# American Institute of Mathematical Sciences

March 2018, 23(2): 785-807. doi: 10.3934/dcdsb.2018043

## How seasonal forcing influences the complexity of a predator-prey system

 1 School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China 2 Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada 3 Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada 4 Laboratory of Mathematical Parallel Systems (Lamps), Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada

* Corresponding author: renjl@zzu.edu.cn

Received  November 2016 Revised  July 2017 Published  December 2017

Almost all population communities are strongly influenced by their seasonally varying living environments. We investigate the influence of seasons on populations via a periodically forced predator-prey system with a nonmonotonic functional response. We study four seasonality mechanisms via a continuation technique. When the natural death rate is periodically varied, we get six different bifurcation diagrams corresponding to different bifurcation cases of the unforced system. If the carrying capacity is periodic, two different bifurcation diagrams are obtained. Here we cannot get a "universal diagram" like the one in the periodically forced system with monotonic Holling type Ⅱ functional response; that is, the two elementary seasonality mechanisms have different effects on the population. When both the natural death rate and the carrying capacity are forced with two different seasonality mechanisms, the phenomena that arise are to some extent different. The bifurcation results also show that each seasonality mechanism can display complex dynamics such as multiple attractors including stable cycles of different periods, quasi-periodic solutions, chaos, switching between these attractors and catastrophic transitions. In addition, we give some orbits in phase space and corresponding Poincaré sections to illustrate different attractors.

