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Stochastic three species systems

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Abstract

This paper examines the coexistence of three species. In particular, stochastic models for (i) two predators and one prey, (ii) two prey and one predator, and (iii) a prey, a predator and a parasite to the predator are considered. It is found that coexistence is possible in each case. A large population approximation is developed which enables accurate description of the long run behaviour of stochastic models.

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References

  • Barbour, A. D.: Equilibrium distributions Markov population process. Adv. Appl. Prob. 12, 591–614 (1980).

    Google Scholar 

  • Bartlett, M. S.: Deterministic and stochastic models for recurrent epidemics. In: Proceedings, Third Berkeley Symposium on Mathematical Statistics and Probability, vol. 4, pp. 81–109. University of California Press (1956).

  • Bartlett, M. S.: On theoretical models for competitive and predatory biological systems. Biometrika 44, 27–42 (1957).

    Google Scholar 

  • Bhat, N., Pande, L. K.: Three species ecosystems in a solvable model. J. Theor. Biol. 83, 321–344 (1980).

    Google Scholar 

  • Billard, L.: Generalized two-dimensional bounded birth and death processes and some applications. J. Appl. Probab. 18, 335–347 (1981).

    Google Scholar 

  • Billard, L.: On Lotka-Volterra predator prey models. J. Appl. Probab. 14, 375–381 (1977).

    Google Scholar 

  • Chiang, C. L.: In: Kempthorne, O. et al. (eds.). Statistics and mathematics in biology, pp. 197–215. Iowa State College Press (1954).

  • Cramer, N. F., May, R. M.: Interspecific competition predation and species diversity: a comment. J. Theor. Biol. 34, 289–293 (1972).

    Google Scholar 

  • Freedman, H. I., Waltman, P.: Mathematical analysis of some three-species food-chain models. Math. Biosci. 33, 257–276 (1977).

    Google Scholar 

  • Freedman, H. I., Waltman, P.: Persistence in models of three interacting predator-prey populations. Math. Biosci. 68, 213–231 (1984).

    Google Scholar 

  • Gard, T. C., Kannan, D.: On a stochastic differential equation modelling of prey-predator evolution. J. Appl. Probab. 13, 429–443 (1976).

    Google Scholar 

  • Goel, N. S., Maitra, S. C., Montroll, E. W.: On the Volterra and other non-linear models of interacting populations. Reviews of Modern Physics Monographs. New York London: Academic Press 1971.

    Google Scholar 

  • Hallam, T. G., Svoboda, L. J., Gard T. C.: Persistence and extinction in three species Lotka-Volterra competitive systems. Math. Biosci. 46, 117–124 (1979).

    Google Scholar 

  • Hitchcock, S. E.: Extinction probabilities in predator-prey models. J. Appl. Probab. 23, 1–13 (1986).

    Google Scholar 

  • Hutson, V., Law, R.: Permanent coexistence in general models of three interacting species. J. Theor. Biol. 44, 285–298 (1985).

    Google Scholar 

  • Hutson, V., Vickers, G. T.: A criterion for permanent co-existence of species, with an application to a two-prey one-predator system. Math. Biosci. 63, 253–269 (1983).

    Google Scholar 

  • Koch, A. L.: Competitive coexistence of two predators utilizing the same prey under constant environmental conditions. J. Theor. Biol. 44, 387–395 (1974).

    Google Scholar 

  • Leslie, P. H.: A stochastic model for studying the properties of certain biological systems by numerical methods. Biometrika 45, 16–31 (1958).

    Google Scholar 

  • Leslie, P. H., Gower, J. C.: The properties of a stochastic model for the predator-prey type of interaction between two species. Biometrika 47, 219–234 (1960).

    Google Scholar 

  • Lin J., Kahn, P. B.: Qualitative dynamics of three species predator prey systems. J. Math. Biol. 5, 257–268 (1978).

    Google Scholar 

  • Loman, J.: Influence of territoriality on the stability and co-existence of competing predators—a simulation study. Ecol. Modelling 27, 95–108 (1985).

    Google Scholar 

  • May, R. M., Leonard, W. J.: Nonlinear aspects of competition between three species. SIAM J. Appl. Math. 29, 243–253 (1975).

    Google Scholar 

  • Nisbet, R. M., Gurney, W. S. C.: Modelling fluctuating populations. New York: Wiley 1982.

    Google Scholar 

  • Pande, L. K.: Ecosystems with three species: one-prey-and-two-predator system in an exactly solvable model. J. Theor. Biol. 74, 591–598 (1978).

    Google Scholar 

  • Parrish, J. D., Saila, S. B.: Interspecific competition predation and species diversity. J. Theor. Biol. 27, 207–220 (1970).

    Google Scholar 

  • Roozen, H.: Equilibrium and extinction in stochastic population dynamics. Bull. Math. Biol. 49, 671–696 (1987).

    Google Scholar 

  • Roy, A. B., Solimano, F.: Global stability and oscillations in classical Lotka-Volterra loops. J. Math. Biol. 24, 603–616 (1987).

    Google Scholar 

  • Shukla, V. P., Das, P. C.: Effects of dispersion on stability of multispecies prey-predator systems. Bull. Math. Biol. 44, 571–578 (1982).

    Google Scholar 

  • Smith, R. H., Mead, R.: On predicting extinction in simple population models i. stochastic linearization. J. Theor. Biol. 80, 189–203 (1979).

    Google Scholar 

  • Takeuchi, Y., Adachi, N.: Existence of bifurcation of stable equilibrium in two-prey, one-predator communities. Bull. Math. Biol. 45, 877–900 (1983).

    Google Scholar 

  • Tsokos, C. P., Hinkley, S. W.: A stochastic bivariate ecology model for competing species. Math. Biosci. 16, 191–208 (1973).

    Google Scholar 

  • Volterra, V.: Variazioni e fluttuazioni del numero d'individui in specie animali conviventi. Mem. Acad. Lincei Roma 2, 31–113 (1926).

    Google Scholar 

  • Whittle, P.: On the use of the normal approximation in the treatment of stochastic processes. J. R. Stat. Soc., Ser. B 19, 268–281 (1957).

    Google Scholar 

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Sridhara, R., Watson, R. Stochastic three species systems. J. Math. Biol. 28, 595–607 (1990). https://doi.org/10.1007/BF00164165

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  • DOI: https://doi.org/10.1007/BF00164165

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