Dynamical complexity of a spatial phytoplankton-zooplankton model with an alternative prey and refuge effect. (English) Zbl 1266.92070

Summary: The spatiotemporal dynamics of a phytoplankton-zooplankton model with an alternative prey and refuge effect is investigated mathematically and numerically. The stability of the equilibrium point and the traveling wave solution of the phytoplankton-zooplankton model are described based on theoretical mathematical work, which provides the basis of the numerical simulations. The numerical analysis shows that refuges have a strong effect on the spatiotemporal dynamics of the model according to the pattern formation. These results may help us to understand prey-predator interactions in water ecosystems. They are also relevant to research into phytoplankton-zooplankton ecosystems.


92D40 Ecology
92-08 Computational methods for problems pertaining to biology
35Q92 PDEs in connection with biology, chemistry and other natural sciences
37N25 Dynamical systems in biology
Full Text: DOI


[1] K. Das and N. H. Gazi, “Random excitations in modelling of algal blooms in estuarine systems,” Ecological Modelling, vol. 222, no. 14, pp. 2495-2501, 2011. · doi:10.1016/j.ecolmodel.2010.11.022
[2] E. Beltrami, “Unusual algal blooms as excitable systems: the case of “brown-tides,” Environmental Modeling and Assessment, vol. 1, pp. 19-24, 1996.
[3] C. Dai and M. Zhao, “Mathematical and dynamic analysis of a prey-predator model in the presence of alternative prey with impulsive state feedback control,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 724014, 19 pages, 2012. · Zbl 1244.92052 · doi:10.1155/2012/724014
[4] R. May, Stability and Complexity in Model Ecosystems with a New Introduction by the Author, Princeton University Press, Princeton, NJ, USA, 1973.
[5] H. I. Freedman, Deterministic Mathematical Models in Population Ecology, vol. 57, Marcel Dekker, New York, NY, USA, 1980. · Zbl 0448.92023
[6] M. Gatto, “Some remarks of models of piankton densities in lake,” The American Naturalist, vol. 137, no. 3, pp. 264-267, 1991.
[7] M. Haque, “A detailed study of the Beddington-DeAngelis predator-prey model,” Mathematical Biosciences, vol. 234, no. 1, pp. 1-16, 2011. · Zbl 1402.92345 · doi:10.1016/j.mbs.2011.07.003
[8] H. R. Akcakaya, R. Arditi, and L. R. Ginzburg, “Ratio-dependent predation: an abstraction that works,” Ecology, vol. 76, no. 3, pp. 995-1004, 1995.
[9] C. Cosner, D. L. Deangelis, J. S. Ault, and D. B. Olson, “Effects of spatial grouping on the functional response of predators,” Theoretical Population Biology, vol. 56, no. 1, pp. 65-75, 1999. · Zbl 0928.92031 · doi:10.1006/tpbi.1999.1414
[10] A. P. Gutierrez, “Physiological basis of ratio-dependent predator-prey theory: the metabolic pool model as a paradigm,” Ecology, vol. 73, no. 5, pp. 1552-1563, 1992.
[11] P. A. Abrams, “The fallacies of \(^{\prime\prime}\)ratio-dependent\(^{\prime\prime}\) predation,” Ecology, vol. 75, no. 6, pp. 1842-1850, 1994.
[12] S. B. Hsu, T. W. Hwang, and Y. Kuang, “Rich dynamics of a ratio-dependent one-prey two-predators model,” Journal of Mathematical Biology, vol. 43, no. 5, pp. 377-396, 2001. · Zbl 1007.34054 · doi:10.1007/s002850100100
[13] Y. Kuang, “Rich dynamics of Gause-type ratio-dependent predator-prey system,” The Fields Institute Communications, vol. 21, pp. 325-337, 1999. · Zbl 0920.92032
[14] C. Duque and M. Lizana, “On the dynamics of a predator-prey model with nonconstant death rate and diffusion,” Nonlinear Analysis, vol. 12, no. 4, pp. 2198-2210, 2011. · Zbl 1219.35125 · doi:10.1016/j.nonrwa.2011.01.002
[15] B. I. Camara, “Waves analysis and spatiotemporal pattern formation of an ecosystem model,” Nonlinear Analysis, vol. 12, no. 5, pp. 2511-2528, 2011. · Zbl 1228.35033 · doi:10.1016/j.nonrwa.2011.02.020
[16] X. Guan, W. Wang, and Y. Cai, “Spatiotemporal dynamics of a Leslie-Gower predator-prey model incorporating a prey refuge,” Nonlinear Analysis, vol. 12, no. 4, pp. 2385-2395, 2011. · Zbl 1225.49038 · doi:10.1016/j.nonrwa.2011.02.011
[17] T. K. Kar and S. K. Chattopadhyay, “A focus on long-run sustainability of a harvested prey predator system in the presence of alternative prey,” Comptes Rendus, vol. 333, no. 11-12, pp. 841-849, 2010. · doi:10.1016/j.crvi.2010.09.001
