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Existence of chaos in two-prey, one-predator system. (English) Zbl 1034.92033

From the introduction: We investigate a logically consistent, continuous time, food web model, consisting of two competing preys and one predator. The model satisfies a simple set of criteria for logically credible food web models. It incorporates the modified Holling type II functional response in each prey or predator equation. The local stability conditions for the system have been obtained and analyzed. Numerical simulations have been carried out to study the complex behavior of the system for biologically reasonable ranges of parameters.

MSC:

92D40 Ecology
92D25 Population dynamics (general)
37N25 Dynamical systems in biology
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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[1] Segel, L., Modeling dynamic phenomena in molecular and cellular biology (1984), Cambridge University Press: Cambridge University Press Cambridge, England · Zbl 0639.92001
[2] Price, P. W.; Bouton, C. E.; Gross, P.; Mcpheropn, B. A.; Thompson, J. N.; Weis, A. E., Interactions among three trophic levels: influence of plants on interactions between insect herbivores and natural enemies, Annu. Rev. Ecol. Syst., 11, 41-65 (1980)
[3] Schaffer, W. M., Order and chaos in ecological systems, Ecology, 66, 93-106 (1985)
[4] Upadhyay, R. K.; Rai, V., Crisis-limited chaotic dynamics in ecological system, Chaos, Solitons and Fractals, 12, 205-218 (2001) · Zbl 0977.92033
[5] Hastings, A.; Powell, T., Chaos in three species food chain, Ecology, 72, 896-903 (1991)
[6] Rinaldi, S.; Casagrandi, R.; Gragnani, A., Reduced order models for the prediction of the time of occurrence of extreme episodes, Chaos, Solitons and Fractals, 12, 313-320 (2001) · Zbl 0976.92030
[7] Klebanoff, A.; Hastings, A., Chaos in three species food chains, J. Math. Biol., 32, 427-451 (1993) · Zbl 0823.92030
[8] Edwards, A. M.; Bees, M. A., Generic dynamics of a simple plankton population model with a non-integer exponent at closure, Chaos, Solitons and Fractals, 12, 289-300 (2001) · Zbl 0977.92027
[9] Letellier, C.; Aziz-Alaoui, M. A., Analysis of the dynamics of a realistic ecological model, Chaos, Solitons and Fractals, 13, 95-107 (2002) · Zbl 0977.92029
[10] Gilpin, M. E., Spiral chaos in a predator-prey model, Am. Nat., 113, 306-308 (1979)
[11] Klebanoff, A.; Hastings, A., Chaos in one predator two prey model: general results from bifurcation theory, Math. Bios., 122, 221-223 (1994) · Zbl 0802.92017
[12] Fujii, K., Relationship of two-prey, one-predator species model: local and global stability, J. Theor. Biol., 69, 613-623 (1977)
[13] Vance, R. R., Predation and resource partitioning in one predator, two prey model communities, Am. Nat., 112, 797-813 (1978)
[14] Arditi, R.; Michalski, J., Nonlinear food web models and their response to increased basal productivity, (Polis, G. A.; Winemillr, K. O., Food webs: integration of patterns and dynamics (1996), Chapman and Hall: Chapman and Hall New York), 122-133
[15] Kuang Y. Basic properties of mathematical population models, preprints. J. Biomath., in press; Kuang Y. Basic properties of mathematical population models, preprints. J. Biomath., in press
[16] May, R. M., Stability and complexity in model ecosystem (1973), Princeton university press: Princeton university press Princeton, NJ, USA
[17] Freedman, H. I., Deterministic mathematical models in population ecology (1980), Marcal Dekker, Inc: Marcal Dekker, Inc New York, USA · Zbl 0448.92023
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