Persistence in stochastic food web models. (English) Zbl 0533.92028

Author’s summary: A sufficient condition is given for stochastic boundedness persistence of a top predator in generalized Lotka-Volterra- type stochastic food web models in arbitrary bounded regions of state space. The main result indicates that persistence in the corresponding deterministic system is preserved in the stochastic system if the intensities of random fluctuations are not too large.
Reviewer: W.J.Padgett


92D40 Ecology
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Full Text: DOI


[1] Barra, M., G. Del Grosso, A. Gerardi, G. Koch and F. Marchetti. 1979. ”Some Basic Properties of Stochastic Population Models.” InSystems Theory in Immunology, Proceedings of the Working Conference, Rome, 1978, Ed. C. Bruni, Lecture Notes in Biomath 32. New York: Springer-Verlag.
[2] Billingsley, P. 1968.Convergence of Probability Measures. New York: Wiley. · Zbl 0172.21201
[3] Chesson, P. 1978. ”Predator-Prey Theory and Variability.”Ann. Rev. ecol. Syst. 9, 323–347.
[4] – 1982. ”The Stabilizing Effect of a Random Environment.”J. math Biol. 15, 1–36. · Zbl 0505.92021
[5] Gard, T. C. 1980. ”Persistence in Food Chains with General Interactions.”Math. Biosci. 51, 165–174. · Zbl 0453.92017
[6] – 1982. ”Top Predator Persistence in Differential Equation Models of Food Chains: the Effects of Omnivory and External Forcing of Lower Trophic Levels.”J. math. Biol. 14, 285–299. · Zbl 0494.92022
[7] – 1984. ”Persistence in Food Webs.” InAutumn Course in Ecology, Proceedings of the International Conference, Trieste, 1982, Ed. S. A. Levin, Lecture Notes in Biomathematics. New York: Springer-Verlag (in press).
[8] – and T. G. Hallam. 1979. ”Persistence in Food Webs–I. Lotka-Volterra Food Chains.”Bull. math. Biol. 41, 877–891. · Zbl 0422.92017
[9] – and D. Kannan. 1976. ”On a Stochastic Differential Equation Modeling of Prey-Predator Evolution.”J. appl. Prob. 13, 429–443. · Zbl 0352.92013
[10] Gihman, I. I. and A. V. Skorohod. 1972.Stochastic Differential Equations. New York: Springer-Verlag. · Zbl 0242.60003
[11] Ikeda, N. and S. Watanabe. 1981.Stochastic Differential Equations and Diffusion Processes. Amsterdam: North-Holland. · Zbl 0495.60005
[12] Khas’minskii, R. Z. 1980.Stochastic Stability of Differential Equations. The Hague: Sijthoff and Noordhoff.
[13] May, R. M. and R. H. MacArthur. 1972. ”Niche Overlap as a Function of Environmental Variability.”Proc Natl. Acad. Sci. USA 69, 1109–1113.
[14] McKean, H. P. 1969.Stochastic Integrals. New York: Academic Press. · Zbl 0191.46603
[15] Nisbet, R. M. and W. S. C. Gurney. 1982.Modelling Fluctuating Populations. New York: Wiley. · Zbl 0593.92013
[16] Polansky, P. 1979. ”Invariant Distributions for Multi-population Models in Random Environments.”Theor. Pop. Biol. 16, 25–34. · Zbl 0417.92019
[17] Turelli, M. 1977. ”Random Environments and Stochastic Calculus.”Theor. Pop. Biol. 12, 140–178. · Zbl 0444.92013
[18] – 1984. ”Stochastic Community Theory: a Partially-guided Tour.” InA Course in Mathematical Ecology, Ed. T. G. Hallam and S. A. Levin, New York: Springer-Verlag (in press).
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.