Chaos in a periodically forced predator-prey ecosystem model. (English) Zbl 0767.92028

Summary: We subject to periodic forcing the classical Volterra predator-prey ecosystem model, which in its unforced state has a globally stable focus as its equilibrium. The periodic forcing is effected by assuming a periodic variation in the intrinsic growth rate of the prey. In nondimensional form the forced system contains four control parameters, including the forcing amplitude and forcing frequency. Numerical experiments carried out over sections of the parameter space reveal an abundance of steady-state chaotic solutions. We graph Poincaré maps and calculate Lyapunov exponents and fractal dimensions for a representative selection of strange attractors. The transitions to chaos were found to be either via a Feigenbaum cascade of period-doubling bifurcations or via frequency locking.


92D40 Ecology
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
92-08 Computational methods for problems pertaining to biology
Full Text: DOI


[1] Allen, J. C., Chaos and phase-locking in predator-prey models in relation to the functional response, Florida Entomol., 73, 100-110 (1990)
[2] Arneodo, A.; Coullet, P.; Peyraud, J.; Tresser, C., Strange attractors in Volterra equations for species in competition, J. Math Biol., 14, 153-157 (1982) · Zbl 0489.92017
[3] Cvitanović, P., Universality in Chaos (1984), Adam Hilger: Adam Hilger Bristol
[4] D’Humieres, D.; Beasley, M. R.; Huberman, B. A.; Libchaber, A., Chaotic states and routes to chaos in the forced pendulum, Phys. Rev., A26, 3483-3496 (1982)
[5] Farmer, J. D.; Ott, E.; Yorke, J. A., The dimension of chaotic attractors, Physica, 7D, 153-180 (1983)
[6] Gardini, L.; Lupini, R.; Messia, M. G., Hopf bifurcation and transition to chaos in Lotka-Volterra equation, J. Math. Biol., 27, 259-272 (1989) · Zbl 0715.92020
[7] Gause, G. F., The Struggle for Existence (1934), Williams and Wilkins: Williams and Wilkins Baltimore
[8] Gilpin, M. E., Spiral chaos in a predator-prey model, Am. Nat., 113, 306-308 (1979)
[9] Grassberger, P.; Procaccia, I., Characterization of strange attractors, Phys. Rev. Lett., 50, 346-349 (1983)
[10] Grassberger, P.; Procaccia, I., Measuring the strangeness of strange attractors, Physica, 9D, 189-208 (1983) · Zbl 0593.58024
[11] Guckenheimer, J.; Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0515.34001
[12] Gumowski, I.; Mira, C., Recurrences and Discrete Dynamic Systems (1980), Springer-Verlag: Springer-Verlag New York · Zbl 0449.58003
[13] Huberman, B. A.; Crutchfield, J. P.; Packard, N. H., Noise phenomena in Josephson junctions, Appl. Phys. Lett., 37, 750-752 (1980)
[14] Inoue, M.; Kamifukumoto, H., Scenarios leading to chaos in a forced Lotka-Volterra model, Prog. Theor. Phys., 71, 930-937 (1984) · Zbl 1074.37522
[15] Jordan, D. W.; Smith, P., Nonlinear Ordinary Differential Equations (1987), Oxford Univ. Press: Oxford Univ. Press New York · Zbl 0611.34001
[16] Kot, M.; Sayler, G. S.; Schultz, T. W., Complex dynamics in a model microbial system, Bull. Math. Biol., 54, 619-648 (1992) · Zbl 0761.92041
[17] McLaughlin, J. B., Period-doubling bifurcations and chaotic motion for a parametrically forced pendulum, J. Stat. Phys., 24, 375-388 (1981)
[18] Marotto, F. R., The dynamics of a discrete population model with threshold, Math. Biosci., 58, 123-128 (1982) · Zbl 0486.92017
[19] May, R. M., Time-delay versus stability in population models with two and three trophic levels, Ecology, 54, 315-325 (1973)
[20] May, R. M., Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos, Science, 186, 645-647 (15 Nov. 1974)
[21] May, R. M., Simple mathematical models with very complicated dynamics, Nature, 261, 459-467 (1976) · Zbl 1369.37088
[22] May, R. M., Nonlinear phenomena in ecology and epidemiology, Ann. NY Acad. Sci., 357, 267-281 (1980)
[23] May, R. M., When two and two do not make four: nonlinear phenomena in ecology, Proc. Roy. Soc. Lond. B, 228, 241-266 (1986)
[24] May, R. M., The chaotic rhythms of life, New Sci., 124, 1691, 37-41 (1989)
[25] May, R. M.; Leonard, W. J., Nonlinear aspects of competition between three species, SIAM J. Appl. Math., 29, 243-253 (1975) · Zbl 0314.92008
[26] Ott, E., Strange attractors and chaotic motions of dynamical systems, Rev. Mod. Phys., 53, 655-671 (1981) · Zbl 1114.37303
[27] Parker, T. S.; Chua, L. O., Practical Numerical Algorithms for Chaotic Systems (1989), Springer-Verlag: Springer-Verlag New York · Zbl 0692.58001
[28] Pavlou, S.; Kevrekidis, I. G., Microbial predation in a periodically operated chemostat: a global study of the interaction between natural and externally imposed frequencies, Math. Biosci., 108, 1-55 (1992) · Zbl 0729.92522
[29] Rajasekar, S.; Lakshmanan, M., Period-doubling bifurcations, chaos, phase-locking and devil’s staircase in a Bonhoeffer-Van der Pol oscillator, Physica D, 3, 146-152 (1988) · Zbl 0658.58025
[30] Rasband, S. N., Chaotic Dynamics of Nonlinear Systems (1990), Wiley: Wiley New York · Zbl 0863.58053
[31] Schaffer, W. M., Stretching and folding in lynx fur returns: evidence for a strange attractor in nature?, Am. Nat., 124, 798-820 (1984)
[32] Schaffer, W. M., Order and chaos in ecological systems, Ecology, 66, 93-106 (1985)
[33] Schaffer, W. M., Perceiving order in the chaos of nature, (Boyce, M. S., Evolution of Life Histories of Mammals (1988), Yale Univ. Press: Yale Univ. Press New Haven), 313-350
[34] Schaffer, W. M.; Kot, M., Differential systems in ecology and epidemiology, (Holden, A. V., Chaos (1986), Princeton Univ. Press: Princeton Univ. Press Princeton, N.J), 158-178
[35] Schuster, H. G., Deterministic Chaos (1989), VCH: VCH Weinheim
[36] Shibata, A.; Saitô, N., Time delays and chaos in two competing species, Math. Biosci., 51, 199-211 (1980) · Zbl 0455.92011
[37] Thompson, J. M.T.; Stewart, H. B., Nonlinear Dynamics and Chaos (1986), Wiley: Wiley New York · Zbl 0601.58001
[38] Tomita, K., Periodically forced nonlinear oscillators, (Holden, A. V., Chaos (1986), Princeton Univ. Press: Princeton Univ. Press Princeton, N.J), 211-236
[39] Toro, M.; Aracil, J., Qualitative analysis of system dynamics ecological models, Syst. Dynam. Rev., 4, 56-80 (1988)
[40] Volterra, V., Variazione e fluttuazini del numero d’individui in specie animali conviventi, Mem. Accad. Nazionale Lincei, 2, 31-113 (1926), Ser. 6 · JFM 52.0450.06
[41] Wolf, A.; Swift, J. B.; Swimney, H. L.; Vastano, J. A., Determining Lyapunov exponents from a time series, Physica, 16D, 285-317 (1985) · Zbl 0585.58037
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