An Introduction to Scaling, Spatial and Evolutionary ... Introduction to Scaling, Spatial and...

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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PART I: Scaling Patterns in Ecology


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).


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


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 Hui (2008) Unpublished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


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


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


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


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).


0~20% 21~40% 41~60% 61~80% 81~100%

OccupancyHui & McGeoch (2007) Theor Popl BiolHui & McGeoch (2007) OikosHui & McGeoch (2008) Ecology


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


Satellite species

Core species

Hui C ‐Modelling in ecology 15Hui & McGeoch (2007) Oikos


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


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


PART II: Spatial modelling using Mathematica©


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


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


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


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


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


p [ g [ [ ] { }] ] p

PART III: The rise of diversity by adaptive dynamics


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


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?


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)


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


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.

⎟⎠⎞

⎜⎝⎛ −⋅=

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r=0.2; K=10

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t)N(y,αt)N(x,α1t)N(x,rt)dN(x, xyxx

⎟⎟⎠

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K(x)αt)N(y,α1t)N(yrt)dN(y, yxyy

Invasion Fitness Landscape at an Evolutionary Singular

⎟⎟⎠

⎜⎜⎝ K(y)1t)N(y,r

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Mut

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_+

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Mut

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If g(x*)=0, we call x* an evolutionary singular Resident x

_

Resident xResident x

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tant

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_+

tant

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0x *xx

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0y *xy

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Resident xResident xResident xResident xEvolutionary Attractor Fitness Maximum Fitness Minimum

Evolutionary Branching

⎟⎟⎞

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= 2

22α

22α

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00 2σ

)(expKK(x)1.4

σα=0.6; β=0σα=0.6; β=1.5

1 6 β 01000

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0

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⎟⎟⎞

⎜⎜⎛−=

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⎟⎟⎠

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= 1.0Trait

⎟⎟⎠

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0 50000 100 000 150 000 200 0000.0

Time


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.

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