Competitive Coexistence in Stoichiometric Chaos

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Competitive coexistence in stoichiometric chaosBo Deng a and Irakli Loladze b Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588, USA

Received 24 January 2007; accepted 3 June 2007; published online 21 August 2007

Classical predator-prey models, such as Lotka-Volterra, track the abundance of prey, but ignore its

quality. Yet, in the past decade, some new and occasionally counterintuitive effects of prey qualityon food web dynamics emerged from both experiments and mathematical modeling. The underpin-ning of this work is the theory of ecological stoichiometry that is centered on the fact that eachorganism is a mixture of multiple chemical elements such as carbon C , nitrogen N , and phos-phorus P . The ratios of these elements can vary within and among species, providing simple waysto represent prey quality as its C:N or C:P ratios. When these ratios modeled to vary, as theyfrequently do in nature, seemingly paradoxical results can arise such as the extinction of a predatorthat has an abundant and accessible prey. Here, for the rst time, we show analytically that thereduction in prey quality can give rise to chaotic oscillations. In particular, when competing preda-tors differ in their sensitivity to prey quality then all species can coexist via chaotic uctuations.The chaos generating mechanism is based on the existence of a junction-fold point on the nullclinesurfaces of the species. Conditions on parameters are found for such a point, and the singularperturbation method and the kneading sequence analysis are used to demonstrate the existence of aperiod-doubling cascade to chaos as a result of the point. 2007 American Institute of Physics .DOI: 10.1063/1.2752491

Predator-prey or consumer-resource dynamics is centralto ecological complexity. Critical to all biomasses lies themass balance law governing abiotic elements owingthrough the species of ecological systems. Not only are thecomponent uxes in quantity important to the species butalso the relative ratios of the elements of which individualspecies may have their own preferred composition. Thequestion is: To what extent does the elemental composi-tion of a resource inuence its consumers population?Adding to our understanding about ecological complexitywe will demonstrate here that a simple food web can bedriven to chaotic oscillation in population by just oneconsumers sensitivity to the composition of only two el-ements of its resource.

I. INTRODUCTION

Eating more prey never hurts predator growth is oneof the old adages in population dynamics. It goes back toLotka-Volterra predator-prey models. This assumption is of twofold: predation and growth. The Lotka-Volterra modelsassumed that predator captures prey linearly this response isknown as Holling type I . Different predator responses wereintroduced later to account for more realistic situations, suchas the cases where the capture rates can be saturated Hollingtypes II, III, IV . However, in all nonstoichiometric casesconsuming more prey always translates into positive growthrates for predators. Interestingly, Lotka 1 himself hinted thatthis need not be true. As a physical chemist he realized that aprey and a predator do not operate with a single working

substance, but with a complex variety of such substances, afact which has certain important consequences. Examplesof such consequences were found in the past decade, whenthe theory of biological stoichiometry was applied topredator-prey interactions. Biological stoichiometry em-braces and extends Lotkas idea of multiple workingsubstancesit considers species not just as biomasses butas assemblages of multiple essential elements. The fact thatthe proportions of essential elements can vary within andbetween species can have some profound effects on speciesinteractions.

For example, both experiments 24 and theoreticalmodels 58 showed that when the quality of phytoplanktondrops often expressed as phosphorus to carbon, P:C, ratio ,the growth of its predator, zooplankton, declines even as itconsumes more algae. The reason is that the low quality of prey, as measured by its P content, hampers the growth of zooplankton that has a higher P:C ratio than its food. Thiseffect can lead to unusual dynamical outcomes, which in-clude deterministic extinction of the predator at a very high

prey density and a very low prey quality , stabilization of predator-prey oscillations, an d coexistence of two predatorson one prey at a steady state. 9 Stoichiometric models reectthe fact that prey is an assemblage of multiple elements re-quired by its consumers albeit in different ratios. The sameprey can be of good quality to one consumer and of badquality to another. This interplay between quantity and qual-ity should expand possibilities of coexistence beyond stableequilibriums.

Here, for the rst time to our knowledge, we prove thatstoichiometry can lead to coexistence via chaotic uctua-tions. The idea that uctuations can enhance coexistence is

a Electronic mail: [email protected] Electronic mail: [email protected]

CHAOS 17, 033108 2007

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not new. Armstrong and McGehee 10,11 showed that n con-sumers can coexist on fewer than n resources via periodicuctuations. Huisman and Weissing 12 numerically showedthat the competition of phytoplankton species for three ormore abiotic resources can give rise to chaos, where manyspecies coexist on a handful of abiotic resources. We showhere both analytically and numerically that chaos can arisewhen two consumers compete for just one biotic resource

and it happens when the resource is of bad quality to at leastone of the consumers.

II. MODEL

For a detailed derivation and justication of the model,we refer readers to Ref. 9. Here we briey describe themodel of two consumers exploiting one prey in a systemwith no spatial heterogeneity or external variability. We donot assume, a priori , that higher prey density should alwaysbenet consumer growth; instead, mass balance laws willdetermine consumer growth in our model. To have some

concrete system in mind, we can think of the prey as a singlespecies of alga, while the consumers are two distinct zoop-lankton species, all placed in a well-mixed system open onlyto light and air.

For simplicity, we will model only two chemical ele-ments, P and C. The choice of C is clear, because this ele-ment comprises the bulk of the dry weight of most organ-isms. Instead of P, however, one can choose any otherelement as long as it is essential to all species in the systeme.g., N, S, or Ca . Our model will reect the following sto-

ichiometric assumptions:

1 The preys P: C ratio varies, but never falls below a

minimum q mg P/mg C .2 The two consumers maintain a constant P:C ratio, s1and s2 mg P/mg C , respectively.

3 The system is closed for P, with a total of P concentra-tion in mg P/L, which is divided into two pools: P in theconsumers and the rest as potentially available for theprey.

From these assumptions it follows that P available forthe prey at any given time is

P s1Y 1 s2Y 2

in mg P/L, where Y 1 , Y 2 denote the consumer densities inmilligrams of C per liter, i.e., mg C/L. Recalling that P:C inthe prey should be at least q mg P/mg C , one obtains thatthe prey density cannot exceed P s1Y 1 s2Y 2 / q mg C/L .

In addition, prey growth will be limited by light. Let K represent light limitation in the following way: Suppose thatwe x light intensity at a certain value, then let the preywhich is a photoautotroph grow with no consumers but

with ample nutrients. The resource density will increase untilself-shading ultimately stabilizes it at some value, K . Thus,every K value corresponds to a specic limiting light inten-sity and we can model the inuence of higher light in tensityas having the effect of raising K , all else being equal. 6 Fol-

lowing Liebigs law of the minimum, the combination of light and P limits the carrying capacity of the prey to

min K ,P s1Y 1 s2Y 2

q.

To determine the preys P:C at any given time, we followAndersen 1 by assuming that the prey can absorb all poten-tially available P, which means that the preys P: Cmg P:mg C at any time is

P s1Y

1 s

2Y

2 / X , 1

where X denotes the prey density in mg C/L.Note that we do not impose the maximal P quota on the

prey, thus its P :C can become potentially unbounded forsmall X according to 1 . As we will show later, this propertydoes not have undesirable effects on the dynamics, becausewhen the preys P:C exceeds that of the consumers i.e.,

max s1 , s2 , the consumers become limited by C and thepreys P:C, however large it is, becomes dynamically irrel-evant.

