On a nonlinear nonautonomous predator–prey model with diffusion and distributed delay. (English) Zbl 1061.92058

Summary: A nonlinear nonautonomous predator-prey model with diffusion and continuous distributed delay is studied, where all the parameters are time-dependent. The system, which is composed of two patches, has two species: the prey can diffuse between the two patches, but the predator is confined to one patch. We first discuss the uniform persistence and global asymptotic stability of the model; after that, by constructing a suitable Lyapunov functional, some sufficient conditions for the existence of a unique almost periodic solution of the system are obtained. An example shows the feasibility of our main results.


92D40 Ecology
34D23 Global stability of solutions to ordinary differential equations
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
Full Text: DOI


[1] Aiello, W. G.; Freedman, H. I., A time-delay model of single-species growth with stage structure, Math. Biosci., 101, 2, 139-153 (1990) · Zbl 0719.92017
[2] Ayala, F. J.; Gilpin, M. E.; Eherenfeld, J. G., Competition between speciestheoretical models and experimental tests, Theor. Popul. Biol., 4, 331-356 (1973)
[3] Barbaˇlat, I., Systems d’equations differential d’oscillations nonlinearies, Rev. Roumaine Math. Pure Appl., 4, 2, 267-270 (1959) · Zbl 0090.06601
[4] Berryman, A. A., The origins and evolution of predator-prey theory, Ecology, 75, 1530-1535 (1992)
[5] Chen, L. S., Mathematical Models and Methods in Ecology (1988), Science Press: Science Press Beijing, (in Chinese)
[7] Chen, F. D.; Lin, S. J., Periodicity in a Logistic type system with several delays, Comput. Math. Appl., 48, 1-2, 35-44 (2004) · Zbl 1061.34050
[8] Chen, F. D.; Lin, F. X.; Chen, X. X., Sufficient conditions for the existence positive periodic solutions of a class of neutral delay models with feedback control, Appl. Math. Comput., 158, 1, 45-68 (2004) · Zbl 1096.93017
[9] Chen, S. H.; Wang, F.; Young, T., Existence of positive periodic solution for nonautonomous predator-prey system with diffusion and time delay, J. Comput. Appl. Math., 159, 375-386 (2003) · Zbl 1039.34061
[10] Chen, F. D., Periodicity in a food-limited population model with toxicants and state dependent delays, J. Math. Anal. Appl., 288, 1, 132-142 (2003)
[12] Edelstein-Keshet, L., Mathematical Models in Biology (1988), Random House: Random House New York · Zbl 0674.92001
[13] Fan, M.; Wang, K., Global periodic solutions of a generalized \(n\)-species Gilpin-Ayala competition model, Comput. Math. Appl., 40, 1141-1151 (2000) · Zbl 0954.92027
[14] Feng, C. H., On the existence and uniqueness of almost periodic solutions for delay Logistic equations, Appl. Math. Comput., 136, 487-494 (2003) · Zbl 1047.34083
[15] Foryś, U., Hopf bifurcation in Marchuk’s model of immune reactions, Math. Comput. Modelling, 34, 7-8, 725-735 (2001) · Zbl 0999.92023
[16] Gilpin, M. E.; Ayala, F. J., Global models of growth and competition, Proc. Nat. Acad. Sci. USA, 70, 3590-3593 (1973) · Zbl 0272.92016
[17] Gilpin, M. E.; Ayala, F. J., Schoener’s model and Drosophila competition, Theor. Popul. Biol., 9, 12-14 (1976)
[18] Levin, S. A., Dispersion and population interactions, Am. Nat., 108, 207-228 (1974)
[19] Li, C. R.; Lu, S. J., The qualitative analysis of \(N\)-species periodic coefficient, nonlinear relation, prey-competition systems, Appl. Math-JCU, 12, 2, 147-156 (1997), (in Chinese) · Zbl 0880.34042
[21] Song, X. Y.; Chen, L. S., Persistence and global stability for nonautonomous predator-prey system with diffusion and time delay, Comput. Math. Appl., 35, 6, 33-40 (1998) · Zbl 0903.92029
[22] Song, X. Y.; Chen, L. S., Persistence and periodic orbits for two-species predator-prey system with diffusion, Can. Appl. Math. Quart., 6, 3, 233-244 (1998) · Zbl 0941.92032
[23] Xu, R.; Rui, M. A.; Chaplain, J.; Davidson, F. A., Periodic solutions for a delayed predator-prey model of prey dispersal in two-patch environments, Nonlinear Anal. Real World Appl., 5, 1, 183-206 (2004) · Zbl 1066.92059
[24] Xu, R.; Rui, M. A.; Chaplain, J.; Davidson, F. A., Periodic solution of a Lotka-Volterra predator-prey model with dispersion and time delays, Appl. Math. Comput., 148, 2, 537-560 (2004) · Zbl 1048.34119
[25] Zheng, Z. X., Theory of Functional Differential Equations (1994), Anhui Education Press
[26] Zhang, Z. Q.; Wang, Z. C., Periodic solutions for nonautonomous predator-prey system with diffusion and time delay, Hiroshima Math. J., 31, 3, 371-381 (2001) · Zbl 1052.34077
[27] Zhao, J. D.; Chen, W. C., The qualitative analysis of \(N\)-species nonlinear prey-competition systems, Appl. Math. Comput., 149, 2, 567-576 (2004) · Zbl 1045.92038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.