# American Institute of Mathematical Sciences

December  2019, 6(2): 469-483. doi: 10.3934/jcd.2019024

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

 Analysis and Geometry Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jl. Ganesha no 10 Bandung, Indonesia

Received  February 2019 Revised  July 2019 Published  November 2019

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.

Citation: Johan Matheus Tuwankotta, Eric Harjanto. Strange attractors in a predator–prey system with non-monotonic response function and periodic perturbation. Journal of Computational Dynamics, 2019, 6 (2) : 469-483. doi: 10.3934/jcd.2019024
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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 $\textsf{F}'_2$
. 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 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
. 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 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
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
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 $\textsf{K}$
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)
The chaotic transient behaviour in the neighborhood of the fold of limit cycles bifurcation
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 $0.10448$ $0.292264 \cdot 10^{-6}$ $1.0000759023$
 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 $0.10448$ $0.292264 \cdot 10^{-6}$ $1.0000759023$
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