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Systems Ecology: Principles and Modelling

Spatial Models in Ecology

Oliver Jakoby

Introduction

Task:

Develop a spatial model of Zürich

Introduction

Introduction “Spatial models of Zurich”

Model types

General classification of models:

• purpose: generic (strategic / theoretical) versus specific (tactical / applied) models

• methodical: mathematical equations versus computer simulation models

• complexity: (how much detail do we need) stochasticity – space – individual variability

Why are ecologists interested in space? – Spatial pattern

Importance of space in ecology

Why are ecologists interested in space? – Spatial pattern (patchiness)


Why are ecologists interested in space? – Spatial pattern – Individual interaction in space


Why are ecologists interested in space? – Spatial pattern – Individual interaction in space (swarm behaviour)



Why are ecologists interested in space? – Spatial pattern – Individual interaction in space – Species distribution (differs in space)


Why are ecologists interested in space? – Spatial pattern – Individual interaction in space – Species distribution (differs in space and time)

Why do we need to incorporate space in ecological models? Spatial heterogeneity

– habitat vs. non-habitat

Space in ecological models


– habitat vs. non-habitat – fragmentation (barriers / connectivity)

© Benjamin Pennington



– habitat vs. non-habitat – fragmentation – environmental gradients (temperature, acidity, precipitation, …)



– habitat vs. non-habitat – fragmentation – environmental gradients (temperature, acidity, precipitation, …) – land-use / management


• What are important ecological processes in spatial modelling? – dispersal / migration


• What are important ecological processes in spatial modelling? – dispersal / migration – small scale movement


• What are important ecological processes in spatial modelling? – dispersal / migration – small scale movement

– local interactions • competition for space, nutrients/food or light • predation • territorial behaviour in animals • disease transmission


Classification of spatial models

• Representation of space: – implicit vs. explicit

Spatially implicit models include assumptions how spatial structures affect dynamics in the model, but they do not explicitly including geographic space (no explicit position)

Spatially explicit models include explicit representations of a geographic space. Among spatial entities (patches, cells, points) exist defined spatial relations. However, dynamics within entities (patches, cells) can be assumed to be independent of spatial affects.

y - p

ositi

on

x - position , , , ,


• Representation of space: – implicit vs. explicit – discrete vs. continuous

Discrete space is divided into cells / patches / sites, neighbourhood relations within these sites homogeneous

Continuous space is referenced via a Cartesian coordinate system (x, y)

y - p

ositi

on

x - position

y - p

ositi

on

x - position


• Representation of space: – implicit vs. explicit – discrete vs. continuous – basic units:

• abundance-based (population / individual density at one point)

• site/patch/grid-based (discrete cells) • individual-based (individuals)



• abundance-based (population) • site/patch/grid-based (discrete cells) • individual-based (Individuals)

– spatial dimensions: 1, 2, 3

0 0 4 0 3 3 0 6 8 2 7 1 0 0 1 0 0 1 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19



• abundance-based (population) • site/patch/grid-based (discrete cells) • individual-based (Individuals)

– spatial dimensions: 1, 2, 3 – environmental heterogeneities: static vs. dynamic

Grid based

Spatially implicit models

Spatially explicit models

Patch models

Abundance based

Individual based

Discrete space Continuous space

Patch models

Classification of spatial models Focus of the talk

Representation of space

Grid based



Patch models

Abundance based

Individual based


Patch models


10 minutes break


Grid based



Patch models

Abundance based

Individual based


Patch models

Metapopulation models

Spatial implicit models Patch models

abiotic influences, e.g: Topography, water availability

biotic influences, e.g: Spatial distribution of predator or prey

Habitat suitability is determined by:

Space is divided into two categories: 1. suitable habitat allows survival and reproduction 2. matrix only suitable for dispersal

Levins’ Metapopulation • Local populations live in different patches • Each patch can be occupied or empty • Local populations can go extinct • Empty patches can be re-colonized from other local populations

Spatial implicit models Patch models

Assumptions: – homogeneous and equally connected patches

(same size, habitat quality, connectivity) – no spatial structure / no explicit patch location – (infinite) large number of identical patches – no local population dynamics – constant extinction and colonisation rates

Spatial implicit models Levins‘ Metapopulation

EIdtdP

−=

Entities: discrete patches Processes: colonisation and extinction Question: persistence of metapopulation

P = proportion of occupied patches I = immigration process E = extinction process t = time

Spatial implicit models Levins‘ Metapopulation

Entities: discrete patches Processes: colonisation and extinction Question: persistence of metapopulation

PePPcdtdP

⋅−−⋅⋅= )1(

P = proportion of occupied patches c = colonization rate (constant) e = extinction rate (constant) t = time

What will be the proportion of occupied patches that we expect in the long term? Under which conditions will a metapopulation persist?

