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Global stability and persistence in diffusive food chains

Published online by Cambridge University Press:  17 February 2009

Yang Kuang
Yang Kuang
Affiliation:
Department of Mathematics, Arizona State University, Tempe, AZ 85287, USA. e-mail: kuang@asu.edu
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Abstract

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In this paper, the results of Freedman and So [13] on global stability and persistence of simple food chains are extended to general diffusive food chains. For global stability of the unique homogeneous positive steady state, our approach involves an application of the invariance principle of reaction-diffusion equations and the construction of a Liapunov functional. For persistence, we use the dynamical system results of Dunbar et al. [11] and Hutson and Moran [29].

Type
Research Article
Information
The ANZIAM Journal , Volume 43 , Issue 2 , October 2001 , pp. 247 - 268
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Alikakos, N., “An application of the invariance principle to reaction-diffusion equations”, J. Diff. Equations 33 (1979) 301–325.CrossRefGoogle Scholar
[2]Beddington, J. R. and Hammond, P. S., “On the dynamics of host-parasite-hyperparasite interactions”, J. Anim. Ecol. 46 (1977) 811–821.Google Scholar
[3]Burton, T. and Hutson, V., “Repellers in systems with infinite delay”, J. Math. Anal. Appl. 137 (1989) 240–263.CrossRefGoogle Scholar
[4]Butler, G., Freedman, H. I. and Waltman, P., “Uniformly persistent systems”, Proc. Amer Math. Soc. 96 (1986) 425–430.Google Scholar
[5]Butler, G. and Waltman, P., “Persistence in dynamical systems”, J. Differential Equations 63 (1986) 255–263.Google Scholar
[6]Case, T. J. and Casten, R. G., “Global stability and multiple domains of attraction in ecological systems”, Amer. Nat. 113 (1979) 705–714.CrossRefGoogle Scholar
[7]Cheng, K.-S., Hsu, S.-B. and Lin, S.-S., “Some results on global stability of a predator-prey system”, J. Math. Biol. 12 (1981) 115–126.CrossRefGoogle Scholar
[8]Conrad, M., “Stability of foodwebs and its relation to species diversity”, J. Theoret. Biol. 34 (1972) 325–335.CrossRefGoogle ScholarPubMed
[9]DeAngelis, D. L., “Stability and connectance in food web models”, Ecology 56 (1975) 238–243.CrossRefGoogle Scholar
[10]DeAngelis, D. L. and Goldstein, R. A., “Criteria that forbid a large, nonlinear food-web model from having more than one equilibrium point”, Math. Biosci. 41 (1978) 81–90.CrossRefGoogle Scholar
[11]Dunbar, S. R., Rybakowski, K.P. and Schmitt, K.,“Persistence in models of predator-prey populations with diffusion”, J. Differential Equations 65 (1986) 117–138.CrossRefGoogle Scholar
[12]Freedman, H. I., Deterministic mathematical models in population ecology (Marcel Dekker, New York, 1980).Google Scholar
[13]Freedman, H. I. and So, J. W.-H., “Global stability and persistence of simple food chains”, Math. Biosci. 76 (1985) 69–86.CrossRefGoogle Scholar
[14]Freedman, H. I. and Waltman, P., “Persistence in models of three interacting predator-prey populations”, Math. Biosci. 68 (1984) 213–231.Google Scholar
[15]Freedman, H. I. and Waltman, P., “Persistence in a model of three competitive populations”, Math. Biosci. 73 (1985) 89–101.Google Scholar
[16]Gard, T. C., “Persistence in food webs: Holling-type food chains”, Math. Biosci. 49 (1980) 61–67.CrossRefGoogle Scholar
[17]Gard, T. C., “Persistence in food webs with general interactions”, Math. Biosci. 51 (1980) 165–174.CrossRefGoogle Scholar
[18]Gard, T. C., “Persistence for ecosystem microcosm models”, Ecol. Model. 12 (1981) 221–229.Google Scholar
[19]Gard, T. C., “Top predator persistence in differential equation models of food chains: The effect of omnivory and external forcing of lower trophic levels”, J. Math. Biol. 14 (1982) 285–299.Google Scholar
[20]Gard, T. C., “Uniform persistence in multispecies population models”, Math. Biosci. 85 (1987) 93–104.CrossRefGoogle Scholar
[21]Gard, T. C. and Hallam, T. G., “Persistence in food webs–1. Lotka-Volterra food chains”, Bull. Math. Biol. 41 (1979) 877–891.Google Scholar
