| |
| Due to the fear to predation risk preys may decrease birth rate or flee high level of predators patch to prey-only patch at a cost of decreased resources. In this paper a predator-prey model with fear effect and patch structure is constructed to study how the fear and diffusion affect predator-prey dynamics. The stability of equilibria and existence of Hopf bifurcation are studied. The fear to predation risk can be modeled by two types of parameters: $k$, $a_{21}$ and $a_{12}$, where high level of fear leads to large $k$ and thus low level of prey's birth rate; high level of fear results in large prey's diffusive rate $a_{21}$ from predator-prey patch 1 to prey-only patch 2, and great hunger and bad ability of remembering fear in patch 1 cause large prey's diffusive rate $a_{12}$ from patch 2 to patch 1. Numerical simulations are as follows. (1) In some cases, large $k$ can stabilize the predator-prey system by excluding the existence of periodic solutions when $a_{21}$ is small. However, when $a_{21}$ is large the change of $k$ can not lead to periodic oscillations. In addition, when $a_{21}$ is larger, the predators will die out. Thus, the oscillation behavior or the persistence of predators may be overestimated if the diffusive behaviors $a_{21}$ is weakened or ignored. (2) Under some situations, the change of $a_{12}$ causes Hopf bifurcations twice. This implies that the oscillation behavior may be underestimated or overestimated if the diffusive behavior $a_{12}$ is weakened. %Numerical simulations show that high levels of fear (or low birth rate of preys) can stabilize the predator-prey system by excluding the existence of periodic solutions. However, high level of dispersal caused by fear from predator-prey patch to predator-free patch can stabilize the predator-prey system. On the contrary, high level of dispersal caused by starvation from predator-free patch to predator-prey patch can induce periodic oscillations. % These conclusions imply that the oscillation behavior may be underestimated or overestimated according to whether the fear effect to predation risk (or low birth rate of preys) or dispersal caused by starvation dominates if the dispersal is ignored. |
|
| Keywords:Applied Mathematics,Fear effect, Dispersal, Stability, Hopf bifurcation |
|
