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 y\in\Omega\,, k(x,y)\, is a

^ 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. 38:1:162169.

^ Kot, Mark and William M Schaffer. 1986. DiscreteTime Growth Dispersal Models. Mathematical Biosciences. 80:109136
References
Other types of equations used to model population dynamics through space include reactiondiffusion 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.^{[2]} Metapopulation equations are different from integrodifference equations in the fact that they break the population down into discrete patches rather than a continuous landscape.

c^* = \min_{ w > 0 } \left[\frac{1}{w} \ln \left( R \int_{\infty}^{\infty} k(s) e^{w s} ds \right) \right]
it has been shown that the critical wave speed

M(s) = \int_{\infty}^{\infty} e^{sx} n(x) dx
Using a momentgeneratingfunction transformation

n_{t+1} = f'(0) k * n_t
where R = df/dn(n=0). This can be written as the convoluion

n_{t+1} = \int_{\infty}^{\infty} k(xy) R n_t(y) dy
In one spatial dimension, the dispersal kernel often depends only on the distance between the source and the destination, and can be written as k(xy). In this case, some natural conditions on f and k imply that there is a welldefined spreading speed for waves of invasion generated from compact initial conditions. The wave speed is often calculated by studying the linearized equation
Convolution Kernels and Invasion Speeds
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