Lecture 4: Spatial Models in Ecology - · PDF fileforest succession, wildfires cellular automata ... Secondary mangrove succession ... Lecture 4 „Spatial Models in Ecology“

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Nadja Rüger; Lecture 4 „Spatial Models in Ecology“ (Part I; Day 4)

Lecture 4:Spatial Models in Ecology

Nadja RügerHelmholtz Centre for Environmental Research – UFZ, Leipzig

Department of Ecological Modeling

Part I: From spatial patterns and processes

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Outline

1. Why explicitly consider space?

2. Classification of spatial models in ecology

3. Five ways to spacialize an ecological model

4. Quiz

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Heterogeneity in

time → stochastic modelsspace → spatial modelsindividual traits → individual-based model

How can heterogeneity in space arise and when does it matter?

Abiotic spatial heterogeneityspatial heterogeneity/structure of habitat or landscape influences relevant ecological processes

Biotic spatial heterogeneityIndividuals create heterogeneity through their interaction, relevant ecological processes act on a certain spatial scale, self-organized spatial heterogeneity


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Abiotic spatial heterogeneityhabitat qualityresource availabilitypredation riskdispersal barriers

Biotic spatial heterogeneityterritorial behaviour in animalslocal interaction between individuals

competition for space, nutrients or light in plantscompetition for food, shelter etc. in animalspredationdisease transmission

local dispersal

affects ecological processesdispersal probabilitymortality ratecolonization probabilitybehaviour

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2. Classification of spatial models

1. Cause for heterogeneity: abiotic vs. biotic

2. Representation of space I: implicit vs. explicit

Spatially implicit modelsincorporate assumptions about spatial structure of biotic interactions, but do not include geographical space

Spatially explicit modelsrepresent a heterogeneous space which iscontinuous or discrete.

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3. Representation of space II: discrete vs. continuous

DiscreteSpace is divided into cells/patches/sites,neighbourhood relations, within patches/cells/sites homogeneous

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

x

y

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4. Basic unit: abundance-based vs. site-based vs. individual-based

abundance-based: basic unit = population (e.g. PDE)

site/grid/patch-based: basic unit = cell (e.g. CA)

individual-based: basic unit = individual (e.g. distance models)

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4. Basic unit: abundance-based vs. site-based vs. individual-based

5. Number of dimensions: 1, 2, 3

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Nadja Rüger; Lecture „Spatial Models in Ecology“ (Day 4)


forest succession,wildfires

cellular automata

discrete

site-based neighbourhood models

relevant ecological processes act on certain spatial scale, self-organized spatial heterogeneity

discrete

individual- and grid-based models

competition, facilitation between plants

territory building, disease transmission in animals

metapopulationdynamics,host-parasitoiddynmaics

dispersal and foraging in animals,dispersal in plants

continuousdiscretespatially implicitcontinuous/ discrete

tesselationmodels

individual-based neighbourhood models

distance models

metapopulationmodels

dispersal models

patch-based ‘movement’models

individual-based ‘movement’models

spatial structure influences ecological processes

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3. Five ways to spacialize an ecological model

3.1 Wildfires in boreal forests – Cellular Automaton

3.2 Plant competition and self-thinning – Distance models

3.3 Sustainable management of species-rich tropical forests –

Individual- and grid-based ‘interaction’ model

3.4 Population dynamics of a territorial bird with social breeding –

Individual- and grid-based ‘movement’ model

3.5 Metapopulation models and Meta-X (-> Part II)

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3.1 Wildfires in boreal forests (Ratz 1995)

Spatial processes

fire spreads from one location to another depending on the amount of flammable material

Non-spatial processes

aging of the forest; the amount of flammable material depends on the age of the forest

Consequence

a complex mosaic of forest patches in different successionalstages is created (self-organized spatial heterogeneity)

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Questions

Is there a general size distribution of wildfires?Is there a typical average fire size?Are there specific reasons for large fires?

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0 10 20 30 40 50

Feuer

200

0Klimax

Sukzessionstufen

cell size = 5 ha (forest stand)cells age in steps of 10 yearsflashes at random locations

Fire Climax

Successional Stages


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2)10/(agecityflammabili ⋅+=

c = 0 : f constant

0 < c < 1: f increases with stand age

flammability f = ignition probability of stand


Form und Größe des Feuers ergebensich aus dem Modell.size and shape of the fires emerge from simple model rules

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size distribution is power law → self organized criticalityno typical average fire sizeno specific reasons for large fires

msasD ⋅=

+=

)(log(s) m a log log(D(s))


-4 -3 -2 -1 0log(s)

0

-1

-2

-3

-4

-5

log(

D)

s fire sizeD(s) number of fires of size s

power law

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3.1 Cellular Automatadiscrete space, regular griddiscrete time stepsfinite (small) set of states of cellsneighbourhood size and shapesite-basedtransition rules (‘updating rules’,‘next-state-function’) describeneighbourhood interactions

Advantageswithin-site dynamicslocal interactions simple rules reflect ecological knowledge single species and community dynamics can be modelledcomputationally efficient

Disadvantagesregular and equally-sized shape of sites

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3.2 Plant competition (review Berger et al. 2008)

Ecological questionsHow does inter- and intraspecific competition affect plant population dynamics (succession, self-thinning)?

Spatial processescompetition

is a function of how space is divided or shared among individualsinteraction strength is a function of distance

mortalityseed dispersal

Non-spatial processesI can’t think of any, maybe you can?

