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The motion of weakly interacting localized patterns for reaction-diffusion systems with nonlocal effect
Chaotic dynamics in a simple predator-prey model with discrete delay
1. | Department of Mathematics, Columbus State University, Columbus, Georgia 31907, USA |
2. | Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1, Canda |
A discrete delay is included to model the time between the capture of the prey and its conversion to viable biomass in the simplest classical Gause type predator-prey model that has equilibrium dynamics without delay. As the delay increases from zero, the coexistence equilibrium undergoes a supercritical Hopf bifurcation, two saddle-node bifurcations of limit cycles, and a cascade of period doublings, eventually leading to chaos. The resulting periodic orbits and the strange attractor resemble their counterparts for the Mackey-Glass equation. Due to the global stability of the system without delay, this complicated dynamics can be solely attributed to the introduction of the delay. Since many models include predator-prey like interactions as submodels, this study emphasizes the importance of understanding the implications of overlooking delay in such models on the reliability of the model-based predictions, especially since temperature is known to have an effect on the length of certain delays.
References:
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J. Arino, L. Wang and G. S. K. Wolkowicz,
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B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, SIAM, 2002.
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G. Fan and G. S. K. Wolkowicz, A predator-prey model in the chemostat with time delay, Int. J. Differ. Equ., (2010), Art. ID 287969, 41pp.
doi: 10.1155/2010/287969. |
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J. E. Forde, Delay Differential Equation Models in Mathematical Biology, PhD thesis, University of Michigan, 2005. |
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A.-M. Ginoux, B. Rossetto and J.-L. Jamet,
Chaos in a three-dimensional Volterra-Gause model of predator-prey type, Internat. J. of Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 1689-1708.
doi: 10.1142/S0218127405012934. |
[12] |
S. A. Gourley and Y. Kuang,
A stage structured predator-prey model and its dependence on maturation delay and death rate, J. Math. Biol., 49 (2004), 188-200.
doi: 10.1007/s00285-004-0278-2. |
[13] |
J. K. Hale and V. L. S. M., Introduction to Functional-Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[14] |
M. A. Haque, A predator-prey model with discrete time delay considering different growth function of prey, Adv. Appl. Math. Biosci., 2 (2011), 1-16. Google Scholar |
[15] |
A. Hastings and T. Powell, Chaos in a three-species food chain, Ecology, 72 (1991), 896-903. Google Scholar |
[16] |
W. M. Hirsch, H. Hanisch and J.-P. Gabriel,
Differential equation models of some parasitic infections: methods for the study of asymptotic behavior, Comm. Pure Appl. Math., 38 (1985), 733-753.
doi: 10.1002/cpa.3160380607. |
[17] |
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press Inc., Boston, MA, 1993. |
[18] |
V. Lakshmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications, vol. 55-Ⅰ, Academic Press, New York, 1969. |
[19] |
M. Y. Li, X. Lin and H. Wang,
Global Hopf branches and multiple limit cycles in a delayed Lotka-Volterra predator-prey model, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 747-760.
doi: 10.3934/dcdsb.2014.19.747. |
[20] |
M. C. Mackey and L. Glass, Oscillations and chaos in physiological control, Science, 197 (1977), 287-289. Google Scholar |
[21] |
_____, Mackey-Glass equation, Scholarpedia, 4 (2009), p. 6908. Google Scholar |
[22] |
MAPLE, Maplesoft, A Division of Waterloo Maple Inc., Waterloo, Ontario, 2017. Google Scholar |
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MATLAB, Version 9.5.0 (R2018b), The MathWorks Inc., Natick, Massachusetts, 2018. Google Scholar |
[24] |
A. Morozov, S. Petrovskii and B.-L. Li, Bifurcations and chaos in a predator-prey system with the allee effect, P Roy. Soc. B-Biol. Sci., 271 (2004), 1407-1414. Google Scholar |
[25] |
M. L. Rosenzweig, Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time, Science, 171 (1971), 385-387. Google Scholar |
[26] |
H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, vol. 57 of Texts in Applied Mathematics, Springer, New York, 2011.
doi: 10.1007/978-1-4419-7646-8. |
[27] |
H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, vol. 118 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2011. |
[28] |
J. Wang and W. Jiang,
Bifurcation and chaos of a delayed predator-prey model with dormancy of predators, Nonlinear Dynam., 69 (2012), 1541-1558.
doi: 10.1007/s11071-012-0368-4. |
show all references
References:
[1] |
J. Arino, L. Wang and G. S. K. Wolkowicz,
An alternative formulation for a delayed logistic equation, J. Theoret. Biol., 241 (2006), 109-119.
