Abstract In this paper, we consider a diffusive predator–prey model with time delay and prey toxicity. The effect of on the stability positive equilibrium is studied by analyzing eigenvalue spectrum. Delay-induced Hopf bifurcation also investigated. By utilizing normal form method center manifold reduction for partial functional differential equations, formulas determining property are given.

In this paper, we consider the following three-species predator-prey problem with prey-taxis and competition:$ \begin{equation*} \left\{ \begin{aligned} &u_t = d_1\Delta u + \chi_1 \nabla \cdot (u\nabla v) r_1u(1-u-kv-b_1w),&x\in \Omega, t>0,\\ &v_t d_2\Delta v r_2 v(1-hu-v-b_2w),&x\in &w_t d_3\Delta w -\chi_2 (w\nabla u)- \chi_3 v)\\ &\qquad\, +r_3 (-1+au+av -w^\sigm...