While prey stoichiometry varies according to 1 , eachconsumer maintains its constant homeostatic P:C, s i. If the

preys P:C ratio is s i, then the ith consumer converts con-sumed prey with the maximal in C terms efciency, whichwe denote as e i, and egests or excretes any excess of ingestedP. If the preys P:C is s i, then the ith consumer wastes theexcess of ingested C. This waste is assumed to be propor-tional to the ratio of preys P :C to ith consumers P :C Ref.1 , which reduces the growth efciency in C terms. The fol-lowing minimum function provides the simplest way, basedstrictly on mass balance laws, to capture such effects of vari-able prey quality on consumer growth efciency:

e i min 1,P s1Y 1 s2Y 2 / X

s i. 2

Thus, e1 and e2 are the maximal growth efciencies conver-sion rates or yield constants of converting ingested preybiomass into consumer biomasses. They are achieved onlywhen prey quality is good relative to consumer needs. Themass balance law requires that e1 1 and e2 1. An alter-native formulation that does not use minimum f unctions ispossible using the concept of synthesizing units. 7

Incorporating these assumptions above and others to bespecied below, we consider in this paper the followingmodel:

dX

dt = rX 1

X

min K , P s1Y 1 s2Y 2 / q

c1 X

a 1 + X Y

1

c2 X

a 2 + X Y 2 ,

dY 1dt

= e1 min 1,P s1Y 1 s2Y 2 / X

s1

c1 X

a 1 + X Y 1 d 1Y 1 ,

3

dY 2dt

= e2 min 1,P s1Y 1 s2Y 2 / X

s2

c2 X

a 2 + X Y 2 d 2Y 2 ,

where again X , Y 1, and Y 2 are the densities of the prey andthe two consumers, respectively in mg C/L . Also, param-

033108-2 B. Deng and I. Loladze Chaos 17 , 033108 2007



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eter r is the intrinsic growth rate of the prey day 1 . Param-eters d 1 and d 2 are the specic loss rates of the consumersthat include respiration and death day 1 . Holling type IIfunctional response is assumed for consumer ingestion rates,with c i being the saturation ingestion rates and a i being thehalf-saturation densities.

Figure 1 gives a numerical diagnosis of the chaos attrac-tor. The attractor shown in Fig. 1 a has two distinct features:spirals around a horn and the returning of the spirals from thetip of the horn to the opening of the horn. Without the re-turning of the spirals the attractor would settle down to aperiodic oscillation qualitatively similarly to a classical, two-dimensional 2D predator-prey oscillation: more predators

follow more prey. The stoichiometric effect of the dynamicslies precisely in this returning feature of the attractor: fewer

predators follow more prey. That is, in the tip-to-opening journey, the system has an excessive carbon in X for con-sumer Y 2 and the excess negatively impacts on Y 2s popula-tion density. Figures 1 b and 1 c are return maps of theattractors in variable Y 2 , Y 1, respectively. They are denedwhenever the orbit hits a local minimum in X , showing asdots in Fig. 1 a . Notice that the return maps are approxi-mately one-dimensional 1D and have a unique criticalpoint. The critical point separates each consumer variablerange into two phases. In the case of Y 2 from Fig. 1 b , thesubinterval left of the maximum point corresponds to theclassical predator-prey oscillation, whereas the subintervalright of the maximum point corresponds to the stoichio-

metric effect of the system. The stoichiometric phase for Y 1from Fig. 1 c is the subinterval left of the minimum point in

FIG. 1. Color online a A chaotic attractor of Eq. 3 projected to the XY 2 plane with parameter values: r =1.4, K =0.64, c1 =0.81, c2 =0.75, a 1 =0.25, a 2=0.28, e1 =0.7, e2 =0.85, P =0.026, q =0.0038, s1 =0.025, s2 =0.037, d 1 =0.2, d 2 =0.18. b The return map in variable Y 2. c The same return map in variableY 1. All variables shown are scaled according to 5 .

033108-3 Stoichiometric chaos Chaos 17 , 033108 2007



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which the surge in Y 1s density is the direct result of a greaterresource in X made available by Y 2s decline; one competi-tors loss is another competitors gain.

The return map plots also suggest a near two-dimensional ow structure of the attractor. The analyticalobjective of this paper is to understand the stoichiometriceffect that leads to such low-dimensional chaotic attractors inthe model. The main theoretical result of this paper can be

stated roughly as follows:Theorem 1 . There is a parameter region through which

a one-parameter bifurcation, primarily in varying parameter 1 = d 1 / e1c1 , gives rise to a period-doubling cascade sce-

nario of chaos generation, and the corresponding dynamicsmust go through a phase in which increasing X decreases Y 2.

A precise restatement of this theorem is given later intheorem 2.

III. ANALYSIS

We will use the method of multi-time-scale analysis,which is technically known as the singular perturbationmethod in the eld of dynamical systems. The method isgeometrical in nature. It facilitates a geometric constructionof the chaotic attractor. As a result, we will have to specifythe corresponding parameter region for the attractor. An out-line of the analysis is as follows. In Sec. III A a furthersimplication of the model, as well as its nondimensional-ization, are carried out. To keep track of the many geometricobjects throughout the paper, a useful notational taxonomy isintroduced in Sec. III B. The special cases of two-dimensional subsystems in X , Y 1 and X , Y 2, respectively, aresummarized from the literature in Sec. III C to gradually in-troduce some of the terminology and more importantly arudimentary treatment of the singular perturbation method.Various conditions are specied along the way to zoom in onthe parameter region wanted. In Sec. III D the full three-dimensional 3D system is pieced together from lower-dimensional subsystems in XY 1 and XY 2, respectively. Last, aone-dimensional return map is constructed in Sec. III E andits dynamics is shown to be chaotic. A precise statement of the main result as well as a proof are also given there.

A. Model simplication

Although the stoichiometric inclusion of phosphorus inthe model is absolutely essential for its chaotic dynamics, notall phosphorus related effects are equally important. One task is to nd out the minimum ingredients for the underliningdynamics. It turns out that it only sufces to consider thecase in which only one consumer is truly affected by thestoichiometric consideration. As a result, in the effective re-gion of the stoichiometric chaos under consideration, themodel Eq. 3 has a constant carrying capacity for the re-source and a constant efciency factor for one of the con-sumers, which we will take to be Y 1. Because of this, we willremove these nonlinearities from the equations for simplicity

and only consider the following equivalent equations fromnow on:

dX

dt = rX 1

X

K

c1 X

a 1 + X Y 1

c2 X

a 2 + X Y 2

dY 1dt

= e1c1 X

a 1 + X Y 1 d 1Y 1 4

dY 2dt = e2 min 1,

P s1Y 1 s2Y 2 / X

s2

c2 X

a 2 + X Y 2 d 2Y 2 .

As a necessary rst step for mathematical analysis, wenondimensionalize Eq. 4 so that the new dimensionlesssystem contains a minimum number of parameters. Follow-ing the same scaling idea of Ref. 13, the following changesof variables and parameters are used for the nondimension-alization:

t e2c2t , x = X

K , y1 =

Y 1rK / c1

, y2 =Y 2

rK / c2, 5

i =a iK

, i =d i

e ic i i =

s irK c iP

, = Ks 2P

, = e2c2r

,

=e1c1e2c2

.

