ Bd(x∞, ε). For example an =

(

1 − 1n , 3−n

)

∈ R2 converges to (1,0).A sequence {xn} is said to be a Cauchy sequence if given ε > 0, there is N > 0 such that

d(xn, xm) < ε for all n,m ≥ N .

Exercise. Show that in any metric space a convergent sequence is Cauchy.The converse is not true in general. For example, consider Q as a metric space with the

Euclidean metric. If x1 = 3, x2 = 3.1, x3 = 3.14, x4 = 3.141 · · · etc. is a Cauchy sequence in Q

that does not converge to an element in Q, although it converges to the element π ∈ R.

Definition. A metric space is said to be complete if every Cauchy sequence in the metric spaceconverges. (R, d) is complete but not (Q, d) as seen above, where d(x, y) = |x − y| is the usualEuclidean metric in R.

Definition. Let (X, d) be a metric space and S ⊆ X. A point x0 ∈ X is called a closure point ofS if there is a sequence in S converging to x0. Furthermore, x0 ∈ X is called a limit point of S inX if there is a sequence of distinct points of S converging to x0.

If S is a subset of X, the collection of limit points is denoted by S ′ and closure of S, denotedS is defined as S ∪ S ′.

Exercise. Show that S is closed and it is the smallest closed set in X containing S.

Definition. A set S ⊂ X is called compact, if every infinite sequence in S has a convergentsubsequence converging to an element in S.

Theorem. If f : X → Y is continuous and S ⊂ X is compact, then f(S) ⊂ Y is compact.

Proof. Let {yn} be a sequence in f(S). Hence for each n, there is xn ∈ S such that yn = f(xn).Thus, we have a sequence {xn} in S. Since S is compact, it follows we have a convergent subsequence{xnk

} converging to say x∞ in S. Since f is continuous, it follows that ynk= f(xnk

) → f(x∞) ∈f(S). Thus {yn} has a subsequence converging to f(x∞) in f(S). Hence f(S) is compact.

Definition. Let (X, d) be a metric space. A subset S ⊂ X is said to be bounded if there existsx0 ∈ X and R > 0 such that S ⊆ Bd(x0, R). The subset S is said to be totally bounded if for anygiven ε > 0 we can find x1, x2 . . . xn ∈ X such that for any x ∈ S, there is xi such that x ∈ Bd(xi, ε).

In other words X is totally bounded if for every ε one has X =n⋃

i=1Bd(xi, ε). The set {x1x2, . . . ,

xn} is called an ε-net for S.

Theorem. Let (X, d) be a complete metric space. Then, a subset S ⊂ X is compact if and only if

it is closed and totally bounded.


Proof. Suppose S is compact. We show that S is closed and totally bounded. First we show thatS is closed. Let x0 be a limit point of S. Then there is a sequence {xn} in S such that xn → x0.However S is compact and {xn} ⊂ S. Hence there is a subsequence xnk

of xn which converges toa point x ∈ S. Again, as {xnk

} is a subsequence of xn, it follows that xnkalso converges to x0.

Hence x0 = x ∈ S. Hence S is closed.

Now we show that S is totally bounded. Let ε > 0 be any number. Pick any x1 ∈ S. IfB(x1, ε) = S we are through. If not there is x2 ∈ S but not in B(x1, ε). If S = B(x1, ε) ∪B(x2, ε),again we are done. If not we can continue. If the process terminates after finite number of steps,

S =m⋃

i=1B(xi, ε) and again we can stop. Otherwise we obtain an infinite sequence xn such that

xn ∈ S∖

(

n−1⋃

i=1B(xi, ε)

)

. Since {xn} is a sequence in S, which by assumption is compact, {xn} has

a convergent subsequence {xnk} which is therefore Cauchy. However, for different k, `

d(xnk, xn`

) ≥ ε.

Thus, {xnk} cannot converge and we have a contradiction. Hence S is a finite union of balls B(xi, ε)

and is therefore totally bounded.

