An Introduction to Scaling, Spatial and Evolutionary Modelling in Ecology Evolutionary Modelling in Ecology CANG HUI Centre for Invasion Biology Department of Botany & Zoology Stellenbosch University E‐Mail: [email protected]
An Introduction to Scaling, Spatial and Evolutionary Modelling in EcologyEvolutionary Modelling in Ecology
CANG HUICentre for Invasion Biology
Department of Botany & Zoology
Stellenbosch University
E‐Mail: [email protected]
‘WHY’ is a more important question than ‘HOW’ in science.p q
I was like a boy playing on the sea‐shore, and diverting myself now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.
Isaac Newton
Hui C ‐Modelling in ecology 2
Scientific Research Procedure: Observe, Explain & Apply
1. Pattern recognition; 2. Pattern explanation; 3. Application
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Hui C ‐Modelling in ecology 3
Scaling Patterns in Ecology: What is scaling & Why we study scaling patterns?
Ecological measurements (species distributions and interactions) are hardly random, i.e. they are spatially and temporally (auto) correlated. Independence and
d ft f ll t ti ( d l) f th b d t t drandomness often forms a null expectation (model) of the observed structures and patterns.
Scaling patterns describe how ecological measurements change across scales (i.e. unit size, resolution, grain and extent of measurements). g )
The problem of relating phenomena [patterns] across scales is the central problemThe problem of relating phenomena [patterns] across scales is the central problem in biology and in all of sciences (Levin 1992 Ecology).
Hui C ‐Modelling in ecology 5
Pattern 1. Fractal shape of natural objects.
Example 2: Fractal landscape
Example 1: The length of British coastal lineExample 1: The length of British coastal line
A fractal landscape generated by the mid‐point algorithmHui et al. (2010) In: Fifty Years of Invasion Ecology. Wiley‐Blackwell
Example 3: Fractal plant life‐forms
Mandelbrot (1967) Science
Hui C ‐Modelling in ecology 6
Fractal trees. Hui (2008) Unpublished
Pattern 2. Multi‐scale processes and interactions
Example 4: Ecological processes in forests
Example 5: The impact of invasive plants on species richness in Mediterranean‐type ecosystems
Holling (1992) Ecological Monographs
Gaertner et al (2009) Progress in Physical Geography
Hui C ‐Modelling in ecology 7
Gaertner et al. (2009) Progress in Physical Geography
Pattern 2. Multi‐scale processes and interactions
Example 6: Ecological and environmental determinants of the distributions of Argentine ants (Linepithema humile) at local, regional and global scales
L l 125 t t R i lLocal: 125m transects
Aggregation difference of Argentine antsSpecies turnover of native ants
Regional:
Topographic aspectWinter temperatureAnnual precipitation
Roura‐Pascual et al. (2010) Biological Invasions
Annual precipitation
Roura‐Pascual et al. (2009) Global Change Biology
Global:
Climatic suitabilityHuman footprint
Roura‐Pascual et al. (2011) PNAS
Hui C ‐Modelling in ecology 8
Pattern 3. Size does matter
E l 7 All t i li ( h l i E l 8 N k hi (Example 7: Allometric scaling (whole‐organism biomass production)
Example 8: Network architecture (e.g. nestedness and modularity)
Brown et al. (2004) Ecology Zhang et al. (2011) Ecology Letters
Hui C ‐Modelling in ecology 9
Pattern 4. Species distribution and co‐occurrence
Example 10: Area‐of‐occupancy
Example 9: Species‐area curves p p
Kunin (1998) Science
MacArthur & Wilson (1967) The theory of island biogeography. Princeton U Press.g g p y
2
Hui C ‐Modelling in ecology 10Hui & McGeoch (2007) Ecoscience
Grain m2
Pattern 4. Species distributions and co‐occurrence
Example 11: The scaling patterns of aggregation indices
Example 12: Species association
Hui (2009) Journal of Theoretical Biology
Hui C ‐Modelling in ecology 11Hui et al. (2010) Ecography
Model: A dissection of ecological models
Modelling core
Statistical models (e.g. ANOVA, GLM and BRT)Probability models (e.g. Bayesian and MaxEnt)Dynamic system (e.g. ODE and PDE)Simulations (e.g. IBM and CA)
Di l St t i
Demographic Processes
E i t l
Biotic Interactions
Environmental
Dispersal StrategiesEnvironmental Heterogeneity
Stochasticity
Range DynamicsModelling environments
Modelling output
Hui et al. (2010) In: Fifty Years of Invasion Ecology. Wiley‐Blackwell( ) y gy y
Hui C ‐Modelling in ecology 12
Model. Species‐range size distributions – A probability model
Core‐satellite hypothesis
Christen C. Raunkiær
(1860‐1938)
Ilkka Hanski
( )
60Raunkiear’s law of frequency:
Sampling artifacts from skewed
rank‐abundance distributions
40
50
species Sean Nee
20
30
he num
ber o
f s
A probability property of how species
distribute across scales
0
10
T (i.e. the scaling pattern of species distribution).
