×

zbMATH — the first resource for mathematics

Population models: Stability in one dimension. (English) Zbl 1298.92079
Summary: Some of the simplest models of population growth are one dimensional nonlinear difference equations. While such models can display wild behavior including chaos, the standard biological models have the interesting property that they display global stability if they display local stability. Various researchers have sought a simple explanation for this agreement of local and global stability. Here, we show that enveloping by a linear fractional function is sufficient for global stability. We also show that for seven standard biological models local stability implies enveloping and hence global stability. We derive two methods to demonstrate enveloping and show that these methods can easily be applied to the seven example models. Although enveloping by a linear fractional is a sufficient for global stability, we show by example that such enveloping is not necessary. We extend our results by showing that enveloping implies global stability even when \(f(x)\) is a discontinuous multi-function which might be a more reasonable description of real bilogical data. We show that our techniques can be applied to situations which are not population models. Finally, we give examples of population models which have local stability but not global stability.

MSC:
92D25 Population dynamics (general)
37N25 Dynamical systems in biology
39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Cull, P., 1981. Global stability of population models. Bull. Math. Biol. 43, 47–58. · Zbl 0451.92011
[2] Cull, P., 1986. Local and global stability for population models. Biol. Cybern. 54, 141–149. · Zbl 0607.92018 · doi:10.1007/BF00356852
[3] Cull, P., 1988a. Local and global stability of discrete one-dimensional population models. In: Ricciardi, L.M. (Ed.), Biomathematics and Related Computational Problems. Kluwer, Dordrecht, pp. 271–278. · Zbl 0645.92016
[4] Cull, P., 1988b. Stability of discrete one-dimensional population models. Bull. Math. Biol. 50(1), 67–75. · Zbl 0637.92011
[5] Cull, P., 2002. Linear fractionals—simple models with chaotic-like behavior. In: Dubois, D.M. (Ed.), Computing Anticipatory Systems: CASYS 2001—Fifth International Conference, Conference Proceedings 627, American Institue of Physics, Woodbury, NY, pp. 170–181.
[6] Cull, P., 2003. Stability in one-dimensional models. Scientiae Mathematicae Japonicae 58, 349–357. · Zbl 1059.92042
[7] Cull, P., Chaffee, J., 2000a. Stability in discrete population models. In: Dubois, D.M. (Ed.), Computing Anticipatory Systems: CASYS’99, Conference Proceedings 517, American Institute of Physics, Woodbury, NY, pp. 263–275.
[8] Cull, P., Chaffee, J., 2000b. Stability in simple population models. In: Trappl, R. (Ed.) Cybernetics and Systems 2000, Austrian Society for Cybernetics Studies, pp. 289–294.
[9] Cull, P., Flahive, M., Robson, R., 2005. Difference Equations: From Rabbits to Chaos. Springer, New York. · Zbl 1085.39002
[10] Devaney, R., 1986. An Introduction to Chaotic Dynamical Systems. Benjamin, Redwood City. · Zbl 0632.58005
[11] Dubois, D., 1998. Computing anticipatory systems with incursion and hyperincursion. In: Dubois, D.M. (Ed.), Computing Anticipatory Systems: CASYS’97—First International Conference, Conference Proceedings 437, American Institue of Physics, Woodbury, NY, pp. 3–29.
[12] Fisher, M.E., Goh, B.S., Vincent, T.L., 1979. Some stability conditions for discrete-time single species models. Bull. Math. Biol. 41, 861–875. · Zbl 0418.92014
[13] Goh, B.S., 1979. Management and Analysis of Biological Populations. Elsevier, New York. · Zbl 0497.34060
[14] Hassel, M.P., 1974. Density dependence in single species populations. J. Anim. Ecol. 44, 283–296. · doi:10.2307/3863
[15] Heinschel, N., 1994. Sufficient conditions for global stability in population models. Oregon State University REU Proceedings, pp. 51–67.
[16] Huang, Y.N., 1986. A counterexample for P. Cull’s theorem. Kexue Tongbao 31, 1002–1003.
[17] LaSalle, J.P., 1976. The Stability of Dynamical Systems. SIAM, Philadelphia. · Zbl 0364.93002
[18] Li, T-Y., Yorke, J., 1975. Period three implies chaos. Am. Math. Mon. 82, 985–992. · Zbl 0351.92021 · doi:10.2307/2318254
[19] May, R.M., 1974. Biological populations with nonoverlapping generations: Stable points, stable cycles, and chaos. Science 186, 645–647. · doi:10.1126/science.186.4164.645
[20] May, R.M., 1976. Simple mathematical models with very complicated dynamics. Nature 261, 459–467. · Zbl 1369.37088 · doi:10.1038/261459a0
[21] Moran, P.A.P., 1950. Some remarks on animal population dynamics. Biometrics 6, 250–258. · doi:10.2307/3001822
[22] Nobile, A., Ricciardi, L.M., Sacerdote, L., 1982. On gompertz growth model and related difference equations. Biol. Cybern. 42, 221–229. · Zbl 0478.92013
[23] Pennycuick, C.J., Compton, R.M., Beckingham, L., 1968. A computer model for simulating the growth of a population, or of two interacting populations. J. Theor. Biol. 18, 316–329. · doi:10.1016/0022-5193(68)90081-7
[24] Ricker, W.E., 1954. Stock and recruitment. J. Fisheries Research Board of Canada 11, 559–623.
[25] Rosenkranz, G., 1983. On global stability of discrete population models. Math. Biosci. 64, 227–231. · Zbl 0515.92019 · doi:10.1016/0025-5564(83)90005-6
[26] Sarkovskii, A., 1964. Coexistence of cycles of a continuous map of a line to itself. Ukr. Mat. Z. 16, 61–71.
[27] Singer, D., 1978. Stable orbits and bifurcation of maps of the interval. SIAM J. Appl. Math. 35(2), 260–267. · Zbl 0391.58014 · doi:10.1137/0135020
[28] Smith, J.M., 1968. Mathematical Ideas in Biology. Cambridge University Press, Cambridge.
[29] Smith, J.M., 1974. Models in Ecology. Cambridge University Press, Cambridge. · Zbl 0312.92001
[30] Utida, S., 1957. Population fluctuation, an experimental and theoretical approach. Cold Spring Harbor Symposium on Quantitative Biology 22, 139–151.
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.