The complex dynamics of two types of tritrophic food chain model systems when the species undergo spatial movements, modeling two real situations of marine ecosystem, are investigated in this study analytically and using numerical simulations. The study has been carried out with the objective to explore and compare the competitive effects of fish and molluscs species being the top predators, when phytoplankton and zooplankton species are undergoing spatial movements in the subsurface water. Reaction diffusion systems have been used to represent temporal evolution and spatial interaction among the species. The two model systems differ in an essential way that the top predators are generalist and specialist, respectively, in two models. “Wave of Chaos” mechanism is found to be the responsible factor for the pattern (non-Turing) formation in one dimension seen in the food chain ending with top generalist predator. In the present work we have reported WOC phenomenon, for the first time in the literature, in a three-species spatially extended food chain model system. The numerical simulation leads to spontaneous and interesting pattern formation in two dimensions. Constraints on different parameters under which Turing and non-Turing patterns may be observed are obtained analytically. Diffusion-driven analysis is carried out, and the effect of diffusion on the chaotic dynamics of the model systems is studied. The existence of chaotic attractor and long-term chaotic behavior demonstrate the effect of diffusion on the dynamics of the model systems. It is observed from numerical study that food chain model system with top predator as generalist has very rich dynamics and shows very interesting patterns. An ecosystem having top predator as specialist leads to the stability of the system. 1. Introduction The interest in the study of chaos in ecological systems has increased in the last couple of decades. Interactions of population in ecological systems are modeled by continuous time models and have been studied extensively in the literature [1]. The species interaction models will be close to reality if the dispersal of the population is also considered, along with the time. Diffusion models form a reasonable basis for studying the interaction of population. Diffusion is a phenomenon by which the biological population spreads according to the irregular motion of each individual of the population. Diffusion equations have been used effectively to describe the movement of numerous animals in mark-recapture studies [2, 3]. Another aspect of reaction-diffusion theory
References
[1]
M. Kot, Elements of Mathematical Ecology, Cambridge University Press, Cambridge, UK, 2001.
[2]
R. A. J. Taylor, “A family of regression equations describing the density distribution of dispersing organisms,” Nature, vol. 286, no. 5768, pp. 53–55, 1980.
[3]
P. M. Kareiva, “Local movement in herbivorous insects: applying a passive diffusion model to mark-recapture field experiments,” Oecologia, vol. 57, no. 3, pp. 322–327, 1983.
[4]
J. D. Murray, Mathematical Biology, Springer, New York, NY, USA, 1989.
[5]
A. Turing, “The chemical basis of morphogenesis,” Philosophical Transactions of the Royal Society of London B, vol. 237, pp. 37–72, 1952.
[6]
W. M. Ni and I. Takagi, “On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type,” Transactions of the American Mathematical Society, vol. 297, pp. 351–368, 1986.
[7]
M. Wang, “Non-constant positive steady states of the Sel'kov model,” Journal of Differential Equations, vol. 190, no. 2, pp. 600–620, 2003.
[8]
R. Peng and M. Wang, “Positive steady-state solutions of the Noyes-Field model for Belousov-Zhabotinskii reaction,” Nonlinear Analysis, vol. 56, no. 3, pp. 451–464, 2004.
[9]
J. Wei and M. Winter, “Existence and stability of multiple-spot solutions for the Gray-Scott model in R2,” Physica D, vol. 176, no. 3-4, pp. 147–180, 2003.
[10]
W. M. Ni and J. Wei, “On positive solutions concentrating on spheres for the Gierer-Meinhardt system,” Journal of Differential Equations, vol. 221, no. 1, pp. 158–189, 2006.
[11]
M. A. Lewis and S. Pacala, “Modeling and analysis of stochastic invasion processes,” Journal of Mathematical Biology, vol. 41, no. 5, pp. 387–429, 2000.
[12]
M. Pascual, “Diffusion-induced chaos in a spatial predator-prey system,” Proceedings of the Royal Society B, vol. 251, no. 1330, pp. 1–7, 1993.
[13]
J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer, Berlin, Germany, 2003.
[14]
T. H. Keitt, M. A. Lewis, and R. D. Holt, “Allee effects, invasion pinning, and species' borders,” American Naturalist, vol. 157, no. 2, pp. 203–216, 2001.
[15]
S. Petrovskii, B. L. Li, and H. Malchow, “Quantification of the spatial aspect of chaotic dynamics in biological and chemical systems,” Bulletin of Mathematical Biology, vol. 65, no. 3, pp. 425–446, 2003.
