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December  2020, 25(12): 4641-4657. doi: 10.3934/dcdsb.2020117

Bifurcation analysis and dynamic behavior to a predator-prey model with Beddington-DeAngelis functional response and protection zone

Xiao He 1,2, and  Sining Zheng 3,,

1. 

Department of Mathematics and Institute of Natural Sciences, Shanghai Jiaotong University, Shanghai 200240, China
2. 

Department of Mathematics, Dalian Minzu University, Dalian 116600, China
3. 

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

* Corresponding author: Sining Zheng

Received  October 2019 Published  December 2020 Early access  March 2020

Fund Project: The first author is supported by National Natural Science Foundation of China No.11701067 and Natural Science Foundation of Liaoning 2019-ZD-0180

In this paper we study the protection zone problem to a predator-prey model subject to Beddington-DeAngelis functional responses and small prey growth rate. This is a successive work to a previous paper of the authors [X. He, S. N. Zheng, Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response, J. Math. Biol. 75 (2017) 239-257], where the model with large prey growth rate was considered. At first we establish the existence and multiplicity of positive steady state solutions, and then give the dynamic behavior of the evolution problem. It is proved that there may be no positive steady state, or may have at leat one, two, or even three positive steady states, depending on the parameters involved such as the growth rate, the predation rate, and the food handling time of the predators, the growth rate and the refuge ability of the preys, and the sizes of the habitat with protection zone. In addition, it is shown that the dynamics of the solutions rely on the initial state as well, e.g., though there could be multiple positive steady states, the prey will go to extinction as time tends to infinity if its initial value is small.

Keywords: Protection zone, bifurcation, diffusion, predator-prey, dynamic behavior.
Mathematics Subject Classification: Primary: 35J47, 35K57, 92D40.
Citation: Xiao He, Sining Zheng. Bifurcation analysis and dynamic behavior to a predator-prey model with Beddington-DeAngelis functional response and protection zone. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4641-4657. doi: 10.3934/dcdsb.2020117
References:
[1]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44 (1975), 331-340.  doi: 10.2307/3866.  Google Scholar

[2]

J. Blat and K. J. Brown, Global bifurcation of positive solutions in some systems of elliptic equations, SIAM. J. Math. Anal., 17 (1986), 1339-1353.  doi: 10.1137/0517094.  Google Scholar

[3]

W. Chen and M. Wang, Qualitative analysis of predator-prey model with Beddington-DeAngelis functional response and diffusion, Math. Comput. Modelling, 42 (2005), 31-44.  doi: 10.1016/j.mcm.2005.05.013.  Google Scholar

[4]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[5]

E. N. Dancer, On the indices of fixed points of mapppings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151.  doi: 10.1016/0022-247X(83)90098-7.  Google Scholar

[6]

D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for tropic interaction, Ecology, 56 (1975), 881-892.  doi: 10.2307/1936298.  Google Scholar

[7]

Y. Du and X. Liang, A diffusive competition model with a protection zone, J. Differential Equations, 244 (2008), 61-86.  doi: 10.1016/j.jde.2007.10.005.  Google Scholar

[8]

Y. Du, R. Peng and M. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Differential Equations, 246 (2009), 3932-3956.  doi: 10.1016/j.jde.2008.11.007.  Google Scholar

[9]

Y. Du and J. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations, 229 (2006), 63-91.  doi: 10.1016/j.jde.2006.01.013.  Google Scholar

[10]

Y. Du and J. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc., 359 (2007), 4557-4593.  doi: 10.1090/S0002-9947-07-04262-6.  Google Scholar

[11]

S. Geritz and M. Gyllenberg, A mechanistic derivation of the DeAngelis-Beddington functional response, J. Theoret. Biol., 314 (2012), 106-108.  doi: 10.1016/j.jtbi.2012.08.030.  Google Scholar

[12]

G. Guo and J. Wu, Multiplicity and uniqueness of positive solutions for a predator-prey model with B-D functional response, Nonlinear Anal., 72 (2010), 1632-1646.  doi: 10.1016/j.na.2009.09.003.  Google Scholar

[13]

X. He and S. Zheng, Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response, J. Math. Biol., 75 (2017), 239-257.  doi: 10.1007/s00285-016-1082-5.  Google Scholar

[14]

X. He and S. Zheng, Protection zone in a modified Lotka-Volterra model, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2027-2038.  doi: 10.3934/dcdsb.2015.20.2027.  Google Scholar

[15]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Research Notes Mathematics, Vol. 426, Chapman & Hall/CRC, Boca Ration, FL, 2001. doi: 10.1201/9781420035506.  Google Scholar

