Integrodifference equation
In mathematics, an integrodifference equation is a recurrence relation on a function space, of the following form: where formula_2 is a sequence in the function space and formula_3 is the domain of those functions. In most applications, for any formula_4, formula_5 is a probability density function on formula_3. Note that in the definition above, formula_7 can be vector valued, in which case each element of formula_8 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, formula_9  is the population size or density at location formula_10 at time formula_11, formula_12 describes the local population growth at location formula_10 and formula_14, is the probability of moving from point formula_15 to point formula_10, 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, formula_11 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 formula_18. 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 formula_20. This can be written as the convolution 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.