In the new variables and parameters the equations are trans-formed into the following dimensionless form:

dx

dt = x 1 x

y1 1 + x

y2 2 + x

,

dy 1dt

= y1x

1 + x 1 ,

dy 2dt

= y2 min 1,1 1 y1 2 y2

xx

2 + x 2 .

The rationale for this particular choice of nondimensionaliza-tion is as follows. The dimensionless prey variable, x= X / K ,measures the fraction of the prey density against its bestpossible total K . The scaling of consumer y1 = Y 1 / M 1 with M 1 = rK / c1 is motivated by this relation: rK = c1 M 1. Becauser is the maximum per capita growth rate of the prey, and c1is the maximum per capita capture rate by the predator Y 1,then M 1 can be interpreted as the predation capacity in thesense that the best possible catch c1 M 1 by M 1 amount of thepredator equalizes the best possible amount in prey regenera-tion rK . Thus y1 is the fraction of Y 1s density against itspredation capacity M 1. The same explanation applies to thedimensionless density y2.

Parameters 1 , 2 are the dimensionless semisaturationconstants, which are the fractions of the dimensional semis-aturation constants of the predators against preys carryingcapacity K . Parameters 1 , 2 are the dimensionless or rela-tive death rates; each is the fraction of the corresponding

predators minimum per capita death d i rate to its maximumper capita birth rate e ic i. As a necessary condition for species




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survival, the former must be smaller than the latter. Thus wewill make it as a default assumption that 0 i 1 for non-triviality.

Parameter 1 is the dimensionless P:C ratio for predatorY 1. The interpretation follows by noticing that 1= s1 / c1P / rK and that the ratio c1P / rK is the P:C ratioof predator Y 1s maximum per capita consumption rate of phosphorus c1P against preys maximum per capita regen-

eration rate of carbon. The same explanation applies to 2.Parameter = K / P / s2 is the dimensionless fraction of preys carrying capacity in carbon against predator Y 2s car-rying capacity in carbon.

Parameter = e2c2 / r is the Y 2 X prolicacy, measuringthe maximum growth rate of Y 2 against that of X . Similarly,parameter = e1c1 / e2c2 is the Y 1Y 2-prolicacy parameter.By the theory of allometry, 14,15 these ratios correlate recipro-cally well with the fourth roots of the ratios of X s body massto Y 2s body mass and, respectively, Y 2s body mass to Y 1sbody mass. Thus, is usually of a small order such as in thecase of phytoplankton-zooplankton relation under consider-

ation. By the same reasoning parameter may or may not besmall because Y 1 , Y 2 are in the same trophic level. We alsopoint out that 1 =e2c2t unit in the new dimensionless timescale equals t =1/ e2c2 units in the original time scale,which represents the average time interval between birth of Y 2.

The number of parameters is reduced from 13 to 9 di-mensionless parameters. This means every point from thenine-dimensional dimensionless parameter space corre-sponds to a four-dimensional subspace in the original 13-dimensional system parameter space for which the dynamicsof the system are identical when the density variables andtime are scaled properly. This is a key point of nondimen-sionalization and equivalent dynamics.

To minimize the usage of subscripts, we will use fromnow on

y = y1, z = y2 ,

and the equations are

dx

dt = x 1 x

y

1 + x

z

2 + x xf x, y, z ,

dy

dt = y

x 1 + x

1 yg x, y, z , 6

dz

dt = z min 1,

1 1 y 2 z x

x 2 + x

2 zh x, y, z .

Chaotic coexistence has not been observed in the samesystem without the second element phosphorus in our case .The goal is to understand how stoichiometry leads to com-petitive coexisting chaos. It turns out that such chaotic struc-ture can persist in the limit 0 for which the limitingattractor is referred to as the singular attractor. The impor-tance of studying singular attractors has long been recog-nized in the studies of dynamics systems because they can be

considered as the origins of the underlying dynamics, retain-ing most essential information. The remaining sections are

devoted to the description and analysis of the singular cha-otic attractor, emphasizing the mechanistic role of stoichiom-etry.

To a greater extent, the study of an ecological system isto understand its species temporal population dynamics. Be-ing able to unlock the underlining mechanisms for popula-tion boom and bust is a requisite of any effective method.For continuous models such as the one under consideration,

such a method inevitably has to deal with the nullclines of the models simply because it is through the nullclines thatpopulations switch between being increasing and decreasing.The singular perturbation method that we will use indeedcenters its analysis around the nullclines. It is both geometri-cal and global in nature, and has been proved extremelyeffective in understanding chaotic attractors see Refs. 13and 16 20 .

B. Notation convention

A few notational rules are used for this paper, followinga similar convention adopted in Ref. 13. Letter p is alwaysused for point s . Depending on the context, it can be asingle point, or a set of points, which form a curve or asurface. It is usually identied by a subscript in a string of letters. The rst one or two subscript letters have to be thevariables x , y , z, indicating the nullcline s which the pointbelongs to or is associated with. The remaining subscriptstring is usually an acronym characterizing the point. Forexample, pxdc can be read as a point from the d escendingcarrying-capacity to be explained later of the xnullcline. When the subscript has a two-letter variable pre-x, the order is usually important. For example, pxzgt meansthe intersection of the x nullcline with the plane pzgtthe

growing t hreshold of the z nullcline. Although the operationof intersection is order independent, the analysis that uses thepoint is order dependent. It means the consideration of the xnullcline precedes the consideration of the z nullcline, and itis the restriction of the system on the x nullcline rst that the z nullcline of the restricted system is considered. Transposingthe prex shifts the emphasis and the order of analysis com-pletely. One minor exception to these general rules is thenotation pxyz for the equilibrium point of the system that isthe intersection of all nontrivial nullclines.

C. Multi-time-scale analysis: 2D cases

The dimensionless equations 6 can have three timescales if both and are small: 0 1 ,0 1, inwhich case the time scale is fast for x, intermediate for z, andslow for y. Because of the allometry theory, we will alwaysassume that x is the fastest regardless of the relative scalebetween y and z. In addition, it turns out that the chaoticsingular attractor will exist at the ordered singular limit 0followed by 0. We formalize these two considerations inthe following.

Condition 1 . The trophic time diversication and theasymmetry competition prolicacy condition:e1c1 e2c2 r, equivalently, 0 1, 0 1.

We will begin our analysis by rst considering the 2Dsubsystems with z=0 and y =0 separately, and then use them




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as building blocks to piece together the 3D dynamics. Work-ing with the simpler subsystems should allow us to x the

needed notation and concept quicker.

1. Singular dynamics without stoichiometric limitation

To begin, Fig. 2 illustrates some essential elements of thesingularly perturbed subsystems with z=0 and y =0. The xysystem in the absence of z is a singular perturbed system,

dx

dt = x 1 x

y 1 + x

= xf x, y,0 ,

dy

dt = y

x

1 + x 1 = yg x, y,0 ,

for which x is the fast variable and y is the slow variable. In

the limit = 0 the xy system above becomes 0= xf x, y , 0 , dy / dt = yg x, y , 0 , referred to as the -slow

subsystem . It is a one-dimensional system restricted on the xnullcline, 0= xf x, y , 0 . Under the change of time variablet / , referred to as the - fast time , the system becomesdx / d = xf x, y , 0 , dy / d = yg x, y , 0 . In the limit =0, thesystem becomes dx / d = xf x, y , 0 , dy / d =0, referred to asthe - fast subsystem . It is a one-dimensional system with yfrozen as a parameter. The equilibrium point of this system isdened by 0= xf x, y , 0 , i.e., the x nullcline.

