Competition of Species with Intra-Specific Competition

Abstract

Abstract. Intra-specific competition in population dynamics can be described by integro-differential equations where the integral term corresponds to nonlocal consumption of resources by individuals of the same population. Already the single integro-differential equation can show the emergence of nonhomogeneous in space stationary structures and can be used to model the process of speciation, in particular, the emergence of biological species during evolution [6], [7]. On the other hand, competition of two different species represents a well known and well studied model in population dynamics. In this work we study how the intra-specific competition can influence the competition between species. We will prove the existence of travelling waves for the case where the support of the kernel of the integral is sufficiently narrow. Numerical simulations will be carried out in the case of large supports.

Existence of Waves for a Nonlocal Reaction-Diffusion Equation

“…By virtue of a priori estimates obtained in this section we can use the Leray-Schauder method. As shown in [2], equation (2.6) with conditions (1.4) has a solution in the form of monotone travelling wave when τ = 0. This solution is unique up to translation in space.…”

“…In this case, the properties of the equation become quite different. It possesses an interesting nonlinear dynamics [4,8] but the wave existence can be proved only in the case of functions φ with a small support where the perturbation methods are applicable [1,2,3,5]. In this work we do not assume that the support is small.…”

“…In [2] the authors have proved the existence of solutions in the form of monotone travelling waves for a class of integro-differential equations where the function φ has a small support [−ε 0 , ε 0 ]. This result uses the bistability of the nonlinearity.…”

Preface to the Issue Nonlocal Reaction-Diffusion Equations

“…The proof of wave existence in the case of nonlocal equation becomes much more involved, and there are only partial results [2], [5], [6], [11], [15]. 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].…”

“…Their structure and the patterns formed behind the waves can depend on their speed. Wave propagation is also studied in the multidimensional case [18] and for systems of equations [5], [8], [31].…”