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Preface to the issue nonlocal reaction-diffusion equations. (English) Zbl 1337.92165

From the text: Nonlocal reaction-diffusion equations are intensively studied during the last decade in relation with problems in population dynamics and other applications. In comparison with traditional reaction-diffusion equations they possess new mathematical properties and richer nonlinear dynamics. Many studies are devoted to the nonlocal reaction-diffusion equation
\[ \frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2} + F(u, J(u)), \tag{1} \]
where
\[ F(u, J(u))= au^k(1-J(u))-\sigma u,\quad J(u)= \int_{-\infty}^{\infty}\phi(x-y)u(y,t)dy, \]
which describes the distribution of population density in the case of nonlocal consumption of resources.
One of the most interesting applications of nonlocal reaction-diffusion equations concerns modelling of the emergence and evolution of biological species. If we characterize a species as a group of individuals with a close phenotype, then it can be described as a localized solution (pulse) of equation (1) where the space variable \(x\) is some phenotypical characteristics and \(u(x, t)\) is the density distribution with respect to the phenotype. As we discussed above, such stable solutions exist in the bistable case with global consumption of resources. In order to validate this approach it was used to describe human height distribution with a system of two equations with global consumption.

MSC:

92D25 Population dynamics (general)
35K57 Reaction-diffusion equations
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