Stability and Hopf Bifurcation in a Diffusive Predator-Prey System with Beddington-DeAngelis Functional Response and Time Delay

and Applied Analysis 3 Under A1 , let u u − u∗, v v − v∗ and drop the bars for simplicity of notations, then 1.1 can be transformed into the following equivalent system: ut d1Δu t, x u u∗ 1 − u t − τ, x u∗ − s u u ∗ v v∗ m u u∗ n v v∗ , vt d2Δv t, x r u u∗ v v∗ m u u∗ n v v∗ − d v v∗ . 2.2 Let P u, v uv/ m u nv . By u∗ 1 − u∗ − su∗v∗/ m u∗ nv∗ 0 and −dv∗ ru∗v∗/ m u∗ nv∗ 0 2.2 becomes ut d1Δu t, x 1 − u∗ − a1 u t − u∗u t − τ, x − a2v − uu t − τ, x − f u, v , vt d2Δv t, x b1u t b2 − d v t g u, v , 2.3 where a1 sP10 u∗, v∗ , a2 sP01 u∗, v∗ , b1 rP10 u∗, v∗ , b2 rP01 u∗, v∗ , and