Population spread How fast does a species spread across a landscape? Plant & animal migrations Genes and pathogen Re-colonization Invasive exotic species Invasive 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 + = Two processes Population growth Population spread Second derivatives wrt space

1 dimensional Reaction - Diffusion Model Change in Population Density at spatial location x wrt time Population Growth Diffusion + = Two processes Population growth Population spread Second derivatives wrt space

Population spread  How fast does a structured population grow and spread?  Neubert and Caswell, 2000, Ecology 81:  NCEAS Demography and Dispersal Synthesis working group,

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 analysis c*, wavespeed, from analysis of integrodifference equation model

Individuals were contributed potentially from each point, y How 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

Stage-Structured Integrodifference Model Population vector at x, t+1 Population vector at y, t Matrix of Spatial Redistribution From y to x Population projection matrix at y = y = points in space where population occurs at time t K = dispersal kernel: probability of dispersal to point x from points y x = point in space where individuals end up at time t+1

Neubert-Caswell Model Population Projection Matrix at low density Matrix of moment generating functions Hadamard Product Matrix of Demography & Dispersal wave speed and sensitivity analyses Matrix of Dispersal Kernels

Neubert-Caswell Model Population Projection Matrix at low density Matrix of moment generating functions Hadamard Product Matrix of Demography & Dispersal c* = wave speed and sensitivity analyses Matrix of Dispersal Kernels ρ 1 is dominant eigenvalue of H(s) M(s) H(s) B0B0 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 rubrum Birch Betula allegheniensis Birch Betula lenta White ash Fraxinus Americana Tulip tree Liriodendron tulipifera Northern red oak Quercus rubra Carduus nutans noddingthistle Cytisus scoparius Scotchbroom Ardisia elliptica Cephaloleia fenestrata Calathea micans Calathea ovandensis Trillium grandiflorum Calluna vulgaris CommonHeather Erica cinerea Bellheather Rhinanthus minor Yellowrattle Ulex minor Dwarfgorse Cirsium vulgare Spearthistle Pinus palustris LongleafPine Cattle Egret Bubulcus ibis Egyptian Goose Alopochen aegyptiacus Ruddy Duck Oxyura jamaicensis Red Kite Milvus milvus White-tailed Eagle Haliaetus albicilla Marsh Harrier Circus aeruginosus Goshawk Accipiter gentilis Common Buzzard Buteo buteo Osprey Pandion halitus Peregrine Falcon Falco peregrines Collared Dove Streptopelia decaeocto Pied Flycatcher Ficedula hypoleuca Starling Sturnus vulgaris House Sparrow Passer domesticus Coyote Canis latrans Red Fox Vulpus vulpus Muskrat Ondatra zibethicus