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

1. School of Mathematics and Statistics, Central South University, Changsha, 410083

## Abstract    Related pages

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

References

• 1. University of Chicago Press, Chicago, 1931.
• 2. Journal of Beihua University (Natural Science), 12 (2011), 1-9. (in Chinese)
• 3. Ecol. Model., 213 (2008), 356-364.
• 4. Proc. R. Soc. Lond. B., 266 (1999), 557-563.
• 5. Theor. Popul. Biol., 64 (2003), 1-10.
• 6. Int. J. Biomath., 5 (2012), 1250059, 19pp.
• 7. Hifr Co, Edmonton, 1980.
• 8. Amer. Naturalist, 151 (1998), 487-496.
• 9. Nonlinear Anal. Hybrid Syst., 15 (2015), 98-111.
• 10. SIAM J. Appl. Math., 72 (2012), 1524-1548.
• 11. Nonlinear Dynam., 73 (2013), 815-826.
• 12. Int. J. Biomath., 7 (2014), 1450035, 21pp.
• 13. Oikos, 82 (1998), 384-392, http://www.jstor.org/stable/3546980.
• 14. Ecol. Model., 154 (2002), 1-7.
• 15. Commun. Nonlinear Sci. Numer. Simulat., 23 (2015), 78-88.
• 16. Biosystems, 104 (2011), 77-86.
• 17. Nonlinear Dynam., 65 (2011), 1-10.
• 18. J. Math. Biol., 62 (2011), 291-331.
• 19. Int. J. Biomath., 6 (2013), 1350031, 15pp.
• 20. Appl. Math. Comput., 237 (2014), 282-292.
• 21. Nonlinear Dynam., 76 (2014), 1109-1117.
• 22. Shanghai Science and Technology Press, Shanghai, 1984. (in Chinese)
• 23. Math. Biosci., 238 (2012), 55-64.
• 24. Theor. Popul. Biol., 67 (2005), 23-31.
• 25. Math. Biosci., 189 (2004), 103-113.

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