Global dynamics of a generalist predator–prey model in open advective environments

Bifurcation analysis of a reaction-diffusion-advection predator-prey system with delay

A diffusive predator-prey system with advection and time delay is considered. Choosing the conversion delay $ \tau $ as a bifurcation parameter, we find that as $ \tau $ varies, the system will generate Hopf bifurcation. Then, for the reaction diffusion model proposed in this paper, we use an improved center manifold reduction method and normal form theory to derive an algorithm for determining the direction and stability of Hopf bifurcation. Finally, we provide simulations to illustrate the effects of time delay $ \tau $ and advection $ \alpha $ on system behaviors.

Bifurcation analysis for a single population model with advection

In this paper, the dynamics of a single population model with a general growth function is investigated in an advective environment. We show the existence of a nonconstant positive steady state, and give sufficient conditions for the occurrence of a Hopf bifurcation at the positive steady state. Moreover, the theoretical results are applied to the diffusive Nicholson's blowflies and Mackey-Glass's models with advection and delay, respectively. We numerically show that the population density decreases as the increase of advection rate or death rate, and a delay-induced Hopf bifurcation is more likely to occur with small advection or low mortality rate.