Conversely, suppose S is closed and totally bounded. We show S is compact. In the first placeS itself is complete. To see this let {xn} be any Cauchy sequence in S. Since S is a subset of acomplete metric space, xn → x0 ∈ X. Thus, x0 is a closure point of S and since S is closed x0 ∈ S.Thus {xn} converges in S and therefore, S is complete.

Let {xn} be a sequence of points of S. We show that it has a subsequence which is Cauchy.First cover S by finitely many balls of radius 1. (This is possible as S is totally bounded.) At leastone of these balls say B1 contains xn for infinitely many values of n. Let J1 = {k ∈ N : xk ∈ B1}.Now cover X by 1/2 balls. Since J1 is infinite at least one of these balls, say B2 will contain xn forinfinitely many n with n ∈ J1. Choose J2 ⊂ J1 s.t. xn ∈ Bn for n ∈ J2.

In general after choosing Jm choose Jm+1 to be an infinite subset of Jm such that there isa ball Bm+1 of radius 1

m+1 such that xn ∈ Bm+1 for all n ∈ Jm+1. Choose n1 ∈ J1. Choosen2 ∈ J2 s.t. n2 > n1 and similarly after choosing nm ∈ Jm choose nm+1 > nm, nm+1 ∈ Jm+1. Fori, j ≥ m, the indices ni, nj ∈ Jm. Hence xni

, xnjlie in Bm whose radius is 1

m . Thus if 2m < ε,

d(xni, xnj

) < ε. Hence (xnk) is a Cauchy subsequence in S, which is complete. Hence {xn} has a

convergent subsequence converging in S. This completes the proof.

Remark. In Rn the compact sets are precisely the sets which are closed and bounded.

Note however, that there are metric spaces where closed and bounded sets are not necessarily

compact. Consider the metric space `1, where `1 = {s = (s1, s2 . . .) : sj ∈ R and∞∑

i=1|sj| < ∞}.

This is a metric space with d(s, t) =∑∞

i=1 |sj − tj |. Let B(0, 1) = {s : Σ|sj| ≤ 1}. Then this is aclosed and bounded subset. Consider a sequence {ej} where ej = (0, 0, . . . , 1, . . . , 0). This sequencelies in B(0, 1) but has no convergent subsequence, since d(ei, ej) = 2.

Our aim now is to further study the compact subsets of a given complete metric space. In thestudy of landscapes and habitats one is often concerned with subregions of the habitat (and thesecan be always taken to be compact).

Definition. Let A be a subset of the metric space X with metric d. Then we define diameter ofA, diam A to be the sup{d(x, y) : x, y ∈ A}. If d(A) < ∞, we say A has finite diameter.

9.3. COMPACTNESS 61

Definition. Let (X, d) be a metric space. Let A ⊆ X. Then distance between a point x ∈ X andA is defined to be dist(x,A) = inf{d(x, a) : a ∈ A}.Remark. dist(x,A) is a continuous function of x. Furthermore

(i) d(x,A) = 0 if and only if x ∈ A.

(ii) If A is compact d(x,A) = d(x, a0) for some a0 ∈ A, and in this case we can write

dist(x,A) = min{d(x, a) : a ∈ A}.

Now let (X, d) be a complete metric space and let H(X) denote the set whose elements A arecompact subspaces of X (and hence closed and totally bounded). We want to define a metric onH(X), as well.

A natural approach would be to define d(A,B) = inf{d(a, b) : a ∈ A, b ∈ B}. But, sinced(A,B) = 0 iff A ∩ B 6= ∅, we do not have A = B necessarily if d(A,B) = 0. Hence this conceptfails to define a metric on H(X). Recall that dist(a,B) = min{d(a, b) : b ∈ B} (B is compact).Define

d(A → B) = max{d(a,B) : a ∈ A}d(B → A) = max{d(b, A) : b ∈ B}.

Note that d(A → B) 6= d(B → A) in general. For example, if S1 denotes the unit circle in R2 let

A = {(x, y) ∈ S1 : 0 ≤ y} and

B = {(x, 2) : − 2 ≤ x ≤ 2}b B

d(bA)d(B A)

A a

d(aB)

A

B

-1 0 1

-2

d(A B)

2

Exercise. Show that d(A → B) = 2 and that d(B → A) = 2√

2 − 1.

