1. 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, Size-specific
elasticity: applying a new structured population
model. Ecology 81:

2. The state of the population

3. 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

4. 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)

5. 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.

6. 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.

7. 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.

8. 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.

9. 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

10. 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

11. 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

12. 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