Hui C ‐Modelling in ecology 13
0~20% 21~40% 41~60% 61~80% 81~100%
OccupancyHui & McGeoch (2007) Theor Popl BiolHui & McGeoch (2007) OikosHui & McGeoch (2008) Ecology
Model. Species‐range size distributions – A probability model
Modifiable Areal Unit Problem (MAUP) ‐ Openshaw (1984)
1. The edge effect (i.e. the effect of the perimeter to area ratio)1. The edge effect (i.e. the effect of the perimeter to area ratio) will inflate the occupancy observed under conditions of spatial aggregation, and the density estimates will be lower in transects than in the same‐size square samples.
2. When the spatial autocorrelation is weak, the species Hui 2009 A Bayesian solution to the modifiable areal unit problem. In: Foundations of Computational Intelligence Vol 2. Springer‐Verlag.
p , pdistribution will be strongly scale‐dependent.
Hui C ‐Modelling in ecology 14Hui et al. (2006) Journal of Animal Ecology
Model. Species‐range size distributions – A probability model
Satellite species
Core species
Hui C ‐Modelling in ecology 15Hui & McGeoch (2007) Oikos
Model. Species‐range size distributions – A probability model
A Sudoku puzzle game
The magic square of Meleconlia IThe magic square of Meleconlia I
Hui C ‐Modelling in ecology 16Hui & McGeoch (2008) Ecology
Application 1. Extrapolating population size from the scaling pattern of occupancy
US: 5.66 billion3.78 billion (wintering)7.24 billion (445 breeding)
EU: 1.75 billion (434; EU25)SA: 1 94 billion (610; LLM)SA: 1.94 billion (610; LLM)
2.35 billion (610; BEM)
Hui C ‐Modelling in ecology 17Hui et al. (2009) Ecological Applications
Application 2. Defining optimal sampling effort for large‐scale monitoring of invasive alien plants
Spanish flagOrnamental
T V l
Prickly PearFruit or fencing?T V l
Common floss flowerLawn weed
Fl id C ibbTexas ‐ VenezuelaTexas ‐ Venezuela Florida ‐ Caribbean
Water cabbageAquatic weed
Nil L k Vi t i
Santa Maria FeverfewWeed (disturbed land)
T V l
Water hyacinthWastewater treatment
A b i
50 51 52 53 54 55 56
Species Records Abundance 95%CI 95%CI Moran's I OSEA OSED
Nile, Lake Victoria Texas‐Venezuela Amazon basin
Species Records Abundance 95%CI 95%CI Moran s I OSEA OSEDOpuntia stricta 20029 549243 25.877 30.499 0.0638 21.59 3.0Lantana camara 2332 144885 5.569 9.303 0.0356 35.60 24.7Chromolaena odorata 327 80234 2.271 5.965 ‐0.0007 48.66 81.1Pistia stratiotes 218 76485 2 089 5 762 0 0079 51 81 71 1
50
60
70
80
sive
spe
cies
Pistia stratiotes 218 76485 2.089 5.762 0.0079 51.81 71.1Parthenium hysterophorus 204 72736 1.903 5.563 ‐0.0006 52.63 114.4Eichhornia crassipes 199 78291 2.181 5.855 0.0111 51.76 64.0
10
20
30
40
Num
ber o
f invas
Hui C ‐Modelling in ecology 18Hui et al. (2011) Journal of Applied Ecology
0
10 11 12 13 14 15 16 17 18 19
Log2(Abundance estimation)
Application 3. Macroecology meets invasion ecology: linking native distribution of Australian acacias to invasiveness
Invasive species with low percolation exponent (high population growth rate)
Acacia dealbataAcacia melanoxylonAcacia implexaAcacia victoriaeA i l if liAcacia longifolia
Invasive species with large native range
Acacia melanoxylonAcacia strictaAcacia pycnanthaAcacia salicinaAcacia elata
Hui C ‐Modelling in ecology 19Hui et al. (2011) Diversity and Distributions
Past, Present & Future
Past:Spatial chaosCooperation evolutionNiche construction
Present:Macroecological patterns of Australian acaciasDispersal & range expansion of starlings & mynasE l i l t k ( t li ti & t i ti f d b )
Future:Universal laws of macroecologyAdaptive dynamics of polymorphismNi h ti f i d bi dNiche construction
Allee effectMetapopulation dynamicsEco‐epidmiology
Ecological networks (mutualistic & antagonistic, food webs)Plant invasions in the Kruger National ParkPlant‐soil‐SMC relationships in Fynbos and KarooFish invasion in the Sundays River irrigation networks
Niche conservatism of acacias and birdsAdaptive network dynamicsEcological dynamics on networksDispersal at the advancing range frontInheritable epigenetic variationsInheritable epigenetic variationsShift of range and dispersal strategy in a changing climate
And development from present researchAnd development from present research
Hui C ‐Modelling in ecology 20
A summary of scaling patterns…by William Blake
To see a World in a Grain of SandAnd Heaven in a Wild Flower,Hold Infinity in the palm of your hand And Eternity in an hour.
