Separatrices of competition-diffusion equations (Q1912823)

The Lotka-Volterra competition model with diffusion \[ u_t=d_1u_{xx}+ u(m_1-c_{11}u-c_{12}v), \quad v_t=d_2v_{xx}+ v(m_1-c_{21}v-c_{22}u) \qquad (0<x<1,\;t>0), \] under Neumann boundary conditions is investigated. The constants in the equations are set such that the system has two stable spatially constant equilibria contained on the boundary of the first orthant in the \(u\), \(v\) plane. The author describes the separatrix of the basins of attraction of these equilibria near a third (unstable) equilibrium and discusses conditions on the initial state that determine the asymptotic state of the solution as \(t\to\infty\).