Abstract
To describe population dynamics, it is crucial to take into account jointly evolution mechanisms and spatial motion. However, the models which include these both aspects, are not still well-understood. Can we extend the existing results on type structured populations, to models of populations structured by type and space, considering diffusion and nonlocal competition between individuals? We study a nonlocal competitive Lotka-Volterra type system, describing a spatially structured population which can be either monomorphic or dimorphic. Considering spatial diffusion, intrinsic death and birth rates, together with death rates due to intraspecific and interspecific competition between the individuals, leading to some integral terms, we analyze the long time behavior of the solutions. We first prove existence of steady states and next determine the long time limits, depending on the competition rates and the principal eigenvalues of some operators, corresponding somehow to the strength of traits. Numerical computations illustrate that the introduction of a new mutant population can lead to the long time evolution of the spatial niche.
Highlights
The spatial aspect of populations is an important ecological issue which has been extensively studied
The combination of spatial motion and mutation-selection processes is known for a long time to have important effects on population dynamics ([19], [24])
We study the steady states and the long time behavior of such systems
Summary
The spatial aspect of populations is an important ecological issue which has been extensively studied (see [17], [18], [23], [25], [28]). The main results of the paper (Theorems 2.3 and 2.4) give assumptions based on spectral parameters and competitive kernels under which the solution of the equation converges, as time goes to infinity, to one of these steady states This result is new and interesting by itself but it will be the first step in an adaptive dynamics framework, if we want to understand how mutant individuals invade the population at the evolutive scale (see [11], [13]). The density dynamics is led by a nonlinear partial differential equation of parabolic type with a non-local competition term In this case, we show existence of steady states for a more general competition term:. For a sufficiently smooth function u and x ∈ ∂X , we denote by ∂nu(x) the scalar product ∇u(x).n(x)
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