Stability of positive steady-state solutions in a delayed Lotka-Volterra diffusion system
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This paper considers the stability of positive steady-state so-lutions bifurcating from the trivial solution in a delayed Lotka-Volterra two-species predator-prey diffusion system with a discrete delay and sub-ject to the homogeneous Dirichlet boundary conditions on a general boun-ded open spatial domain with smooth boundary. The existence, unique-ness and asymptotic expressions of small positive steady-sate solutions bifurcating from the trivial solution are given by using the implicit func-tion theorem. By regarding the time delay as the bifurcation parameter and analyzing in detail the eigenvalue problems of system at the positive steady-state solutions, the asymptotic stability of bifurcating steady-state solutions is studied. It is demonstrated that the bifurcating steady-state solutions are asymptotically stable when the delay is less than a certain critical value and is unstable when the delay is greater than this critical value and the system under consideration can undergo a Hopf bifurcation at the bifurcating steady-state solutions when the delay crosses through a sequence of critical values.
Abstract
This paper considers the stability of positive steady-state so-lutions bifurcating from the trivial solution in a delayed Lotka-Volterra two-species predator-prey diffusion system with a discrete delay and sub-ject to the homogeneous Dirichlet boundary conditions on a general boun-ded open spatial domain with smooth boundary. The existence, unique-ness and asymptotic expressions of small positive steady-sate solutions bifurcating from the trivial solution are given by using the implicit func-tion theorem. By regarding the time delay as the bifurcation parameter and analyzing in detail the eigenvalue problems of system at the positive steady-state solutions, the asymptotic stability of bifurcating steady-state solutions is studied. It is demonstrated that the bifurcating steady-state solutions are asymptotically stable when the delay is less than a certain critical value and is unstable when the delay is greater than this critical value and the system under consideration can undergo a Hopf bifurcation at the bifurcating steady-state solutions when the delay crosses through a sequence of critical values.
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