JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2022, Vol. 57 ›› Issue (7): 35-42.doi: 10.6040/j.issn.1671-9352.0.2021.609

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Positive solutions of predator-prey model with spatial heterogeneity and hunting cooperation

HAN Zhuo-ru, LI Shan-bing*   

  1. College of Mathematics and Statistics, Xidian University, Xian 710126, Shaanxi, China
  • Published:2022-06-29

Abstract: The steady-state problem of a predator-prey model with spatial heterogeneity and hunting cooperation is investigated. Firstly, by using the Riesz-Schauder theory, the local asymptotic stability of trivial solution and semi-trivial solutions is obtained. By means of the comparison principle, the global asymptotic stability of trivial solution and semi-trivial solutions is derived. Finally, the sufficient conditions for the existence of positive solutions are established by the fixed point index theory.

Key words: predator-prey model, spatial heterogeneity, hunting cooperations, positive solutions, stability

CLC Number: 

  • O175.26
[1] DUGATKIN L A. Cooperation among animals: an evolutionary perspective[M]. New York: Oxford University Press, 1997.
[2] PACKER C, SCHEEL D, PUSEY A E. Why lions form groups: food is not enough[J]. The American Naturalist, 1990, 136(1):1-19.
[3] SCHEEL D, PACKER C. Group hunting behaviour of lions: a search for cooperation[J]. Animal Behaviour, 1991, 41(4):697-709.
[4] SCHMIDT P A, MECH L D. Wolf pack size and food acquisition[J]. The American Naturalist, 1997, 150(4):513-517.
[5] BOESCH C. Cooperative hunting in wild chimpanzees[J]. Animal Behaviour, 1994, 48(3):653-667.
[6] COSNER C, DEANGELIS D L, AULT J S, et al. Effects of spatial grouping on the functional response of predators[J]. Theoretical Population Biology, 1999, 56(1):65-75.
[7] RYU K, KO W, HAQUE M. Bifurcation analysis in a predator-prey system with a functional response increasing in both predator and prey densities[J]. Nonlinear Dynamics, 2018, 94(3):1639-1656.
[8] ALVES M T, HILKER F M. Hunting cooperation and allee effects in predators[J]. Journal of Theoretical Biology, 2017, 419:13-22.
[9] RYU K, KO W. Asymptotic behavior of positive solutions to a predator-prey elliptic system with strong hunting cooperation in predators[J]. Physica A, 2019, 531:121726.
[10] WU Daiyong, ZHAO Min. Qualitative analysis for a diffusive predator-prey model with hunting cooperative[J]. Physica A, 2019, 515:299-309.
[11] FU Shengmao, ZHANG Huisen. Effect of hunting cooperation on the dynamic behavior for a diffusive Holling type II predator-prey model[J]. Communications in Nonlinear Science and Numerical Simulation, 2021, 99:105807.
[12] ALLEN L J S, BOLKER B M, LOU Y, et al. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model[J]. Discrete and Continuous Dynamical Systems, 2008, 21(1):1-20.
[13] 叶其孝,李正元,王明新,等. 反应扩散方程引论[M]. 北京: 科学出版社, 2011. YE Qixiao, LI Zhengyuan, WANG Mingxin, et al. Introduction to reaction-diffusion equations[M]. Beijing: Science Press, 2011.
[14] RUAN Weihua, FENG Wei. On the fixed point index and multiple steady-state solutions of reaction-diffusion systems[J]. Differential and Integral Equations, 1995, 8(2):371-391.
[15] LI Shanbing, XIAO Yanni, DONG Yaying. Diffusive predator-prey models with fear effect in spatially heterogeneous environment[J]. Electronic Journal of Differential Equations, 2021, 2021(70):1-31.
[16] LI Lige. Coexistence theorems of steady states for predator-prey interacting systems[J]. Transactions of the American Mathematical Society, 1988, 305(1):143-166.
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