where is a sequence in the function space and is the domain of those functions. In most applications, for any , is a probability density function on . Note that in the definition above, can be vector valued, in which case each element of has a scalar valued integrodifference equation associated with it. Integrodifference equations are widely used in mathematical biology, especially theoretical ecology, to model the dispersal and growth of populations. In this case, is the population size or density at location at time , describes the local population growth at location and , is the probability of moving from point to point , often referred to as the dispersal kernel. Integrodifference equations are most commonly used to describe univoltine populations, including, but not limited to, many arthropod, and annual plant species. However, multivoltine populations can also be modeled with integrodifference equations, as long as the organism has non-overlapping generations. In this case, is not measured in years, but rather the time increment between broods.
Convolution Kernels and Invasion Speeds
In one spatial dimension, the dispersal kernel often depends only on the distance between the source and the destination, and can be written as . In this case, some natural conditions on f and k imply that there is a well-defined spreading speed for waves of invasion generated from compact initial conditions. The wave speed is often calculated by studying the linearized equation
where . This can be written as the convoluion
Using a moment-generating-function transformation
it has been shown that the critical wave speed
Other types of equations used to model population dynamics through space include reaction-diffusion equations and metapopulation equations. However, diffusion equations do not as easily allow for the inclusion of explicit dispersal patterns and are only biologically accurate for populations with overlapping generations. Metapopulation equations are different from integrodifference equations in the fact that they break the population down into discrete patches rather than a continuous landscape.
- Kean, John M., and Nigel D. Barlow. 2001. A Spatial Model for the Successful Biological Control of Sitona discoideus by Microctonus aethiopoides. The Journal of Applied Ecology. 38:1:162-169.
- Kot, Mark and William M Schaffer. 1986. Discrete-Time Growth Dispersal Models. Mathematical Biosciences. 80:109-136