Abstract
Two discrete population models, one with stochasticity in the carrying capacity and one with stochasticity in the per capita growth rate, are investigated. Conditions under which the corresponding Markov processes are null recurrent and positively recurrent are derived.
Similar content being viewed by others
References
Brockwell, P. J.: The Extinction Time of A Birth, Death and Catastrophic Process and of A Related Diffusion Model. Adv. Appl. Probab. 17,42–52 (1985)
Capocelli, R. M., Ricciardi, L. M.: A Diffusion Model for Population Growth in Random Environment. Theor. Popul. Biol. 5, 28–41 (1974)
Ellner, S. P.: Asymptotic behaviour of some stochastic difference equation population models. J. Math. Biol. 19, 169–200 (1984)
Ellner, S. P.: Convergence to stationary distributions in two-species stochastic competition models. J. Math. Biol. 27, 451–462 (1989)
Feldman, M., Roughgarden, J.: A Population's Stationary Distribution and Chance of Extinction in a Stochastic Environmental with Remarks on the Theory of Species Packing. Theor. Popul. Biol. 7, 197–207 (1975)
Feller, W.: An Introduction to Probability Theory and Its Applications, vol. II, second ed. New York: Wiley 1971
Gripenberg, G.: A Stationary Distribution for the Growth of a Population Subject to Random Catastrophes. J. Math. Biol. 17, 371–379 (1983)
Guess, H. A., Gillespie, J. H.: Diffusion Approximation to Linear Stochastic Difference Equations with Stationary Coefficients. J. Appl. Probab. 14, 58–74 (1977)
Hanski, I., Woiwod, I. P.: Mean-related Stochasticity and Population Variability. Oikos 67, 29–39 (1993)
Hanson, F.B., Tuckwell, H. C.: Logistic Growth with Random Density Dependent Disasters. Theor. Popul. Biol. 19, 1–18 (1981)
Jones, D. A.: Nonlinear Autoregressive Processes. Proc. R. Soc. Lond., Ser. A 360, 71–95 (1978)
Keiding, N.: Extinction and exponential growth in random environments. Theor. Popul. Biol. 6, 49–63 (1974)
Kiester, A. R., Barakat, R.: Exact Solutions to Certain Stochastic Differential Equation Models of Population Growth. Theor. Popul. Biol. 6, 1–12 (1974)
Laslett, G. M., Pollard, D. B., Tweedie, R. L.: Techniques for Establishing Ergodic and Recurrence Properties of Continuous-Valued Markov Chains. Nav. Res. Logist. 25, 455–472 (1978)
Levins, R.: The Effect of Random Variation of Different Types on Population Growth. Proc. Natl. Acad. Sci. 62, 1061–1065 (1969)
Lewontin, R. C., Cohen, D.: On population growth in a randomly varying environment. Proc. Natl. Acad. Sci. 62, 1056–1060 (1969)
May, R. M.: Stability and complexity in model ecosystems. Princeton, NJ: Princeton University Press (1973)
May, R. M.: Simple mathematical models with very complicated dynamics. Nature 261, 459–467 (1976)
Nummelin, E.: General Irreducible Markov Chains and Nonnegative Operators. Cambridge: Cambridge University Press 1984
Renshaw, E.: Modelling Biological Populations in Space and Time (Camb. Stud. Math. Biol., vol. 11) Cambridge: Cambridge University Press 1991
Rosenblatt, M.: Markov Processes, Structure and Asymptotic Behavior. Berlin Heidelberg New York: Springer 1971
Roughgarden, J.: Population Dynamics in a Stochastic Environment: Spectral Theory for the Linearized N-species Lotka-Volterra Equations. Theor. Popul. Biol. 7, 1–12 (1975)
Roughgarden, J.: A Simple Model for Population Dynamics in Stochastic Environments. Am. Nat. 109, 713–736 (1975)
Smith, R. H., Mead, R.: The dynamics of discrete-time stochastic models of population growth. J. Theor. Biol. 86, 607–627 (1980)
Tjøstheim, D.: Non-linear Time Series and Markov Chains. Adv. Appl. Probab. 22, 587–611 (1990)
Tong, H.: Non-Linear Time Series. A Dynamical System Approach. Oxford: Clarendon Press 1990
Tuckwell, H. C.: A Study of Some Diffusion Models of Population Growth. Theor. Popul. Biol. 5, 1–12 (1974)
Turelli, M.: Random Environments and Stochastic Calculus. Theor. Popul. Biol. 12,140–178 (1977)
Tweedie, R. L.: Sufficient Conditions for Ergodicty and Recurrence of Markov Chains on a General State Space. Stochastic Processes Appl. 3, 385–403 (1975)
Tweedie, R. L.: Criteria for Classifying General Markov Chains. Adv. Appl. Probab. 8, 737–771 (1976)
Tweedie, R. L.: The Existence of Moments for Stationary Markov Chains. J. Appl. Probab. 20, 191–196 (1983)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gyllenberg, M., Högnäs, G. & Koski, T. Population models with environmental stochasticity. J. Math. Biol. 32, 93–108 (1994). https://doi.org/10.1007/BF00163026
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00163026