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Persistence in models of three interacting predator-prey populations. (English) Zbl 0534.92026
The authors consider a class of ecosystems which can be modeled by the Kolmogorov system $(1)\quad u'=uf(u,v,w),\quad v'=vg(u,v,w),\quad w'=wh(u,v,w)\quad with$ $u(0)=u_ 0\geq 0,\quad v(0)=v_ 0\geq 0,\quad w(0)=w_ 0\geq 0,\quad('=d/dt),$ where f,g,h are continuously differentiable. The population described by u(t) will always be a prey population, w(t) will always be a predator feeding exclusively on prey within the system [v(t) or u(t) or both], v(t) will be either a predator or a prey or both.
The question addressed is: when do all of the components of the model ecosystem persist? The authors use a stronger definition of persistence than usual, namely, a population $$\rho(t)$$ is said to persist if $$\rho(0)>0$$ and $$\lim \inf_{t\to \infty}\rho(t)>0$$. A system is said to persist if each component population persists. A persistence theorem (Theorem 2.1) is formulated which essentially says that if feasible limit sets on the boundary are unstable, persistence follows. This is applied to the case of two prey and one predator, to two predators and one prey, and to food webs, to obtain persistence. The following references may be of interest: M. W. Hirsch, SIAM J. Math. Anal. 13, 167-179 (1982; Zbl 0494.34017) and The dynamical systems approach to differential equations. Bull. Am. Math. Soc., New Ser. 11, 1-64 (1984).
Reviewer: B.L.Li

##### MSC:
 92D40 Ecology 92D25 Population dynamics (general) 37-XX Dynamical systems and ergodic theory
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##### References:
 [1] Akre, B.G.; Johnson, D.M., Switching and Sigmoid functional response curves by damselfly naiads with alternative prey available, J. animal ecol., 48, 703-720, (1979) [2] Albrecht, F.; Gatzke, H.; Haddad, A.; Wax, N., The dynamics of two interacting populations, J. math. anal. appl., 46, 658-670, (1974) · Zbl 0281.92012 [3] Albrecht, F.; Gatzke, H.; Wax, N., Stable limit cycles in prey-predator populations, Science, 181, 1073-1074, (1973) [4] Armstrong, R.A., Coexistence of species competing for shared resources, Theoret. population biol., 9, 317-328, (1980) · Zbl 0349.92030 [5] Armstrong, R.A.; McGehee, R., Coexistence of two competitors on one resource, J. theoret. biol., 56, 499-502, (1976) [6] Armstrong, R.A.; McGehee, R., Competitive exclusion, Amer. natur., 115, 151-170, (1980) [7] Birkland, C., Interactions between a sea pen and seven of its predators, Ecolog. monogr., 44, 211-232, (1974) [8] Butler, G.J.; Burton, T.A., Coexistence in predator-prey systems, Modeling and differential equations in biology, 199-207, (1980) [9] Case, T.J.; Casten, R.G., Global stability and multiple domains of attraction in ecological systems, Amer. natur., 113, 705-714, (1979) [10] Cheng, K.-s.; Hsu, S.-B.; Lin, S.-S., Some results on global stability of a predator-prey system, J. math. biol., 12, 115-126, (1981) · Zbl 0464.92021 [11] Conley, C.; Easton, R., Isolated invariant sets and isolating blocks, Trans. amer. math. soc., 158, 35-61, (1971) · Zbl 0223.58011 [12] Cramer, N.F.; May, R.M., Interspecific competition, predation and species diversity: A comment, J. theoret. biol., 34, 289-293, (1972) [13] DeAngelis, D.L.; Goldstein, R.A., Criteria that forbid a large, nonlinear food-web model from having more than one equilibrium point, Math. biosci., 41, 81-90, (1978) · Zbl 0437.92029 [14] Fredrickson, A.G.; Jost, J.L.; Tsuchiya, H.M.; Hsu, P.-H., Predator-prey interactions between Malthusian populations, J. theoret. biol., 38, 487-526, (1973) [15] Freedman, H.I., Deterministic mathematical models in population ecology, (1980), Marcel Dekker New York · Zbl 0448.92023 [16] Freedman, H.I.; Waltman, P., Mathematical analysis of some three-species food-chain models, Math. biosci., 33, 257-276, (1977) · Zbl 0363.92022 [17] Gard, T.C., Persistence in food webs: Holling-type food chains, Math. biosci., 49, 61-67, (1980) · Zbl 0438.92019 [18] Gard, T.C., Persistence in food chains with general interactions, Math. biosci., 51, 165-174, (1980) · Zbl 0453.92017 [19] Gard, T.C., Persistence for ecosystem microcosm models, Ecolog. model., 12, 221-229, (1981) [20] Gard, T.C., Top predator persistence in differential equation models of food chains: the effects of omnivory and external forcing of lower trophic levels, J. math. biol., 14, 285-299, (1982) · Zbl 0494.92022 [21] T. C. Gard, Persistence in food webs, to appear. · Zbl 0534.92025 [22] Gard, T.C.; Hallam, T.G., Persistence in food webs—I. Lotka-Volterra food chains, Bull. math. biol., 41, 877-891, (1979) · Zbl 0422.92017 [23] Gilpin, M.E., Spiral chaos in a predator-prey model, Amer. nat., 113, 306-308, (1979) [24] Goh, B.S., Global stability in a class of prey-predator models, Bull. math. biol., 40, 525-533, (1978) · Zbl 0378.92009 [25] Goh, B.S., Management and analysis of biological populations, (1980), Elsevier Scientific New York · Zbl 0453.92015 [26] Harrison, G.W., Global stability of food chains, Amer. natur., 114, 