Stability of Bifurcated Periodic Solutions in a Delayed Competition System with Diffusion Effects

TLDR: It is shown that the bIfurcated periodic solution occurring at the first bifurcation point is orbitally asymptotically stable on the center manifold while those occurring at other bifURcation points are unstable.

Abstract: In this paper, a delayed Lotka—Volterra two species competition diffusion system with a single discrete delay and subject to the homogeneous Dirichlet boundary conditions is considered. By applying the normal form theory and the center manifold reduction for partial functional differential equations (PFDEs), the stability of bifurcated periodic solutions occurring through Hopf bifurcations is studied. It is shown that the bifurcated periodic solution occurring at the first bifurcation point is orbitally asymptotically stable on the center manifold while those occurring at other bifurcation points are unstable. Finally, some numerical simulations to a special example are included to verify our theoretical predictions.