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## Integral projection models

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**Integral projection models**Continuous variable determines • Survival • Growth • Reproduction Easterling, Ellner and Dixon, 2000. Size-specific elasticity: applying a new structured population model. Ecology 81:694-708.**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 = 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 = 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**survival**s(x) is the probability that size x individual survives logistic regression check for nonlinearity**growth function**g(x,y) is the probability that size x individuals who survive grow to size y mean regression check for nonlinearity variance**Comparison to Matrix Projection**Model Matrix Projection ModelIntegral Projection Model • 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 • 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**Comparison to Matrix Projection**Model Matrix Projection ModelIntegral Projection Model • Recruitment usually to a single stage • Construction from observed counts • Asymptotic growth rate and structure • Recruitment usually to more than one stage • Construction from combining • survival, growth and fertility functions into one integral kernel • Asymptotic growth rate and structure**Comparison to Matrix Projection**Model Matrix Projection ModelIntegral Projection Model • Analysis by matrix methods • Analysis by numerical integration of the kernel • In practice: make a big matrix with small category ranges • Analysis then 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