Permanence in Lotka-Volterra equations: Linked prey-predator systems. (English) Zbl 0607.92022
V. Hutson and G. T. Vickers, Math. Biosci. 63, 253-269 (1983; Zbl 0524.92023), have shown that a bistable Lotka-Volterra system of two competing species can not be stabilized by the introduction of one predator.
In the present paper, two predators are introduced one feeding on one prey, the other one on the second prey. An intuitively meaningful inequality condition is established implying that the four-dimensional system is permanent provided that a fixed point exists in the interior of the positive orthant of the four-dimensional space.
Interesting numerical examples show that the sufficient condition for permanence may hold even when the interior equilibrium is not stable, and conversely, a system may not be permanent in spite of the fact that the interior equilibrium is (locally) asymptotically stable.
The results are extended to six-dimensional systems (three predators feeding on three preys), and to two predators two prey systems with interspecific competition between the predators. The fairly sophisticated proof of the first theorem is lucidly presented.
In the present paper, two predators are introduced one feeding on one prey, the other one on the second prey. An intuitively meaningful inequality condition is established implying that the four-dimensional system is permanent provided that a fixed point exists in the interior of the positive orthant of the four-dimensional space.
Interesting numerical examples show that the sufficient condition for permanence may hold even when the interior equilibrium is not stable, and conversely, a system may not be permanent in spite of the fact that the interior equilibrium is (locally) asymptotically stable.
The results are extended to six-dimensional systems (three predators feeding on three preys), and to two predators two prey systems with interspecific competition between the predators. The fairly sophisticated proof of the first theorem is lucidly presented.
Reviewer: M.Farkas
MSC:
| 92D40 | Ecology |
| 92D25 | Population dynamics (general) |
| 34D20 | Stability of solutions to ordinary differential equations |
Keywords:
linked prey-predator systems; bistable Lotka-Volterra system; two predators; inequality condition; four-dimensional system; fixed point; numerical examples; permanence; interior equilibrium; six-dimensional systems; three predators feeding on three preys; two predators two prey systems; interspecific competitionCitations:
Zbl 0524.92023References:
| [1] | Amann, E.; Hofbauer, J., Permanence in Lotka-Volterra Equations and Replicator Equations, Lotka-Volterra Approach in Dynamic Systems (1984), Proceedings, Conference, Wartburg, DDR · Zbl 0583.34043 |
| [2] | Arnold, V. I., Mathematical Methods of Classical Mechanics, ((1978), Springer: Springer New York), 69 · Zbl 1211.37007 |
| [3] | Birkhoff, G.; Rota, G.-C., Ordinary Differential Equations, ((1982), Ginn), 23 · Zbl 0183.35601 |
| [5] | Cramer, N. F.; May, R. M., Interspecific competition, predation and species diversity: A comment, J. Theoret. Biol., 34, 289-293 (1972) |
| [6] | Freedman, H. I., Deterministic Mathematical Models in Population Ecology, ((1980), Marcel Dekker: Marcel Dekker New York), 150 · Zbl 0448.92023 |
| [7] | Freedman, H. I.; Waltman, P., Mathematical analysis of some three-species food chain models, Math. Biosci., 33, 257-276 (1977) · Zbl 0363.92022 |
| [8] | Freedman, H. I.; Waltman, P., Persistence in models of three interacting predator-prey populations, Math. Biosci., 68, 213-231 (1984) · Zbl 0534.92026 |
| [9] | Freedman, H. I.; Waltman, P., Persistence in a model of three competitive populations, Math. Biosci., 73, 89-101 (1985) · Zbl 0584.92018 |
| [10] | Gantmacher, F. R., Matrizenrechnung II, Spezielle Fragen und Anwendungen, ((1971), VEB Deutscher Verlag der Wissenschaften: VEB Deutscher Verlag der Wissenschaften Berlin), 172 · Zbl 0085.00904 |
| [11] | Gard, T. C.; Hallam, T. G.; Svoboda, L. J., Persistence and extinction in three species Lotka-Volterra competitive systems, Math. Biosci., 46, 117-124 (1979) · Zbl 0413.92013 |
| [12] | Hofbauer, J.; Sigmund, K., Evolutionstheorie und Dynamische Systeme (1984), Paul Parey: Paul Parey Berlin · Zbl 0578.92015 |
| [14] | Hofbauer, J., A general cooperation theorem for hypercycles, Monatsh. Math., 91, 233-240 (1981) · Zbl 0449.34039 |
| [16] | Hsu, S. B.; Hubbell, S. P., Two predators competing for two prey species: An analysis of MacArthur’s model, Math. Biosci., 47, 143-171 (1979) · Zbl 0437.92022 |
| [17] | Hutson, V., A theorem on average Liapunov functions, Monatsh. Math., 98, 4 (1984) · Zbl 0542.34043 |
| [18] | 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) · Zbl 0524.92023 |
| [20] | Kirlinger, G., Permanenz in Volterra-Lotka Gleichungen: Gekoppelte Räuber-Beute Systeme (1985), Diplomarbeit: Diplomarbeit Wien |
| [21] | May, R. M., Stability in multispecies community models, Math. Biosci., 12, 59-79 (1971) · Zbl 0224.92006 |
| [22] | May, R. M., Stability and Complexity in Model Ecosystems, ((1974), Princeton U.P: Princeton U.P Princeton), 58-61 |
| [23] | May, R. M.; Leonard, W. J., Nonlinear aspects of competition between three species, SIAM J. Appl. Math., 29, 243-253 (1975) · Zbl 0314.92008 |
| [24] | Nikaido, H., Introduction to Sets and Mappings in Modern Economics (1970), North Holland: North Holland Amsterdam · Zbl 0246.90002 |
| [25] | Parrish, J. D.; Saila, S. B., Interspecific competition, predation and species diversity, J. Theoret. Biol., 27, 207-220 (1970) |
| [26] | Roughgarden, J., Theory of Population Genetics and Evolutionary Ecology (1979), MacMillan: MacMillan New York |
| [27] | Schuster, P.; Sigmund, K.; Wolff, R., Dynamical systems under constant organisation III, cooperative and competitive behaviour of hypercycles, J. Differential Equations, 32, 357-368 (1979) · Zbl 0384.34029 |
| [28] | Schuster, P.; Sigmund, K.; Wolff, R., On ω-limits for competition between three species, SIAM J. Appl. Math., 37, 49-54 (1979) · Zbl 0418.92016 |
| [29] | Sigmund, K.; Schuster, P., Permanence and uninvadability for deterministic population models, (Stochastic Phenomena and Chaotic Behaviour in Complex Systems. Stochastic Phenomena and Chaotic Behaviour in Complex Systems, Synergetics, 21 (1984), Springer: Springer New York) · Zbl 0538.92018 |
| [30] | Svirezhev, Yu. M.; Logofet, D. O., (Stability of Biological Communities (1983), Mir: Mir Moscow), 302 |
| [31] | Svirezhev, Yu. M., Modern Problems of Mathematic Ecology, Proceedings of the International Congress of Mathematics (1983) · Zbl 0516.92020 |
| [32] | Takeuchi, Y.; Adachi, N., Influence of predation on species coexistence in Volterra models, Math. Biosci., 70, 65-90 (1984) · Zbl 0548.92014 |
| [33] | Takeuchi, Y.; Adachi, N., Existence and bifurcation of stable equilibrium in two-prey, one-predator communities, Bull. Math. Biol., 45, 877-900 (1983) · Zbl 0524.92025 |
| [34] | Vance, R. R., Predation and resource partioning in one predator-two prey model communities, Amer. Natur., 112, 797-813 (1978) |
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