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Article Contents

# Strange attractors in a predator–prey system with non-monotonic response function and periodic perturbation

• A system of ordinary differential equations of a predator–prey type, depending on nine parameters, is studied. We have included in this model a nonmonotonic response function and time periodic perturbation. Using numerical continuation software, we have detected three codimension two bifurcations for the unperturbed system, namely cusp, Bogdanov-Takens and Bautin bifurcations. Furthermore, we concentrate on two regions in the parameter space, the region where the Bogdanov-Takens and the region where Bautin bifurcations occur. As we turn on the time perturbation, we find strange attractors in the neighborhood of invariant tori of the unperturbed system.

Mathematics Subject Classification: 37C60, 37G15, 34A34, 92D25.

 Citation:

• Figure 1.  Two parameter bifurcation diagram ($\beta-\alpha$) of system (1) with initial condition $\delta = 1.1$, $\lambda_0 = 0.01$, $\mu = 0.1$, $\alpha = 0.002$, $\beta = 0.25$, $\omega = 1$, $x = 1.816$, and $y = 1.434$, while $\varepsilon = 0$. The curve labelled by $\textsf{F}_1$, and $\textsf{F}_2$ are the fold curves. The curves $\textsf{H}_1$ and $\textsf{H}_3$ are the Hopf bifurcation curves. The curve $\textsf{H}_1$ and $\textsf{H}_3$ are joint together at the point $\textsf{H}_2$. The curve plotted with a dashed line (labelled by $\textsf{Hom}$) is the curve of Homoclinic bifurcations while the $\textsf{FLC}$ curve is the fold of limit cycle bifurcations curve. The $\textsf{Hom}$ curve coincides with some part of the $\textsf{F}_2$-curve which is labelled by $\textsf{F}'_2$

Figure 2.  In this figure we have plotted the magnification of the region $\textsf{B}$ of Figure 1. We have indicated five regions on that diagram, i.e. $\textsf{B}_1$, $\textsf{B}_2$, $\textsf{B}_3$, $\textsf{B}_4$, and $\textsf{B}_5$. The diagrams on the second and third rows are the phase portraits in each of these regions. The four phase portraits for $(\alpha, \beta)$ in $\textsf{B}_1$, $\textsf{B}_2$, $\textsf{B}_5$, and $\textsf{B}_3$, correspond to the Bogdanov-Takens bifurcations. The transition from phase portrait when $(\beta, \alpha) \in \textsf{B}_3$ to $(\beta, \alpha) \in \textsf{B}_4$, or when $(\beta, \alpha) \in \textsf{B}_5$ to $(\beta, \alpha) \in \textsf{B}_4$ corresponds to fold bifurcation of equilibrium of system (1) for $\varepsilon = 0$; the latter with the creation of orbit homoclinic to the degenerate equilibrium

Figure 3.  We have plotted the magnification of the region $\textsf{A}$ of Figure 1. We have indicated seven regions on that diagram, i.e. $\textsf{A}_j$, $j = 1, 2, \ldots, 7$. These regions are separated from each other by the bifurcation curves: $\textsf{F}_2$ — where a fold bifurcation occurs—, $\textsf{H}_{1,3}$ — where a Hopf bifurcation occurs—, $\textsf{Hom}$ — where a homoclinic bifurcation occurs —, and $\textsf{FLC}$ — where a fold bifurcation of limit cycles occurs

Figure 4.  We have plotted seven diagrams which correspond to the phase portraits of system (1), for parameter value of $(\beta, \alpha)$ in region: $\textsf{A}_j$, $j = 1,2, \ldots, 7$ and $\varepsilon = 0$. The topological changes of these phase portraits when the parameter moves from $\textsf{A}_1$, to $\textsf{A}_2$, to $\textsf{A}_3$, and back, are in agreement with the scenario of Bautin (or degenerate Hopf) bifurcation

Figure 5.  A plot of a negative time attractor in the strobocospic map (4) (also known as the Poincaré section) showing evidence of the existence of a strange repeller (a strange negative-time attractor). The value of $\varepsilon = 0.07$, $\delta = 1.1$, $\lambda_0 = 0.01$, $\mu = 0.1$, $\omega = 1$, $\alpha = 0.007$, and $\beta = 0.08$. On the left diagram we have plotted the cross section of the strange repeller while on the right we have plotted the magnification of a part of the cross section, indicated by the box $\textsf{K}$

Figure 6.  A comparison between the phase portrait of the stroboscopic map of System (2) for $\varepsilon = 0$ (diagrams in the first row) and $\varepsilon = 0.07$ (diagrams in the second column). The value of $(\beta,\alpha) = (0.07,0.005)$ (for the diagrams in the most left), $(\beta,\alpha) = (0.08,0.005)$ (for the diagrams in middle column), and $(\beta,\alpha) = (0.09,0.005)$ (for the diagrams in the most right column)

Figure 7.  The chaotic transient behaviour in the neighborhood of the fold of limit cycles bifurcation

Table 1.  In this table, a positive Lyapunov exponent and Kaplan-Yorke dimension of some of the attractors for various values of $\beta$ and $\alpha = 0.005$ are listed