‐William Blake, from "Auguries of Innocence“ 1874From Wikimedia Commons
Hui C ‐Modelling in ecology 21
Allee effect
Ecological Imprint
Spatial Complexity
Niche Construction
Spatial Chaosp
Hui C ‐Modelling in ecology 23The eco‐epidemiological dynamics on fractal landscapesSu & Hui (in prep)
The effect of time lagged niche construction on metapopulation
Hui C ‐Modelling in ecology 24
Spread of pathogens under different dispersal capacitiesSu et al. (2008) Bulletin of Mathematical Biology
The effect of time‐lagged niche construction on metapopulation dynamics and environmental heterogeneityHan et al. (2009) AMC
Rising sea level in fractal landscape, generated by the mid‐point algorithmHui et al. (2010) In: Fifty Years of Invasion Ecology. Wiley‐Blackwell
Hui C ‐Modelling in ecology 25
Virtual Ecology
Simulated ecological environments by Maria C.R. Harrington (mariacrharrington.blogspot.com/)
A simulated plant communities; a simplified program of this simulation has been presented in the online attachment of Hui et al. (2010) E h
Hui C ‐Modelling in ecology 26
Ecography
An example of a spatially‐structured population (metapopulation)
Four things to learn for programming:1. Assigning values to variables
p p y p p ( p p )
cell state 1/0g g2. Define a loop3. Conditionals by logic 4 Display a matrix4. Display a matrix
t t+1
=1? YN
deathN Y
1 0
birth NY
1 0
Hui C ‐Modelling in ecology 28
c=0.5;e=0.2; 1. Assign values to model parameters
Do[{ m[i,j]=0;},{i,1,50},{j,1,50}];m[25,25]=m[25,24]=m[24,24]=m[24,25]=1;
Do[{m[0 i]=m[50 i];m[51 i]=m[1 i]} {i 1 50}];
2. Define initial conditions
Do[{m[0,i]=m[50,i];m[51,i]=m[1,i]},{i,1,50}];Do[{m[j,0]=m[j,50];m[j,51]=m[j,1]},{j,1,50}];
Do[{
3. Define boundary conditions
Do[{a=Random[];If[m[i,j]>0.5,{If[a<e {p[i j]=0} {p[i j]=1}]}
4. Define transition rules{If[a<e,{p[i,j]=0},{p[i,j]=1}]},{If[a<1.0c (m[i+1,j]+m[i‐1,j]+m[i,j+1]+m[i,j‐1])/4.0,{p[i,j]=1},{p[i,j]=0}]}];},{i,1,50},{j,1,50}];
{ } { } { }Do[{m[i,j]=p[i,j]},{i,1,50},{j,1,50}];
Do[{m[0,i]=m[50,i];m[51,i]=m[1,i]},{i,1,50}];Do[{m[j,0]=m[j,50];m[j,51]=m[j,1]},{j,1,50}]; 4.a2. Define boundary conditions
4.a1. Swap matricesA Loop
Do[{m[j,0] m[j,50];m[j,51] m[j,1]},{j,1,50}];
tu[t]=ListDensityPlot[Table[m[i,j],{i,1,50},{j,1,50}],InterpolationOrder→0,Frame→None,ColorFunction→GrayLevel];
5. Display
},{t,1,100}];
Export["C:\\test1.gif",Table[tu[t],{t,1,100}],"GIF"] 6. Output
Hui C ‐Modelling in ecology 29
p [ g [ [ ] { }] ] p
Adaptive Dynamics
Charles Darwin’s sketch of anCharles Darwin s sketch of an evolutionary tree from his “First Notebook on Transmutation of Species” (1837) is on view at the American Museum of Natural History This phylogenetic tree, created by David Hillis, Derreck Zwickil and Robin Gutell, depicts the
Hui C ‐Modelling in ecology 32
American Museum of Natural History in New York City.