455-457, (1979) [27] Hastings, A., Spatial heterogeneity and the stability of predator-prey systems: predator-mediated coexistence, Theoret. population biol., 14, 380-395, (1978) · Zbl 0392.92012 [28] Hausrath, A.R., Stability properties of a class of differential equations modeling predator-prey relationships, Math. biosci., 26, 267-281, (1975) · Zbl 0326.92001 [29] Holt, R.D., Predation, apparent competition, and the structure of prey communities, Theoret. population biol., 12, 197-229, (1977) [30] Hsu, S.B., On global stability of a predator-prey system, Math. biosci., 39, 1-10, (1978) · Zbl 0383.92014 [31] Hsu, S.B., Predator-mediated coexistence and extinction, Math. biosci., 54, 231-248, (1981) · Zbl 0456.92020 [32] Hsu, S.B.; Hubbell, S.P.; Waltman, P., A contribution to the mathematical theory of competing predators, Ecolog. monog., 48, 337-349, (1978) [33] Hsu, S.B.; Hubbell, S.P.; Waltman, P., Competing predators, SIAM J. appl. math., 35, 617-625, (1978) · Zbl 0394.92025 [34] Inouye, R.S., Stabilization of a predator-prey equilibrium by addition of a second “keystone” victim, Amer. natur., 115, 300-305, (1980) [35] Koch, A.L., Competitive coexistence of two predators utilizing the same prey under constant environment conditions, J. theoret. biol., 44, 387-395, (1974) [36] Kolmogorov, A., Sulla teoria di Volterra Della lotta per l’esistenza, Gior. ist. ital. attuari., 7, 74-80, (1936) · JFM 62.1263.01 [37] Levin, B.R.; Stewart, F.M.; Chao, L., Resource-limited growth, competition, and predation: A model and experimental studies with bacteria and bacteriophage, Amer. natur., 111, 3-24, (1977) [38] Lin, J.; Kahn, P.B., Qualitative dynamics of three species predator-prey systems, J. math. biol., 5, 257-268, (1978) · Zbl 0379.92012 [39] Logofet, D.O., Investigation of a system of n “predator-prey” pairs coupled by competition, Soviet math. dokl., 16, 1246-1249, (1975) · Zbl 0345.92008 [40] Maly, E.J., Interactions among the predatory rotifer asplanchna and two prey, paramecium and euglena, Ecology, 56, 346-358, (1975) [41] McGehee, R.; Armstrong, R.A., Some mathematical problems concerning the ecological principle of competitive exclusion, J. differential equations, 23, 30-52, (1977) · Zbl 0353.92007 [42] Morin, P.J., Predatory salamanders reverse the outcome of competition among three species of anuran tadpoles, Science, 212, 1284-1286, (1981) [43] Murdoch, W.W., Switching in general predators; experiments on predator specificity and stability of prey populations, Ecolog. monog., 39, 335-354, (1969) [44] Murdoch, W.W., Stabilizing effects of spatial heterogeneity in predator-prey systems., Theoret. population biol., 11, 252-273, (1977) · Zbl 0356.92017 [45] Murdoch, W.W.; Avery, S.; Smyth, M.E.B., Switching in predatory fish, Ecology, 56, 1094-1105, (1975) [46] Nemytskii, V.V.; Stepanov, V.V., Qualitative theory of differential equations, (1960), Princeton U.P Princeton, N.J · Zbl 0089.29502 [47] Oaten, A.; Murdoch, W.W., Switching, functional response, and stability in predator-prey systems, Amer. natur., 109, 299-318, (1975) [48] Pande, L.K., Ecosystems with three species: one-prey-and-two-predator systems in an exactly solvable model, J. theoret. biol., 74, 591-598, (1978) [49] Parrish, J.D.; Saila, S.B., Interspecific competition, predation and species diversity, J. theoret. biol., 27, 207-220, (1970) [50] Rescigno, A., The struggle for life—IV. two predators sharing a prey, Bull. math. biol., 39, 179-185, (1977) · Zbl 0363.92023 [51] Rescigno, A.; Jones, K.G., The struggle for life: III. A predator-prey chain, Bull. math. biophys., 34, 521-532, (1972) [52] Rosenzweig, M.L., Exploitation in three trophic levels, Amer. nat., 107, 275-294, (1973) [53] Saunders, P.T.; Bazin, M.J., On the stability of food chains, J. theoret. biol., 52, 121-142, (1975) [54] So, J.W.H., A note on the global stability and bifurcation phenomenon of a Lotka-Volterra food chain, J. theoret. biol., 80, 185-187, (1979) [55] Steele, J., Application of theoretical models in ecology, J. theoret. biol., 63, 443-451, (1976) [56] Takeuchi, Y.; Adachi, N., The existence of globally stable equilibria of ecosystems of the generalized Volterra type, J. math. biol., 10, 401-415, (1980) · Zbl 0458.92019 [57] Takeuchi, Y.; Adachi, N.; Tokumaru, H., Global stability of ecosystems of the generalized Volterra type, Math. biosci., 42, 119-136, (1978) · Zbl 0394.92024 [58] Teramoto, E.; Kawasaki, K.; Shigesada, N., Switching effect of predation on competitive prey species, J. theoret. biol., 79, 303-315, (1979) [59] Vance, R.R., Predation and resource partitioning in one predator – two prey model communities, Amer. natur., 112, 797-813, (1978) [60] Wilken, D.R., Some remarks on a competing predator problem, SIAM J. appl. math., 42, 895-902, (1982) · Zbl 0515.92022 [61] Wollkind, D.J., Exploitation in three trophic levels: an extension allowing intraspecies carnivore interaction, Amer. natur., 110, 431-447, (1976) [62] Zicarelli, J.D., Mathematical analysis of a population model with several predators on a single prey, ()
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