STABILITY OF POSITIVE STEADY-STATE SOLUTIONS IN A DELAYED LOTKA-VOLTERRA DIFFUSION SYSTEM

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. @u ( x;t ) @t = d 1 ∆u ( x; t ) + u ( x; t )( r 1 a 11 u ( x; t � ) a 12 v ( x; t � )) ; x 2 Ω; t > 0 ; @v ( x;t ) @t = d 2 ∆v ( x; t ) + v ( x; t )( r 2 + a 21 u ( x; t � ) a 22 v ( x; t � )) ; x 2 Ω; t > 0 ; u ( x; t ) = v ( x; t ) = 0 ; x 2 @ Ω; t � 0 ; u ( x; t ) = u 0 ( x; t ) ; v ( x; t ) = v 0 ( x; t ) ; ( x; t ) 2 Ω � ( �; 0) ; where u ( x; t ) and v ( x; t ) designate the population densities for a cooperation species and a competition species at time t and space location x , respectively;