×

Categories of chaos and fractal basin boundaries in forced predator-prey models. (English) Zbl 0976.92033

Summary: Biological communities are affected by perturbations that frequently occur in a more-or-less periodic fashion. We use the circle map to summarize the dynamics of one such community – the periodically forced Lotka-Volterra predator-prey system. As might be expected, we show that the latter system generates a classic devil’s staircase and Arnold tongues, similar to that found from a qualitative analysis of the circle map. The circle map has other subtle features that make it useful for explaining the two qualitatively distinct forms of chaos recently noted in numerical studies of the forced Lotka-Volterra system. In the regions of overlapping tongues, coexisting attractors may be found in the Lotka-Volterra system, including at least one example of three alternative attractors, the separatrices of which are fractal and, in one specific case, Wada. The analysis is extended to a periodically forced tritrophic foodweb model that is chaotic. Interestingly, mode-locking Arnold tongue structures are observed in the model’s phase dynamics even though the foodweb equations are chaotic.

MSC:

92D40 Ecology
37N25 Dynamical systems in biology
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Aron, J.L.; Schwartz, I.B., Seasonality and period-doubling bifurcations in an epidemic model, J theor biol, 110, 665-679, (1984)
[2] Inoue, M.; Kamifukumoto, H., Scenarios leading to chaos in a forced lotka – volterra model, Prog theor phys, 71, 930-937, (1984) · Zbl 1074.37522
[3] Kot, M.; Schaffer, W.M., Discrete-time growth-dispersal models, Math biosci, 80, 109-136, (1986) · Zbl 0595.92011
[4] Schaffer WM. Perceiving order in the chaos of nature. In: Boyce MS. editor. Evolution of life histories in mammals. New Haven: Yale University Press, 1988. pp. 313-350
[5] Vandermeer, J.H., Seasonal isochronic forcing of lotka – volterra equations, Prog theor phys, 96, 13-28, (1996)
[6] King, A.A.; Schaffer, W.M.; Gordon, C.; Treat, J.; Kot, M., Weakly dissipative predator – prey systems, Bull math biol, 58, 835-859, (1996) · Zbl 0856.92020
[7] Val J, Villa F, Lika K, Boe C. Nonlinear models of structured populations: Dynamic consequences of stage structure and discrete sampling. In: Tuljapurkar S, Caswell H. editors. Structured-population models in marine, terrestrial, and freshwater systems. New York: Chapman & Hall, 1997. pp. 587-613
[8] Leven RW, Koch BP, Markman GS. Periodic, quasiperiodic and chaotic motion in a forced predator – prey ecosystem. In: Bothe HG, Ebeling W, Kurzhanski AB, Peschel M. editors. Dynamical systems and environmental models. Berlin: Akademie, 1987. pp. 95-104
[9] Kot, M.; Sayler, G.S.; Schultz, T.W., Complex dynamics in a model microbialsystem, Bull math biol, 54, 619-648, (1992) · Zbl 0761.92041
[10] Sabin, G.C.W.; Summers, D., Chaos in a periodically forced predator – prey ecosystem model, Math biosci, 113, 91-113, (1993) · Zbl 0767.92028
[11] Rinaldi, S.; Muratori, S., Conditioned chaos in seasonally perturbed predator – prey models, Ecol model, 69, 79-97, (1993)
[12] Gragnani, A.; Rinaldi, S., A universal bifurcation diagram for seasonally perturbed predator – prey models, Bull math biol, 57, 701-712, (1995) · Zbl 0824.92027
[13] Rinaldi, S.; Muratori, S.; Kuznetsov, Y., Multiple attractors, catastrophes and chaos in seasonally perturbed predator – prey communities, Bull math biol, 55, 15-35, (1993) · Zbl 0756.92026
[14] Blasius, B.; Huppert, A.; Stone, L., Complex dynamics and phase synchronization in spatially extended ecological systems, Nature, 399, 354-359, (1999)
[15] Nusse, H.E.; Yorke, J.A., Basins of attraction, Science, 271, 1376-1380, (1996) · Zbl 1226.37009
[16] Rosenblum, M.G.; Pikovsky, A.S.; Kurths, J., Phase synchronization of chaotic oscillators, Phys rev lett, 76, 1804-1807, (1996)
[17] Bak P. The devil’s staircase. Phys Today, 1986:39-45
[18] Bohr, T.; Bak, P.; Jensen, M.H., Transition to chaos by interaction of resonances in dissipative systems. II. Josephson junctions, charge-density waves, and standard maps, Phys rev A, 30, 1970-1981, (1984)
[19] Cvitanovic, P.; Gunaratne, G.H.; Vinson, M.J., On the mode-locking universality or critical circle maps, Nonlinearity, 3, 873-885, (1990) · Zbl 0712.58042
[20] Vandermeer, J.H., The qualitative behavior of coupled predator – prey oscillations as deduced from simple circle maps, Ecol model, 73, 135-148, (1994)
[21] Schuster, H.G., Deterministic chaos, (1984), Physik-Verlag Weinheim
[22] Abraham RH, Shaw CD. Dynamics – the geometry of behavior. Part 4: Bifurcation behavior. Santa Cruz, CA: Aerial Press, 1988. p. 196
[23] Zeng, W.-.Z.; Glass, L., Symbolic dynamics and skeletons of circle maps, Physica D, 40, 218-234, (1989) · Zbl 0820.58015
[24] Jensen, M.H.; Bak, P.; Bohr, T., Transition to chaos by interaction of resonances in dissipative systems. I. circle maps, Phys rev A, 30, 1960-1969, (1984)
[25] Gilpin, M.E., Spiral chaos in a predator – prey model, Am naturalist, 113, 306-308, (1979)
[26] Hastings, A.; Powell, T., Chaos in a three-species food chain, Ecology, 72, 896-903, (1991)
[27] Blasius B, Stone L. Chaos and phase synchronization in ecological systems. Int J Bif Chaos 2000;10 (in press) · Zbl 0968.92020
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.