We studied a delayed predator–prey model with diffusion and anti-predator behavior. Assume that additional food is provided for predator population. Then the stability of the positive equilibrium is considered. The existence of Hopf bifurcation is also discussed based on the Hopf bifurcation theory. The property of Hopf bifurcation is derived through the theory of center manifold and normal form method. Finally, we analyze the effect of time delay on the model through numerical simulations.

In this paper, a diffusive predator–prey system with a functional response that increases in both predator and prey densities is considered. By analyzing the characteristic roots of the partial differential equation system, the Turing instability and Hopf bifurcation are studied. In order to consider the dynamics of the model where the Turing bifurcation curve and the Hopf bifurcation curve intersect, we chose the diffusion coefficients d1 and β as bifurcating parameters. In particular, the normal form of Turing–Hopf bifurcation was calculated so that we could obtain the phase diagram. For parameters in each region of the phase diagram, there are different types of solutions, and their dynamic properties are extremely rich. In this study, we have used some numerical simulations in order to confirm these ideas.

<abstract><p>A tumor-immune system with diffusion and delays is proposed in this paper. First, we investigate the impact of delay on the stability of nonnegative equilibrium for the model with a single delay, and the system undergoes Hopf bifurcation when delay passes through some critical values. We obtain the normal form of Hopf bifurcation by applying the multiple time scales method for determining the stability and direction of bifurcating periodic solutions. Then, we study the tumor-immune model with two delays, and show the conditions under which the nontrivial equilibria are locally asymptotically stable. Thus, we can restrain the diffusion of tumor cells by controlling the time delay associated with the time of tumor cell proliferation and the time of immune cells recognizing tumor cells. Finally, numerical simulations are presented to illustrate our analytic results.</p></abstract>

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