Ordinary differential equation (ODE)

c > e

ce

−= 1 *p

Classes of spatial models

Grid based



Patch models

Abundance based

Individual based


Patch models


Metapopulation

Incidence function models

Spatial explicit models Patch models

Add relevant sources of spatial heterogeneity: – Explicit spatial configuration and size of the patches – Colonisation rate of a patch depends on distance and size of the other occupied patches – Extinction rate of a patch depends on its size

Incidence function model (IFM)

(e.g. Hanski 1994)

iiiii PEPC

dtdP

⋅−−⋅= )1(


Incidence function model (IFM)

xii AeE −⋅=∑

≠

⋅=ji

ijji dAbC )exp(α

Pi = probability of patch i to be occupied (incidence) Ai = area of patch i e = extinction rate dij = distance between patch i and j α, b, x = species specific parameters

Hanski 1995: Butterfly-Metapopulation (Melitaea cinxia)


Example: Deadwood islands in managed beech forest

Question: How does the amount of deadwood and the configuration of the islands affect the biodiversity in the forest?


• Several grid cells were clustered in patches connected through dispersal

• Characterized by: – size – carrying capacity – stochastic risk of local

extinction distance


The patch quality is changing over time

Dynamic landscape


The patch quality is changing over time

Dynamic landscape

Deterministic extinction of patches


• Osmoderma eremita: – low ability of dispersal – medium local persistence probability


2.0

1.6

0.4

0.8

1.2

incidence

0.2 ha



Grid based



Patch models

Abundance based

Individual based


Patch models


Metapopulation


Cellular automata and interacting particle systems

• discrete space (uniform and averaged within grid cell)

• regular grid (various cell shapes possible, 1,2,3-dimensional)

Spatial explicit models Grid-based models



• Neighbourhood: determines possible interactions

von Neumann Moore extended Moore





• Neighbourhood: determines possible interactions • Boundary conditions

– absorbing boundaries – reflecting boundaries – periodic boundaries (Torus)



• Neighbourhood: determines possible interactions • Boundary conditions (absorbing, reflecting, continuous boundaries)

• Updating of cell states (synchronous or asynchronous)

• Transition rules: deterministic or stochastic • site-based – individual-based • Size of a cell: depends on the system and on the

question


• space is divided into equal sized discrete cells (sites)

• each cell is assigned to a state or occupied by an individual

• state transition in the next step depends on state focal and neighbouring cells

t = 0





t = 1





• example: segregation model by Th.C. Schelling (Nobel prize, 2005)


Cellular Automata – deterministic rules – synchronous updating – two states possible

Example: Game of live (Conway 1970)


Grid- and individual-based model: • Individuals move in the grid • Behaviour of individuals

can depend on state of the cells and behaviour of other individuals

medium forage

availability Barrier

high forage availability


no forage



t = 0




t =1


Can local dispersal promote coexistence of three bacteria strains?

Kerr et al., Nature, 2002

Example: Grid-based model

–

–

– P

R S

P R S

mor

talit

y

P

S

non-hierarchical competition

Under which condition can species with non-hierarchical competition coexist?



Under which condition can species with non-hierarchical competition coexist?

• Diversity is rapidly lost when dispersal and interaction occur over relatively large spatial scales

• Populations coexist when ecological processes are localised



Example: 3D grid- (and individual-) based forest models

Spatial explicit models Grid- and individual-based models


Grid based



Patch models

Abundance based

Individual based


Patch models


Metapopulation



Reaction diffusion models

x

y

Spatial explicit models Abundance-based models

• Individuals are assumed to move in continuous space

• Only abundances of individuals per area are considered, instead of explicitly describing the behaviour of single individuals

• Examples: – Reaction diffusion equation:

– Integro-difference equation:

