Integrodifference equation

In mathematics, an integrodifference equation is a recurrence relation on a function space, of the following form:

:$n\_\{t+1\}(x)\; =\; int\_\{Omega\}\; k(x,\; y),\; f(n\_t(y)),\; dy,$

where $\{n\_t\},$ is a sequence in the function space and $Omega,$ is the domain of those functions. In most applications, for any $yinOmega,$, $k(x,y),$ is a probability density function on $Omega,$. Integrodifference equations are widely used in mathematical biology to model the dispersal of univoltine populations. This includes, but is not limited to, many arthropod, and annual plant species. In fact, multivoltine populations can also be modeled with integrodifference equations (Kean and Barlow 2001), but the organism must have non-overlapping generations. In this case t would not be measured in years, but rather the time increment between broods.

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 (Kot and Schaffer 1986). Metapopulation equations are different from integrodifference equations in the fact that they break the population down into discrete patches rather than a continuous landscape.

References

* 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