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Discrete-time growth-dispersal models. (English) Zbl 0595.92011
A nonlinear integrodifference equation is presented as a model of a population with growth and dispersal in discrete time. The equation takes the form \[ N_{t+1}(x)=\int^{b}_{a}k(x,y)f(N_ t(y))dy. \] The population is viewed as synchronous, with nonoverlapping generations, and having two distinct phases, one sedentary and one dispersing. \(N_ t(x)\) is the population at spatial coordinate x at the start of the \(t^{th}\) sedentary stage and k(x,y)dy is the probability that an individual moves during its dispersal stage from the interval \((y,y+dy]\) to x. In an isotropic environment the kernel k is symmetric and various examples of such kernels are considered.
The stability of equilibria is discussed by means of the method of linearization. A number of examples are given to illustrate various properties of the solutions such as bifurcation of nontrivial equilibria, period-doubling, chaos, and diffusive instability.
Reviewer: G.F.Webb

92D25 Population dynamics (general)
45M10 Stability theory for integral equations
45H05 Integral equations with miscellaneous special kernels
45J05 Integro-ordinary differential equations
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