Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-23T22:45:16.564Z Has data issue: false hasContentIssue false
  • English
  • Français

Comparison principles for impulsive parabolic equations with applications to models of single species growth

Published online by Cambridge University Press:  17 February 2009

L. H. Erbe ,
H. I. Freedman ,
X. Z. Liu  and
J. H. Wu
L. H. Erbe
Affiliation:
Applied Mathematics Institute, University of Alberta, Edmonton, CanadaT6G 2G1.
H. I. Freedman
Affiliation:
Applied Mathematics Institute, University of Alberta, Edmonton, CanadaT6G 2G1.
X. Z. Liu
Affiliation:
Applied Mathematics Institute, University of Alberta, Edmonton, CanadaT6G 2G1. Present address: Department of Applied Mathematics, University of Waterloo, Waterloo, Canada.
J. H. Wu
Affiliation:
Applied Mathematics Institute, University of Alberta, Edmonton, CanadaT6G 2G1. Present address: Department of Mathematics, York University, North York, Canada.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper establishes some maximum and comparison principles relative to lower and upper solutions of nonlinear parabolic partial differential equations with impulsive effects. These principles are applied to obtain some sufficient conditions for the global asymptotic stability of a unique positive equilibrium in a reaction-diffusion equation modeling the growth of a single-species population subject to abrupt changes of certain important system parameters.

Type
Research Article
Information
The ANZIAM Journal , Volume 32 , Issue 4 , April 1991 , pp. 382 - 400
Copyright
Copyright © Australian Mathematical Society 1991

References

[1] Allen, L. J., “Persistence, extinction and critical patch number for isolated populations,” J. Math. Biol. 24 (1987) 617–625.CrossRefGoogle Scholar
[2] Aronson, D. G., “Density dependent interaction-diffusion systems,” in Proceedings of the advanced seminar on dynamics and modeling of reactive systems, Academic Press (1980).Google Scholar
[3] Bainov, D. D., Lakshmikantham, V. and Simeonov, P. S., Theory of impulsive differential equations, Academic Press (1989).Google Scholar
[4] Brauer, F. and Sanchez, D. A., “Constant rate population harvesting: equilibrium and stability,” Theor. Pop. Biol. 8 (1975) 12–30.Google Scholar
[5] Brauer, F. and Sanchez, D. A., “Some models for population growth with harvesting,” in International conference on differential equations (Antosiewicz, H., ed.), Academic Press (1975) 53–64.Google Scholar
[6] Brauer, F. and Soudack, A. C., “Constant-rate stocking of predator-prey systems,” J. Math. Biol. 11 (1981) 1–14.Google Scholar
[7] Britton, N. F., “Aggregation and the completitive exclusion principle,” J. Theor. Biol. 136 (1989) 57–66.Google Scholar
[8] Cantrell, R. S. and Cosner, G. C., “Diffusive logistic equations with indefinite weights: population models in disrupted environments,” Proc. Roy. Soc. Edin. (in press).Google Scholar
[9] Freedman, H. I., “Graphical stability, enrichment and pest control by a natural enemy,” Math. Biosci 31 (1976) 207–225.Google Scholar
[10] Freedman, H. I., Deterministic mathematical models in population ecology, HIFR Consulting Ltd., Edmonton (1987).Google Scholar
[11] Freedman, H. I., Shukla, J. B. and Takeuchi, Y., “Population diffusion in a two-patch environment,” Math. Biosci. 95 (1989) 111–123.Google Scholar
[12] Freedman, H. I. and Wu, J., “Steady state analysis in a model for population diffusion in a multi-patch environment,” preprint.Google Scholar
[13] Furter, J. and Grinfeld, M., “Local vs. non-local interactions in population dynamics,” J. Math. Biol. 27 (1989) 65–80.CrossRefGoogle Scholar
[14] Hallam, T. G., “A temporal study of diffusion effects on a population modelled by quadratic growth,” Nonlin. Anal. Th. Meth. App. 3 (1979) 123–133.Google Scholar
[15] Ladde, G. S., Lakshmikantham, V. and Vatsala, A. S., Monotone iterative techniques for nonlinear differential equations, (Pitman, Boston, 1985).Google Scholar
[16] Ladyzenskaya, O. A., Solonikov, V. A. and Uralawa, N. N., Linear and quasilinear equations of parabolic type, Amer. Math. Soc. Transl. Monogr. 23, Providence, 1968.Google Scholar
[17] Lakshmikantham, V., “Comparison results for reaction diffusion equations in Banach space,” in A survey of theoretical and numerical trends in nonlinear analysis, Gius. Lat-erza, Bari, Italy (1979) 121–156.Google Scholar
[18] Lakshmikantham, V. and Liu, X., “Stability criteria for impulsive differential equations in terms of two measures,” J Math. Anal. Appl. 137 (1989) 591–604.Google Scholar
[19] Liu, X. and Pirapakaran, R., “Asymptotic stability for impulsive differential systems,” J. Appl. Math. Comp. (in press).Google Scholar
[20] Millan, V. D. and Myshkis, A. D., “On the stability of motion in the presence of impulses,” Siber. Math. Zh. 1 (1960) 233–237 (Russian).Google Scholar
[21] Pandit, S. G. and Deo, S. G., Differential systems involving impulses, (Springer-Verlag, New York, 1982).Google Scholar
[22] Perestyuk, N. A. and Samoilenko, A. M., Differential equations with impulses, (Central Publishers, Kiev, USSR, 1987) (Russian).Google Scholar
[23] Protter, M. H. and Weinberger, H. F., Maximum principles in differential equations, (Prentice-Hall, New Jersey, 1967).Google Scholar
[24] Smith, O. L., “The influence of environmental gradients on ecosystem stability,” Amer. Nat. 116 (1980) 1–24.CrossRefGoogle Scholar
[25] Sperb, R., Maximum principles and their applications, (Academic Press, New York, 1981).Google Scholar
[26] Zabreiko, P. P., Bainov, D. D. and Kostadinov, S. I., “Stability of linear equations with impulsive effect,” Tamkang J. of Math. 18 (1987) 57–63.Google Scholar