Solutions of the fast subsystem are referred to as fast solutions or fast orbits . They are horizontal lines movingeither to or away from the x-equilibrium points dened bythe x nullcline. Similarly, solutions of the slow subsystem are

referred to as slow solutions or slow orbits ; however, they areoriented curves on the x nullcline. The solution curves either

FIG. 2. Color online a Typical singular orbits in the case of stable xy-equilibrium state for which xxcf xygt 1. See text for the derivation of yxpd on thephenomenon of Pontryagins delay of loss of stability at which a boom in x population occurs although the recovery starts immediately after its crossing thetranscritical point yxtc. b A typical xy singular limit cycle and its relaxed cycle for 0 1 in the case 0 xygt xxcf . c The xz subsystem in the case of xzgt xzeb xzdc 1. The dynamical consequence is that without the competition from y, z eventually dies off from a detrimentally low P:C ratio whenexposed to the resource capacity on pxdc .




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move to or away from equilibrium points dened jointly byboth the x and y nullclines, or move on a limit singular cycleas shown in Fig. 2. A detailed description follows below.

The x nullcline consists of two branches: the trivialbranch x=0 and the nontrivial branch 0= f x, y , 0 whichcan be explicitly solved for y as a function of x . Point 1 ,0in the nontrivial branch corresponds to the x-carrying capac-ity in the absence of predator y. With the increase of y as aparameter in the fast subsystem, the capacity in x continuesbut decreases. Thus the branch between the maximal pointand the predator-free capacity 1 , 0 is the predator-mediatedcarrying capacity, which satises analytically f x, y , 0=0 , f x x, y , 0 0. We call it the descending capacity branchof the nontrivial x nullcline, and denote it by pxdc . It consistsof stable equilibrium points of the x equation. The maximumpoint is called the x-crash-fold point, denoted by pxcf , be-cause for y immediately above it the fast x solution alwayscollapses down to the extinction branch x=0. Mathemati-cally, the x equation undergoes a saddle-node bifurcation atthe crash-fold point as y changes. It is solved as pxcf

= 1 1 /2, 1 + 12 / 4 .

The remaining branch of the nontrivial x nullcline, leftof pxcf , is unstable for the fast x equation f =0, f x 0 . Itrepresents the predator mediated capacity/extinction thresh-old in x. That is, for a xed y, the prey grows to its predator-adjusted capacity pxdc if it starts above the threshold and goesoff to extinction if it starts below it. As the predation pressureincreases from y the threshold in x increases. Thus we call itthe ascending threshold and denote it by pxat. Since the ca-pacity branch pxdc decreases as y increases, both must coa-lesce at a point in between, which is the crash-fold point pxcf .

The intersection of the ascending threshold pxat and the

trivial x-nullcline branch x=0 is the emerging point of thecapacity/extinction threshold. It is a transcritical bifurcationpoint where two distinct nullcline branches crisscross for thefast x subsystem as y changes. It is denoted by pxtc, which issolved as pxtc = 0, 1 . In general the nontrivial x nullcline,0= f x, y , 0 , is a parabola-like curve consisting of descend-ing capacity, ascending threshold, and crash fold as branchesand points.

Typical slow orbits always move down on the depletedresource branch x=0, but down or up on the capacity andthreshold branch f x, y , 0 =0, depending on whether the re-source amount x is smaller or larger than the amount to equi-libriumize the y-slow equation. Solve the nontrivial ynullcline g x, y , 0 =0 to obtain x= 1 1 / 1 1 , which rep-resents the minimum amount of x required by the predator togrow. For this reason we call it the growing threshold anddenote it by xygt = 1 1 / 1 1 . That is, for x immediately

xygt , predator y increases dy / dt 0 , and for x immedi-ately xygt , it decreases dy / dt 0 . The nontrivial ynullcline should usually be a curve on which x increases as yincreases, representing an ascending y capacity, i.e., a greaterand xed amount of x sustains a greater amount of y at

equilibrium. For our model, however, it becomes a degener-ate vertical line x= xygt, and this is the main reason for itsname. As shown in Fig. 2 a , when ys growing threshold is

greater than the amount to crash the prey but otherwise lessthan the carrying capacity, xxcf xygt 1, then all slow solu-

tions on xs descending capacity branch converge to the xy-equilibrium point, which is the intersection of pygt and pxdc .

The concatenation of fast and slow orbits are referred toas singular orbits . For the case shown in Fig. 2 a , it is awell-known fact that all singular orbits except for those fromthe axes will converge to the equilibrium point pygt pxdc .Naturally, the attracting equilibrium point is referred to as

an equilibrium singular attractor . Understanding this phe-nomenon is useful for understanding the singular chaotic at-tractor under consideration. Before we do this, we rst col-lect some terminology that will be used throughout the paper.

From the expression xxcf = 1 1 /2 we see that the for-mation of the x-crash-fold pxcf takes place if, and only if,

1 1. Since 1 = a 1 / K , 1 1 means that the predator isable to reach half of its maximum predation rate at a preydensity a 1 smaller than the preys carrying capacity. For thisreason predator y is said to be predatory efcient if indeed itholds that 1 1. Whether predator y is predatory efcient ornot, is not essential for this paper. But it is the case for the z

predator, which will be discussed later. Predator y is said tohave a low predatory-mortality rate if it is predatory efcientand can actually crash the prey. That is, at the crash-fold state pxcf the predator can grow in per capita: g xxcf , yxcf , 0 0,which is solved as

xxcf = 1 1 /2 1 1 / 1 1 = xygt .

It is easy to see that the predators dimensionless mortalityrate 1 should be relatively small as part of the requirement.Predator y is said to have a high predatory-mortality rate if otherwise g xxcf , yxcf , 0 0.

The main reason that all singular orbits except for those

on the axes will converge to the xy-equilibrium point when y is high in predatory-mortality rate Fig. 2 a is counterin-tuitive in the case where singular orbits have a part on theaxes. This in fact gave rise to a classic phenomenon in sin-gular perturbations refer red to as Pontryagins delay of lossof stability PDLS .2125

2. Bust and boom dynamics Take the case of singular orbits, which have a part on the

y axis. More specically, above the transcritical point0 , yxtc , the trivial equilibrium point x=0 of the x-fast equa-

tion is stable and below it; it is unstable. The PDLS phenom-enon deals with the manner by which fast singular orbits jump away from the unstable trivial branch of the preynullclines. In practical terms, the predator declines in a diresituation of depleted prey x 0, followed by a surge in thepreys recovery after the predator reaches a sufciently lowdensity to allow it to happen. The mathematical question ishow the critical predation density is determined. More spe-cically, let x , y t be a solution to the perturbed equationwith 0 and an initial a , b satisfying 0 a xxcf and babove the ascending threshold pxat Fig. 2 a . In the singularlimit =0, the orbit converges to a concatenation of two fastorbits and one slow orbit. The rst fast orbit on y = b col-

lapses onto x=0; the second on y = c for some c explodestoward pxdc . According to the theory see Refs. 19, 23, and




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26 for a derivation , the critical amount c that allows x torecover is determined by the following integral equation:

c

b f 0, s,0

sg 0, s,0ds = 0 .

