Bifurcation Analysis in a Discrete-Time Prey-Predator System With Crowly- Martin Functional Response
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Abstract
In this paper, a discrete time prey-predator system with Crowly- Martin functional response was studied. The fixed points of the model are obtained, and their stability is analyzed. Further existence of bifurcation analysis at each fixed points and Hopf bifurcation are demonstrated. Numerical simulation show that the model perhaps have more complex dynamical behaviors, such as the period-5,10,20 and 35 orbits, including the periodic doubling bifurcation in period-2,4,8 and 16 orbits and chaotic set.
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References
[1] Verhulst, P.F., 1838. Notice sur la loi que la population suit dans son accroissement. Corresp. Math. Phys., 10, pp.113-126.
[2] Pearl, R. and Reed, L.J., 1920. On the rate of growth of the population of the United States since 1790 and its mathematical representation. Proceedings of the national academy of sciences, 6(6), pp.275-288.
[3] Fazly, M. and Hesaaraki, M., 2007. Periodic solutions for a discrete time predator–prey system with monotone functional responses. Comptes Rendus Mathématique, 345(4), pp.199-202.
[4] Hu, D. and Zhang, Z., 2008. Four positive periodic solutions of a discrete time delayed predator–prey system with nonmonotonic functional response and harvesting. Computers & Mathematics with Applications, 56(12), pp.3015-3022.
[5] Liu, X., 2010. A note on the existence of periodic solutions in discrete predator–prey models. Applied Mathematical Modelling, 34(9), pp.2477-2483.
[6] Xia Y, Cao J, Lin M (2007) Discrete-time analogues of predator-prey models with monotonic or nonmonotonic functional responses. Nonlinear Anal RWA 8:1079–1095
[7] Yang, W. and Li, X., 2009. Permanence for a delayed discrete ratio-dependent predator–
prey model with monotonic functional responses. Nonlinear Analysis: Real World Applications, 10(2), pp.1068-1072.
[8] Holling, C.S., 1959. Some characteristics of simple types of predation and parasitism1. The canadian entomologist, 91(7), pp.385-398.
[9] Holling, C.S., 1965. The functional response of predators to prey density and its role in mimicry and population regulation. The Memoirs of the Entomological Society of Canada, 97(S45), pp.5-60.
[10] Ko, W. and Ryu, K., 2006. Qualitative analysis of a predator–prey model with Holling type II functional response incorporating a prey refuge. Journal of Differential Equations, 231(2), pp.534-550.
[11] Huang, Y., Chen, F. and Zhong, L., 2006. Stability analysis of a prey–predator model with Holling type III response function incorporating a prey refuge. Applied Mathematics and Computation, 182(1), pp.672-683.
[12] Huang, J., Ruan, S. and Song, J., 2014. Bifurcations in a predator–prey system of Leslie type with generalized Holling type III functional response. Journal of Differential Equations, 257(6), pp.1721-1752.
[13] Jing, Z. and Yang, J., 2006. Bifurcation and chaos in discrete-time predator–prey system. Chaos, Solitons & Fractals, 27(1), pp.259-277.
[14] Cui, Q., Zhang, Q., Qiu, Z. and Hu, Z., 2016. Complex dynamics of a discrete-time predator-prey system with Holling IV functional response. Chaos, Solitons & Fractals, 87, pp.158-171.
[15] Guan, X. and Chen, F., 2019. Dynamical analysis of a two species amensalism model with Beddington–DeAngelis functional response and Allee effect on the second species. Nonlinear Analysis: Real World Applications, 48, pp.71-93.
[16] Liu, X. and Xiao, D., 2007. Complex dynamic behaviors of a discrete-time predator–prey system. Chaos, Solitons & Fractals, 32(1), pp.80-94.
[17] He, Z. and Lai, X., 2011. Bifurcation and chaotic behavior of a discrete-time predator–prey system. Nonlinear Analysis: Real World Applications, 12(1), pp.403-417.
[18] Lotka, A.J., 1956. Elements of mathematical biology. Dover Publications.
[19]Volterra, V., 1962. Opere Matematiche: Memorie e note, vol. V, 1926–1940. Accad. Naz. Lincei. Roma.
[20] Agiza, H.N., Elabbasy, E.M., El-Metwally, H. and Elsadany, A.A., 2009. Chaotic dynamics of a discrete prey– predator model with Holling type II. Nonlinear Analysis: Real World Applications, 10(1), pp.116-129.
[21] Kazarinoff, N.D. and Van Den Driessche, P., 1978. A model predator-prey system with functional response. Mathematical Biosciences, 39(1-2), pp.125-134.
[22] Hainzl, J., 1988. Stability and Hopf bifurcation in a predator–prey system with several parameters. SIAM Journal on Applied Mathematics, 48(1), pp.170-190.
[23] Saber N.E., 1996. An Introduction to Difference Equations, Springer-New York
[24] Guckenheimer, J. and Holmes, P. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, NY