### Mathematical Biosciences and Engineering

2015, Issue 5: 1065-1081. doi: 10.3934/mbe.2015.12.1065

# Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy

• Received: 01 September 2014 Accepted: 29 June 2018 Published: 01 June 2015
• MSC : Primary: 34A37, 34C23, 34C37, 34D10.

• In this paper, we analyze a general predator-prey modelwith state feedback impulsive harvesting strategies in which the prey species displays a strongAllee effect. We firstly show the existence of order-$1$ heteroclinic cycle and order-$1$ positive periodic solutions by using the geometric theory of differential equations for the unperturbed system. Based on the theory of rotated vector fields, the order-$1$ positive periodic solutions and heteroclinic bifurcation are studied for the perturbed system. Finally, some numerical simulations are provided to illustrate our main results. All the results indicate that the harvesting rate should be maintained at a reasonable range to keep the sustainable development of ecological systems.

Citation: Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy[J]. Mathematical Biosciences and Engineering, 2015, 12(5): 1065-1081. doi: 10.3934/mbe.2015.12.1065

### Related Papers:

• In this paper, we analyze a general predator-prey modelwith state feedback impulsive harvesting strategies in which the prey species displays a strongAllee effect. We firstly show the existence of order-$1$ heteroclinic cycle and order-$1$ positive periodic solutions by using the geometric theory of differential equations for the unperturbed system. Based on the theory of rotated vector fields, the order-$1$ positive periodic solutions and heteroclinic bifurcation are studied for the perturbed system. Finally, some numerical simulations are provided to illustrate our main results. All the results indicate that the harvesting rate should be maintained at a reasonable range to keep the sustainable development of ecological systems.
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沈阳化工大学材料科学与工程学院 沈阳 110142

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