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Persistence and global stability in a delayed predator-prey system with Holling-type functional response

Published online by Cambridge University Press:  17 February 2009

Rui Xu
Affiliation:
Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, Hebei Province, P.R. China; e-mail: rxu88@yahoo.com.cn. Department of Mathematics, University of Dundee, Dundee DD1 4HN, Scotland, U.K.
Lansun Chen
Affiliation:
Institute of Mathematics, Academia Sinica, Beijing 100080, P.R. China.
M. A. J. Chaplain
Affiliation:
Department of Mathematics, University of Dundee, Dundee DD1 4HN, Scotland, U.K.
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Abstract

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A delayed predator-prey system with Holling type III functional response is investigated. It is proved that the system is uniformly persistent under some appropriate conditions. By means of suitable Lyapunov functionals, sufficient conditions are derived for the local and global asymptotic stability of a positive equilibrium of the system. Numerical simulations are presented to illustrate the feasibility of our main results.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Bailey, K. M., “Predation on juvenile flatfish and recruitment variability”, Neth. J. Sea Res. 32 (1994) 175189.Google Scholar
[2]Bazykin, A. D., Structural and dynamic stability of model predator-prey systems (Int. Inst. Appl. Syst. Analysis, Laxenburg, 1976).Google Scholar
[3]Bazykin, A. D., Berezovskaya, F. S., Denisov, G. A. and Kuznetzov, Yu A., “The influence of predator saturation effect and competition among predators on predator-prey system dynamics”, Ecol. Model. 14 (1981) 3957.Google Scholar
[4]Beretta, E. and Kuang, Y., “Convergence results in a well-known delayed predator-prey system”, J. Math. Anal. Appl. 204 (1996) 840853.Google Scholar
[5]Cushing, J. M., Integro-differential equations and delay models in population dynamics (Springer, Heidelberg, 1977).Google Scholar
[6]Freedman, H. I., Deterministic mathematical models in population ecology, Monogr. Textbooks Pure Appl. Math. 57 (Marcel Dekker, New York, 1980).Google Scholar
[7]Freedman, H. I. and Ruan, S., “Uniform persistence in functional differential equations”, J. Differential Equations 115 (1995) 173192.Google Scholar
[8]Gopalsamy, K., “Harmless delay in model systems”, Bull. Math. Biol. 45 (1983) 295309.Google Scholar
[9]Gopalsamy, K., “Delayed responses and stability in two-species systems”, J. Austral. Math. Soc. Ser. B 25(1984) 473500.Google Scholar
[10]Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics (Kluwer, Dordrecht/Norwell, MA, 1992).Google Scholar
[11]Hainzl, J., “Stability and Hopf bifurcation in a predator-prey system with several parameters”, SIAM J. Appl. Math. 48 (1988) 170190.CrossRefGoogle Scholar
[12]Hassell, M. P. (Ed.), The dynamics of arthropod predator-prey systems (Princeton University Press, Princeton, NJ, 1978) 237.Google Scholar
[13]Hastings, A., “Delays in recruitment at different trophic levels: effects on stability”, J. Math. Biol. 21 (1984) 3544.Google Scholar
[14]He, X. Z., “Stability and delays in a predator-prey system”, J. Math. Anal. Appl. 198 (1996) 355370.Google Scholar
[15]He, X. Z., “Degenerate Lyapunov functionals of a well-known predator-prey model with discrete delays”, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 755771.Google Scholar
[16]Hesaaraki, M. and Moghadas, S. M., “Existence of limit cycles for predator-prey systems with a class of functional responses”, Ecol. Model. 142 (2001) 19.Google Scholar
[17]Hixon, M. and Carr, M., “Synergistic predation, density dependence and population regulation in marine fish”, Science 277 (1997) 946949.Google Scholar
[18]Hofbauer, J. and Sigmund, K., The theory of evolution and dynamical systems (Cambridge University Press, New York, 1988).Google Scholar
[19]Holling, C. S., “The components of predation as revealed by a study of small mammal predation of the European pine sawfly”, Can. Entomol. 91 (1959) 293320.Google Scholar
[20]Holling, C. S., “Some characteristics of simple types of predation and parasitism”, Can. Entomol. 91(1959) 385398.Google Scholar
[21]Holling, C. S., “The functional response of predators to prey density and its role in mimicry and population regulation”, Mem. Entomolog. Soc. Can. 45 (1965) 360.Google Scholar
[22]Hutson, V., “The existence of an equilibrium for permanent systems”, Rocky Mountain J. Math. 20 (1990) 10331040.Google Scholar
[23]Kazarinoff, N. D. and Van Den Driessche, P., “A model predator-prey system with functional response”, Math. Biosci. 39 (1978) 125134.Google Scholar
[24]Kooij, R. E. and Zegeling, A., “Qualitative properties of two-dimensional predator-prey systems”, Nonlinear Anal. 29 (1997) 693715.Google Scholar
[25]Kuang, Y., Delay differential equations with applications in population dynamics (Academic Press, New York, 1993).Google Scholar
[26]Kuang, Y., “Global stability in delay differential systems without dominating instantaneous negative feedbacks”, J. Differential Equations 119 (1995) 503532.Google Scholar
[27]Kuang, Y. and Freedman, H. I., “Uniqueness of limit cycles in Gauss-type models of predator-prey systems”, Math. Biosci. 88 (1988) 6784.Google Scholar
[28]MacDonald, N., Time lags in biological models (Springer, Heidelberg, 1978).Google Scholar
[29]May, R. M., “Limit cycles in predator-prey communities”, Science 177 (1972) 900902.Google Scholar
[30]May, R. M., “Time delay versus stability in population models with two and three trophic levels”, Ecology 4 (1973) 315325.Google Scholar
[31]Murdoch, W. W., “Switching in general predators: experiments on predator specificity and stability of prey populations”, Ecol. Monogr. 39 (1969) 335354.Google Scholar
[32]Murdoch, W. W., Avery, S. and Smyth, M. E. B., “Switching in predatory fish”, Ecology 56 (1975) 10941105.CrossRefGoogle Scholar
[33]Murdoch, W. W. and Marks, R. J., “Predation by coccinellid beetles: experiments on switching”, Ecology 54(1973)160167.Google Scholar
[34]Ruan, S., “Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays”, Quart. Appl. Math. 59 (2001) 159173.Google Scholar
[35]Shampine, L. F. and Thompson, S., “Solving DDEs in MATLAB”, Appl. Numer Math. 37 (2002) 441458.Google Scholar
[36]Sugie, J., Kohno, R. and Miyazaki, R., “On a predator-prey system of Holling type”, Proc. Amer Math. Soc. 125 (1997) 20412050.Google Scholar
[37]Wang, W. and Ma, Z., “Harmless delays for uniform persistence”, J. Math. Anal. Appl. 158 (1991) 256268.Google Scholar
[38]Wright, E. M., “A nonlinear differential difference equation”, J. Reine Angew. Math. 194 (1955) 6687.Google Scholar
[39]Xu, R. and Yang, P., “Persistence and stability in a three species food-chain system with time delays”, in Proceedings of the International Conference on Mathematical Biology, Advanced Topics in Biomathematics (ed. Chen, L. S. et al. ), (World Scientific, Singapore, 1998) 277283.Google Scholar