Population spread
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Population spread
How fast does a species spread across a landscape?
Plant & animal migrations Genes and pathogenRe-colonization Invasive exotic speciesInvasive exotic species
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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
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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
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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
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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
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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
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Two parameters
from projection matrix analysisc*, wavespeed, from analysis of integrodifference equation model
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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?
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Dispersal Kernel: probability density for
distance traveled during one time-step
Distance
Frequency
Normal
Laplace
Kot et al. 1996
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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
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Neubert-Caswell Model
PopulationProjectionMatrix at
low density
Matrix ofmoment generatin
g functions
Hadamard Product Matrix of
Demography& Dispersal
wave speedand
sensitivity analyses
Matrix ofDispersalKernels
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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
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Examples from working group: 37 species
temperate trees shrubs annual herbs perennial herbs birds muskrats a hispine beetle
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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