In the example above one has d(A → B) 6= d(B → A). In general, one defines the Hausdorffdistance for A,B ∈ H(X) by

h(A,B) = max{d(B → A), d(A → B)}.Trivially h(A,B) ≥ 0 and

h(A,B) = h(B,A).

If A = B then both d(B → A) and d(A → B) are zero and as such h(A,B) = 0. Conversely,suppose h(A,B) = 0. Without loss in generality suppose h(A,B) = d(A → B). Then

h(A,B) = 0 ⇒ d(A → B) = 0

⇒ max{d(a,B) = 0}⇒ d(a,B) = 0 ∀A

∴ A ⊆ B.


Similarly as d(A → B) ≥ d(B → A), B ⊆ A. Hence A = B. Finally we prove triangle inequality

d(a,B) = min{d(a, b) : b ∈ B}≤ min{d(a, c) + d(c, b) : b ∈ B} for all c ∈ C;

= d(a, c) + d(c,B) for all c ∈ C

≤ d(a, c) + max{d(c,B) : c ∈ C}= d(a, c) + d(C → B)

∴ d(a,B) ≤ min{d(a, c) : c ∈ C} + d(C → B)

= d(a,C) + d(C → B)

≤ d(A → C) + d(C → B)

∴ max{d(a,B) : a ∈ A} ≤ d(A → C) + d(C → B)

∴ d(A → B) ≤ d(A → C) + d(C → B).

Similarly

d(B → A) ≤ d(B → C) + d(C → A)

∴ h(A,B) ≤ h(A,C) + h(B,C).

This makes H(X) into a metric space. The following result highlights some of the importantfeatures of this metric.

Theorem. If (X, d) is a complete metric space, then so is (H(X), h). Also, if (X, d) is compact,

then so is (H(X), h).

Example. Let {Ai}i∈N be a sequence of subsets of R2 defined as follows: A1 is the closed regionbounded by an equilateral triangle with side length 1.

A 1

Now, A2 is the region obtained from A1 by removing the central triangle when A1 is dividedinto 4 equilateral triangles, obtained by joining midpoints the sides of A1.

9.3. COMPACTNESS 63

A2

Now, continue subdividing the remaining three triangles into 4 parts each and drop the centraltriangle from each of them to obtain the region A3.

A3

We continue this process adinfinitum. Observe that each An is a compact set. (As it is closed


and bounded subset of R2.)

d(A2 → A1) = max{d(a,A1) : a ∈ A2}= 0 as A2 ⊂ A1

A

M

B 1 C

r

BM =

√3

2

r =2

3

√3

2=

1√3

d(A1 → A2) = max{d(a,A2) : a ∈ A1}

=1√3

∴ h(A1, A2) =1√3

similarly h(A2, A3) = 12√

3and in general h(An, An+1) = 1√

31

2n−1 . Thus {An} is a Cauchy sequence.

Since H(R2, h) is complete, {An} converges to say A∞ ∈ H(R2).

One can imagine A1 as representing a certain landscape. The process of removing the middletriangle can be compared to a process of depletion or erosion. Ultimately, one ends with smallspecks of land scattered all around buy with no measurable area left.

9.4 Parametric Layout of Landscapes

Using the basic topological concepts introduced in the previous section, we are capable of analyzinganalyzing habitats taking into consideration certain parameters that are tailored to specific classesof species under study. In particular, one can extend the notion of patch (path component) toincorporate clusters of patches that are interrelated according to specified parameters determinedby the species.

The starting point is a determined habitat - for example, the forested region of a specificgeographical location - which comes with specific data describing the distance between its points.One can encode this information by saying that the forest F is a compact subspace of a metric space(X, d). The first piece of information is a given real-valued function f : X → R which determines,for every point x ∈ X, a local value f(x) measuring some property of the habitat at the locationx ∈ X. There are many examples of such functions, that can be tailored to specific studies.

Example. 1. The temperature function f : X → R, assigning the average annual temperaturef(x) at the location x in X.

