Abstract
The question of persistence of interacting species is one of the most important in theoretical ecology; when the system is governed by difference equations this question is particularly difficult to resolve because of the complicated dynamics of the model. The problem has usually been tackled via the concepts of asymptotic stability and global asymptotic stability, however the first is not a strong enough restriction since only orbits starting near a rest point are guaranteed to converge to the rest point, while the second as well as usually being extremely difficult to establish is surely too strong a condition, since it rules out for example a stable limit cycle. It is proposed here that a more biologically realistic criterion, and incidentally one which turns out to be more tractable, is that of cooperativeness, where all orbits are required eventually to enter and remain in a region at a non-zero distance from the boundary (corresponding to a zero value of at least one population). A theorem is proved giving conditions for cooperativeness, and is applied to some examples of predator-prey interactions, simple conditions on the parameters being obtained even for some ranges of the parameters where the dynamics are “chaotic”.
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References
Albrecht F., Gatzke, H., Haddad, A., Wax, N.: The dynamics of two interacting populations. J. Math. Anal. Appl. 46, 658–670 (1974)
Brauer, F.: Boundedness of solutions of predator-prey systems. Theoret. Population Biology 15, 268–273 (1979)
Goh, B.: Management and analysis of biological populations. Amsterdam: Elsevier 1980
Gumowski, I., Mira, C.: Recurrences and discrete dynamical systems. Lecture notes in Math., Vol. 809. Berlin: Springer 1980
Hale, J.: Theory of functional differential equations. New York: Springer 1977
Hasseil, M. P.: Anthropod predator-prey systems. Princeton, New Jersey: Princeton University Press 1978
Hofbauer, J.: General cooperation theorem for hypercycles. Monatsh. Math. 91, 233–240 (1981)
Lasota, A.: Ergodic problems in biology. Asterisque 50, 239–250 (1977)
Levin, S. A. (ed.): Ecosystem analysis and prediction. Proceedings of a SIAM-SIMS Conference held at Alta, Utah; 1974 (SIAM 1976)
Marotto, F. R.: Snap-back repellers imply chaos in ℝn. J. Math. Anal. Appl. 63, 199–223 (1978)
May, R. M.: Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos. Science 18, 645–647 (1974)
May, R. M.: 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. American Naturalist 110, 573–599 (1976)
Roughgarden, J.: Theory of population genetics and evolutionary ecology: An introduction. New York: MacMillan 1979
Schuster, P., Sigmund, K., Wolff, R.: Dynamical systems under constant organization. III. Cooperative and competitive behaviour of hypercycles. J. Diff. Equns. 32, 357–368 (1979)
Smale, S., Williams, R. F.: The qualitative analysis of a difference equation of population growth. J. Math. Biol. 3, 1–4 (1976)
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Hutson, V., Moran, W. Persistence of species obeying difference equations. J. Math. Biol. 15, 203–213 (1982). https://doi.org/10.1007/BF00275073
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DOI: https://doi.org/10.1007/BF00275073