On long-time dynamics for competition-diffusion systems with inhomogeneous Dirichlet boundary conditions (original) (raw)
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We consider a competition-diffusion system for two competing species; the density of the first species satisfies a parabolic equation together with a inhomogeneous Dirichlet boundary condition whereas the second one either satisfies a parabolic equation with a homogeneous Neumann boundary condition, or an ordinary differential equation. Under the situation where the two species spatially segregate as the interspecific competition rate becomes large, we show that the resulting limit problem turns out to be a free boundary problem. We focus on the singular limit of the interspecific reaction term, which involves a measure located on the free boundary.
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Journal of Mathematical Biology, 1981
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Zeitschrift für Analysis und ihre Anwendungen, 1983
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We discuss the generation and the motion of internal layers for a Lotka-Volterra competition-diffusion system with spatially inhomogeneous coefficients. We assume that the corresponding ODE system has two stable equilibria ū 0 and 0 v with equal strength of attraction in the sense to be specified later. The equation involves a small parameter , which reflects the fact that the diffusion is very small compared with the reaction terms. When the parameter is very small, the solution develops a clear transition layer between the region where the u species is dominant and the one where the v species is dominant. As tends to zero, the transition layer becomes a sharp interface, whose motion is subject to a certain law of motion, which is called the "interface equation". A formal asymptotic analysis suggests that the interface equation is the motion by mean curvature coupled with a drift term.
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Nonlinear Analysis-real World Applications, 2012
A class of reaction-diffusion systems modeling plant growth with spatial competition in saturated media is presented. We show, in this context, that standard diffusion can not lead to pattern formation (Diffusion Driven Instability of Turing). Degenerated nonlinear coupled diffusions inducing free boundaries and exclusive spatial diffusions are proposed. Local and global existence results are proved for smooth approximations of the degenerated nonlinear diffusions systems which give rise to long-time pattern formations. Numerical simulations of a competition model with degenerate/non degenerate nonlinear coupled diffusions are performed and we carry out the effect of the these diffusions on pattern formation and on the change of basins of attraction.
Nonlinear Analysis: Real World Applications, 2009
Spatial distribution of interacting chemical or biological species is usually described by a system of reaction-diffusion equations. In this work we consider a system of two reactiondiffusion equations with spatially varying diffusion coefficients which are different for different species and with forcing terms which are the gradient of a spatially varying potential. Such a system describes two competing biological species. We are interested in the possibility of long-term coexistence of the species in a bounded domain. Such long-term coexistence may be associated either with a periodic in time solution (usually associated with a Hopf bifurcation), or with time-independent solutions. We prove that no periodic solution exists for the system. We also consider some steady-states (the time-independent solutions) and examine their stability and bifurcations.