2. The biomass function f : X → R, assigning the average annual biomass density f(x) at thelocation x in X.

9.4. PARAMETRIC LAYOUT OF LANDSCAPES 65

3. The patch core radius function f : F → R which, roughly speaking, assigns to x ∈ F thelargest radius f(x) of a metric ball B containing x and totally contained in the patch (path-component) of F containing x. More precisely, if F denotes the forested portion of a regionX, seen as a subspace of a metric space (X, d), define

f(x) =: sup{ε | x ∈ B(y; ε) ⊂ C(x), for some y ∈ C(x)}, (1)

where C(x) is the connected component of F containing x.

Let F be a subspace of a metric space and fix a function f : F → R. Given a patch P in F , anda real number t the t-core Cf,t(P ) of the patch - with respect to f - is set of points in P where fhas valued greater or equal to t, i.e.

Cf,t := {x ∈ P | f(x) ≥ t}.

Whenever the function f is evident, we simply denote Ct(P ) instead of Cf,t(P ).

The concept of t-core gives a precise way to describe the suitability of a patch to sustainpopulations of a certain species, based on the criteria determined by the function f. In other words,we say that a species S has a minimal core radius t - with respect to f - if it requires a patch tohave a non-empty t-core Cf,t in order to be capable of sustaining a population of the species S.

For example, if f is the patch core radius function defined above, then to say that a speciesS has a minimal core radius t with respect to f is equivalent to say that the species S requires apatch to contain a ball of radius t in order to sustain population. Note that this is a more refinedway of describing the minimal area need of a species. A very narrow but extremely long patch canhave an arbitrarily large area and yet, due to being very narrow, be unsuitable for certain speciesof large mammals that requires shade and high vegetation biomass density to survive.

Definition. Fix a function f , as above. A τ -δ class of species - with respect to f - consists of acollection of species whose minimal core radius is less or equal to τ and whose dispersal range isgreater or equal to δ.

Given a function f and parameters τ and δ, one can subdivide a habitat in such a way that cer-tain clusters of patches work as a single patch for a τ -δ class of species. This yields the fundamentalnotion in this section. First, define a δ-sequence of subregions of F to be a labelled collection ofpath-connected subregions {P1, . . . , Pn} of F such that the distance

dist(Pi, Pi+1) = inf{d(x, y) | x ∈ Pi, y ∈ Pi+1}

between two consecutive subregions Pi, Pi+1 is less or equal than δ.

Definition. Given a function f : F → R, defined on a subspace F of a metric space (X, d), aτ -δ patch P is a subregion (not necessarily connected) of F satisfying the following conditions:

1. The τ -core Cf,τ (P ) of P is non-empty;

2. One can label the path-components of the τ -core Cf,τ (P ) = ∪ni=1Pi in such a way that the

sequence {Pi | i = 1, . . . , n} is a δ-sequence.

3. P is not contained in any subregion P ′ of F satisfying the two conditions above. In otherwords, P is a maximal subregion of F with respect to these two properties.


The meaning of these requirements is that a τ -δ patch is a cluster of subregions each of whichcapable of sustaining a population requiring a non-empty τ -core to survive; and that a specieswith dispersal range δ can migrate from subregion to subregion travelling along a sustaining pathbetween any two points in that cluster. In other words, a τ -δ patch works as a classical patch fromthe point of view of a τ -δ class of species.

There are several subtleties in this definition. For example, one is tempted to consider a singlepatch with non-empty τ -core as a τ -δ patch, for any value of δ. An example showing that this isnot the case is a region in a dumbbell shape, with a long corridor connecting the two ends. This isa single patch, by definition. However, for large enough τ, the connected components of the τ -corecan be quite apart. In particular, this single patch cannot be considered as a τ -δ patch for smallvalues of δ.

Given fixed parameters τ and δ, along with a function f : F → R, we can partition the habitat Finto distinct τ -δ patches, forming a layout of the landscape that depends on the given parameters.For example, consider the patch core radius function and δ < τ . Using a satellite with a resolutionof 1-pixel per τ 2 units of area, the regions with empty τ -core will not appear in the image and aτ -δ patches will appear as a single blob in the image, a single path-component in the picture.