By substituting s = y t , y 0 = b, the slow orbit on the trivialbranch x=0, and observing that ds / sg 0 , s , 0 = dt , the

equation changes to T c0

f 0 , y t , 0 dt =0, where T c is the cor-responding duration of ight y T c = c. Because f is the percapita growth rate of the prey, which has opposite signsaround the transcritical point yxtc, and because the integral of f represents the accumulative per capita growth during thetransition, this equation simply means that the critical point cis such that the preys accumulative per capita growth rateover the growing phase y yxtc cancels it out over the de-clining phase y yxtc. This is the so-called PDLS phenom-enon and the critical amount c is denoted by yxpd = c. ThePDLS quantity yxpd depends on the initial b, which will betaken as the x-crash-fold level b = yxcf most of the time, fromnow on. The case illustrated is for the type of transcriticalpoints at which the fast variable goes through a phase of crash-recovery outbreak. All PDLS points of our xyz modelare of the crash-recovery-outbreak type. We can now seefrom the phase portrait Fig. 2 a that all singular orbits notfrom the axes will converge to the coexisting xy-equilibriumpoint.

The case of low predatory mortality xygt xxcf is illus-trated in Fig. 2 b . In this case, all singular orbits not fromthe x axis will converge to the singular attractor of the limitcycle ABCD , for which B is the crash-fold point and D is thePDLS point. Its existence owes to the property that thepredator can still grow at the crash-fold point xygt xxcf . We

point out that both the equilibrium and limit cycle singularattractors will persist for small 0 1 because of theirhyperbolicity see Refs. 13 and 16 for references on thisquestion as well as the geometric method of singular pertur-bations in general . We will denote throughout this paper thelanding point A of the x-fast orbit from D by pxpd = A andrefer to it as the PDLS point or the PDLS capacity point, and D the PDLS outbreak point instead to make the distinction.

3. Singular dynamics with stoichiometric limitation

We now consider the stoichiometrically mediated xz sys-

tem with y =0: dx / dt = xf x, 0 , z , dz / dt = z E x, 0 , z x / 2 + x 2 , where

E x, y, z = min 1,1 1 y 2 z

x.

The x-fast subsystem has the same structure as the xy sub-system when y is substituted by z. However, unlike the xysubsystem the rst condition we impose on the xz subsystemassumes the following.

Condition 2 . z is predatory efcient 0 2 1.Under this condition, there is a transcritical point pxtc, a

crash-fold point pxcf , and a PDLS point pxpd , respectively.

The main difference lies in the z-slow subsystem, in particu-lar the z nullcline.

The trivial branch z=0 is always there as with any othersubsystem. The nontrivial branch g x, 0 , z = E x, 0 , z x / 2+ x 2 can be very different due to the stoichiometric factor E x, y , z , which is a variable reproductive efciency factor.It divides the phase space into two parts: the subsaturationP: C region E x, y , z 1 and the saturation P: C region E x, y , z =1. The dening relations are equivalent to 1 1 y 2 z / x 1 and 1 1 y 2 z / x 1, respectively. Re-call that the quantity 1 1 y 2 z / x corresponds to the di-mensionless P:C ratio for the growth of predator z. Moreimportantly, being 1 of the ratio means that the predatorsgrowth rate is below optimal for not having enough phos-phorus or too much carbon. The line 1 1 y 2 z / x=1 isthe boundary of these two regions and we call it the ef-ciency boundary and denote it by pzeb; see the dashed line inFig. 2 c through the point xzeb , 0 ,0 with xzeb y= z=0 =1/ .It is easy to see the ecological interpretation: below the ef-ciency boundary, x in carbon is relatively small and there ismore than enough phosphorus to give z a saturated efciencygrowth. The efciency boundary decreases in x as z increasesbecause more consumer z means less nutrient in phosphorusper consumer and thus it takes less resource in carbon to ipthe consumer from a saturated efciency growth to a sub-saturated efciency growth.

In the saturated efciency region E =1, the nontrivial znullcline is x= xzgt = 2 2 / 1 2 provided xzgt xzeb =1/ ,which are solved from the equation g x, 0 , z = x / 2 + x 2 =0 with E =1. Similar to the predator y without stoichio-metric limitation, it represents the growing threshold for z:dz / dt 0 for x immediately greater than xzgt and dz / dt 0for x immediately smaller.

In the subsaturation efciency region E 1, the non-

trivial z nullcline is another line 1 2 z / 2 + x 2 =0.It is an x-supported capacity equilibrium for the z equation:below it dz / dt 0 and above it dz / dt 0. It decreases in z as x increases because it is in the subsaturated efciency regionfor which more carbon in the resource is less desirable,meaning low P:C ratio for the consumer. For a similar rea-son as for pxdc we call it the descending capacity of the zpredator and denote it by pzdc . The x intercept is xzdc y= z=0=1 / 2 2, which is automatically xzeb y= z=0 =1/ be-cause of the same condition xzgt = 2 2 / 1 2 xzeb y= z=0=1 / .

The signicance of this branch of the z nullcline lies in

the fact that as x increases across the branch, z experiences asubsaturation efciency growth dz / dt 0 , E 1 to a com-plete halt on the branch to a subsaturation efciency declinedz / dt 0 , E 1 all because the resource carbon x becomes

critically excessive. Because pzgt is a vertical line and pzdcdecreases in x as z increases, they must intersect, and theintersection is the maximum point of the z nullcline, denotedby pzcf , called the capacity fold point. In fact, both intersecton the efciency boundary pzeb and a simple calculationyields pzcf y=0 = xzgt , 0 , 1 xzgt / 2 . The region to the leftof the growing threshold pzgt is referred to as theP(phosphorus)-excessive region or the C (carbon)-poor re-

gion . The region between pzgt and pzdc is referred to as theP:C- balanced region . The region to the right of the descend-




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ing capacity pzdc is referred to as the P- poor region orC- excessive region . The P:C-balanced region is the onlyregion where z can grow dz / dt 0 .

The nontrivial z nullcline may or may not intersect thenontrivial x nullcline. More specically, the growing thresh-old may not intersect the x nullcline more than once since itis a vertical line segment and for the latter z is a function of x. We will assume for this paper the following.

Condition 3 . The unique existence of an xz -equilibrium point pxzdt on predators growing threshold and preys as-cending threshold: x zgt xzeb , xzgt xxcf , f xzgt , 0 , zzcf 0.

Here the rst inequality guarantees that the existence of predators growing threshold is in the saturated efciencyregion. The second inequality is needed for the point to lie on xs ascending threshold. The third inequality indeed guaran-tees such an existence because zs capacity-fold point pzcf lies above the x nullcline.

Because the nontrivial x -nullcline pxat pxdc is a pa-rabola and zs descending capacity curve pzdc is a negative-sloped line, they can intersect at most at two points over theinterval 0 x 1. If there is no intersection, then the singularorbit structure for the xz system is qualitatively the same asthe xy system without the stoichiometric effect. The originalsystem with the parameter values discussed in Sec. II be-longs to the case of having two intersections of pxdc and pzdc.However, the mechanism for chaos generation is almostidentical to the case of having just one intersection, which isthe case we will consider from now on because of its relativesimplicity. The precise condition for this case is as follows:

Condition 4 . The unique existence of an xz -equilibrium point on predators and preys descending capacities: xzdc y= z=0 =1/ 2 2 1 , g xxcf , 0 , zxcf 0.