[16]

H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci., 15 (1979), 401-454.  doi: 10.2977/prims/1195188180.  Google Scholar

[17]

K. Oeda, Effect of cross-diffusion on the stationary problem of a prey-predator model with a protection zone, J. Differential Equations, 250 (2011), 3988-4009.  doi: 10.1016/j.jde.2011.01.026.  Google Scholar

[18]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[19]

J. Shi, Persistence and bifurcation of degerate solutions, J. Functional Analysis, 169 (1999), 494-531.  doi: 10.1006/jfan.1999.3483.  Google Scholar

[20]

Y.-X. Wang and W.-T. Li, Effect of cross-diffusion on the stationary problem of a diffusive competition model with a protection zone, Nonlinear Anal. Real World Appl., 14 (2013), 224-245.  doi: 10.1016/j.nonrwa.2012.06.001.  Google Scholar

show all references

References:
[1]

J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44 (1975), 331-340.  doi: 10.2307/3866.  Google Scholar

[2]

J. Blat and K. J. Brown, Global bifurcation of positive solutions in some systems of elliptic equations, SIAM. J. Math. Anal., 17 (1986), 1339-1353.  doi: 10.1137/0517094.  Google Scholar

[3]

W. Chen and M. Wang, Qualitative analysis of predator-prey model with Beddington-DeAngelis functional response and diffusion, Math. Comput. Modelling, 42 (2005), 31-44.  doi: 10.1016/j.mcm.2005.05.013.  Google Scholar

[4]

M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340.  doi: 10.1016/0022-1236(71)90015-2.  Google Scholar

[5]

E. N. Dancer, On the indices of fixed points of mapppings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151.  doi: 10.1016/0022-247X(83)90098-7.  Google Scholar

[6]

D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for tropic interaction, Ecology, 56 (1975), 881-892.  doi: 10.2307/1936298.  Google Scholar

[7]

Y. Du and X. Liang, A diffusive competition model with a protection zone, J. Differential Equations, 244 (2008), 61-86.  doi: 10.1016/j.jde.2007.10.005.  Google Scholar

[8]

Y. Du, R. Peng and M. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Differential Equations, 246 (2009), 3932-3956.  doi: 10.1016/j.jde.2008.11.007.  Google Scholar

[9]

Y. Du and J. Shi, A diffusive predator-prey model with a protection zone, J. Differential Equations, 229 (2006), 63-91.  doi: 10.1016/j.jde.2006.01.013.  Google Scholar

[10]

Y. Du and J. Shi, Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc., 359 (2007), 4557-4593.  doi: 10.1090/S0002-9947-07-04262-6.  Google Scholar

[11]

S. Geritz and M. Gyllenberg, A mechanistic derivation of the DeAngelis-Beddington functional response, J. Theoret. Biol., 314 (2012), 106-108.  doi: 10.1016/j.jtbi.2012.08.030.  Google Scholar

[12]

G. Guo and J. Wu, Multiplicity and uniqueness of positive solutions for a predator-prey model with B-D functional response, Nonlinear Anal., 72 (2010), 1632-1646.  doi: 10.1016/j.na.2009.09.003.  Google Scholar

[13]

X. He and S. Zheng, Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response, J. Math. Biol., 75 (2017), 239-257.  doi: 10.1007/s00285-016-1082-5.  Google Scholar

[14]

X. He and S. Zheng, Protection zone in a modified Lotka-Volterra model, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2027-2038.  doi: 10.3934/dcdsb.2015.20.2027.  Google Scholar

[15]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Research Notes Mathematics, Vol. 426, Chapman & Hall/CRC, Boca Ration, FL, 2001. doi: 10.1201/9781420035506.  Google Scholar

[16]

H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci., 15 (1979), 401-454.  doi: 10.2977/prims/1195188180.  Google Scholar

[17]

K. Oeda, Effect of cross-diffusion on the stationary problem of a prey-predator model with a protection zone, J. Differential Equations, 250 (2011), 3988-4009.  doi: 10.1016/j.jde.2011.01.026.  Google Scholar

[18]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis, 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[19]

J. Shi, Persistence and bifurcation of degerate solutions, J. Functional Analysis, 169 (1999), 494-531.  doi: 10.1006/jfan.1999.3483.  Google Scholar

[20]

Y.-X. Wang and W.-T. Li, Effect of cross-diffusion on the stationary problem of a diffusive competition model with a protection zone, Nonlinear Anal. Real World Appl., 14 (2013), 224-245.  doi: 10.1016/j.nonrwa.2012.06.001.  Google Scholar

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