2

2 ),(),()(),(x

txpDtxpprt

txp∂

∂+=

∂∂

∫ −=+ O tt dzzNfxzkxN ))(()()(1

Abundance-based models Reaction-diffusion equations

Reaction diffusion equation (RDE) • Calculating the density of individuals at each point in space over time

Mass flow = (undirected) migration

Point x

Reaction = birth and death

Popu

latio

n de

nsity

Space

Reaction diffusion equation for populations

NNrxND

tN

⋅+∂∂⋅=

∂∂ )(2

2

Change of population size N over time at

point x

Immigration and emigration Population growth + =

Reaction term: e.g. population growth

Diffusion term: e.g. migration / movement

Reaction diffusion equation (RDE) Mass flow = (undirected) migration

Point x

Reaction = birth and death

abundance-based models Example: Range expansion by sea otter using integro-difference equations

Tinker et al. (2008)

Spatial explicit models Abundance-based models


Grid based



Patch models

Abundance based

Individual based


Patch models


Metapopulation




Distance models

x

y

x

y

Spatially continuous individual based models

Resource use and competition between individuals usually occur on a restricted spatial scale distance between individuals is important to determine competition

Zone of influence (ZOI)

Fixed radius neighborhood (FRN)

Field of neighborhood (FON) models:

Spatial explicit models Distance models

Fixed radius neighborhood (FRN) models:

• fixed circular area assigned to each individual

• interaction between all individuals whose centers are located within this area

• e.g. survival or fecundity decreases with the density of neighbours


Zone of influence (ZOI) models:

• individuals are characterized by centre in continuous space and a circular ZOI

• radius depends on size of individual

• size of overlapping area determine resource competition between both individuals



Field of neighbourhood (FON) models:

• potential impact to neighbours decreases conical from center to boundary

• radius depends on individual size

• competition strength decreases with distance

• competition affecting focal individual = integral over the FONs of other individuals in its area

Spatial explicit models Tessellation models

• Assume direct linkage between individual performance and amount of resources at individuals disposal

• Resource use and competition between individuals only on a certain spatial scale (also named Voronoi or Thiessen diagrams) • Number and proximity of neighbours determine area and

shape of a polygon (influencing e.g. fecundity and survival)


Grid based



Patch models

Abundance based

Individual based


Patch models


Metapopulation




Distance models

x

y

x

y

x

y

Movement models

Spatial explicit models Movement models

Modelling individual movement • foraging strategies, searching for mating partners, habitat, … • random walk (Brownian motion):

– individuals floating through the environment, without any directionality (passive diffusion)

– parameters: • step length • turning angle / direction

Modelling individual movement • random walk (Brownian motion):






• correlated random walk – random walk with correlated direction (turning angles are limited to a

certain range or follow a non-uniform distribution)

α


Modelling individual movement • correlated random walk

– random walk with correlated direction (turning angles are limited to a certain range or follow a non-uniform distribution)

α




• correlated random walk – random walk with correlated direction (turning angles are limited to a certain

range or follow a non-uniform distribution)

• Levy walk / flight – involve a uniform distribution for the turning angles, but a heavy-

tailed distribution (e.g. power-law) for the step lengths


Modelling individual movement • Levy walk / flight

– involve a uniform distribution for the turning angles, but a heavy-tailed distribution (e.g. power-law) for the step lengths

Example: • Hilltopping: “Males and females of a certain butterfly

species seek a topographic summit to mate”

Peer et al., 2006


Final remarks

• Many different approaches to represent space in ecological models

• Combination of different model types is possible and useful

• Final models structure has to be based on the question and (to some extend) on available data

Final remarks

Acknowledgements

Many thanks to:

Karin Johst Mathias Franz Nadja Rüger Mira Kattwinkel

for contributing material to this lecture!

Some further reading

Tamás Czárán (1997), Spatiotemporal Models of Population and Community Dynamics

Jordi Bascompte and Ricard V. Solé (1998), Modelling Spatiotemporal Dynamics in Ecology

Ilkka Hanski (1999), Metapopulation Ecology

Berger et al. (2008), Competition among plants: Concepts, individual-based modelling approaches, and a proposal for a future research strategy. Perspect. Plant Ecol. Evol. Syst.

Tamás Czárán and Sàndor Bartha (1992), Spatiotemporal Dynamics of Plant Population Models. TREE

Contact: [email protected]

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