Integral projection models Continuous variable determines Survival Growth Reproduction Invasion wave speed is the speed with which a species spreads or migrates across a landscape. It has been studied in several different venues, including plant and animal migrations following deglaciation events, the spread of selectively favorable genes and pathogens through a population, and recolonization of habitat following catastrophic disturbances. But more recently, invasion biologists have become interested in the rate of spread of invasive exotic species. Easterling, Ellner and Dixon, 2000. Size-specific elasticity: applying a new structured population model. Ecology 81:694-708.

The state of the population

Integral Projection Model Integrate over all possible sizes Number of size x individuals at time t Number of size y individuals at time t+1 = This is Neubert-Caswell’s integrodifference equation for modeling population size and shape at point x as a function of population size and shape at all other points y. At each point the population grows according the matrix B and some individuals are redistributed across space according to the matrix of dispersal kernels K. Babies of size y made by size x individuals Probability size x individuals Will survive and become size y individuals

Integral Projection Model Integrate over all possible sizes Number of size x individuals at time t Number of size y individuals at time t+1 = This is Neubert-Caswell’s integrodifference equation for modeling population size and shape at point x as a function of population size and shape at all other points y. At each point the population grows according the matrix B and some individuals are redistributed across space according to the matrix of dispersal kernels K. The kernel (a non-negative surface representing All possible transitions from size x to size y)

survival and growth functions s(x) is the probability that size x individual survives g(x,y) is the probability that size x individuals who survive grow to size y This is Neubert-Caswell’s integrodifference equation for modeling population size and shape at point x as a function of population size and shape at all other points y. At each point the population grows according the matrix B and some individuals are redistributed across space according to the matrix of dispersal kernels K.

survival s(x) is the probability that size x individual survives logistic regression check for nonlinearity This is Neubert-Caswell’s integrodifference equation for modeling population size and shape at point x as a function of population size and shape at all other points y. At each point the population grows according the matrix B and some individuals are redistributed across space according to the matrix of dispersal kernels K.

growth function g(x,y) is the probability that size x individuals who survive grow to size y mean regression check for nonlinearity variance This is Neubert-Caswell’s integrodifference equation for modeling population size and shape at point x as a function of population size and shape at all other points y. At each point the population grows according the matrix B and some individuals are redistributed across space according to the matrix of dispersal kernels K.

growth function This is Neubert-Caswell’s integrodifference equation for modeling population size and shape at point x as a function of population size and shape at all other points y. At each point the population grows according the matrix B and some individuals are redistributed across space according to the matrix of dispersal kernels K.

Comparison to Matrix Projection Model Matrix Projection Model Integral Projection Model Populations are structured Discrete time model Population characterized by a continuous distribution Parameters are estimated statistically for relationships: few parameters are needed Parameters estimated by regression analysis Populations are structured Discrete time model Population divided into discrete stages Parameters are estimated for each cell of the matrix: many parameters needed Parameters estimated by counts of transitions

Comparison to Matrix Projection Model Matrix Projection Model Integral Projection Model Recruitment usually to more than one stage Construction from combining survival, growth and fertility functions into one integral kernel Asymptotic growth rate and structure Recruitment usually to a single stage Construction from observed counts Asymptotic growth rate and structure

Comparison to Matrix Projection Model Matrix Projection Model Integral Projection Model Analysis by numerical integration of the kernel In practice: make a big matrix with small category ranges Analysis then by matrix methods Analysis by matrix methods

Steps read in the data statistically fit the model components combine the components to compute the kernel construct the "big matrix“ analyze the matrix draw the surfaces