The rst inequality guarantees that there is only one xz-equilibrium point on zs capacity pzdc and the second in-equality makes sure the point is also on xs capacity pxdcwhen z is low in predatory-mortality rate. We note thatwhether there is one or two xz-equilibrium points on preda-tors descending capacity pzdc , the important point is that partof the x-carrying capacity pxdc must lie in predators P-poorregion in which dz / dt 0. It is this unique stoichiometricproperty that ultimately determines the existence of the sin-gular chaos attractor under consideration as we will demon-strate later.

The case satisfying all the conditions 14 so far is

illustrated in Fig. 2 c . Both xz equilibrium points are un-stable for the z subsystem. The singular attractor has twogeneric kinds. The equilibrium point pxzdc = pxdc pzdc is al-ways repelling. If it lies above the PDLS point, pxpd , allsingular orbits converge to the predator-free x-capacity equi-librium point 1 ,0 . In this case, the P-poor x-capacity regionis substantial to be dynamically distinctive from thestoichiometry-free xy subsystem. If it lies below pxpd , allsingular orbits above the line z= zxzdc converge to a singularlimit cycle through the x-crash-fold point, and all singularorbits below the line z= zxzdc converge to the predator-free xequilibrium. The latter case qualitatively resembles the xy

dynamics as if there is no stoichiometric limitation on z aslong as solutions do not start below the line z= zxzdc . In other

words, the P-poor x-capacity region is too small to make asubstantial difference. It is the former case that we will con-sider in this paper as formalized below.

Condition 5 . The P-poor x -capacity region is relativelysubstantial to capture the PDLS capacity point: z xpd zxzdc .

We end this section by commenting on whether or notthese conditions can satisfy simultaneously. Note rst, thatthe singular parameters , have nothing to do with the

nullcline structures. So condition 1 does not affect the ques-tion one way or another. Second, the remaining conditions sofar concern the xz subsystem and parameters 2 , 2 , 2 , only. The number of dening equations is 4: 2 1, xzgt

xzeb, xzdc y= z=0 1, g xxcf , 0 , zxcf 0 with the last imply-ing f xzgt , 0 , zzcf 0 . The number of parameters is also 4.Generically, the equations dene a nonempty parameter re-gion. For example, making 2 small will make the secondinequality hold. Making large will make the third inequal-ity to hold. Finally, decreasing 2 and 2 if needed willmake the z-capacity-fold point pzcf to slide up on the verticalline x= xzgt, hence making the last inequality to hold.

D. Multi-time-scale analysis: 3D case

We are now ready to consider the full system. Figure 3highlights the essential congurations of the nullclines thatgive rise to the singular chaos attractor. We will build up the3D congurations by piecing together properties from thelower-dimensional subsystems as building blocks. For ex-ample, on the y =0 section, the illustration simply reproducesthat of Fig. 2 c for the xz subsystem. Similarly, on the z=0 section, it reproduces that of Figs. 2 a or 2 b . To ll inthe remaining space between these coordinate planes, we

will use the xz subsystems parametrized by variable y tosweep the effective rst octant.We rst describe the full nontrivial x nullcline, which

now is a cylindrical parabola-like surface, f x, y , z =0. At y=0 we already have its full description as a parabola goingthrough 1 ,0 ,0 with the crash-fold point pxcf , the transcriti-cal point pxtc, and the PDLS point pxpd due to condition 2. Allthese points will continue as parametrized by y. For example,increasing the density of y a little reduces the resource xoriginally allocated to z, and consequently it only takes asmaller amount of z to crash x. Therefore the continued x-crash-fold curve pxcf decreases in z as y increases. This iseasily seen when it is projected down to the yz plane as inFig. 3 b . This conclusion also follows by analyzing thedening equations f x, y , z =0, f x x, y , z =0 as in Ref. 16.The same yz-reciprocal relation also holds for the x-transcritical line pxtc, which is explicitly expressed as 1 y / 1 z / 2 =0. The regions bounded by these curves andthe coordinate planes, respectively, are the x-capacity surface pxdc : f x, y , z =0, f x x, y , z 0, which decreases in x as y or z increases; the x-threshold surface pxat : f x, y , z=0, f x x, y , z 0, which increases in x as y or z increases.

Last, the PDLS outbreak points are dened by two inte-gral equations, one for each of the predator variables. Again,the projected points on xs descending capacity surface pxdcare denoted by pxpd , referred to as the PDLS points as before.For the type of nullcline congurations we need, we want the




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PDLS curve to decrease in z when y increases, and viceversa. This reciprocal relation holds at the parameter values

1 = 2 =1, which can be directly veried that the corre-sponding transcritical line f 0 , y , z =1 y z=0 and thecrash fold f x, y , z =0, f x x, y , z =0 coincide. Hence, thePDLS curve coincides with the transcritical line and thecrash fold as well, for which the wanted reciprocal relationholds. Since the equations are innitely smooth in the satu-rated efciency region E =1, where the PDLS jump is de-

ned, the same reciprocal relation holds for i near 1. Weformalize this by the following condition.

Condition 6 . The PDLS curve on the x -capacity surfacedecreases in z as y increases .

We consider next the y nullcline, which is a plane, x= xygt , parallel to the yz plane. Because singular orbits willeventually visit either the x-trivial branch x=0 or the capac-ity surface pxdc , it is important to consider its intersectionwith the nontrivial x nullcline. Its intersection with xs as-cending threshold pxat has little consequence to this paper; itsintersection with the descending capacity pxdc does. The in-tersection is a curve continued from the xy-equilibrium pointfor the xy subsystem. Its projection to the yz plane is a line,since it is dened by f xygt , y , z =0 for which y and z arereciprocally and linearly related. It represents a descendingcapacity or threshold for y according to our terminology con-vention: For xed z it attracts the y solutions if it is on xsdescending capacity pxdc , and it repels the y solutions if it ison xs ascending threshold pxat. In either case, it decreases in y as z increases due to zs competitive pressure see Figs.3 b and 3 c . Part of the curve should be denoted by pxydcwith the subscript standing for the y-equations descending

capacity equilibrium on xs carrying capacity pxdc and theother part by pxydt , interpreted accordingly. However, thisner distinction will not be used further and for this reasonwe will use pygt and pxydc interchangeably, should it causelittle confusion. We will come back for more discussion onwhere to position this curve to generate the singular chaosattractor we are after.