doi: 10.1016/j.jtbi.2005.11.007. |
[2] |
A. D. Bazykin, Nonlinear Dynamics of Interacting Populations, vol. 11 of A, World Scientific Series on Nonlinear Science, Singapore, 1998.
doi: 10.1142/9789812798725. |
[3] |
E. Beretta and Y. Kuang,
Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165.
doi: 10.1137/S0036141000376086. |
[4] |
K. L. Cooke, R. H. Elderkin and W. Huang,
Predator-prey interactions with delays due to juvenile maturation, SIAM J. Appl. Math., 66 (2006), 1050-1079.
doi: 10.1137/05063135. |
[5] |
R. Driver,
Existence and stability of solutions of a delay-differential system, Arch. Ration. Mech. Anal., 10 (1962), 401-426.
doi: 10.1007/BF00281203. |
[6] |
B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, SIAM, 2002.
doi: 10.1137/1.9780898718195. |
[7] |
G. Fan and G. S. K. Wolkowicz, A predator-prey model in the chemostat with time delay, Int. J. Differ. Equ., (2010), Art. ID 287969, 41pp.
doi: 10.1155/2010/287969. |
[8] |
J. E. Forde, Delay Differential Equation Models in Mathematical Biology, PhD thesis, University of Michigan, 2005. |
[9] |
H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York, 1980. |
[10] |
F. R. Gantmacher, Applications of the Theory of Matrices, Trans. J. L. Brenner et al., New York: Interscience, 1959. |
[11] |
A.-M. Ginoux, B. Rossetto and J.-L. Jamet,
Chaos in a three-dimensional Volterra-Gause model of predator-prey type, Internat. J. of Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 1689-1708.
doi: 10.1142/S0218127405012934. |
[12] |
S. A. Gourley and Y. Kuang,
A stage structured predator-prey model and its dependence on maturation delay and death rate, J. Math. Biol., 49 (2004), 188-200.
doi: 10.1007/s00285-004-0278-2. |
[13] |
J. K. Hale and V. L. S. M., Introduction to Functional-Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[14] |
M. A. Haque, A predator-prey model with discrete time delay considering different growth function of prey, Adv. Appl. Math. Biosci., 2 (2011), 1-16. Google Scholar |
[15] |
A. Hastings and T. Powell, Chaos in a three-species food chain, Ecology, 72 (1991), 896-903. Google Scholar |
[16] |
W. M. Hirsch, H. Hanisch and J.-P. Gabriel,
Differential equation models of some parasitic infections: methods for the study of asymptotic behavior, Comm. Pure Appl. Math., 38 (1985), 733-753.
doi: 10.1002/cpa.3160380607. |
[17] |
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press Inc., Boston, MA, 1993. |
[18] |
V. Lakshmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications, vol. 55-Ⅰ, Academic Press, New York, 1969. |
[19] |
M. Y. Li, X. Lin and H. Wang,
Global Hopf branches and multiple limit cycles in a delayed Lotka-Volterra predator-prey model, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 747-760.
doi: 10.3934/dcdsb.2014.19.747. |
[20] |
M. C. Mackey and L. Glass, Oscillations and chaos in physiological control, Science, 197 (1977), 287-289. Google Scholar |
[21] |
_____, Mackey-Glass equation, Scholarpedia, 4 (2009), p. 6908. Google Scholar |
[22] |
MAPLE, Maplesoft, A Division of Waterloo Maple Inc., Waterloo, Ontario, 2017. Google Scholar |
[23] |
MATLAB, Version 9.5.0 (R2018b), The MathWorks Inc., Natick, Massachusetts, 2018. Google Scholar |
[24] |
A. Morozov, S. Petrovskii and B.-L. Li, Bifurcations and chaos in a predator-prey system with the allee effect, P Roy. Soc. B-Biol. Sci., 271 (2004), 1407-1414. Google Scholar |
[25] |
M. L. Rosenzweig, Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time, Science, 171 (1971), 385-387. Google Scholar |
[26] |
H. L. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, vol. 57 of Texts in Applied Mathematics, Springer, New York, 2011.
doi: 10.1007/978-1-4419-7646-8. |
[27] |
H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, vol. 118 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2011. |
[28] |
J. Wang and W. Jiang,
Bifurcation and chaos of a delayed predator-prey model with dormancy of predators, Nonlinear Dynam., 69 (2012), 1541-1558.
doi: 10.1007/s11071-012-0368-4. |










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