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Integrodifference equation

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Title: Integrodifference equation  
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Subject: Recurrence relation, Recurrence relations, Mathematical and theoretical biology, Theoretical ecology, Integro-differential equation
Collection: Mathematical and Theoretical Biology, Recurrence Relations
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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 y\in\Omega\,, k(x,y)\, is a

  1. ^ 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.
  2. ^ Kot, Mark and William M Schaffer. 1986. Discrete-Time Growth Dispersal Models. Mathematical Biosciences. 80:109-136

References

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.[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 moment-generating-function 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(x-y) 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(x-y). 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

Convolution Kernels and Invasion Speeds
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