×

zbMATH — the first resource for mathematics

The stabilizing effect of a random environment. (English) Zbl 0505.92021

MSC:
92D40 Ecology
92D25 Population dynamics (general)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Armstrong, R. A., McGehee, R.: Coexistence of species competing for shared resources. Theoret. Population Biology 9, 317-328 (1976) · Zbl 0349.92030
[2] Barbour, A. D.: Quasi-stationary distributions in Markov population processes. Adv. Appl. Probab. 8, 296-314 (1976) · Zbl 0337.60069
[3] Bartlett, M. S.: On theoretical models for competitive and predatory biological systems. Biometrika 44, 27-42 (1957) · Zbl 0080.36301
[4] Bartlett, M. S., Gower, J. C., Leslie, P. H.: A comparison of theoretical and empirical results for some stochastic population models. Biometrika 47, 1-11 (1960) · Zbl 0097.14601
[5] Beddington, J. R., Free, C. A., Lawton, J. H.: Concepts of stability and resilience in predator-prey models. J. Anim. Ecol. 45, 791-816 (1976) · Zbl 0341.92018
[6] Billingsley, P.: Convergence of probability measures. New York: Wiley 1968 · Zbl 0172.21201
[7] Billingsley, P.: Weak convergence of measures. Philadelphia: SIAM 1971 · Zbl 0271.60009
[8] Botkin, D. B., Sobel, M. J.: Stability in time-varying ecosystems. Am. Nat. 109, 625-646 (1975)
[9] Breiman, L.: Probability. Menlo Park, CA: Addison-Wesley 1968
[10] Chesson, P. L.: Predator-prey theory and variability. Ann. Rev. Ecol. Syst. 9, 323-347 (1978)
[11] Chesson, P. L.: Models for spatially distributed populations: The effect of within-patch variability. Theoret. Population Biology 19, 288-325 (1981) · Zbl 0472.92015
[12] Chesson, P. L., Warner, R. R.: Environmental variability promotes coexistence in lottery competitive systems. Am. Nat. 117, 923-943 (1981)
[13] Cushing, J. M.: Two species competition in a periodic environment. J. Math. Biol. 10, 385-400 (1980) · Zbl 0455.92012
[14] de Mottoni, P., Schiaffino, A.: Competition systems with periodic coefficients: A geometric approach. J. Math. Biol. 11, 319-335 (1981) · Zbl 0474.92015
[15] Feller, W.: An introduction to probability theory and its applications, Vol. II, second edition. New York: Wiley 1971 · Zbl 0219.60003
[16] Goh, B. S.: Stability, vulnerability, and persistence of complex ecosystems. Ecological Modelling 1, 105-116 (1975)
[17] Goh, B. S.: Nonvulnerability of ecosystems in unpredictable environments. Theoret. Population Biology 10, 83-95 (1976) · Zbl 0335.92017
[18] Heyde, C. C.: On the central limit theorem for stationary processes. Z. Wahrscheinlichkeitstheorie 30, 315-320 (1974) · Zbl 0297.60014
[19] Holling, C. S.: Resilience and stability of ecological systems. Ann. Rev. Ecol. Syst. 4, 1-23 (1973)
[20] Ibragimov, I. A.: A note on the central limit theorem for dependent random variables. Theoret. Prob. Appl. 20, 135-141 (1975) · Zbl 0335.60023
[21] Karlin, S., Liberman, U.: Random temporal variation in selection intensities: Case of large population size?I. Theoret. Population Biology 6, 355-382 (1974) · Zbl 0289.92025
[22] Koch, A. L.: Coexistence resulting from an alternation of density independent and density dependent growth. J. Theoret. Biol. 44, 373-386 (1974)
[23] Kurtz, T. G.: Solutions of ordinary differential equations as limits of pure jump Markov processes. J. Appl. Probab. 7, 49-58 (1970) · Zbl 0191.47301
[24] Leigh, E. G.: Population fluctuations, community stability, and environmental variability. In: (M. L. Cody, J. M. Diamond, eds.) Ecology and evolution of communities, pp. 51-73. Cambridge MA: Harvard University Press 1975
[25] Leslie, P. H.: A stochastic model for studying the properties of certain biological systems by numerical methods. Biometrika 45, 16-31 (1958) · Zbl 0089.15803
[26] Leslie, P. H., Gower, J. C.: The properties of a stochastic model for two competing species. Biometrika 45, 316-330 (1958) · Zbl 0087.34501
[27] Leslie, P. H., Gower, J. G.: The properties of a stochastic model for the predator-prey type of interaction between two species. Biometrika 47, 219-234 (1960) · Zbl 0103.12502
[28] Levins, R.: Coexistence in a variable environment. Am. Nat. 114, 765-783 (1979)
[29] Lewontin, R. C.: The meaning of stability. In: Diversity and stability in ecological systems. Brookhaven Symposium in Biology No. 22, 13-24 (1969)
[30] Ludwig, D.: A singular perturbation problem in the theory of population extinction. Soc. Ind. Appl. Math.-Am. Math. Soc. Proc. 10, 87-104 (1976) · Zbl 0351.92023
[31] May, R. M.: Stability and complexity in model ecosystems, second edition. Princeton, N. J.: Princeton Univ. Press 1974
[32] Norman, F.: An ergodic theorem for evolution in a random environment. J. Appl. Prob. 12, 661-672 (1975) · Zbl 0324.60065
[33] Revuz, D.: Markov chains. Amsterdam: North-Holland Publication Co. 1975 · Zbl 0332.60045
[34] Sale, P. F.: Maintenance of high diversity in coral reef fish communities. Am. Nat. 111, 337-359 (1977)
[35] Stewart, F. M., Levin, B. R.: Partitioning of resources and the outcome of interspecific competition: A model and some general considerations. Am. Nat. 107, 171-198 (1973)
[36] Turelli, M.: Random environments and stochastic calculus. Theoret. Population Biology 12, 140-178 (1977) · Zbl 0444.92013
[37] Turelli, M.: A reexamination of stability in randomly fluctuating versus deterministic environments. Theoret. Population Biology 13, 244-267 (1978a) · Zbl 0407.92019
[38] Turelli, M.: Does environmental variability limit niche overlap? Proc. Natl. Acad. Sci. USA 75, 5065-5089 (1978b) · Zbl 0395.92022
[39] Turelli, M.: Niche overlap and invasion of competitors in random environments I. Models without demographic stochasticity. Theoret. Population Biology 20, 1-56 (1981) · Zbl 0494.92023
[40] Turelli, M., Gillespie, J. H.: Conditions for the existence of stationary densities for some two dimensional diffusion processes with applications in population biology. Theoret. Population Biology 17, 167-189 (1980) · Zbl 0441.60081
[41] Wang, F. J. S.: Limit theorems for age and density dependent stochastic population models. J. Math. Biol. 2, 373-400 (1975) · Zbl 0324.60064
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.