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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 |
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.
References:
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. |
show all references
References:
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. |







Type of attractor | Positive exp. | Kaplan-Yorke dimension | |
Negative | |||
Negative | |||
Positive | |||
Positive | |||
Type of attractor | Positive exp. | Kaplan-Yorke dimension | |
Negative | |||
Negative | |||
Positive | |||
Positive | |||
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