Preface to the Issue Nonlocal Reaction-Diffusion Equations

Abstract

Nonlocal reaction-diffusion equations in population dynamicsNonlocal 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 equationwherewhich describes the distribution of population density in the case of nonlocal consumption of resources.Here k is a positive integer, k = 1 corresponds to asexual and k = 2 to sexual reproduction. If the kernel φ of the integral is replaced by the δ-function, then we obtain conventional reaction-diffusion equation.In the case k = 1, it is the logistic equation with the reproduction term au(1 − u) proportional to the population density u and to available resources (1 − u). In the case of nonlocal consumption of resources, the integral J(u) describes consumption of resources at the space point x by individuals located in some area around this point. The function φ(x − y) determines the efficiency of such consumption. Introduction of nonlocal consumption of resources changes the properties of solutions of this equation. Consider for certainty the case where k = 1 and σ = 0. Suppose that ∞ −∞ φ(y)dy = 1. Then u = 1 is a stationary solution of this equation. It is stable in the case of the local equation but it can lose its stability for the nonlocal equation. If it becomes unstable, then a periodic in space stationary solution bifurcates from it [13], [19], [21]. This simple result of the linear stability analysis has important consequences from the mathematical point of view and for the applications.

Doubly nonlocal reaction–diffusion equations and the emergence of species

“…Then in the limit of large h 2 we obtain global consumption of resources with the integral I(u) = ∞ −∞ u(y, t)dy in the consumption term. Various particular cases of this equation are studied in the literature (see [1,27,31] and the references therein). In the limit of small h 1 and h 2 we obtain the classical reactiondiffusion equation…”

Global boundedness and Allee effect for a nonlocal time fractional p-Laplacian reaction-diffusion equation

“…Therefore, many researches are devoted to the nonlocal reaction-diffusion equation. In [3], the proposed nonlocal reaction-diffusion equation describes the population intensive distribution under the assumption of non-local sources. Its nonlocalization form is F (u, J(u)) = au k (1 − J(u)) − σu, J(u) = +∞ −∞ φ(x − y)u(y, t)dy.…”

Global boundedness and Allee effect for a nonlocal time fractional reaction-diffusion equation

“…Therefore, many researches are devoted to the nonlocal reaction-diffusion equation. In [8], in order to describe the distribution of population density in the case of nonlocal consumption of resources, the following nonlocal reaction-diffusion equation is obtained. ∂u ∂t = D ∂ 2 u ∂x 2 + F (u, J(u)), where F (u, J(u)) = au k (1 − J(u)) − σu, J(u) = +∞ −∞ φ(x − y)u(y, t)dy.…”

“…It is stable in the case of local equations, but may lose stability to nonlocal equations. Therefore, in [8], it can be obtained that the local equations have stable monotonic waves and unstable pulses, while the nonlocal equations have simple periodic waves and stable or unstable pulses. As discussed above, the stable solution exists in a bistable situation with global resource consumption.…”