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2015, 12(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

1. 

School of Mathematics and Statistics, Central South University, Changsha, 410083, China, China

Received  September 2014 Revised  January 2015 Published  June 2015

In this paper, we analyze a general predator-prey model with state feedback impulsive harvesting strategies in which the prey species displays a strong Allee 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. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065
References:
[1]

W. C. Allee, Animal Aggregations: A Study in General Sociology,, University of Chicago Press, (1931).   Google Scholar

[2]

L. S. Chen, Pest control and geometric theory of semicontinuous dynamical system,, Journal of Beihua University (Natural Science), 12 (2011), 1.  doi: 10.3969/j.issn.1009-4822.2011.01.001.  Google Scholar

[3]

L. L. Chen and Z. S. Lin, The effect of habitat destruction on metapopulations with the Allee-like effect: A study case of yancheng in Jiangsu province, China,, Ecol. Model., 213 (2008), 356.  doi: 10.1016/j.ecolmodel.2007.12.016.  Google Scholar

[4]

F. Courchamp, B. Grenfell and T. Clutton-Brock, Population dynamics of obligate cooperators,, Proc. R. Soc. Lond. B., 266 (1999), 557.  doi: 10.1098/rspb.1999.0672.  Google Scholar

[5]

M. G. Correigh, Habitat selection reduces extinction of populations subject to Allee effects,, Theor. Popul. Biol., 64 (2003), 1.  doi: 10.1016/S0040-5809(03)00025-X.  Google Scholar

[6]

C. J. Dai, M. Zhao and L. S. Chen, Homoclinic bifurcation in semi-continuous dynamic systems,, Int. J. Biomath., 5 (2012).  doi: 10.1142/S1793524512500593.  Google Scholar

[7]

H. I. Freedman, Deterministic Mathematical Models in Population Ecology,, Hifr Co, (1980).  doi: 10.2307/3556198.  Google Scholar

[8]

M. J. Groom, Allee effects limit population viability of an annual plant,, Amer. Naturalist, 151 (1998), 487.  doi: 10.1086/286135.  Google Scholar

[9]

H. J. Guo, L. S. Chen and X. Y. Song, Geometric properties of solution of a cylindrical dynamic system with impulsive state feedback control,, Nonlinear Anal. Hybrid Syst., 15 (2015), 98.  doi: 10.1016/j.nahs.2014.08.002.  Google Scholar

[10]

M. Z. Huang, J. X. Li, X. Y. Song and H. J. Guo, Modeling impulsive injections of insulin: Towards artificial pancreas,, SIAM J. Appl. Math., 72 (2012), 1524.  doi: 10.1137/110860306.  Google Scholar

[11]

M. Z. Huang, S. Z. Liu, X. Y. Song and L. S. Chen, Periodic solutions and homoclinic bifurcation of a predator-prey system with two types of harvesting,, Nonlinear Dynam., 73 (2013), 815.  doi: 10.1007/s11071-013-0834-7.  Google Scholar

[12]

M. Z. Huang and X. Y. Song, Modeling and qualitative analysis of diabetes therapies with state feedback control,, Int. J. Biomath., 7 (2014).  doi: 10.1142/S1793524514500351.  Google Scholar

[13]

M. Kuussaari, I. Saccheri, M. Camara and I. Hanski, Allee effect and population dynamics in the Glanville fritillary butterfly,, Oikos, 82 (1998), 384.  doi: 10.2307/3546980.  Google Scholar

[14]

Z. S. Lin and B. L. Li, The maximum sustainable yield of Allee dynamic system,, Ecol. Model., 154 (2002), 1.  doi: 10.1016/S0304-3800(01)00479-3.  Google Scholar

[15]

G. P. Pang, L. S. Chen, W. J. Xu and G. Fu, A stage structure pest management model with impulsive state feedback control,, Commun. Nonlinear Sci. Numer. Simulat., 23 (2015), 78.  doi: 10.1016/j.cnsns.2014.10.033.  Google Scholar

[16]

Y. Tian, A. Kasperski, K. B. Sun and L. S. Chen, Theoretical approach to modelling and analysis of the bioprocess with product inhibition and impulse effect,, Biosystems, 104 (2011), 77.  doi: 10.1016/j.biosystems.2011.01.003.  Google Scholar

