zbMATH — the first resource for mathematics

Evolving to the edge of chaos: Chance or necessity? (English) Zbl 1142.92344
Summary: We show that ecological systems evolve to edges of chaos (EOC). This has been demonstrated by analyzing three diverse model ecosystems using numerical simulations in combination with analytical procedures. It has been found that all these systems reside on EOC and display short-term recurrent chaos (strc). The first two are non-linear food chains and the third one is a linear food chain. The dynamics of first two is dictated by deterministic changes in system parameters. In contrast to this, dynamics of the third model system (the linear food chain) is governed by both deterministic changes in system parameters as well as exogenous stochastic perturbations (unforeseen changes in initial conditions) of these dynamical systems.

92D40 Ecology
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
Full Text: DOI
[1] May, R.M., Biological populations with non-overlapping generations: stable points, stable cycles and chaos, Science, 186, 645-647, (1974)
[2] May, R.M., Simple mathematical models with very complicated dynamics, Nature (London), 261, 459-467, (1976) · Zbl 1369.37088
[3] Beddington, J.R.; Free, C.A.; Lawton, J.H., Dynamic complexity of predator – prey models framed in difference equations, Nature, 255, 58-60, (1975)
[4] Neubert, M.G.; Kot, M., The subcritical collapse of predator populations in discrete-time predator – prey models, Math biosci, 110, 45-66, (1992) · Zbl 0747.92024
[5] Gilpin, M.E., Spiral chaos in a predator – prey model, Am naturalist, 113, 306-308, (1979)
[6] Schaffer, W.M., Order and chaos in ecological systems, Ecology, 66, 93-106, (1985)
[7] Schaffer, W.M.; Kot, M., Do strange attractors govern ecological systems?, Bioscience, 36, 342-350, (1985)
[8] Hastings, A.; Powell, T., Chaos in a three-species food chain, Ecology, 72, 896-903, (1991)
[9] McCann, K.; Yodzis, P., Biological conditions for chaos in a three-species food chain, Ecology, 75, 561-564, (1994)
[10] McCann, K.; Yodzis, P., Bifurcation structure of a three-species food chain model, Theor popul biol, 48, 93-125, (1995) · Zbl 0854.92022
[11] Costantino, R.F.; Desharnais, R.A.; Cushing, J.M.; Dennis, B., Chaotic dynamics in an insect population, Science, 275, 389-391, (1997) · Zbl 1225.37103
[12] Upadhyay, R.K.; Rai, V., Why chaos is rarely observed in natural populations?, Chaos, solitons & fractals, 8, 1933-1939, (1997)
[13] Upadhyay, R.K.; Iyengar, S.R.K.; Rai, V., Chaos: an ecological reality?, Int J bifurc chaos, 8, 1325-1333, (1998) · Zbl 0935.92037
[14] Gilpin, M.E., Group selection in predator – prey communities, (1975), Princeton University Press Princeton, NJ, USA
[15] Thomas, W.R.; Pomerantz, M.J.; Gilpin, M.E., Chaos, asymmetric growth and group selection for dynamical stability, Ecology, 61, 1312-1320, (1980)
[16] Berryman, A.A.; Millstein, J.A., Are ecological systems chaotic and if not, why not?, Trends ecol evol, 4, 26-28, (1989)
[17] Shertzer, K.W.; Ellner, S.P., Energy storage and the evolution of population dynamics, J theor biol, 215, 183-200, (2002)
[18] Stone, L., Period-doubling reversal and chaos in simple ecological models, Nature, 365, 617-620, (1993)
[19] McCann, K.; Hastings, A.; Huxel, G.R., Weak trophic interactions and the balance of nature, Nature, 395, 794-798, (1998)
[20] Hastings, A.; Hom, C.L.; Ellner, S.; Turchin, P.; Godfray, H.C.J., Chaos in ecology: Is mother nature a strange attractor?, Ann rev ecol syst, 24, 1-33, (1993)
[21] Sugihara, G.; May, R.M., Nonlinear forecasting as a way of distinguishing chaos from measurement error in time-series, Nature, 344, 734-741, (1991)
[22] Casdagli, M., Chaos and deterministic versus stochastic modeling, J roy stat soc B, 303-328, (1992)
[23] Ellner, S.P.; Turchin, P., Chaos in a noisy world: new methods and evidence from time-series analysis, Am naturalist, 143, 343-375, (1995)
[24] Turchin, P.; Ellner, S.P., Living on the edge of chaos: population dynamics of Fennoscandian voles, Ecology, 81, 3099-3116, (2000)
[25] Rai, V.; Schaffer, W.M., Chaos in ecology, Chaos, solitons & fractals, 12, 197-203, (2001) · Zbl 0973.00028
[26] Rosenzweig, M.L.; MacArthur, R.H., Graphical representation and stability conditions of predator – prey interactions, Am naturalist, 97, 209-223, (1963)
[27] Pielou, E.C., Mathematical ecology: an introduction, (1997), Wiley New York · Zbl 0259.92001
[28] May, R.M., Stability and complexity in model ecosystems, (2001), Princeton University Press Princeton, NJ, USA
[29] Rai V, Anand M, Upadhyay RK. Trophic structure and dynamical complexity in predator – prey models. Ecol Compl, submitted for publication.
[30] Rai, V., Chaos in natural populations: edge or wedge?, Ecol compl, 1, 127-138, (2004)
[31] Anderson, M., Sexual selection, (1994), Princeton University Press Princeton, NJ, USA
[32] Moller, A.P.; Legendre, S., Allee effect, sexual selection and demographic stochasticity, Oikos, 92, 27-34, (2001)
[33] Rai, V.; Ananad, M., Is dynamic complexity of ecological systems quantifiable?, Int J ecol environ sci, 30, 123-129, (2004)
[34] Rai, V.; Upadhyay, R.K., Chaotic population dynamics and biology of the top-predator, Chaos, solitons & fractals, 21, 1195-1204, (2004) · Zbl 1057.92056
[35] Yodzis, P.; Innes, S., Body size and consumer-resource dynamics, Am naturalist, 139, 1151-1175, (1992)
[36] Jorgensen, S.E., Handbook of environmental data and ecological parameters, (1979), Pergamon Press Amsterdam
[37] Berg’, P.; Pomeau, Y.; Vidal, C., Order within chaos: towards a deterministic approach to turbulence, (1986), John Wiley & Sons New York
[38] Vandermeer, J., Loose coupling of predator – prey cycles: entrainment, chaos and intermittency in the classical macarthur consumer-resource equations, Am naturalist, 141, 687-716, (1993)
[39] Langton, C.G., Computation at the edge of chaos: phase transitions and emergent computation, Physica D, 42, 12-37, (1990)
[40] Letellier, C.; Aziz-Alaoui, M.A., Analysis of the dynamics of a realistic ecological model, Chaos, solitons & fractals, 13, 95-107, (2002) · Zbl 0977.92029
[41] Letellier, C.; Aguirre, L.A.; Maquet, J.; Aziz-Alaoui, M.A., Should all the species of a food chain be counted to investigate the global dynamics?, Chaos, solitons & fractals, 13, 1099-1113, (2002) · Zbl 1004.92039
[42] Aziz-Alaoui, M.A., Study of a leslie – gower type tritrophic population model, Chaos, solitons & fractals, 13, 1275-1293, (2002) · Zbl 1031.92027
[43] Upadhyay, R.K.; Rai, V., Crisis-limited chaotic dynamics in ecological systems, Chaos, solitons & fractals, 12, 205-218, (2001) · Zbl 0977.92033
[44] Peter A. Private communications.
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.