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Persistence and extinction in single-species reaction-diffusion models. (English) Zbl 0543.92020

This report studies the effects of three different dispersal mechanisms on species survival by analyzing reaction-diffusion models. Models of the discrete type in the form of ordinary differential equations and their continuous analogues in the form of partial differential equations are studied. For the purpose of comparison it is assumed that the species biological reaction mechanism is governed by the logistic equation: $\left(du/dt\right)=u\left[a-bu\right]$. The effects of various methods of dispersal on a logistic population are then examined.

Three types of diffusion mechanisms are presented which are referred to as linear diffusion, biased diffusion and directed diffusion. A population modelled by the logistic equation represents a persistent population; however, population extinction can result if the population disperses over a region and dispersion is modelled by linear diffusion. A population modelled by either biased diffusion or directed diffusion cannot result in population extinction. In fact, the chances of population survival may be increased.

##### MSC:
 92D25 Population dynamics (general) 35B35 Stability of solutions of PDE 92D40 Ecology 35K60 Nonlinear initial value problems for linear parabolic equations