p y g y pevolutionary relationships of about 3,000 species throughout the Tree of Life. Less than 1 percent of known species are depicted. Pennis (2003)
Can speciation or diversification happen in a single population?Can speciation or diversification happen in a single population?
Hui C ‐Modelling in ecology 33
Darwin’s Finches
The derivation of thirteen or more species on the Galapagos from a single l i h d f d li i f i l liancestral species was the product of repeated splitting of single lineages
into two or more non‐interbreeding lines of descent. How did these events take place?
Stresemann (1936) ‘one of separate populations of a single species on different islands evolving in different directions.’ (also Lack 1945)g ( )
Geographical opportunity: Archipelagos are suitable environments for speciation because they provide conditions for the establishment of manyspeciation because they provide conditions for the establishment of many populations of a species, and the opportunity for them to evolve independently.
‐ Grant & Grant (2008)
Hui C ‐Modelling in ecology 34
Evolutionary branching & speciationDoebeli & Dieckmann 2000 Am Nat 156, S77‐S101.
Assumptions:(1) Ecological equilibiriummutations are sufficiently rare so that mutants encounter monomorphic resident populations that are at their ecological equilibrium. This corresponds to assuming a separation of ecological and evolutionary timescales, with the ecological dynamics occurring faster than the evolutionary dynamics. Under the further assumption that mutants whose invasion fitness is >0 not only can invade (with some probability) but also can replace the former resident and thus become the new resident, it is possible to study the evolutionary dynamics by analyzing a function f (y, x) describing the invasion fitness of a mutant y in a resident population x. Here x may be a multidimensional vector, either because the trait under studyh h b h h l d l d h f ll lhas more than one component or because there are more than one species involved. Evolutionary dynamics then follow selection gradients determined by derivatives of the invasion fitness function f (y, x).
The definition of invasion fitnessM t Ni b t & G it 1992 TREE 7 198 202Metz, Nisbet & Geritz 1992 TREE 7, 198‐202
Phenotypes of special interest are those where the selection gradient is 0, and the first question is whether these points actually are evolutionary attractors. In classical optimization models of evolution, reaching such attractors implies that evolution comes to a halt because evolutionary attractors only occur at fitness maximabecause evolutionary attractors only occur at fitness maxima.
When frequency‐dependent ecological interactions drive the evolutionary process, it is possible that an evolutionary attractor represents a fitness minimum at which the population experiences disruptive selection. Adaptive dynamics takes these analyses one step further by asking what happens after the fitness minimum has been reached.asking what happens after the fitness minimum has been reached.
(2) Quantitative traitsThe quantitative characters influencing ecological interactions are determined by many additive diploid loci.
(3) Assortative mating in sexual populationsThe framework of adaptive dynamics has so far mainly been developed as an asexual theory that lacks population genetic considerations. In fact, there is a valid caveat against considering evolutionary branching in asexual models as a basis for understanding aspects of speciation in sexual populations, branching could be prevented by the continual production of intermediate offspring phenotypes
Hui C ‐Modelling in ecology 35
through recombination between incipient branches. If mating is random, this indeed is the case. However, if mating is assortative with respect to the ecological characters under study, evolutionary branching ispossible in sexual populations and can lead to speciation.
