Predator's invasion into an isolated patch with spatially heterogeneous prey distribution (Q1175544)

Let \(S\) be a subinterval of the unit interval \(I\) in \(R\), and let \(a-d\) be positive real constants. The author studies two-component predator- prey systems of the form \((t>0)\): \[ \begin{aligned} u_ t &= D(v)u_{xx}+a(v- b)u\quad\hbox{in }I,\\ v_ t &= c(1-v-u)v \hbox{in }S,\qquad v=0\quad\hbox{in }I-S,\\ \end{aligned} \] with diffusion coefficient \(D(v)\) given by \(D\)=\(d\) in \(S\) and \(D\)=\(1\) in \(I-S\). The boundary conditions for \(u\) are homogeneous and of Dirichlet or Neumann type. Using a linearization argument, the author derives necessary and sufficient conditions for the (local) asymptotic stability or instability of the equilibrium solution \(u=0\), \(v=1\) in \(S\) and \(v=0\) in \(I-S\). A similar problem for two competing and diffusing species is also treated.