Abstract
A non-smooth switched harvest on predators is introduced into a simple predator-prey model with logistical growth of the prey and a bilinear functional response. If the density of the predator is below a switched value, the harvesting rate is linear; otherwise, it is constant. The model links the well studied predator-prey model with constant harvesting to that with a proportional harvesting rate. It is shown that when the net reproductive number for the predator is greater than unity, the system is permanent and there may exist multiple positive equilibria due to the effects of the switched harvest, a saddle-node bifurcation, a limit cycle, and the coexistence of a stable equilibrium and a unstable circled inside limit cycle and a stable circled outside limit cycle. When the net reproductive number is less than unity, a backward bifurcation from a positive equilibrium occurs, which implies that the stable predator-extinct equilibrium may coexist with two coexistence equilibria. In this situation, reducing the net reproductive number to less than unity is not enough to enable the predator to go extinct. Numerical simulations are provided to illustrate the theoretical results. It seems that the model possesses new complex dynamics compared to the existing harvesting models.
Highlights
Mathematical modeling of predator-prey interactions have attracted wide attention since the original work by Lotka and Volterra in 1920s, and there have been extensively studied for their rich dynamics [1,2,3]
With switched predator harvest strategy; that is, when the density of predator is below harvest level P0, the linear harvesting rate is applied to the system, whereas when the density of predators is higher than harvest level, the system adopts nonzero constant harvesting rate, even if n < 1, the prey and predator may coexist
We proposed and studied a new predator-prey model with non-smooth switched harvest on the predator
Summary
Mathematical modeling of predator-prey interactions have attracted wide attention since the original work by Lotka and Volterra in 1920s, and there have been extensively studied for their rich dynamics [1,2,3]. Xiao and Jennings [16] considered the dynamical properties of the ratio-dependent predator-prey model with constant prey harvesting. Philip et al [26] discussed two predator-prey models with linear or nonzero constant predator harvesting. It is interesting to construct a new kind of harvesting rate that combines the advantages from both linear and constant harvesting rates Motivated by these ideas, in this paper, we consider a predator-prey model with a novel harvesting rate. Following the methods in [14,15,16,17,30], we investigate the existence and stability of multiple equilibria, bifurcations, and limit cycles, and study the effects of switched harvest on the dynamics of the predator-prey model.
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