Coexistence and Segregation for Strongly Competing Species in Special Domains

Coexistence-segregation dichotomy in the full cross-diffusion limit of the stationary SKT model

This paper studies the Lotka-Volterra competition model with cross-diffusion terms under homogeneous Dirichlet boundary conditions. We consider the asymptotic behavior of positive steady-states as equal two cross-diffusion coefficients tend to infinity (so-called the full cross-diffusion limit). The first result shows that, at the full cross-diffusion limit, the set of positive steady-state solutions can be classified into two types: the small coexistence or the complete segregation. The small coexistence can be characterized by the set of positive solutions of a nonlinear elliptic equation, w…

Global minimizers of coexistence for competing species

A class of variational models describing ecological systems of k species competing for the same resources is investigated. The occurrence of coexistence in minimal energy solutions is discussed and positive results are proven for suitably differentiated internal dynamics.

Some remarks on segregation of k species in strongly competing system

Spatial segregation occurs in population dynamics when $k$ species interact in a highly competitive way. As a model for the study of this phenomenon, we consider the competition-diffusion system of $k$ differential equations \[ -Δu_i(x)=-μu_i (x)\displaystyle{\sum_{j\neq i}} u_j (x) \quad i=1,...,k \] in a domain $D$ with appropriate boundary conditions. Any $u_i$ represents a population density and the parameter $μ$ determines the interaction strength between the populations. The purpose of this paper is to study the geometry of the limiting configuration as $μ\longrightarrow+\infty$ on a pla…

Minimal coexistence configurations for multispecies systems

We deal with strongly competing multispecies systems of Lotka-Volterra type with homogeneous Neumann boundary conditions in dumbbell-like domains. Under suitable non-degeneracy assumptions, we show that, as the competition rate grows indefinitely, the system reaches a state of coexistence of all the species in spatial segregation. Furthermore, the limit configuration is a local minimizer for the associated free energy.

On the limit configuration of four species strongly competing systems

We analysed some qualitative properties of the limit configuration of the solutions of a reaction-diffusion system of four competing species as the competition rate tends to infinity. Large interaction induces the spatial segregation of the species and only two limit configurations are possible: either there is a point where four species concur, a 4-point, or there are two points where only three species concur. We characterized, for a given datum, the possible 4-point configuration by means of the solution of a Dirichlet problem for the Laplace equation.

二维强竞争系统中的一般构型

Generic configurations in 2D strongly competing systems

研究平面区域中任意数量竞争物种隔离问题，运用到奇异空间的调和映射形式体系及霍普夫微分引入复结构，证明一般情况下区域划分为仅含三重连接点的子区域，即自由边界上最多三个种群相遇。