In this manuscript, we first consider the diffusive competition and cooperation system subject to Neumann boundary conditions without delay terms and get the conclusion that the unique positive constant equilibrium is locally asymptotically stable. Then, we study the diffusive delayed competition and cooperation system subject to Neumann boundary conditions, and the existence of Hopf bifurcation at the positive equilibrium is obtained by regarding delay term as the parameter. By the theory of center manifold and normal form, an algorithm for determining the direction and stability of Hopf bifurcation is derived. Finally, some numerical simulations and summarizations are carried out for illustrating the theoretical analytic results.

In this article, a nonautonomous stochastic modified Bazykin model is introduced. The positive solution is proved to be unique and global for any initial data via stochastic comparison theorem and Itô formula. Stochastic ultimate boundedness and stochastic permanence are also considered. Finally, some simulations are performed to verify the validity of results.

This article is devoted to the dynamical behavior of a stochastic modified Bazykin predator–prey model under regime switching. Some sufficient conditions are derived to guarantee the asymptotic properties, persistent and extinct of solutions by using the stochastic comparison theorem, Itô formula and exponential martingale inequality. At last, some simulations are given to illustrate our main results.