Pattern and Waves for a Model in Population Dynamics with Nonlocal Consumption of Resources

Abstract

Abstract. We study a reaction-diffusion equation with an integral term describing nonlocal consumption of resources in population dynamics. We show that a homogeneous equilibrium can lose its stability resulting in appearance of stationary spatial structures. They can be related to the emergence of biological species due to the intra-specific competition and random mutations. Various types of travelling waves are observed.

Preface to the Issue Nonlocal Reaction-Diffusion Equations

“…If the speed of this wave is less than the speed of the wave between two constant solutions at infinity, then they propagate one after another with the distance between them increasing in time. If the speed of the periodic wave is greater than the other one, they merge forming a single periodic wave [19], [22], [34], [36], [40]. Such behavior is observed both in the bistable and in the monostable cases.…”

“…The notion of generalized travelling waves, which can be characterized as propagating solutions existing for all times from −∞ to ∞ [32], becomes appropriate here and allows the proof of wave existence without the assumption that the support of the kernel is sufficiently narrow [4], [11]. Numerical simulations show that nonmonotone travelling waves can be stable, and there exist periodic travelling waves [19], [22], [36].…”

“…Periodic travelling waves describe the process of speciation [19], [20], [38]. They exist for both, bistable and monostable cases.…”

“…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.…”

Concentration in the Nonlocal Fisher Equation: the Hamilton-Jacobi Limit

“…by Génieys, Volpert and Auger in [1]. The authors interpret the variable x as a morphological trait, the diffusion term models mutations and the convolution term mimics competition for resources between individuals whose traits are close enough.…”