Citation: Xueping Li, Jingli Ren, Sue Ann Campbell, Gail S. K. Wolkowicz, Huaiping Zhu. How seasonal forcing influences the complexity of a predator-prey system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 785-807. doi: 10.3934/dcdsb.2018043
##### References:
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##### References:
 [1] J. F. Andrews, A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnol. Bioeng., 10 (1968), 707-723. doi: 10.1002/bit.260100602. [2] E. Benincá, B. Ballantine, S. P. Ellner and J. Huisman, Species fluctuations sustained by a cyclic succession at the edge of chaos, P. Natl. Acad. Sci. USA, 112 (2015), 6389-6394. [3] J. B. Collings, The effects of the functional response on the bifurcation behavior of a mite predator-prey interaction model, J. Math. Biol., 36 (1997), 149-168. doi: 10.1007/s002850050095. [4] E. J. Doedel and B. E. Oldeman, AUTO-07P: Continuation and bifurcation software for ordinary differential equations, http://cmvl.cs.concordia.ca/auto, 2012. [5] H. I. Freedman and G. S. K. Wolkowicz, Predator-prey systems with group defence: The paradox of enrichment revisited, Bull. Math. Biol., 48 (1986), 493-508. doi: 10.1007/BF02462320. [6] A. Gragnani and S. Rinaldi, A universal bifurcation diagram for seasonally perturbed predator-prey models, Bull. Math. Biol., 57 (1995), 701-712. [7] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. [8] C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the European pine sawfly, Can. Entomol., 91 (1959), 293-320. doi: 10.4039/Ent91293-5. [9] C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Can., 97 (1965), 5-60. doi: 10.4039/entm9745fv. [10] Y. A. Kuznetsov, S. Muratori and S. Rinaldi, Bifurcations and chaos in a periodic predator-prey model, Int. J. Bifurcat. Chaos, 2 (1992), 117-128. doi: 10.1142/S0218127492000112. [11] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 1998. [12] D. Levy, H. A. Harrington and R. A. Van Gorder, Role of seasonality on predator-prey-subsidy population dynamics, J. Theoret. Biol., 396 (2016), 163-181. doi: 10.1016/j.jtbi.2016.02.005. [13] Y. Li and D. Xiao, Bifurcations of a predator-prey system of Holling and Leslie types, Chaos Solitons Fractals, 34 (2007), 606-620. doi: 10.1016/j.chaos.2006.03.068. [14] R. M. May, Limit cycles in predator-prey communities, Science, 177 (1972), 900-902. doi: 10.1126/science.177.4052.900. [15] J. L. Ren and P. Li, Bifurcations in a seasonally forced predator-prey model with generalized Holling type Ⅳ functional response, Int. J. Bifurcat. Chaos, 26 (2016), 1650203, 19pp. [16] S. Rinaldi, S. Muratori and Y. A. Kuznetsov, Multiple attractors, catastrophes and chaos in seasonally perturbed predator-prey communities, Bull. Math. Biol., 55 (1993), 15-35. [17] S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2001), 1445-1472. [18] M. Scheffer, S. Rinaldi, Y. A. Kuznetsov and E. H. van Nes, Seasonal dynamics of Daphnia and algae explained as a periodically forced predator-prey system, OIKOS, 80 (1997), 519-532. doi: 10.2307/3546625. [19] R. A. Taylor, J. A. Sherratt and A. White, Seasonal forcing and multi-year cycles in interacting populations: Lessons from a predator-prey model, J. Math. Biol., 67 (2013), 1741-1764. doi: 10.1007/s00285-012-0612-z. [20] R. A. Taylor, A. White and A. Sherratt, How do variations in seasonality affect population cycles?, Proc. R. Soc. B, 280 (2013), 20122714-1764. doi: 10.1098/rspb.2012.2714. [21] G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system involving group defence, SIAM J. Appl. Math., 48 (1988), 592-606. doi: 10.1137/0148033. [22] H. Zhu, S. A. Campbell and G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 63 (2002), 636-682. [23] MATLAB Version 8. 1 (R2013a). The MathWorks Inc., Natick, Massachusetts, 2013.
Bifurcation diagrams of the forced system with $a = 4$, $b = 0$, $c = 1$, $m = 5\pi$, $r = 4.8\pi$. (C) and (D) are partial enlargements of (A) and (B) respectively.
Bifurcation diagrams of the forced system with $a = 4$, $b = 0$, $c = 1$, $m = 5\pi$
Local enlargements of Fig. 2: (A)-amplification of Fig. 2a; (B) and (C)-amplification of Fig. 2b; (D)-amplification of Fig. 2c; (E)-amplification of Fig. 2d
Bifurcation diagrams of the forced system with $a = 4$, $b = 0$, $c = 1$, $m = 5\pi$, $r = 3\pi$
Bifurcation diagrams of the forced system with $a = 4$, $b = -1.5$, $c = 1$, $m = 5\pi$, $r = 4\pi$. The two diagrams in (C) and (D) are partial enlargements of (A)
Bifurcation diagrams of the forced system with $a = 4$, $b = 2$, $c = 1$, $m = 5\pi$, $r = 7.5\pi$. The two diagrams in (C) and (D) are partial enlargements of (B)
Bifurcation diagrams of the system (6) with $a = 4$, $b = -1.5$, $c = 1$, $m = 5\pi$, $r = 4\pi$, $d_0 = 1.95\pi$, $K_0 = 1.01$. (B), (C) and (D) are partial enlargements of (A)
Bifurcation diagrams of the system (6) with $a = 4$, $b = 0$, $c = 1$, $m = 5\pi$, $r = 2.15\pi$, $d_0 = 0.85\pi$, $K_0 = 0.92$. (B) is a partial enlargement of (A)
Bifurcation diagrams of the system (7) with $a = 4$, $b = -1.5$, $c = 1$, $m = 5\pi$, $r = 4\pi$, $d_0 = 1.95\pi$, $K_0 = 1.01$. (B) is a partial enlargement of (A)
Bifurcation diagrams of the system (10) with $a = 4$, $b = 0$, $c = 1$, $m = 5\pi$, $r = 2.15\pi$, $d_0 = 0.85\pi$, $K_0 = 0.92$
Obits in phase space (left) and corresponding Poincaré sections (right) for the case in Fig. 2a. (A) and (B)-quasiperiodic solution, (C) and (D)-chaotic solution through torus destruction, (E) and (F)-chaotic solution through cascade of period doublings
Solutions for the case in Fig. 4a. (A)-orbits in phase space of a torus; (B)-Poincaré section corresponding to (A); (C)-orbits in phase space of a chaotic solution through torus destruction; (D)-Poincaré section corresponding to (C). Note that in (D), we have set the time $t$ large enough, yielding the chaotic solution consisting of several irregular segments. And in this case the torus is the stable one outside the unstable one arising from the subcitical Hopf bifurcation
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