 Type of attractor $\beta$ Positive exp. Kaplan-Yorke dimension Negative $0.08$ $0.718621 \cdot 10^{-6}$ $1.0000772161$ Negative $0.0505$ $0.378245 \cdot 10^{-6}$ $1.0000802951$ Positive $0.08$ $0.506676 \cdot 10^{-6}$ $1.0000015131$ Positive $0.10448$ $0.292264 \cdot 10^{-6}$ $1.0000759023$
•  [1] F. K. Balagaddé, H. Song, J. Ozaki, C. H. Collins, M. Barnet, F. H. Arnold, S. R. Quake and L. You, A synthetic escherichia coli predator-prey ecosystem, Molecular Systems Biology, 4 (2008), 187, 1–8. [2] A. A. Berryman, The orgins and evolution of predator-prey theory, Ecology, 73 (1992), 1530-1535.  doi: 10.2307/1940005. [3] G. E. Briggs and J. B. S. Haldane, A note on the kinetics of enzyme action, Biochemical Journal, 19 (1925), 338-339.  doi: 10.1042/bj0190338. [4] H. W. Broer, K. Saleh, V. Naudot and R. Roussarie, Dynamics of a predator-prey model with non-monotonic response function, Discrete & Continuous Dynamical Systems-A, 18 (2007), 221-251.  doi: 10.3934/dcds.2007.18.221. [5] Z. H. Cai, Q. Wang and G. Q. Liu, Modeling the natural capital investment on tourism industry using a predator-prey model, in Advances in Computer Science and its Applications, (2014), 751–756. doi: 10.1007/978-3-642-41674-3_107. [6] E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede, X. Wang and et al., Continuation and bifurcation software for ordinary differential equations (with homcont), AUTO97, Concordia University, Canada. [7] A. Fenton and S. E. Perkins, Applying predator-prey theory to modelling immune-mediated, within-host interspecific parasite interactions, Parasitology, 137 (2010), 1027-1038.  doi: 10.1017/S0031182009991788. [8] R. M. Goodwin, A growth cycle, Essays in Economic Dynamics, (1967), 165–170. doi: 10.1007/978-1-349-05504-3_12. [9] C. Grimme and J. Lepping, Integrating niching into the predator-prey model using epsilon-constraints, in Proceedings of the 13th Annual Conference Companion on Genetic and Evolutionary Computation, ACM, (2011), 109–110. doi: 10.1145/2001858.2001920. [10] E. Harjanto and J.M. Tuwankotta, Vanishing two folds without cusp bifurcation in a predator-prey type of systems with group defense mechanism and seasonal variation (in bahasa indonesia), Prosiding Konferensi Nasional Matematika, Indonesian Mathematical Society, 17 (2014), 767-772. [11] E. Harjanto and J. M. Tuwankotta, Bifurcation of periodic solution in a Predator-Prey type of systems with non-monotonic response function and periodic perturbation, International Journal of Non-Linear Mechanics, 85 (2016), 188-196.  doi: 10.1016/j.ijnonlinmec.2016.06.011. [12] C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the european pine sawfly, The Canadian Entomologist, 91 (1959), 293-320.  doi: 10.4039/Ent91293-5. [13] C. S. Holling, Some characteristics of simple types of predation and parasitism, The Canadian Entomologist, 91 (1959), 385-398.  doi: 10.4039/Ent91385-7. [14] Y. X. Huang and O. Diekmann, Predator migration in response to prey density: What are the consequences?, Journal of Mathematical Biology, 43 (2001), 561-581.  doi: 10.1007/s002850100107. [15] I. Koren and G. Feingold, Aerosol-cloud-precipitation system as a predator-prey problem, Proceedings of the National Academy of Sciences, 108 (2011), 12227-12232.  doi: 10.1073/pnas.1101777108. [16] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4757-2421-9. [17] S. Nagano and Y. Maeda, Phase transitions in predator-prey systems, Physical Review E, 85 (2012), 011915. doi: 10.1103/PhysRevE.85.011915. [18] S. Rinaldi, S. Muratori and Y. Kuznetsov, Multiple attractors, catastrophes and chaos in seasonally perturbed predator-prey communities, Bulletin of mathematical Biology, 55 (1993), 15-35. [19] A. Sharma and N. Singh, Object detection in image using predator-prey optimization, Signal & Image Processing, 2 (2011), 205-221.  doi: 10.5121/sipij.2011.2115. [20] J. M. Tuwankotta, Chaos in a coupled oscillators system with widely spaced frequencies and energy-preserving non-linearity, International Journal of Non-Linear Mechanics, 41 (2006), 180-191.  doi: 10.1016/j.ijnonlinmec.2005.02.007. [21] T. H. Zhang, Y. P. Xing, H. Zang and M. A. Han, Spatio-temporal dynamics of a reaction-diffusion system for a predator-prey model with hyperbolic mortality, Nonlinear Dynamics, 78 (2014), 265-277.  doi: 10.1007/s11071-014-1438-6. [22] T. H. Zhang and H. Zang, Delay-induced turing instability in reaction-diffusion equations, Physical Review E, 90 (2014), 052908. doi: 10.1103/PhysRevE.90.052908. [23] H. P. Zhu, S. A. Campbell and G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM Journal on Applied Mathematics, 63 (2002), 636-682.  doi: 10.1137/S0036139901397285.

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