Some remarks on segregation of k species in strongly competing system

Coexistence and Segregation for Strongly Competing Species in Special Domains

We deal with strongly competing multispecies systems of Lotka-Volterra type with homogeneous Dirichlet boundary conditions. For a class of nonconvex domains composed by balls connected with thin corridors, we show the occurrence of pattern formation (coexistence and spatial segregation of all the species), as the competition grows indefinitely. As a result we prove the existence and uniqueness of solutions for a remarkable system of differential inequalities involved in segregation phenomena and optimal partition problems.

On the long term spatial segregation for a competition-diffusion system

We investigate the long term behavior for a class of competition-diffusion systems of Lotka-Volterra type for two competing species in the case of low regularity assumptions on the data. Due to the coupling that we consider the system cannot be reduced to a single equation yielding uniform estimates with respect to the inter-specific competition rate parameter. Moreover, in the particular but meaningful case of initial data with disjoint support and Dirichlet boundary data which are time-independent, we prove that as the competition rate goes to infinity the solution converges, along with suit…

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。

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.