⎟⎠⎞
⎜⎝⎛ −⋅=
KN1Nr
dtdN
10
⎠⎝ Kdt
t)N(x, N(t)→
8
4
6
ulati
onsiz
e
2
4
Popu
0 10 20 30 40 500
r=0.2; K=10
0 10 20 30 40 50T ime
Hui C ‐Modelling in ecology 36
⎟⎞
⎜⎛ t)(x,Nt)dN(x,
⎟⎟⎠
⎞⎜⎜⎝
⎛−⋅=
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dtt)dN(x, e
⎟⎟⎞
⎜⎜⎛ +−⋅=
t)N(y,αt)N(x,α1t)N(x,rt)dN(x, xyxx
⎟⎟⎠
⎜⎜⎝ K(x)1t)N(x,r
dt
Hui C ‐Modelling in ecology 37
⎟⎟⎞
⎜⎜⎛ +−⋅=
K(x)αt)N(y,α1t)N(yrt)dN(y, yxyy
Invasion Fitness Landscape at an Evolutionary Singular
⎟⎟⎠
⎜⎜⎝ K(y)1t)N(y,r
dt
⎟⎞
⎜⎛ K(x)αt)dN(y, yx an
t y
+_
ant y
ant y
+_
⎟⎟⎠
⎞⎜⎜⎝
⎛−⋅=
K(y)( )
1t)N(y,rdt
t)dN(y, yx
Mut
a
+
_Mut
aM
uta
+
_
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
K(y)K(x)α
1rx)f(y, yx Invasion FitnessResident xResident xResident x
_+
_+⎠⎝ (y)
x)f(y,g(x)∂
∂= Selection Gradient M
utan
t y
+Mut
ant y
Mut
ant y
+
xyy =∂
If g(x*)=0, we call x* an evolutionary singular Resident x
_
Resident xResident x
_
tant
y
_+
tant
y
_+
tant
yta
nt y
_+
0g(x)<
∂ 0x)f(y,2
<∂ 0x)f(y,2
>∂
Mu
_
+Mu
_
+Mu
Mu
_
+
Evolutionary Attractor
0x *xx
<∂ =
0y *xy
2 <∂
=
Fitness Maximum
0y *xy
2 >∂
=
Fitness Minimum
Hui C ‐Modelling in ecology 38
Resident xResident xResident xResident xEvolutionary Attractor Fitness Maximum Fitness Minimum
Evolutionary Branching
⎟⎟⎞
⎜⎜⎛ +−−
⎟⎟⎞
⎜⎜⎛
= 2
22α
22α
xyβ)σy(xexpβσexpα ⎟⎟
⎠⎜⎜⎝
⎟⎟⎠
⎜⎜⎝
2α
xy 2σe p
2e pα
⎟⎟⎞
⎜⎜⎛ − 2
0 )x(xKK( ) ⎟⎟⎠
⎜⎜⎝−⋅= 2
K
00 2σ
)(expKK(x)1.4
σα=0.6; β=0σα=0.6; β=1.5
1 6 β 01000
1.0
1.2
ngth
σα=1.6; β=0σα=0.6; β=‐1
800
0.8
etiti
onSt
ren
600
Capa
city
0.4
0.6
Com
pe
400
Carry
ing
0.2200
K0=1000; x0=0K0=1000; x0=1
3 2 1 0 1 2 30.0
Trait difference, x y0.0 0.5 1.0 1.5 2.0 2.5 3.0
0
Trait, x
Hui C ‐Modelling in ecology 39
⎟⎟⎞
⎜⎜⎛−=
K(x)α1rx)f(y yx
⎟⎟⎠
⎜⎜⎝ K(y)1rx)f(y,
⎟⎞
⎜⎛ −∂ xxx)f(y
⎟⎟⎠
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⎛−−=
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==
βσ
xxry
x)f(y,g(x) 2K
0
xy
2 0 2
K0 σβxx* ⋅+=
rg(x)∂ 1.5
2.0
0
σr
xg(x)
2K*xx
<−=∂
∂
= 1.0Trait
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
∂∂
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2α*
2
2
σ1
σ1r
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0.5⎠⎝= Kα*xyy
Kα σσ <
0 50000 100 000 150 000 200 0000.0
Time
Hui C ‐Modelling in ecology 40
Ad ti d i f lti l i
Evolutionary trap Evolutionary branching
Adaptive dynamics for multiple species
Red Queen dynamics – Type I Red Queen dynamics – Type II
Hui C ‐Modelling in ecology 41Zhang, Hui, Pauw (in prep)
Mathematics is biology's next microscope, only better. Conversely, mathematics will benefit increasingly from its involvement with biology.
‐ Joel E. Cohen
Hui C ‐Modelling in ecology [email protected]
Hui, C. (2011) An Introduction to Scaling, Spatial and Evolutionary Modelling in Ecology. Centre for Invasion Biology, Department of Botany and Zoology, Stellenbosch University.
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Photos of species in slides 18 to 20 are public domain pictures collected from internet. Copyright of these pictures reserve by their owners.