We now consider the z nullcline, which has twobranches: the degenerate, unstable branch of the growingthreshold pzgt, and the stable, descending capacity branch pzdc . The former lies in the saturated efciency region E =1and the latter in the subsaturation region E 1. Without com-

petitor y, it intersects the x nullcline at two equilibriumpoints by conditions 24. These two points continue as acurve, each as y increases. By making 1 small enough wecan make the intersection curve with zs growing thresholdinitially sufciently away from xs crash-fold curve so thatwe do not need to deal with its inconsequential role later. Theimportant feature of the z nullcline instead is the intersectionof its descending capacity pzdc with xs descending capacity pxdc . Its projection to the yz plane is h y , z , y , z= 1 1 y 2 z / 2 + y , z 2 =0, with E 1 and x= y , z solved from f x, y , z =0. Using implicity differen-tiation, it is easy to show that for i / modestly large y and z are reciprocal in relation on the curve by using the fact that E 1 , y 0 , z 0. It is actually easier to see this ecologi-cally. The intersection represents an unstable threshold equi-librium for the slow z equation with y xed on xs capacitysurface pxdc . This is the case when y =0 by condition 4 forwhich more of x means more excessive carbon for z, thusdriving down zs density further from the equilibrium, givingrise to an unstable, threshold equilibrium. With a small in-creasing in y, the excessive amount of x is removed by yaccordingly, becoming less excessive for z. Hence it takes agreater x amount to drive down z. That is, zs descendingcapacity is reduced on pxdc . The condition that i is largesimply guarantees that this z descending threshold continues

down to z=0 as y increases. By our terminology convention,it is denoted by pxzdt , standing for zs descending threshold

FIG. 3. Color online a A 3D view of the nullclines satisfying all theconditions 18, and a typical singular orbit. b The projected view onto the yz plane and the corresponding -singular return map with 0 1. cThe -limiting view of the perturbed case b . Marker designation: Open

circles are xyz equilibrium points; half-lled circles are transcritical points;open squares are PDLS points; lled-squares are junction-fold points.




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on xs capacity surface, even though the original branch for zis the descending capacity. We will use pxzdt and pzdc inter-changeably since our analysis is entirely framed on xs ca-pacity surface. The descending rate depends on ys dimen-sionless P:C ratio 1: the higher the ratio is the faster thedescent becomes. This is an important property to be usedbelow. In addition, the intersection of pxzdt with the trivial znullcline z=0 is a transcritical point pxztc as shown in Fig. 3,

for which the PDLS phenomenon takes place for the sin-gularly perturbed yz system on the resource capacity surface;seeFig. 3 c for its ramication.

One of the key congurations we are after is for zs x-supported descending threshold pxzdt to intersect xs PDLScurve pxpd . By condition 5, zs P-poor, x-capacity regionshaded in Fig. 3 captures the PDLS point when y =0, i.e.,

zxpd y=0 zxzdt y=0 . For the intersection to take place weonly need to show that the other end of the PDLS curve pxpdat z=0 is outside the region or to the opposite side of itsboundary pxzdt , i.e.,

yxzdt z=0 yxpd z=0 . 7

This indeed is the case because of the following. First wechoose 1 large enough so that the transcritical point pxztc z=0 lies below the x crash fold on the y axis, i.e., yxzdt z=0 yxcf z=0 . To be precise, we can rst pick 1, sothat yxztc z=0 1 + s 2 /4 with s =0 and with the right hand of the inequality from the expression of yxcf z=0 = 1+ 1 2 /4,which is independent of 1. After xing this 1, we thenincrease 1 i.e., decrease the predatory efciency of y , if necessary, so that the y end of xs transcritical point pxtc is

pushed towards xs crash-fold point. As a result, the y end of xs PDLS curve is pushed closer to xs crash-fold point. Infact, at 1 =1 the y end of the crash-fold point pxcf , the tran-scritical point pxtc, and the PDLS point pxdp , all merge as onepoint for the xy subsystem because xxcf z=0 = 1 1 /2, andthe relation yxzdt z=0 1/ 4 yxcf z=0 = yxtc z=0 = yxpd z=0 =1,holds. It will hold for a range of 1 away from 1 =1 bycontinuity, and hence the relation 7 will hold. Note furtherthat by increasing 1 further we can make sure the intersec-tion is unique because we can make the descending rate of the threshold curve pxzdt faster than the descending rate of the PDLS curve pxpd condition 6 . This property is formal-ized as follows.

Condition 7 . The PDLS curve p xpd intersects zs x-supported descending threshold p xzdt at a unique point, de-noted by p xzxpd = pxzdt pxpd , in the manner so that (7) holds(see Fig. 3).

One important consequence of this conguration is thatin zs P-poor region on xs capacity surface z decreases whenstarted from that part of the PDLS curve, whereas in zsP:C-balanced region on pxdc , z increases when started fromthis part of the PDLS curve. It is this dichotomy that essen-tially drives the -singular orbits to chaos as we demonstratenow.

Figure 3 a illustrates this -singular chaos in a prelimi-

nary way. The initial point of the yz-slow orbit 1 is on thePDLS curve pxpd and zs descending threshold pxzdt on the

x-capacity surface. Hence it develops parallel with the y axisinitially, away from zs P-poor region and into itsP :C-balance region, in which it moves up and hits the x-crash-fold line in a nite time. At that point the x-fast orbit2 takes it to the resource depleted branch x=0, followed bythe yz-slow orbit 3, which decreases in both y and z, to xsPDLS outbreak jump line. The x-fast orbit 4 then takes it tothe PDLS curve on xs capacity surface again. Assume it

lands left, where it started from orbit 1; that is, in zs P-poorregion a condition that guarantees this assumption will begiven in the next section . Then the yz-slow orbit 5 will rstdecrease in z before horizontally crossing the P-poor regionsboundary pxzdt and then increases in z in the P:C-balancedregion until hitting xs crash-fold again. This is to be fol-lowed by singular orbits 6, 7, 8, 9, and so on. Again, assumethe fast x-PDLS orbit 8 lands left where it started from orbit5 and the same for all subsequent singular cycles. Then, atrain of such singular oscillations in the manner of Fig. 2 bdevelops and moves leftward until hitting the P-poor regionagain, switching its direction southward like orbit 5 to start

another round of such bursts of oscillation. The task thatremains is to make this description precise.

E. Singular return maps

We will use Poincar return maps to capture the dynam-ics of the singular attractor. Since the -singular dynamics is2D, consisting of orbits of the reduced 2D, , slow yz equa-tions on the x-capacity surface pxdc and the trivial surface x=0, and orbits of the 1D, -fast x orbits, the return map is of 1D, which can be dened on any curve on the x-capacitysurface pxdc or the trivial surface x=0 that is transversal to

the reducedyz

-slow ows. For deniteness we will take the x-crash-fold curve as the domain of the return map, and de-note the domain by I and the return map by , which is dependent. The domain I can be thought as an interval, pa-rametrized by the y coordinate on the fold, and the returnmap as yn+1 = yn for yi I . It is easier to describe thereturn map by looking into the x axis and view the 3D struc-ture of Fig. 3 a in its projection to the yz plane as in Figs.3 b and 3 c . The return map is a unimodal map as de-picted in Fig. 3 b with one minimum point at y = b as theunique critical point. To the left of b, is decreasing and tothe right it is increasing but below the diagonal line yn+1= yn. This property is due to the existence of a junction fold point, 13 which by denition is on the PDLS curve pxpd and atwhich the reduced yz-vector eld is tangent to the PDLScurve. A sufcient condition for the existence of such a pointis the following.

Condition 8 . Let the x -supported y nullcline be p xydc= pygt pxdc . Then it lies above the point p xzxpd , dened fromcondition 7, i.e., dy / dt 0 at p xzxpd , and if p xydc intersectseither p xpd or p xzdt then it does so that y xzdt z=0 yxydc z=0

yxpd z=0 ; see Fig. 3.A geometric paraphrasing of this condition goes as fol-

lows. That dy / dt 0 at pxzxpd holds automatically for 1 =0because pxydc coincides with pxtc and because of condition 7.

