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doi: 10.3934/dcdsb.2021252
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## Dynamics of a Leslie-Gower predator-prey system with hunting cooperation and prey harvesting

 1 School of Mathematics and Physics, Wuhan Institute of Technology, Wuhan, Hubei 430205, China 2 School of Sciences, Southwest Petroleum University, Chengdu, Sichuan 610500, China, Institute for Artificial Intelligence, Southwest Petroleum University, Chengdu, Sichuan 610500, China

* Corresponding author: Lingling Liu

Received  March 2021 Revised  July 2021 Early access October 2021

Fund Project: The first author is supported by NSFC #12101470, WIT #K2021077. The second author is supported by NSFC #11771308, #11871041, SWPU #2017CXTD02, #18TD0013, #2019CXTD08

In this paper, we study the dynamics of a Leslie-Gower predator-prey system with hunting cooperation among predator population and constant-rate harvesting for prey population. It is shown that there are a weak focus of multiplicity up to three and a cusp of codimension at most two for various parameter values, and the system exhibits two saddle-node bifurcations, a Bogdanov-Takens bifurcation of codimension two and a Hopf bifurcation as the bifurcation parameters vary. The results developed in this article reveal far more complex dynamics compared to the Leslie-Gower system and show how the prey harvesting and the hunting cooperation affect the dynamics of the system. In particular, there exist some critical values of prey harvesting and hunting cooperation such that the predator and prey populations are at risk of extinction if the intensities of harvesting and hunting cooperation are greater than these critical values. Moreover, numerical simulations are presented to illustrate our theoretical results.

Citation: Yong Yao, Lingling Liu. Dynamics of a Leslie-Gower predator-prey system with hunting cooperation and prey harvesting. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021252
##### References:

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##### References:
Saddle-nodes of system (5) when $\alpha = \beta = \gamma = 1$. (A) Saddle-nodes $E_{1*}$ when $h = 0.25$. (B) Saddle-nodes $E_{2*}$ when $h = 0.1126$
Codimension 2 cusp when $\beta = 1$, $\gamma = 0.3$, $\alpha = 0.82645$ and $h = 0.1144$
The Bogdanov-Takens bifurcation diagram and corresponding phase portraits of system (5) when $\beta = 1$ and $\gamma = 0.3$. (A) Bifurcation diagram. (B) No equilibrium when $(\alpha, h) = (1.2, 0.1108)\in\mathcal{I}_1$. (C) An unstable focus $E_{22}$ and a saddle $E_{21}$ when $(\alpha, h) = (1.5, 0.108)\in\mathcal{I}_2$. (D) An unstable limit cycle around a stable focus $E_{22}$ and a saddle $E_{21}$ when $(\alpha, h) = (1.5, 0.1078)\in\mathcal{I}_3$. (E) A homoclinic orbit when $(\alpha, h) = (1.2, 0.11057)\in\mathcal{HL}$. (F) A stable focus $E_{22}$ and a saddle $E_{21}$ when $(\alpha, h) = (1.5, 0.107)\in\mathcal{I}_4$
Limit cycles induced by Hopf bifurcation at $E_{22}$ in system (5). (A) An unstable limit cycle around $E_{22}$ with $\beta = 1.8$, $\gamma = 0.3$, $\alpha = 10.49382716$ and $h = 0.038$. (B) Two limit cycles with $\beta = 8$, $\gamma = 0.1$, $\alpha = 86.65647719836$ and $h = 0.0013277768$. (C) The orbit from $P_2$ spirals outward. (D) The orbit from $P_3$ spirals inward
Dynamical behaviors near $E_{2*}$
 Parameters Equilibria and properties Closed orbits and homoclinic orbits $(\beta, \gamma)$ $(h, \alpha)$ $(0,+\infty)$ $\times[\frac{1}{2}, \frac{2}{3})$ or $(0,\frac{2\gamma}{1-2\gamma})$ $\times(0, \frac{1}{2})$ $\mathcal{I}_{1}$ No equilibria No $\mathcal{SN}^+\cup\mathcal{SN}^-$ $E_{2*}$(saddle node) No $\mathcal{I}_{2}$ $E_{21}$(saddle) $E_{22}$(unstable focus or node) No $\mathcal{H}$ $E_{21}$(saddle) $E_{22}$(unstable weak focus) No $\mathcal{I}_3$ $E_{21}$(saddle) $E_{22}$(stable focus) A unstable limit cycle $\mathcal{HL}$ $E_{21}$(saddle) $E_{22}$(stable focus) A homoclinic orbit $\mathcal{I}_{4}$ $E_{21}$(saddle) $E_{22}$(stable focus or node) No $(h_*, \alpha_*)$ $E_{2*}$(cusp) No
 Parameters Equilibria and properties Closed orbits and homoclinic orbits $(\beta, \gamma)$ $(h, \alpha)$ $(0,+\infty)$ $\times[\frac{1}{2}, \frac{2}{3})$ or $(0,\frac{2\gamma}{1-2\gamma})$ $\times(0, \frac{1}{2})$ $\mathcal{I}_{1}$ No equilibria No $\mathcal{SN}^+\cup\mathcal{SN}^-$ $E_{2*}$(saddle node) No $\mathcal{I}_{2}$ $E_{21}$(saddle) $E_{22}$(unstable focus or node) No $\mathcal{H}$ $E_{21}$(saddle) $E_{22}$(unstable weak focus) No $\mathcal{I}_3$ $E_{21}$(saddle) $E_{22}$(stable focus) A unstable limit cycle $\mathcal{HL}$ $E_{21}$(saddle) $E_{22}$(stable focus) A homoclinic orbit $\mathcal{I}_{4}$ $E_{21}$(saddle) $E_{22}$(stable focus or node) No $(h_*, \alpha_*)$ $E_{2*}$(cusp) No
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