Global Hopf Bifurcation of a Diffusive Modified Leslie–Gower Predator–Prey Model with Delay and Michaelis–Menten Type Prey Harvesting

Abstract

In this paper, we investigate the global Hopf bifurcation of a diffusive modified Leslie–Gower predator–prey model with delay and Michaelis–Menten type prey harvesting. First, we obtain the stability of positive steady state and the existence of local Hopf bifurcation under certain conditions. Second, we get the permanence of the system by using the comparison theorem. Moreover, by constructing a suitable Lyapunov function, we derive sufficient conditions for the global attractivity of the unique positive steady state for the system without delay. Then, the global existence of positive periodic solutions is established by using the global Hopf bifurcation result of Wu. Finally, the results are verified by numerical simulation.

Appendix

In this appendix, we provide the details of the proof of Theorem 2.1. To prove the existence and uniqueness of positive constant steady state, we only need to prove that $\phi (u)=0$ has a unique positive root. If $\mathbb {E}_1>0$, we have $\phi (0)>0$. Since $\phi (\infty )=-\infty$, $\phi (u)=0$ has at least one positive root. From $\phi '(0)=\mathbb {D}_{1}$ and $\phi ''(0)=\mathbb {C}_{1}$, we know that if one of the following cases holds for $u\in [0,+\infty )$, $\phi (u)=0$ has a unique positive root.

$(s_{1})$:

: $\phi (u)$ monotonically decreasing;

$(s_{2})$:

: $\phi (u)$ increases and then decreases;

$(s_{3})$:

: $\phi (u)$ decreases and then increases and then decreases and either $\phi (u_{1}^{2})>0$ or $\phi (u_{1}^{1})<0$;

$(s_{4})$:

: $\phi (u)$ increases and then decreases and then increases and then decreases and either $\phi (u_{1}^{2})>0$ or $\phi (u_{1}^{3})>0$ and $\phi (u_{1}^{1})<0$.

Next, we analyze $(H_{1a})$-$(H_{1l})$ in turn. The following discussion satisfies $u\in [0,+\infty )$.

$(H_{1a})$. If $\triangle \le 0$, than we have $\phi ''(u)<0$, which implies that $\phi '(u)$ monotonically decreasing, and $\phi (u)$ monotonically decreasing or increases and then decreases.

$(H_{1b})$. If $\triangle >0$, $\mathbb {B}_{1}\le 0$, $\mathbb {C}_1>0$ and $\mathbb {D}_{1}\ge 0$, this means $\phi ''(u)=0$ has a positive root, $\phi '(u)$ increases and then decreases and $\phi (u)$ increases and then decreases.

$(H_{1c})$. If $\triangle >0$, $\mathbb {B}_{1}\le 0$ and $\mathbb {C}_1>0$, than $\phi ''(u)=0$ has a positive root, which implies $\phi '(u)$ increases and then decreases. Under the conditions that $\mathbb {D}_{1}<0$ and $\phi '(u_{2}^{+})\le 0$, we have $\phi '(u)<0$ and $\phi (u)$ monotonically decreasing.

$(H_{1d})$. Under the conditions of $\triangle >0$, $\mathbb {B}_{1}\le 0$ and $\mathbb {C}_1>0$, we obtain $\phi ''(u)=0$ has a positive root, $\phi '(u)$ increases and then decreases. If $\mathbb {D}_{1}<0$ and $\phi '(u_{2}^{+})>0$ hold, the maximum point of $\phi '(u)$ to be greater than 0, $\phi (u)$ decreases and then increases and then decreases.

$(H_{1e})$. In the case of $\triangle >0$, $\mathbb {B}_{1}\le 0$ and $\mathbb {C}_1\le 0$, we have $\phi ''(u)=0$ has no positive root, $\phi '(u)$ monotonically decreasing and $\phi (u)$ monotonically decreasing or increases and then decreases.

$(H_{1f})$. If $\triangle >0$, $\mathbb {B}_{1}>0$ and $\mathbb {C}_1\ge 0$, than $\phi ''(u)=0$ has a positive root and $\phi '(u)$ increases and then decreases. Under the conditions of $\mathbb {D}_1\ge 0$, $\phi (u)$ increases and then decreases.

$(H_{1g})$. Under the conditions that $\triangle >0$, $\mathbb {B}_{1}>0$, $\mathbb {C}_1\ge 0$, $\mathbb {D}_{1}<0$ and $\phi '(u_{2}^{+})\le 0$, we obtain $\phi '(u)\le 0$, and $\phi (u)$ monotonically decreasing.

$(H_{1h})$. If $\triangle >0$, $\mathbb {B}_{1}>0$, $\mathbb {C}_1\ge 0$, $\mathbb {D}_{1}<0$ and $\phi '(u_{2}^{+})>0$, than the maximum point of $\phi '(u)$ to be greater than 0. So $\phi (u)$ decreases and then increases and then decreases.

$(H_{1i})$. In the case of $\triangle >0$, $\mathbb {B}_{1}>0$ and $\mathbb {C}_1<0$, we have $\phi '(u)$ decreases and then increases and then decreases. If $\mathbb {D}_{1}>0$ and $\phi '(u_{2}^{-})\ge 0$, the minimum point of $\phi '(u)$ to be greater than 0, which implies that $\phi (u)$ increases and then decreases.

$(H_{1j})$. Under the conditions that $\triangle >0$, $\mathbb {B}_{1}>0$, $\mathbb {C}_1<0$, $\mathbb {D}_{1}>0$, $\phi '(u_{2}^{+})>0$ and $\phi '(u_{2}^{-})<0$, we obtain the maximum point of $\phi '(u)$ to be greater than 0, and the minimum point of $\phi '(u)$ to be less than 0, this means $\phi (u)$ increases and then decreases and then increases and then decreases.

$(H_{1k})$. If $\triangle >0$, $\mathbb {B}_{1}>0$ and $\mathbb {C}_1<0$, this means $\phi '(u)$ decreases and then increases and then decreases. In the case of $\phi '(u_{2}^{+})\le 0$, we have the maximum point of $\phi '(u)$ to be less than 0, and $\phi (u)$ monotonically decreasing or increases and then decreases.

$(H_{1l})$. Under the conditions of $\triangle >0$, $\mathbb {B}_{1}>0$, $\mathbb {C}_1<0$, $\mathbb {D}_{1}\le 0$ and $\phi '(u_{2}^{+})>0$, which inplies the maximum point of $\phi '(u)$ to be greater than 0, and $\phi (u)$ decreases and then increases and then decreases.

The proof is complete.