Population spread

How fast does a species spread across a landscape?

Plant & animal migrations Genes and pathogenRe-colonization Invasive exotic speciesInvasive exotic species

Two types of models of spread

Dispersal based on mean value All individuals disperse Populations are not structured Results provide only wave

speed

Continuous time model

• Dispersal based on entire distribution

• Only some stage classes disperse

• Populations are structured

• Perturbation analyses also possible

• Discrete time model

Reaction-Diffusion Structured

Integrodifference

2 dimensional Reaction - Diffusion Model

Change in Population Density at Spatial location (x,y)wrt time

Population Growth

Diffusion +

rny

n

x

nD

t

n

2

2

2

2

=

Two processesPopulation growthPopulation spread

Second derivatives wrt space

1 dimensional Reaction - Diffusion Model

Change in Population Densityat spatial location x wrt time

Population Growth

Diffusion +

rnx

nD

t

n

2

2

=

Two processesPopulation growthPopulation spread

Second derivatives wrt space

Population spread

How fast does a structured population grow and spread?

Neubert and Caswell, 2000, Ecology 81:1613-1628

NCEAS Demography and Dispersal Synthesis working group, 2001-2002

Two dynamic processes

In situ population growth each time step

Population projection matrix

Redistribution of dispersing stages over space each time step

Dispersal kernel: probability density function

Two parameters

from projection matrix analysisc*, wavespeed, from analysis of integrodifference equation model

Individuals were contributed potentially from each point, yHow many? Number produced at y, probability of redistribution to given distance, for each stage

Individuals of a given stage seen at point x at a certain time, where did they come from?

Dispersal Kernel: probability density for

distance traveled during one time-step

Distance

Frequency

Normal

Laplace

Kot et al. 1996

Stage-Structured Integrodifference Model

Population vector at

x, t+1

Population

vector at y, t

Matrix of Spatial

Redistribution

From y to x

dytyyxtx n ),(])([)1,( nBKn

Population projection

matrix at y

=

y = points in space where population occurs at time tK = dispersal kernel: probability of dispersal to point x from points yx = point in space where individuals end up at time t+1

Neubert-Caswell Model

PopulationProjectionMatrix at

low density

Matrix ofmoment generatin

g functions

Hadamard Product Matrix of

Demography& Dispersal

wave speedand

sensitivity analyses

Matrix ofDispersalKernels

Neubert-Caswell Model

PopulationProjectionMatrix at

low density

Matrix ofmoment generatin

g functions

Hadamard Product Matrix of

Demography& Dispersal

c* = wave speedand

sensitivity analyses

Matrix ofDispersalKernels

(s)ρlns

1minc* 1

ρ1 is dominant eigenvalue of H(s)

M(s)

H(s)

B0

K

s is a wave shape parameter,(try more than one to look for the one of interest, s*)

s* is the value corresponding to c*

λ is the dominant eigenvalue of B

Examples from working group: 37 species

temperate trees shrubs annual herbs perennial herbs birds muskrats a hispine beetle

Striped maple Acer pennsylvanicum Red maple Acer rubrumBirch Betula allegheniensisBirch Betula lenta White ash Fraxinus AmericanaTulip tree Liriodendron tulipiferaNorthern red oak Quercus rubra Carduus nutans noddingthistleCytisus scoparius ScotchbroomArdisia ellipticaCephaloleia fenestrataCalathea micansCalathea ovandensisTrillium grandiflorumCalluna vulgaris CommonHeatherErica cinerea BellheatherRhinanthus minor YellowrattleUlex minor DwarfgorseCirsium vulgare SpearthistlePinus palustris LongleafPine

Cattle Egret Bubulcus ibisEgyptian Goose Alopochen aegyptiacusRuddy Duck Oxyura jamaicensis Red Kite Milvus milvusWhite-tailed Eagle Haliaetus albicillaMarsh Harrier Circus aeruginosusGoshawk Accipiter gentilisCommon Buzzard Buteo buteoOsprey Pandion halitusPeregrine Falcon Falco peregrinesCollared Dove Streptopelia decaeoctoPied Flycatcher Ficedula hypoleucaStarling Sturnus vulgarisHouse Sparrow Passer domesticusCoyote Canis latransRed Fox Vulpus vulpusMuskrat Ondatra zibethicus