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The motion of weakly interacting localized patterns for reactiondiffusion systems with nonlocal effect
Chaotic dynamics in a simple predatorprey 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 predatorprey model that has equilibrium dynamics without delay. As the delay increases from zero, the coexistence equilibrium undergoes a supercritical Hopf bifurcation, two saddlenode 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 MackeyGlass 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 predatorprey like interactions as submodels, this study emphasizes the importance of understanding the implications of overlooking delay in such models on the reliability of the modelbased predictions, especially since temperature is known to have an effect on the length of certain delays.
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A.M. Ginoux, B. Rossetto and J.L. Jamet, Chaos in a threedimensional VolterraGause model of predatorprey type, Internat. J. of Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 16891708. doi: 10.1142/S0218127405012934. 
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S. A. Gourley and Y. Kuang, A stage structured predatorprey model and its dependence on maturation delay and death rate, J. Math. Biol., 49 (2004), 188200. doi: 10.1007/s0028500402782. 
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M. A. Haque, A predatorprey model with discrete time delay considering different growth function of prey, Adv. Appl. Math. Biosci., 2 (2011), 116. 
[15] 
A. Hastings and T. Powell, Chaos in a threespecies food chain, Ecology, 72 (1991), 896903. 
[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), 733753. doi: 10.1002/cpa.3160380607. 
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Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press Inc., Boston, MA, 1993. 
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V. Lakshmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications, vol. 55Ⅰ, Academic Press, New York, 1969. 
[19] 
M. Y. Li, X. Lin and H. Wang, Global Hopf branches and multiple limit cycles in a delayed LotkaVolterra predatorprey model, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 747760. doi: 10.3934/dcdsb.2014.19.747. 
[20] 
M. C. Mackey and L. Glass, Oscillations and chaos in physiological control, Science, 197 (1977), 287289. 
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A. Morozov, S. Petrovskii and B.L. Li, Bifurcations and chaos in a predatorprey system with the allee effect, P Roy. Soc. BBiol. Sci., 271 (2004), 14071414. 
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M. L. Rosenzweig, Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time, Science, 171 (1971), 385387. 
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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: 10.1007/9781441976468. 
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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. 
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J. Wang and W. Jiang, Bifurcation and chaos of a delayed predatorprey model with dormancy of predators, Nonlinear Dynam., 69 (2012), 15411558. doi: 10.1007/s1107101203684. 
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), 109119. 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), 11441165. doi: 10.1137/S0036141000376086. 
[4] 
K. L. Cooke, R. H. Elderkin and W. Huang, Predatorprey interactions with delays due to juvenile maturation, SIAM J. Appl. Math., 66 (2006), 10501079. doi: 10.1137/05063135. 
[5] 
R. Driver, Existence and stability of solutions of a delaydifferential system, Arch. Ration. Mech. Anal., 10 (1962), 401426. 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 predatorprey 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 threedimensional VolterraGause model of predatorprey type, Internat. J. of Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 16891708. doi: 10.1142/S0218127405012934. 
[12] 
S. A. Gourley and Y. Kuang, A stage structured predatorprey model and its dependence on maturation delay and death rate, J. Math. Biol., 49 (2004), 188200. doi: 10.1007/s0028500402782. 
[13] 
J. K. Hale and V. L. S. M., Introduction to FunctionalDifferential Equations, vol. 99 of Applied Mathematical Sciences, SpringerVerlag, New York, 1993. doi: 10.1007/9781461243427. 
[14] 
M. A. Haque, A predatorprey model with discrete time delay considering different growth function of prey, Adv. Appl. Math. Biosci., 2 (2011), 116. 
[15] 
A. Hastings and T. Powell, Chaos in a threespecies food chain, Ecology, 72 (1991), 896903. 
[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), 733753. 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. 55Ⅰ, Academic Press, New York, 1969. 
[19] 
M. Y. Li, X. Lin and H. Wang, Global Hopf branches and multiple limit cycles in a delayed LotkaVolterra predatorprey model, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 747760. doi: 10.3934/dcdsb.2014.19.747. 
[20] 
M. C. Mackey and L. Glass, Oscillations and chaos in physiological control, Science, 197 (1977), 287289. 
[21] 
_____, MackeyGlass equation, Scholarpedia, 4 (2009), p. 6908. 
[22] 
MAPLE, Maplesoft, A Division of Waterloo Maple Inc., Waterloo, Ontario, 2017. 
[23] 
MATLAB, Version 9.5.0 (R2018b), The MathWorks Inc., Natick, Massachusetts, 2018. 
[24] 
A. Morozov, S. Petrovskii and B.L. Li, Bifurcations and chaos in a predatorprey system with the allee effect, P Roy. Soc. BBiol. Sci., 271 (2004), 14071414. 
[25] 
M. L. Rosenzweig, Paradox of enrichment: Destabilization of exploitation ecosystems in ecological time, Science, 171 (1971), 385387. 
[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: 10.1007/9781441976468. 
[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 predatorprey model with dormancy of predators, Nonlinear Dynam., 69 (2012), 15411558. doi: 10.1007/s1107101203684. 
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