[22]Goh, B.-S., “Global stability in a class of predator-prey models”, Bull. Math. Biol. 40 (1978) 525–533.Google Scholar
[23]Hale, J. K., “Large diffusivity and asymptotic behavior in parabolic systems”, J. Math. Anal. Appl. 118 (1986) 455–466.CrossRefGoogle Scholar
[24]Hale, J. K. and Waltman, P., “Persistence in infinite-dimensional systems”, SIAM J. Math. Anal. 20 (1989) 388–395.Google Scholar
[25]Harrison, G. W., “Global stability of food chains”, Amer. Nat. 114 (1979) 455–457.CrossRefGoogle Scholar
[26]Hastings, A., “Global stability in Lotka-Volterra systems with diffusion”, J. Math. Biol. 6 (1978) 163–168.CrossRefGoogle Scholar
[27]Henry, D., Geometric theory of semilinear parabolic equations, Lecture Notes 840 (1981).Google Scholar
[28]Hofbauer, J. and Sigmund, K., The theory of evolution and dynamical systems (Cambridge University Press, 1988).Google Scholar
[29]Hutson, V. and Moran, W., “Repellers in reaction-diffusion systems”, Rocky Mountain J. Math. 17 (1987) 301–314.CrossRefGoogle Scholar
[30]Hutson, V. and Schmitt, K., “Permanence in dynamical systems”, preprint.Google Scholar
[31]Hutson, V. and Vickers, T., “A criterion for permanent coexistence of species, with an application to a two-prey one-predator system”, Math. Biosci. 63 (1983) 253–269.Google Scholar
[32]Kaung, Y., “Global stability of Gause-type predator-prey systems”, J. Math. Biol. 28 (1990) 463–474.Google Scholar
[33]Kuang, Y. and Freedman, H. I., “Uniqueness of limit cycles in Gause-type models of predator-prey systems”, Math. Biosci. 88 (1988) 67–84.CrossRefGoogle Scholar
[34]Kuang, Y., Martin, R. H. and Smith, H. L., “Global stability for infinite delay, dispersive Lotka-Volterra systems; weakly interacting populations in nearly identical patches”, J. Dynamics and Differential Equations 3 (1991) 339–360.CrossRefGoogle Scholar
[35]Kuang, Y. and Smith, H. L., “Global stability in diffusive delay Lotka-Volterra systems”, Differential and Integral Equations 4 (1991) 117–128.Google Scholar
[36]Kuang, Y. and Smith, H. L., “Global stability for infinite delay Lotka-Volterra type systems”, J. Diff. Eqns. 103 (1992) 221–246.CrossRefGoogle Scholar
[37]Martin, R. H. and Smith, H. L., “Convergence in Lotka-Volterra systems with diffusion and delay”, in Diff. Eqns with Appl. in Biology, Physics and Engineering (ed. Goldstein, J. A. et al. ), (Marcel Dekker, 1989).Google Scholar
[38]Mimura, M. and Murray, J. D., “On a diffusive prey-predator model which exhibits patchiness”, J. Theor. Biol. 75 (1978) 249–262.Google Scholar
[39]Murray, J. D., Mathematical biology, Biomathematics Text 19 (Springer, 1989).CrossRefGoogle Scholar
[40]Redheffer, R., Redlinger, R. and Walter, W., “A theorem of LaSalle-Lyapunov type for parabolic systems”, SIAM J. Math. Anal. 19 (1982) 99–103.Google Scholar
[41]Redlinger, R., “Über die C 2–Kompaktheit der Bahn von Lösungen semilinearer parabolischer Systeme”, Proc. Roy. Soc. Edinburgh, Sect. A 93 (1982) 99–103.CrossRefGoogle Scholar
[42]Redlinger, R., “Compactness results for time-dependent parabolic systems”, J. Differential Equations 64 (1986) 133–153.Google Scholar
[43]Rescigno, A. and Jones, K.G., “The struggle for life: III. A predator-prey food chain”, Bull. Math. Biophys. 34 (1972) 521–532.CrossRefGoogle Scholar
[44]Rogers, D. J. and Hassell, M. P., “General models for insect parasite and predator searching behaviour: Interference”, J. Anim. Ecol. 43 (1974) 239–253.CrossRefGoogle Scholar
[45]Saunders, P. T. and Bazin, M. J., “On the stability of food chains”, J. Theoret. Biol. 52 (1975) 121–142.CrossRefGoogle ScholarPubMed
[46]Smoller, J., Shock waves and reaction diffusion equations (Springer, New York, 1982).Google Scholar
[47]Takeuchi, Y. and Adachi, N., “The existence of globally stable equilibria of ecosystems of the generalized Volterra type”, J. Math. Biol. 10 (1980) 401–415.Google Scholar
[48]Takeuchi, Y., Adachi, N. and Tokumaru, H., “The stability of generalized Volterra equations”, J. Math. Anal. Appl. 62 (1978) 453–473.Google Scholar
[49]Volterra, V., “Variazionie fluttuazioni del numero d'individui in specie animali conviventi”, Mem. Acad. Lincei. 2 (1926) 31–113.Google Scholar