Journal of Mathematical Biology

, Volume 27, Issue 5, pp 491–506

A nonautonomous model of population growth

  • R. R. Vance
  • E. A. Coddington
Article

Abstract

With x = population size, the nonautonomous equation x = xf(t,x) provides a very general description of population growth in which any of the many factors that influence the growth rate may vary through time. If there is some fixed length of time (usually long) such that during any interval of this length the population experiences environmental variability representative of the variation that occurs in all time, then definite conclusions about the population's long-term behavior apply. Specifically, conditions that produce population persistence can be distinguished from conditions that cause extinction, and the difference between any pair of solutions eventually converges to zero. These attributes resemble corresponding features of the related autonomous population growth model x = xf(x).

Key words

Environmental variation Nonautonomous Population growth 

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References

  1. Andrewartha, H. G., Birch, L. C.: The distribution and abundance of animals. Chicago: University of Chicago Press 1954Google Scholar
  2. Arrigoni, M., Steiner, A.: Logistisches Wachstum in fluktuierender Umwelt [Logistic growth in a fluctuating environment]. J. Math. Biol. 21, 237–241 (1985)Google Scholar
  3. Åström, K. J.: Introduction to stochastic control theory. New York: Academic Press 1970Google Scholar
  4. Boyce, M. S., Daley, D. J.: Population tracking of fluctuating environments and natural selection for tracking ability. Am. Nat. 115, 480–491 (1980)Google Scholar
  5. Brauer, F.: Periodic solutions of some ecological models. J. Theor. Biol. 69, 143–152 (1977)Google Scholar
  6. Coddington, E. A., Levinson, N.: Theory of ordinary differential equations. Malabar, Florida: Krieger 1984Google Scholar
  7. Coleman, B. D.: Nonautonomous logistic equations as models of the adjustment of populations to environmental change. Math. Biosci. 45, 159–173 (1979)Google Scholar
  8. Cushing, J. M.: Oscillatory population growth in periodic environments. Theor. Popul. Biol. 30, 289–308 (1986)Google Scholar
  9. de Mottoni, P., Schiaffino, L.: Bifurcation of periodic solutions for some systems with periodic coefficients. In: de Mottoni, P., Schiaffino, L. (eds.) Nonlinear differential equations: invariance, stability, and bifurcation, pp. 327–338. New York: Academic Press 1981aGoogle Scholar
  10. de Mottoni, P., Schiaffino, A.: Competition systems with periodic coefficients: a geometric approach. J. Math. Biol. 11, 319–335 (1981b)Google Scholar
  11. Hallam, T. G., Clark, C. E.: Non-autonomous logistic equations as models of populations in a deteriorating environment. J. Theor. Biol. 93, 303–311 (1981)Google Scholar
  12. Hallam, T. G., Ma, Z.: Persistence in population models with demographic fluctuations. J. Math. Biol. 24, 327–339 (1986)Google Scholar
  13. Hallam, T. G., Ma, Z.: On density and extinction in continuous population models. J. Math. Biol. 25, 191–201 (1987)Google Scholar
  14. Hanson, F. B., Tuckwell, H. C.: Persistence times of populations with large random fluctuations. Theor. Popul. Biol. 14, 46–61 (1978)Google Scholar
  15. Hanson, F. B., Tuckwell, H. C.: Logistic growth with random density independent disasters. Theor. Popul. Biol. 19, 1–18 (1981)Google Scholar
  16. Leigh, E. G., Jr.: The average lifetime of a population in a varying environment. J. Theor. Biol. 90, 213–239 (1981)Google Scholar
  17. May, R. M.: Stability and complexity in model ecosystems. Princeton, NJ: Princeton University Press 1973Google Scholar
  18. May, R. M., Oster, G. F.: Bifurcations and dynamic complexity in simple ecological models. Am. Nat. 110, 573–599 (1976)Google Scholar
  19. Pickett, S. T. A., White, P. S. (eds.): The ecology of natural disturbance and patch dynamics. New York: Academic Press 1985Google Scholar
  20. Rosenblatt, S.: Population models in a periodically fluctuating environment. J. Math. Biol. 9, 23–36 (1980)Google Scholar
  21. Strebel, D. E.: Logistic growth in the presence of non-white environmental noise. J. Theor. Biol. 85, 713–738 (1980)Google Scholar
  22. Strong, D. R., Jr., Simberloff, D., Abele, L.G., Thistle, A. B. (eds.): Ecological communities: conceptual issues and the evidence. Princeton, NJ: Princeton University Press 1984Google Scholar
  23. Tuckwell, H. C., Koziol, J. A.: Logistic population growth under random dispersal. Bull. Math. Biol. 49, 495–506 (1987)Google Scholar
  24. Turelli, M.: Random environments and stochastic calculus. Theor. Popul. Biol. 12, 140–178 (1977)Google Scholar
  25. Turelli, M.: Niche overlap and invasion of competitors in random environments I. Models without demographic stochasticity. Theor. Popul. Biol. 20, 1–56 (1981)Google Scholar
  26. Turelli, M., Gillespie, J. H.: Conditions for the existence of stationary densities for some two-dimensional diffusion processes with applications in population biology. Theor. Popul. Biol. 17, 167–189 (1980)Google Scholar
  27. Turelli, M., Petry, D.: Density-dependent selection in a random environment: an evolutionary process that can maintain stable population dynamics. Proc. Natl. Acad. Sci. USA 77, 7501–7505 (1980)Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • R. R. Vance
    • 1
  • E. A. Coddington
    • 2
  1. 1.Department of BiologyUniversity of CaliforniaLos AngelesUSA
  2. 2.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

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