It is important to keep in mind that the construction depends on the “suitability” functionf being used, and that one can certainly refine the function to incorporate sophisticated andcomprehensive measures of suitability.

The τ -δ parametric layout provides a very flexible tool in the study of conservation issues.For example, by adding corridors at crucial locations in a landsacpe, one can drastically changethe layout by connecting initially distinct τ -δ patches to form a single habitat for a τ -δ class ofspecies. This can be used in many ways, incorporating topological techniques to identify locationswhere the addition of corridors (of a certain size) would optimize the size of a habitat to sustainminimal biodiversity needs. Also, one can use the same modelling techniques to assess the impactof destruction of certain corridors and regions in the landscape.

9.5. HASTINGS MODEL 67

9.5 Hastings Model

Recently Alan Hastings proposed an alternative approach to the diffusion partial differential equa-tions setting for modeling the movement of populations on flat landscapes. His approach involvesdiscrete time continuous spatial integrals rather than continuous time and spatial partial deriva-tives. We describe his approach below and then generalize it to non-flat landscapes.Some Basic Assumptions:

In practice, observational data of a population is gathered at fixed, discrete intervals of time.The time period of observation might be determined by the life span of the species or age requiredto mature or the duration of a reproductive cycle and so on. For this reason, discrete time modelsare commonly used in practice.

Hastings’ model is predicated upon a number of important assumptions which were intendedto hold for certain mollusks living off the North American Pacific coast.

1. Adults of these creatures do not move, but their juveniles do drift in the currents after birthand gradually settle back to the ocean floor.

2. Each adult of the species produces a fixed number of juveniles per unit time. Let us denotethis number by m.

3. The maturity age for a specimen is say j years.

4. The survival rate of the species per cycle is a constant which we denote by a.

Remark: For a more realistic model, we should incorporate age structure into the model. However,for simplicity we shall ignore this issue.

The important quantity we measure in our model is nt(x), which is the density of the popu-lation at location x at time t. Then the total population in a region R equals

∫∫

R nt(x)dAx Weassume the following:

1. The density of juveniles trying to settle can be justifiably considered as mnt(x) .

2. The density of juveniles reaching adult age in the next cycle is assumed to be a functionof mnt(x) , f(mnt(x) ). Note that this function must obviously depend upon the particularspecies and environmental factors. However, for simplicity we take it in the form f(mnt(x) ) =cmnt(x) , where c is a constant.

Remarks:(i) Suppose the population is evenly distributed so that the density is independent of the posi-

tion( i.e., x). Then the value of nt(x) for the next cycle can be given by

nt+1(x) = f(mnt(x) ) + ant(x)

and since by assumption this does not depend on x, we can write

nt+1 = −f(mnt) + ant.

(ii) Suppose the species is harvested with H denoting the percentage of harvested population .Then we have

nt+1(x) = (1 − H){f(mnt(x) ) + ant(x) }.(iii) Spatial dependence can be incorporated through the following construction. Specifically,

we assume here that there is a function, denoted k(x, y), called the probability density function


describing the proportion of the population of juveniles near y that will migrate to x and reachadult age in the next cycle. We then have

nt+1(x) = ant(x) +

∫∫

Rf(mnt(y))k(x, y)dAy

More generally, one can also incorporate age stratification by introducing age classes for adultsDefine ni

t(x) to be the density of population with age i, near x at time t. Let mi denote the fertilityrate at age i and let ai denote the survival rate at age i. Then we have

nt+1(x) =

N∑

i=1

ainit(x) +

∫∫

Rf(m1n

1t (y),m2n

2t (y), ...mNnN

t (y))k(x, y)dAy

As an illustration assume a = 0,m = 1, i.e., one offspring per creature per life cycle. Assumethat every juvenile reaches the adult age. It follows that f(mnt(x)) = nt(x). Further supposeR = R and take k(x, y) = k(x, y) = ce−β|x−y|. For the initial distribution, we consider n0(x)defined as

n0(x) =

{

1, if |x| < 1;

0, if |x| > 1

It follows that

nt+1(x) =

∫ ∞

−∞nt(y)ce−β|x−y|dy.

