Abstract
The question of the long term survival of species in models governed by Lotka-Volterra difference equations is considered. The criterion used is the biologically realistic one of permanence, that is populations with all initial values positive must eventually all become greater than some fixed positive number. We show that in spite of the complex dynamics associated even with the simplest of such systems, it is possible to obtain readily applicable criteria for permanence in a wide range of cases.
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Amann, E., Hofbauer, J.: Permanence in Lotka-Volterra and replicator equations. In: Ebeling, W., Peschel, M. (eds.) Lotka-Volterra-approach to cooperation and competition in dynamic systems. Berlin: Akademieverlag 1985
Butler, G., Freedman, H. I., Waltman, P.: Uniformly persistent systems. Proc. Am. Math. Soc. 96, 425–430 (1986)
Hassell, M. P.: Anthropod predator-prey systems. Princeton: Princeton University Press 1978
Hofbauer, J.: A general cooperation theorem for hypercycles. Monatsh. Math. 91, 233–240 (1981)
Hofbauer, J.: A difference equation model for the hypercycle. Siam J. Appl. Math. 44, 762–772 (1984)
Hofbauer, J.: B-matrices, Lotka-Volterra equations and the linear complementarity problem. To appear
Hofbauer, J., Sigmund, K.: Dynamical systems and the theory of evolution. Cambridge: Cambridge University Press (1987)
Hofbauer, J., Sigmund, K.: Permanence for replicator equations. In: Kurzhanski, A. B., Sigmund, K. (eds.) Dynamical Systems. Proceedings, Sopron 1985 (Lect. Notes Econ. Math. Syst., vol. 287, pp. 70–91) Berlin Heidelberg New York: Springer 1987
Hutson, V., Vickers, G. T.: A criterion for permanent coexistence of species, with an application to a two-prey one-predator system. Math. Biosci. 63, 253–269 (1983)
Hutson, V.: Predator mediated coexistence with a switching predator. Math. Biosci. 68, 233–246 (1984)
Hutson, V.: A theorem on average Liapunov functions. Monatsh. Math. 98, 267–275 (1984)
Hutson, V., Law, R.: Permanent coexistence in general models of three interacting species. J. Math. Biol. 21, 285–298 (1985)
Hutson, V., Moran, W.: Persistence in systems with diffusion. In: Aubin, J.-P., Sarri, D., Sigmund, K. (eds.) Dynamics of macrosystems (Lect. Notes Econ. Math. Syst., vol. 257, pp. 43–48) Berlin Heidelberg New York: Springer 1985
Hutson, V., Moran, W.: Persistence of species obeying difference equations. J. Math. Biol. 15, 203–213 (1982)
Hutson, V., Pym, J.: Repellers for generalized semidynamical systems. In: Kurzhanski, A. B., Sigmund, K. (eds.) Dynamical systems. Proceedings, Sopron 1985 (Lect. Notes Econ. Math. Syst., vol. 287, pp. 39–49) Berlin Heidelberg New York: Springer 1987
Jansen, W.: A permanence theorem for replicator and Lotka-Volterra systems. J. Math. Biol. 25, 411–422 (1987)
Kirlinger, G.: Permanence in Lotka-Volterra-equations: Linked predator prey systems. Math. Biosci. 82, 165–191 (1986)
May, R. M., Leonard, W. J.: Nonlinear aspects of competition between three species. SIAM J. Appl. Math. 29, 243–253 (1975)
May, R. M., Oster, G. F.: Bifurcations and dynamic complexity in simple ecological models. Am. Nat. 110, 573–599 (1976)
Schuster, P., Sigmund, K., Wolff, R.: Dynamical systems under constant organization III. Cooperative and competitive behaviour of hypercycles. J. Diff. Equations 32, 357–368 (1979)
Schuster, P., Sigmund, K., Wolff, R.: On ω-limits for competition between three species. SIAM J. Appl. Math. 37, 49–54 (1979)
Schuster, P., Sigmund, K., Wolff, R.: Mass action kinetics of self replication in flow reactors. J. Math. Anal. Appl. 78, 88–112 (1980)
Scudo, F. M., Ziegler, J. R.: The golden age of theoretical ecology: 1923–1940. Lect. Notes Biomath., vol. 22. Berlin Heidelberg New York: Springer 1978
Sigmund, K., Schuster, P.: Permanence and uninvadability for deterministic population models. In: Schuster, P. (ed.) Stochastic phenomena and chaotic behaviour in complex systems. Springer Series in Synergetics, vol. 21. (1984)
Strobeck, C.: N-species competition. Ecology, 54, 650–654 (1973)
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Hofbauer, J., Hutson, V. & Jansen, W. Coexistence for systems governed by difference equations of Lotka-Volterra type. J. Math. Biology 25, 553–570 (1987). https://doi.org/10.1007/BF00276199
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DOI: https://doi.org/10.1007/BF00276199