[18] P. A. Abrams and H. Matsuda, “Positive indirect effects between prey species that share predators,” Ecology, vol. 77, no. 2, pp. 610-616, 1996.
[19] W. T. Li and S. L. Wu, “Traveling waves in a diffusive predator-prey model with Holling type-III functional response,” Chaos, Solitons & Fractals, vol. 37, no. 2, pp. 476-486, 2008. · Zbl 1155.37046 · doi:10.1016/j.chaos.2006.09.039
[20] S. R. Dunbar, “Traveling waves in diffusive predator-prey equations: periodic orbits and point-to-periodic heteroclinic orbits,” SIAM Journal on Applied Mathematics, vol. 46, no. 6, pp. 1057-1078, 1986. · Zbl 0617.92020 · doi:10.1137/0146063
[21] R. A. Gardner, “Existence of travelling wave solutions of predator-prey systems via the connection index,” SIAM Journal on Applied Mathematics, vol. 44, no. 1, pp. 56-79, 1984. · Zbl 0541.35044 · doi:10.1137/0144006
[22] K. Mischaikow and J. F. Reineck, “Travelling waves in predator-prey systems,” SIAM Journal on Mathematical Analysis, vol. 24, no. 5, pp. 1179-1214, 1993. · Zbl 0815.35045 · doi:10.1137/0524068
[23] M. R. Owen and M. A. Lewis, “How predation can slow, stop or reverse a prey invasion,” Bulletin of Mathematical Biology, vol. 63, no. 4, pp. 655-684, 2001. · Zbl 1323.92181 · doi:10.1006/bulm.2001.0239
[24] A. I. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, vol. 140 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1994. · Zbl 0805.35143
[25] F. F. Zhang, G. Huo, Q. X. Liu, G. Q. Sun, and Z. Jin, “Existence of travelling waves in nonlinear SI epidemic models,” Journal of Biological Systems, vol. 17, no. 4, pp. 643-657, 2009. · Zbl 1342.92293 · doi:10.1142/S0218339009003101
[26] J. Yang and M. Zhao, “A mathematical model for the dynamics of a fish algae consumption model with impulsive control strategy,” Journal of Applied Mathematics, vol. 2012, Article ID 575047, 17 pages, 2012. · Zbl 1244.93115 · doi:10.1155/2012/452789
[27] H. Yu and M. Zhao, “Seasonally perturbed prey-predator ecological system with the Beddington-DeAngelis functional response,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 150359, 12 pages, 2012. · Zbl 1233.92081 · doi:10.1155/2012/150359
[28] M. Zhao, X. Wang, H. Yu, and J. Zhu, “Dynamics of an ecological model with impulsive control strategy and distributed time delay,” Mathematics and Computers in Simulation, vol. 82, no. 8, pp. 1432-1444, 2012. · Zbl 1251.92049 · doi:10.1016/j.matcom.2011.08.009
[29] P. P. Liu, “An analysis of a predator-prey model with both diffusion and migration,” Mathematical and Computer Modelling, vol. 51, no. 9-10, pp. 1064-1070, 2010. · Zbl 1198.35124 · doi:10.1016/j.mcm.2009.12.010
[30] N. H. Gazi and K. Das, “Structural stability analysis of an algal bloom mathematical model in tropic interaction,” Nonlinear Analysis, vol. 11, no. 4, pp. 2191-2206, 2010. · Zbl 1201.34072 · doi:10.1016/j.nonrwa.2009.06.009
[31] C. S. Holling, “The components of predation as revealed by a study of small mammal predation on the European pine sawfly,” The Canadian Entomologist, vol. 91, no. 5, pp. 293-320, 1989.
[32] W. Wang, Y. Lin, F. Yang, L. Zhang, and Y. Tan, “Numerical study of pattern formation in an extended Gray-Scott model,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 4, pp. 2016-2026, 2011. · Zbl 1221.35456 · doi:10.1016/j.cnsns.2010.09.002
[33] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, 2003. · Zbl 1059.92051 · doi:10.1002/0470871296
[34] W. Wang, Y. Cai, M. Wu, K. Wang, and Z. Li, “Complex dynamics of a reaction-diffusion epidemic model,” Nonlinear Analysis, vol. 13, no. 5, pp. 2240-2258, 2012. · Zbl 1327.92069 · doi:10.1016/j.nonrwa.2012.01.018
[35] E. E. Holmes, M. A. Lewis, J. E. Banks, and R. R. Veit, “Partial differential equations in ecology: spatial interactions and population dynamics,” Ecology, vol. 75, no. 1, pp. 17-29, 1994.
[36] H. K. Khalil, Nonlinear Systems, Macmillan Publishing Company, New York, NY, USA, 1992. · Zbl 0969.34001
[37] P. Hartman, Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 1973. · Zbl 0281.34001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.