Thus,

n1(x) = c

∫ ∞

−∞n0(y)e−β|x−y|dy

= c

∫ 1

−1(1)e−β|x−y|dy

When |x| < 1,

n1(x) = c[

∫ x

−1e−β|x−y|dy +

∫ 1

xe−β|x−y|dy]

= c[e−β(x)(eβ(x) − e−β)

β+

eβ(x)(e−β − e−β(x))

−β]

=c

β[eβx(e−βx − e−β) + e−βx(eβx − e−β)]

=c

β[2 − e−β+βx − e−βx−β]

=2c

β− ce−β

β[eβx + e−βx]

=2c

β− 2ce−β

β

[eβx + e−βx]

2

=2c

β[1 − cosh βx

eβ].

9.5. HASTINGS MODEL 69

Now let |x| > 1. If x > 1, |x − y| = x − y and when x < −1, |x − y| = −(x + y). In the firstcase

n1(x) = c

∫ 1

−1e−β(x−y)dy

= ce−βx

∫ 1

−1eβydy

= ce−βx eβ − e−β

β

= ce−β|x| eβ − e−β

β,

and in the second case

n1(x) = c

∫ 1

−1eβ(x+y)dy

= ceβx

∫ 1

−1eβydy


β.

Thus we have

n1(x) =

=2c

β[1 − cosh βx

eβ] if |x| < 1


βif |x| > 1.

As an exercise calculate n2(x), n3(x) using a computer algebra system.Observe that in the above illustration the population exhibits exponential spatial decay. A

situation wherein the total population remains constant is called the Conservative State. In theexample just discussed this would mean that

∫ ∞

−∞nt+1(x)dx =

∫ ∞

−∞nt(x)dx.

To see the effect of this assumption we first find∫∞−∞ e−|x−y|dx. This can be written as

∫ y−∞ e−|x−y|dx+

∫∞y e−|x−y|dx.

In the first integral |x − y| = −(x − y), and as a consequence the first integral reduces to∫ y−∞ e(x−y)dx. To evaluate this integral we substitute x − y = u which changes the integral to∫ 0−∞ eudu = 1.

Similarly putting x − y = u in the second, we get∫∞y e−|x−y|dx =

∫∞0 e−udu = 1 again. Thus

∫∞−∞ e−|x−y|dx = 2 and hence

∫∞−∞ e−β|x−y|dx =

2

β. It follows that

∫ ∞

−∞nt+1(x)dx =

∫ ∞

−∞{∫ ∞

−∞nt(y)ce−β|x−y|dy}dx

= c

∫ ∞

−∞nt(y)dy{

∫ ∞

−∞e−β|x−y|dx}

=2c

β

∫ ∞

−∞nt(y)dy

=2c

β

∫ ∞

−∞nt(x)dx.


Since we want∫ ∞

−∞nt+1(x)dx =

∫ ∞

−∞nt(x)dx,

we conclude that we must take 2c = β. Thus the system is conservative if c = 1/2β.Sometimes it is useful to apply iterative processes to solve such problems and for that it is

necessary to define operators. For example we can define T : f(x) 7→∫∫

R f(y)k(x, y)dAy =T (f)(x). In the previous example,we had nt+1(x) = T (nt(x)). As an exercise take R = R, and

k(x, y) =β

2e−β|x−y| and also assume that f is smooth with compact support and show that the

function y(x) = T (f)(x) satisfies the following Initial Value Problemy′ ′ − β2y = β2f ;

y(0) =β

2

∫ ∞

−∞f(y)e−β|y|dy, and

y ′(0) =β2

2

∫ ∞

0(f(y) + f(−y))e−β|y|dy. Also find which functions will satisfy T (f) = f.

Paulo Lima-Filho and Jay R. Walton 2004Topics in Mathematical Ecology1 Paulo Lima-Filho and Jay R. Walton 2004 1Copyrigh tc 2004 by P. Lima-Filho and J. R. Walton.All righ reserved.

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