[17]

T. Y. Wang and L. S. Chen, Nonlinear analysis of a microbial pesticide model with impulsive state feedback control,, Nonlinear Dynam., 65 (2011), 1.  doi: 10.1007/s11071-010-9828-x.  Google Scholar

[18]

J. F. Wang, J. P. Shi and J. J. Wei, Predator-prey system with strong Allee effect in prey,, J. Math. Biol., 62 (2011), 291.  doi: 10.1007/s00285-010-0332-1.  Google Scholar

[19]

C. J. Wei and L. S. Chen, Heteroclinic bifurcation of a prey-predator fishery model with impulsive harvesting,, Int. J. Biomath., 6 (2013).  doi: 10.1142/S1793524513500319.  Google Scholar

[20]

C. J. Wei and L. S. Chen, Homoclinic bifurcation of prey-predator model with impulsive state feedback control,, Appl. Math. Comput., 237 (2014), 282.  doi: 10.1016/j.amc.2014.03.124.  Google Scholar

[21]

C. J. Wei and L. S. Chen, Periodic solution and heteroclinic bifurcation in a predator-prey system with Allee effect and impulsive harvesting,, Nonlinear Dynam., 76 (2014), 1109.  doi: 10.1007/s11071-013-1194-z.  Google Scholar

[22]

Y. Q. Ye, Limit Cycle Theory,, Shanghai Science and Technology Press, (1984).   Google Scholar

[23]

L. C. Zhao, L. S. Chen and Q. L. Zhang, The geometrical analysis of a predator-prey model with two state impulses,, Math. Biosci., 238 (2012), 55.  doi: 10.1016/j.mbs.2012.03.011.  Google Scholar

[24]

S. R. Zhou, Y. F. Liu and G. Wang, The stability of predator-prey systems subject to the Allee effects,, Theor. Popul. Biol., 67 (2005), 23.  doi: 10.1016/j.tpb.2004.06.007.  Google Scholar

[25]

S. R. Zhou and G. Wang, Allee-like effects in metapopulation dynamics,, Math. Biosci., 189 (2004), 103.  doi: 10.1016/j.mbs.2003.06.001.  Google Scholar

show all references

References:
[1]

W. C. Allee, Animal Aggregations: A Study in General Sociology,, University of Chicago Press, (1931).   Google Scholar

[2]

L. S. Chen, Pest control and geometric theory of semicontinuous dynamical system,, Journal of Beihua University (Natural Science), 12 (2011), 1.  doi: 10.3969/j.issn.1009-4822.2011.01.001.  Google Scholar

[3]

L. L. Chen and Z. S. Lin, The effect of habitat destruction on metapopulations with the Allee-like effect: A study case of yancheng in Jiangsu province, China,, Ecol. Model., 213 (2008), 356.  doi: 10.1016/j.ecolmodel.2007.12.016.  Google Scholar

[4]

F. Courchamp, B. Grenfell and T. Clutton-Brock, Population dynamics of obligate cooperators,, Proc. R. Soc. Lond. B., 266 (1999), 557.  doi: 10.1098/rspb.1999.0672.  Google Scholar

[5]

M. G. Correigh, Habitat selection reduces extinction of populations subject to Allee effects,, Theor. Popul. Biol., 64 (2003), 1.  doi: 10.1016/S0040-5809(03)00025-X.  Google Scholar

[6]

C. J. Dai, M. Zhao and L. S. Chen, Homoclinic bifurcation in semi-continuous dynamic systems,, Int. J. Biomath., 5 (2012).  doi: 10.1142/S1793524512500593.  Google Scholar

[7]

H. I. Freedman, Deterministic Mathematical Models in Population Ecology,, Hifr Co, (1980).  doi: 10.2307/3556198.  Google Scholar

[8]

M. J. Groom, Allee effects limit population viability of an annual plant,, Amer. Naturalist, 151 (1998), 487.  doi: 10.1086/286135.  Google Scholar

[9]

H. J. Guo, L. S. Chen and X. Y. Song, Geometric properties of solution of a cylindrical dynamic system with impulsive state feedback control,, Nonlinear Anal. Hybrid Syst., 15 (2015), 98.  doi: 10.1016/j.nahs.2014.08.002.  Google Scholar

