Journal of Guangxi Normal University(Natural Science Edition) ›› 2022, Vol. 40 ›› Issue (2): 103-115.doi: 10.16088/j.issn.1001-6600.2021052801

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Stability of a Prey-predator Model with Migration and Allee Effects

XU Wangjun1, CAO Jinde2, WU Daiyong1, SHEN Chuansheng1*   

  1. 1. School of Mathematics and Physics, Anqing Normal University, Anqing Anhui 246133, China;
    2. School of Mathematics, Southeast University, Nanjing Jiangsu 211189, China
  • Received:2021-05-02 Revised:2021-06-28 Published:2022-05-31

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Abstract: A kind of prey-predator system with Allee effect and artificially controlled migration of predators is studied. The system has a square root functional response function. Firstly, by qualitative analysis of the model, the boundedness of the solution is proved, and the existence of the equilibrium point is analyzed. Sufficient conditions for the local stability of the equilibrium point of the system are obtained. Then, the existence of the Hopf-bifurcation of the equilibrium point is discussed, and the stability and direction of the equilibrium Hopf-bifurcation are studied by calculating the first Lyapunov coefficient. Finally, the correctness of the conclusion is verified by numerical simulation. The results indicate that the Allee effect and artificially controlled migration rate are important for the survival and extinction of prey and predator populations.

Key words: prey-predator model, Allee effect, migration rate, Hopf bifurcation, stability

CLC Number: 

  • Q141
[1] MA Z H, WANG S F, WANG T T, et al. Stability analysis of prey-predator system with Holling type functional response and prey refuge[J]. Advances in Difference Equations, 2017, 2017: 243.
[2] LIU B, ZHANG Y J, CHEN L S. Dynamic complexities of a Holling I predator-prey model concerning periodic biological and chemical control[J]. Chaos, Solitons & Fractals, 2004, 22(1): 123-134.
[3] ZHANG Y J, XU Z L, LIU B, et al. Dynamic analysis of a Holling I predator-prey system with mutual interference concerning pest control[J]. Journal of Biological Systems, 2005, 13(1): 45-58.
[4] ZU J, MIMURA M. The impact of Allee effect on a predator-prey system with Holling type II functional response[J]. Applied Mathematics and Computation, 2010, 217(7): 3542-3556.
[5] SUN G, SARWARDI S, PAL P J, et al. The spatial patterns through diffusion driven instability in modified Leslie-Gower and Holling-type II predator-prey model[J]. Journal of Biological Systems, 2010, 18(3): 593-603.
[6] YI F Q, WEI J J, SHI J P. Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system[J]. Journal of Differential Equations, 2009, 246(5): 1944-1977.
[7] FAN M, KUANG Y. Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response[J]. Journal of Mathematical Analysis and Applications, 2004, 295(1): 15-39.
[8] AJRALDI V, PITTAVINO M, VENTURINO E. Modeling herd behavior in population systems[J]. Nonlinear Analysis: Real World Applications, 2011, 12(4): 2319-2338.
[9] KOSOBUD R F, O'NEILL W D. On the dependence of population growth on income: New results in a ricardian-Malthus model[J]. De Economist, 1981, 129(2): 206-223.
[10] TSOULARIS A, WALLACE J. Analysis of logistic growth models[J]. Mathematical Biosciences, 2002, 179(1): 21-55.
[11] BOUKAL D S, SABELIS M W, BEREC L. How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses[J]. Theoretical Population Biology, 2007, 72(1): 136-147.
[12] KRAMER A M, DENNIS B, LIEBHOLD A M, et al. The evidence for Allee effects[J]. Population Ecology, 2009, 51(3): 341-354.
[13] BEREC L, ANGULO E, COURCHAMP F. Multiple Allee effects and population management[J]. Trends in Ecology & Evolution, 2007, 22(4): 185-191.
[14] BISWAS S, SAIFUDDIN M, SASMAL S K, et al. A delayed prey-predator system with prey subject to the strong Allee effect and disease[J]. Nonlinear Dynamics, 2016, 84(3): 1569-1594.
[15] COURCHAMP F, BEREC L, GASCOIGNE J. Allee effects in ecology and conservation[J]. Environmental Conservation, 2008, 36(1): 80-85.
[16] WU D Y, ZHAO H Y. Complex dynamics of a discrete predator-prey model with the prey subject to the Allee effect[J]. Journal of Difference Equations Applications, 2017, 23(11): 1765-1806.
[17] PETROVSKII S, MOROZOV A, LI B L. Regimes of biological invasion in a predator-prey system with the Allee effect[J]. Bulletin of Mathematical Biology, 2005, 67(3): 637-661.
[18] RAO F, KANG Y. The complex dynamics of a diffusive prey-predator model with an Allee effect in prey[J]. Ecological Complexity, 2016, 28: 123-144.
[19] BARCLAY H J. Models for pest control using predator release, habitat management and pesticide release in combination[J]. Journal of Applied Ecology, 1982, 19(2): 337-348.
[20] TANG S Y, TANG G Y, CHEKE R A. Optimum timing for integrated pest management: Modelling rates of pesticide application and natural enemy releases[J]. Journal of Theoretical Biology, 2010, 264(2): 623-638.
[21] 成定平. 鼠类-天敌系统渐近稳定性的数学分析[J]. 生物数学学报, 2003, 18(3): 283-286.
[22] CHEN Y M, ZHANG F Q. Dynamics of a delayed predator-prey model with predator migration[J]. Applied Mathematical Modelling, 2013, 37(3): 1400-1412.
[23] JIN M, XU F, SHEN C S, et al. Nontrivial effect of time varying migration on the three species prey-predator system[J]. Communications in Theoretical Physics, 2019, 71(1): 127-131.
[24] ABDLLAOUI A E, AUGER P, KOOI B W, et al. Effects of density-dependent migrations on stability of a two-patch predator-prey model[J]. Mathematical Biosciences, 2007, 210(1): 335-354.
[25] HUANG Y, DIEKMANN O. Predator migration in response to prey density: What are the consequences?[J]. Journal of Mathematical Biology, 2001, 43(6): 561-581.
[26] SUN G Q, JIN Z, LIU Q X, et al. Dynamical complexity of a spatial predator-prey model with migration[J]. Ecological Modelling, 2008, 219(1/2): 248-255.
[27] PETROVSKII S, LI B L. An exactly solvable model of population dynamics with density-dependent migrations and the Allee effect[J]. Mathematical Biosciences, 2003, 186(1): 79-91.
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