
-
Previous Article
Using Lie group integrators to solve two and higher dimensional variational problems with symmetry
- JCD Home
- This Issue
-
Next Article
Linear degree growth in lattice equations
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. |
[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. |
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. |
[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. |







Type of attractor | Positive exp. | Kaplan-Yorke dimension | |
Negative | |||
Negative | |||
Positive | |||
Positive | |||
Type of attractor | Positive exp. | Kaplan-Yorke dimension | |
Negative | |||
Negative | |||
Positive | |||
Positive | |||
[1] |
Carmen Núñez, Rafael Obaya. A non-autonomous bifurcation theory for deterministic scalar differential equations. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 701-730. doi: 10.3934/dcdsb.2008.9.701 |
[2] |
Yang Lu, Xia Wang, Shengqiang Liu. A non-autonomous predator-prey model with infected prey. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3817-3836. doi: 10.3934/dcdsb.2018082 |
[3] |
Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Amelia G. Nobile. A non-autonomous stochastic predator-prey model. Mathematical Biosciences & Engineering, 2014, 11 (2) : 167-188. doi: 10.3934/mbe.2014.11.167 |
[4] |
T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037 |
[5] |
Sampurna Sengupta, Pritha Das, Debasis Mukherjee. Stochastic non-autonomous Holling type-Ⅲ prey-predator model with predator's intra-specific competition. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3275-3296. doi: 10.3934/dcdsb.2018244 |
[6] |
Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17 |
[7] |
José A. Langa, James C. Robinson, Aníbal Rodríguez-Bernal, A. Suárez, A. Vidal-López. Existence and nonexistence of unbounded forwards attractor for a class of non-autonomous reaction diffusion equations. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 483-497. doi: 10.3934/dcds.2007.18.483 |
[8] |
V. V. Chepyzhov, M. I. Vishik, W. L. Wendland. On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 27-38. doi: 10.3934/dcds.2005.12.27 |
[9] |
T. Tachim Medjo. Non-autonomous 3D primitive equations with oscillating external force and its global attractor. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 265-291. doi: 10.3934/dcds.2012.32.265 |
[10] |
Chunyou Sun, Daomin Cao, Jinqiao Duan. Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 743-761. doi: 10.3934/dcdsb.2008.9.743 |
[11] |
Xiaolin Jia, Caidi Zhao, Juan Cao. Uniform attractor of the non-autonomous discrete Selkov model. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 229-248. doi: 10.3934/dcds.2014.34.229 |
[12] |
Olivier Goubet, Wided Kechiche. Uniform attractor for non-autonomous nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2011, 10 (2) : 639-651. doi: 10.3934/cpaa.2011.10.639 |
[13] |
Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195 |
[14] |
Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure and Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809 |
[15] |
Iacopo P. Longo, Sylvia Novo, Rafael Obaya. Topologies of continuity for Carathéodory delay differential equations with applications in non-autonomous dynamics. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5491-5520. doi: 10.3934/dcds.2019224 |
[16] |
Rafael Obaya, Ana M. Sanz. Persistence in non-autonomous quasimonotone parabolic partial functional differential equations with delay. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3947-3970. doi: 10.3934/dcdsb.2018338 |
[17] |
Sylvia Novo, Carmen Núñez, Rafael Obaya, Ana M. Sanz. Skew-product semiflows for non-autonomous partial functional differential equations with delay. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4291-4321. doi: 10.3934/dcds.2014.34.4291 |
[18] |
Eduardo Hernández, Donal O'Regan. $C^{\alpha}$-Hölder classical solutions for non-autonomous neutral differential equations. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 241-260. doi: 10.3934/dcds.2011.29.241 |
[19] |
Lucile Mégret, Jacques Demongeot. Gevrey solutions of singularly perturbed differential equations, an extension to the non-autonomous case. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2145-2163. doi: 10.3934/dcdss.2020183 |
[20] |
Yanan Li, Alexandre N. Carvalho, Tito L. M. Luna, Estefani M. Moreira. A non-autonomous bifurcation problem for a non-local scalar one-dimensional parabolic equation. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5181-5196. doi: 10.3934/cpaa.2020232 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]