[16]
S. Petrovskii, H. Malchow, and B. L. Li, “An exact solution of a diffusive predator-prey system,” Proceedings of the Royal Society A, vol. 461, no. 2056, pp. 1029–1053, 2005.
[17]
J. G. Lambrinos, “How interactions between ecology and evolution influence contemporary invasion dynamics,” Ecology, vol. 85, no. 8, pp. 2061–2070, 2004.
[18]
W. T. Li and S. L. Wu, “Traveling waves in a diffusive predator-prey model with holling type-III functional response,” Chaos, Solitons and Fractals, vol. 37, no. 2, pp. 476–486, 2008.
[19]
G. T. Stamov, “Almost periodic models in impulsive ecological systems with variable diffusion,” Journal of Applied Mathematics and Computing, vol. 27, no. 1-2, pp. 243–255, 2008.
[20]
A. Klebanoff and A. Hastings, “Chaos in one-predator, two-prey models: general results from bifurcation theory,” Mathematical Biosciences, vol. 122, no. 2, pp. 221–233, 1994.
[21]
B. Deng and G. Hines, “Food chain chaos due to Shilnikov's orbit,” Chaos, vol. 12, no. 3, pp. 533–538, 2002.
[22]
M. Zhao and S. Lv, “Chaos in a three-species food chain model with a Beddington-DeAngelis functional response,” Chaos, Solitons and Fractals, vol. 40, no. 5, pp. 2305–2316, 2009.
[23]
X. J. Wu, J. Li, and R. K. Upadhyay, “Chaos control and synchronization of a three-species food chain model via Holling functional response,” International Journal of Computer Mathematics, vol. 87, no. 1, pp. 199–214, 2010.
[24]
R. K. Naji, R. K. Upadhyay, and V. Rai, “Dynamical consequences of predator interference in a tri-trophic model food chain,” Nonlinear Analysis: Real World Applications, vol. 11, no. 2, pp. 809–818, 2010.
[25]
D. O. Maionchi, S. F. dos Reis, and M. A. M. de Aguiar, “Chaos and pattern formation in a spatial tritrophic food chain,” Ecological Modelling, vol. 191, no. 2, pp. 291–303, 2006.
[26]
C. Shen and M. You, “Permanence and extinction of a three-species ratio-dependent food chain model with delay and prey diffusion,” Applied Mathematics and Computation, vol. 217, no. 5, pp. 1825–1830, 2010.
[27]
Z. Xie, “Cross diffusion induced turing instability for a three species food chain model,” Journal of Mathematical Analysis and Applications, vol. 388, no. 1, pp. 539–547, 2012.
[28]
R. K. Upadhyay, R. K. Naji, and N. Kumari, “Dynamical complexity in some ecological models: effect of toxin production by phytoplankton,” Nonlinear Analysis: Modelling and Control, vol. 12, no. 1, pp. 123–138, 2007.
[29]
R. K. Upadhyay and J. Chattopadhyay, “Chaos to order: role of toxin producing phytoplankton in aquatic systems,” Nonlinear Analysis: Modeling and Control, vol. 10, no. 4, pp. 383–396, 2005.
[30]
A. Hastings and T. Powell, “Chaos in a three-species food chain,” Ecology, vol. 72, no. 3, pp. 896–903, 1991.
[31]
V. Rai, “Chaos in natural populations: edge or wedge?” Ecological Complexity, vol. 1, no. 2, pp. 127–138, 2004.
[32]
C. Letellier and M. A. Aziz-Alaoui, “Analysis of the dynamics of a realistic ecological model,” Chaos, Solitons and Fractals, vol. 13, no. 1, pp. 95–107, 2002.
[33]
S. V. Petrovskii and H. Malchow, “Wave of chaos: new mechanism of pattern formation in spatio-temporal population dynamics,” Theoretical Population Biology, vol. 59, no. 2, pp. 157–174, 2001.
[34]
R. K. Upadhyay, N. Kumari, and V. Rai, “Wave of chaos in a diffusive system: generating realistic patterns of patchiness in plankton-fish dynamics,” Chaos, Solitons and Fractals, vol. 40, no. 1, pp. 262–276, 2009.
[35]
C. L. Wood, K. D. Lafferty, and F. Micheli, “Fishing out marine parasites? Impacts of fishing on rates of parasitism in the ocean,” Ecology Letters, vol. 13, no. 6, pp. 761–775, 2010.