Increasing 1 pushes the curve pxydc closer and closer to thePDLS point pxzxpd on pxzdc from above, maintaining the re-




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lation dy / dt 0 at pxzxpd . By making the x supported znullcline pxzdt steeper and pxpd atter if necessary we canmake pxydc approach the point pxzxpd in the way required bycondition 8 for all 1 from an interval 0 , 1

* for which pxzxpd pxydc when 1 = 1

*. In fact, as 1 tends to 1* from

below, pxydc must intersect pxzdt from some 1 value on andthe intersection is a unique unstable xyz-equilibrium point pxyz = pxydc pxzdt . This point approaches pxzxpd from above.

As a result the return map must goes through a cascade of period-doubling bifurcations to chaos. The precise statementis as follows.

Theorem 2 . If conditions 18 are satised, then for =0 and each 0 1 a cascade of period-doubling bifur-cations must take place for the perturbed return map as

1 changes over a subinterval of 0 , 1* in which the

x-capacity supported y nullcline p xydc = pygt pxdc lies above pxzxpd , and pxyz pxzxpd = O 1/ ln .

A proof can now be constructed by modifying the proof of Theorem 6.1 of Ref. 13 or the proof of Theorem 1.1 of Ref. 17. In fact, the geometric conguration established byconditions 18 as well as the manner by which the bifurca-tion takes place is qualitatively identical to both cited theo-rems. Here below we demonstrate why is a unimodal mapas we claimed in the last section.

Note that below the curve pxydc , dy / dt 0 and above it,

dy / dt 0. It is easy to show now the existence of a junction-fold point on the PDLS curve pxpd by the intermediate valuetheorem. Specically, at the y =0 end of the PDLS curve, the yz-vector eld points to the down side of the PDLS curve. Incontrast, at pxzxpd , the pxzdt end of the PDLS curve, the yz-vector eld is horizontal, pointing to the opposite side of the PDLS curve because the point lies in the dy / dt 0 re-gion by condition 8 above and the fact that pxpd has a nega-tive slope everywhere by condition 6. Hence there must be

FIG. 4. Color online Parameter values for a and b : =0.01, =0.12, 1 =1.1, 2 =0.33, 1 =0.368 403, 2 =0.25, 1 =0.2, 2 =4/6, =20/6. a A 3Dphase portrait. b The Poincar return map in coordinate y when z reaches a local maximum. c A bifurcation diagram for the same parameter value exceptfor a range of 1 from 0.35,0.372 .




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such a point q in between inside the P-poor region at whichthe yz-vector eld is tangent with the PDLS curve, pointingto neither side of pxpd .

The existence of the junction-fold point q implies thefollowing. For points on the PDLS curve and left of q , the yzorbit moves below the curve rst and then across it on theright of q. For PDLS points right of q , the yz orbit movesabove the curve, never intersecting it again before hitting the x-crash-fold curve pxcf . This effect leads to the unimodalproperty of the return map at the minimum point b. To bemore specic, point b is chosen so that its -singular orbitlands precisely on the junction-fold point q . Its returningimage is a . Following any two points left of b , they all returnto I with their relative position reversed because their yz-slow ows swing them half turn about the junction-foldpoint. Because the return map reverses the orientation of the subinterval left of b, it is monotone decreasing in thesubinterval. In contrast, for any pair of points right of b, theirimages by the return map preserve their relative orderingbecause of the absence of a half twist by the yz ow aroundthe junction-fold point q. Hence the map is monotone in-creasing on the right subinterval of b , implying that b is aglobal minimum for .

The reason that the minimum point b , a = b , b liesbelow the diagonal for each 0 1 so that pxyz pxzxpd O 1/ ln can be easily seen at the end point

1 = 1* at which pxyz = pxzxpd and the junction fold q coincides

with pxzxpd as well. As a consequence, all the yz orbits com-ing out right of the point q drift leftward without exception,so that the corresponding image of is lower than its pre-image, i.e., the graph of over the right of q = pxzxpd liesbelow the diagonal. All points left of q = pxzxpd return to the

right of the point.On the other hand, when 1 is below and away from 1

*,the junction-fold q moves into the P-poor region and part of the subinterval right of q falls in the region where dy / dt

0, just like the left subinterval always does. The yz solu-tion from q will rst drift right, go vertical on pxydc , and thendrift left. Thus, if it accumulates more rightward drift thanleftward drift, the minimum value of a = b will lie abovewhere it started, i.e., b a . In this situation, a stable xedpoint emerges in the right subinterval of b , giving rise to thestarting cycle of the period-doubling cascade. This will hap-pen for 1 sufciently below 1

* so that pxyz pxzxpd

= O 1/ ln .Finally, the proof of the period-doubling cascade from

the cited references was done by the theory of kneading se-quences. In a nutshell, we calculate the symbol sequencekneading sequence of the minimum point b . Specically, in

one extreme when 1 = 1*, the symbol sequence is LRR . . . ,

with L for the left interval and R for the right interval of b,and at least one R after L. In the other extreme when pxyz pxzxpd = O 1/ ln for 1 sufciently below 1

* forwhich has an attracting xed point in the right subintervalas explained above, the kneading sequence of b is R = RR. . . ,a sequence of repeating Rs. From these two properties it is

sufciently to conclude by the theory of unimodal maps thatin the intervening interval of 1 all the kneading sequences

characteristic of the period-doubling cascade must take placeand in the order prescribed by the cascade. Figure 4 gives anexperimental validation to the scenario.

IV. DISCUSSION

We analyzed a chaos generating mechanism in a system,where two predators compete for one prey. A distinct featureof the system is its ability to track both prey quantity andprey quality expressed as P:C content, i.e., stoichiometry, of prey . The variation in prey quality results in an unusual,hump-shaped consumer null surface. Yet, the particular inter-section of all three species null surfaces considered is quali-tatively similar in conguration to a type found in a simpletritrophic food chain model for which at least four distincttypes o f chao s generation mechanisms have beenclassied. 13,18 20 This type is the simplest kind resulted fromthe existence of a junction-fold point on the consumer-mediated prey capacity. We nd that both consumers cancoexist on one prey in an attractor that, as we have shown

both numerically and analytically, is chaotic. The chaoswould not be possible without the stoichiometric mediationwhen prey quality negatively limits the growth of at least oneconsumer. However, other chaos generating mechanisms arepossible in this stoichiometric competition model.

We note that the system exhibits stabilization mecha-nisms in which chaotic attractor collapses to a stable equilib-rium. The manner by which this stabilization occurs isinterestingwhen a predator becomes more efcient withlowering death rate. We know that for a tritrophic food chainmodel, top consumers reproductive ef ciency plays an im-portant role in similar stabilizations. 27 Likewise, we should

expect the same stabilizing principle to prevail in the consid-ered model. That is, increasing the relative efciency rates and of the consumers against the prey should change thechaotic oscillations to small periodic oscillations and even-tually to coexisting equilibrium states. Proving this stabiliza-tion principle as well as classifying all stoichiometricallygenerated chaos remains to be worked out.

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Competitive Coexistence in Stoichiometric Chaos

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