Spatial segregation limit of a competition-diffusion system with Dirichlet boundary conditions (original) (raw)
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Spatial segregation limit of a competition–diffusion system
European Journal of Applied Mathematics, 1999
We consider a competition–diffusion system and study its singular limit as the interspecific competition rate tend to infinity. We prove the convergence to a Stefan problem with zero latent heat.
Spatial segregation limit of a non-autonomous competition–diffusion system
2012
This paper is concerned with the spatial behavior of the non-autonomous competitiondiffusion system arising in population ecology. The limiting profile of the system is given as the competition rate tends to infinity. Our result shows that two competing species spatially segregate as the competition rates become large. Moreover, for the case of the same non-autonomous terms, we obtain the uniform convergence result.
Asymptotic estimates for the spatial segregation of competitive systems
Advances in Mathematics, 2005
For a class of population models of competitive type, we study the asymptotic behavior of the positive solutions as the competition rate tends to infinity. We show that the limiting problem is a remarkable system of differential inequalities, which defines the functional class S in (2). By exploiting the regularity theory recently developed in [10] for the elements of functional classes of the form S, we provide some qualitative and regularity property of the limiting configurations. Besides, for the case of two competing species, we obtain a full description of the limiting states and we prove some quantitative estimates for the rate of convergence. Finally, we prove some new Liouville type results which allow to have uniform regularity estimates of the solutions.
Singular limit of a competition–diffusion system with large interspecific interaction
Journal of Mathematical Analysis and Applications, 2012
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.
2007
We consider a two-component competition-diffusion system with equal diffusion coefficients and inhomogeneous Dirichlet boundary conditions. When the interspecific competition parameter tends to infinity, the system solution converges to that of a freeboundary problem. If all stationary solutions of this limit problem are non-degenerate and if a certain linear combination of the boundary data does not identically vanish, then for sufficiently large interspecific competition, all non-negative solutions of the competition-diffusion system converge to stationary states as time tends to infinity. Such dynamics are much simpler than those found for the corresponding system with either homogeneous Neumann or homogeneous Dirichlet boundary conditions.
On a competitive system with ideal free dispersal
Journal of Differential Equations, 2018
In this article we study the long term behavior of the competitive system ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂u ∂t = ∇ • α(x)∇ u m + u(m(x) − u − bv) in , t > 0, ∂u ∂t = ∇ • [β(x)∇v] + v(m(x) − cu − v) in , t > 0, ∇ u m •n = ∇v •n = 0 on ∂ , t > 0, which supports for the first species an ideal free distribution, that is a positive steady state which matches the per-capita growth rate. Previous results have stated that when b = c = 1 the ideal free distribution is an evolutionarily stable and neighborhood invader strategy, that is the species with density v always goes extinct. Thus, of particular interest will be to study the interplay between the inter-specific competition coefficients b, c and the diffusion coefficients α(x) and β(x) on the critical values for stability of semi-trivial steady states, and the structure of bifurcation branches of positive equilibria arising from these equilibria. We will also show that under certain regimes the system sustains multiple positive steady states.
Singular Limit of a Spatially Inhomogeneous Lotka–Volterra Competition–Diffusion System
Communications in Partial Differential Equations, 2007
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.
A reaction-diffusion-advection competition model with a free boundary
In this paper, we study a competitive diffusion quasilinear system with a free boundary. First, we investigate the mathematical questions of the problem. A priori estimates of Schauder type are established, which are necessary for the solvability of the problem. One of two competing species is an invader, which initially exists on a certain sub-interval. The other is initially distributed throughout the area under consideration. Examining the influence of baseline data on the success or failure of the first invasion. We conclude that there is a dichotomy of spread and extinction and give precise criteria for spread and extinction in this model.