[10]

M. Z. Huang, J. X. Li, X. Y. Song and H. J. Guo, Modeling impulsive injections of insulin: Towards artificial pancreas,, SIAM J. Appl. Math., 72 (2012), 1524.  doi: 10.1137/110860306.  Google Scholar

[11]

M. Z. Huang, S. Z. Liu, X. Y. Song and L. S. Chen, Periodic solutions and homoclinic bifurcation of a predator-prey system with two types of harvesting,, Nonlinear Dynam., 73 (2013), 815.  doi: 10.1007/s11071-013-0834-7.  Google Scholar

[12]

M. Z. Huang and X. Y. Song, Modeling and qualitative analysis of diabetes therapies with state feedback control,, Int. J. Biomath., 7 (2014).  doi: 10.1142/S1793524514500351.  Google Scholar

[13]

M. Kuussaari, I. Saccheri, M. Camara and I. Hanski, Allee effect and population dynamics in the Glanville fritillary butterfly,, Oikos, 82 (1998), 384.  doi: 10.2307/3546980.  Google Scholar

[14]

Z. S. Lin and B. L. Li, The maximum sustainable yield of Allee dynamic system,, Ecol. Model., 154 (2002), 1.  doi: 10.1016/S0304-3800(01)00479-3.  Google Scholar

[15]

G. P. Pang, L. S. Chen, W. J. Xu and G. Fu, A stage structure pest management model with impulsive state feedback control,, Commun. Nonlinear Sci. Numer. Simulat., 23 (2015), 78.  doi: 10.1016/j.cnsns.2014.10.033.  Google Scholar

[16]

Y. Tian, A. Kasperski, K. B. Sun and L. S. Chen, Theoretical approach to modelling and analysis of the bioprocess with product inhibition and impulse effect,, Biosystems, 104 (2011), 77.  doi: 10.1016/j.biosystems.2011.01.003.  Google Scholar

[17]

T. Y. Wang and L. S. Chen, Nonlinear analysis of a microbial pesticide model with impulsive state feedback control,, Nonlinear Dynam., 65 (2011), 1.  doi: 10.1007/s11071-010-9828-x.  Google Scholar

[18]

J. F. Wang, J. P. Shi and J. J. Wei, Predator-prey system with strong Allee effect in prey,, J. Math. Biol., 62 (2011), 291.  doi: 10.1007/s00285-010-0332-1.  Google Scholar

[19]

C. J. Wei and L. S. Chen, Heteroclinic bifurcation of a prey-predator fishery model with impulsive harvesting,, Int. J. Biomath., 6 (2013).  doi: 10.1142/S1793524513500319.  Google Scholar

[20]

C. J. Wei and L. S. Chen, Homoclinic bifurcation of prey-predator model with impulsive state feedback control,, Appl. Math. Comput., 237 (2014), 282.  doi: 10.1016/j.amc.2014.03.124.  Google Scholar

[21]

C. J. Wei and L. S. Chen, Periodic solution and heteroclinic bifurcation in a predator-prey system with Allee effect and impulsive harvesting,, Nonlinear Dynam., 76 (2014), 1109.  doi: 10.1007/s11071-013-1194-z.  Google Scholar

[22]

Y. Q. Ye, Limit Cycle Theory,, Shanghai Science and Technology Press, (1984).   Google Scholar

[23]

L. C. Zhao, L. S. Chen and Q. L. Zhang, The geometrical analysis of a predator-prey model with two state impulses,, Math. Biosci., 238 (2012), 55.  doi: 10.1016/j.mbs.2012.03.011.  Google Scholar

[24]

S. R. Zhou, Y. F. Liu and G. Wang, The stability of predator-prey systems subject to the Allee effects,, Theor. Popul. Biol., 67 (2005), 23.  doi: 10.1016/j.tpb.2004.06.007.  Google Scholar

[25]

S. R. Zhou and G. Wang, Allee-like effects in metapopulation dynamics,, Math. Biosci., 189 (2004), 103.  doi: 10.1016/j.mbs.2003.06.001.  Google Scholar

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