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Fixed-radius neighbourhood (FRN)

fixed radius of interaction for all individualsradius defined ad hoc or based onfield data

Advantageconceptually simple

Disadvantageno size(or age)-dependency


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Zone-of-influence (ZOI)

radius depends on sizeZOI represents resource uptake, i.e.performanceoverlap of ZOIs representscompetition → reduced performanceused to understand size distributionsof even-aged monocultures

Advantagesconceptually simple

Disadvantagecomputationally non-trivialmany studies ignore mortality


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parameters defined ad hoc or based on field datamortality, growth and recruitment taken into account

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R

dbh( ) ∑=

NnFONyxF ,

FONminFON (r)

bdbhaR )2/(⋅=

Field-of-neighbourhood (FON)

Berger and Hildenbrandt (2000)radius depends on sizecompetition strength depends on distancecompetition affecting focal plant = integral over the FONs of otherplants in its area


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R

dbh( ) ∑=

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FONminFON (r)

bdbhaR )2/(⋅=

Field-of-neighbourhood (FON)

Advantagesrepresents interaction strengthsimple way to model effect of competition on seedlingestablishment

Disadvantagecomputationally ‘expensive’ and non-trivial

ExampleSecondary mangrove succession (Berger et al. 2006)


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3.2 Distance models

continuous spaceindividual-basedlocal competition, (dispersal)single or multiple species

Advantagesvariable (except FRN) scale of interaction

Disadvantagesmechanisms of interaction are highly simplified =phenomenological description of competitioncomputationally expensive

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Question How can species-rich rain forests be managed in a sustainable way ?

Spatial processescompetition for light

tree growth affects competitionfor light and vice versa (vertical)harvesting affects competitionfor light (vertical)gap creation (horizontal)

competition for space(seed dispersal)

3.3 Sustainable management of tropical forests (e.g. Köhler and Huth 1998, Köhler et al. 2001, 2003, Rüger et al. in press)

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The model (FORMIND)plant functional types (PFT)individual-basedprocess-based: regeneration, growth, mortality, competition for light and space, gap creation, loggingtree growth modelled on a physiological basishorizontal space: 20 m × 20 m gridsvertical space: 0.5 m height layersannual time steps


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Light absorption

Photosynthesis

Biomass increment


Competition for light

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Why so detailed?species-richnessuneven age structureobjectives: quantitative long-term information on detailed management strategies

howeverspatial heterogeneity within grid cells and height layers are ignored


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Reality test


(Muñiz-Castro et al. 2006)

range of model results

Time (years)

Time (years)Time (years)

Time (years)

Basal areaStem number

Mean height Maximum height

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Exemplary results


logging intensity: 5 – 100 m³/ha stem volumelogging cycle: 10 yearssimulation time: 400 years

12 m³/ha40 – 100 cm4, 5, 6S44.5 m³/ha40 – 60 cm4, 5, 6S33 m³/ha40 – 100 cm 4, 6S22 m³/ha40 – 60 cm4, 6S1

Maximumharvest

Diameterrange

Logged PFTsScenario

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Nadja Rüger; Lecture „Spatial Models in Ecology“ (Day 4)


increase of small treesdrastic decrease of large old treesmore homogeneous and younger forest structure

Stem diameter

Ste

m n

umbe

r (in

d/ha

)

Annual wood extraction (m³/ha)




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Advantagesof IBMs

make testable predictionscan be parameterized with individual-level information

of grid-based modelsefficient way to represent spatial heterogeneity at relevant scale

single and multiple species flexible

Disadvantagesoften complex = labour-intensive to parameterize and analyzecomputationally expensive

3.3 Individual- and grid-based ‘interaction’ model

discrete space, regular gridindividual-basedmechanistic description of localinteractions (competition for light)

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Three models of forest dynamics

How do they differ any why?

What is the appropriate way to represent space?

What is the appropriate spatial resolution?

00.20.40.60.81

R

dbh( ) ∑=

NnFONyxF ,

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3.4 Woodhoopoe population dynamics (Neuert et al. 1995; Grimm and Railsback 2005)

Green woodhoopoe(Phoeniculus purpureus)lives in riparian forests in Africaterritorialsocial hierarchysocial breeding: only alpha couplereproduces and helperslong-distance scouting forays

Spatial processlong-distance scouting forays

Non-spatial processesreproduction, mortality

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The model1 grid cell = 1 territoryalpha couple reproduces helpers decide for scoutingforays depending on their ageand social rankthe older individuals are andthe lower their social rank thehigher is the probability forlong-distance foraystime step = 1 month

Ecological question:How do long-distance scouting forays affect the spatialcoherence and the persistence of the population?

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Ecological question:How do long-distance scouting forays affect the spatialcoherence and the persistence of the population?

→ long-distance scouting forays maintain the spatial coherence→ if ≥ 15 territories are connected by scouting birds, the

metapopulation persists

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3.4 Individual- and grid-based ‘movement’ model

individual-baseddiscrete space, regular grid‘movement’ model (dispersal model)no local interactions

Advantagesof IBMs

make testable predictionscan be parameterized with individual-level information

of grid-based modelsefficient way to represent spatial heterogeneity at relevant scale

single and multiple species flexible

Disadvantagesoften complex = labour-intensive to parameterize and analyzecomputationally expensive

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Acknowledgements

Uta Berger, Karin Frank, Volker Grimm, Karin Johst, Michael Müller, Guy Pe’er for providing material

Literature:Berger et al. 2006, 2008Berger and Hildenbrandt 2000Czárán 1998 Grimm and Railsback 2005Köhler and Huth 1998Köhler et al. 2001, 2003Rüger et al. 2007Neuert et al. 1995 Ratz 1995

Documents

Lecture 4: Spatial Models in Ecology - · PDF fileforest succession, wildfires cellular automata ... Secondary mangrove succession ... Lecture 4 „Spatial Models in Ecology“