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Qualitative analysis of Beddington-DeAngelis type impulsive predator-prey models. (English) Zbl 1204.34062

This paper discusses the dynamics of a predator-prey model which is subject to periodic impulsive perturbations, understood to describe the effect of predator immigration or stocking and of pesticide spraying. Regarding the characteristics of the predation process, it is assumed that the functional response of the predator is of Beddington-deAngelis type.

First, the model is shown to be biologically well posed, the local stability of the prey-free periodic solution is then discussed by using the Floquet theory of impulsively perturbed systems. It is proved that, if a certain inequality holds, then the prey-free periodic solution is locally stable, while if the converse inequality holds, then the model is shown to be permanent. In the limit case (that is, if the corresponding equality holds instead of the previously mentioned inequalities), a nontrivial periodic solution is shown to bifurcate from the prey-free periodic solution.

A slightly more general model, in which the “proportional” and the “constant” impulsive perturbations occur at different times, but with the same periodicity, has earlier been considered from a similar viewpoint in [H. Zhang, P. Georgescu and L. Chen, On the impulsive controllability and bifurcation of a predator-pest model of IPM, Biosystems 93, No. 3, 151–171 (2009)], where the model discussed by the author is obtained for $\stackrel{˜}{l}=1$. The present paper puts more emphasis on the practical consequences of the bifurcation result, it presents a somewhat different approach towards the proof of the permanence of the model.

##### MSC:
 34C60 Qualitative investigation and simulation of models (ODE) 34A37 Differential equations with impulses 92D25 Population dynamics (general) 34C25 Periodic solutions of ODE 34D20 Stability of ODE
##### References:
 [1] Beddington, J. R.: Mutual interference between parasites or predator and its effect on searching efficiency, J. animal ecol. 44, 331-340 (1975) [2] Deangelis, D. L.; Goldstein, R. A.; O’neill, R. V.: A model for trophic interaction, Ecology 56, 881-892 (1975) [3] Skalski, G. T.; Gilliam, J. F.: Functional responses with predator interference: viable alternatives to the Holling type II mode, Ecology 82, 3083-3092 (2001) [4] Fan, M.; Kuang, Y.: Dynamics of a nonautonomous predator–prey system with the beddington–deangelis functional response, J. math. Anal. appl. 295, 15-39 (2004) · Zbl 1051.34033 · doi:10.1016/j.jmaa.2004.02.038 [5] Hwang, T. -W.: Uniqueness of limit cycles of the predator–prey system with beddington–deangelis functional response, J. math. Anal. appl. 290, 113-122 (2004) · Zbl 1086.34028 · doi:10.1016/j.jmaa.2003.09.073 [6] Cushing, J. M.: Periodic time-dependent predator–prey systems, SIAM J. Appl. math. 32, 82-95 (1977) · Zbl 0348.34031 · doi:10.1137/0132006 [7] Bainov, D. D.; Simeonov, P. S.: Impulsive differential equations: periodic solutions and application, Pitman monographs and surveys in pure and applied mathematics 66 (1993) · Zbl 0815.34001 [8] Lakshmikantham, V.; Bainov, D.; Simeonov, P.: Theory of impulsive differential equations, (1989) · Zbl 0718.34011 [9] Liu, B.; Zhang, Y.; Chen, L.: Dynamic complexities in a Lotka–Volterra predator–prey model concerning impulsive control strategy, Internat. J. Bifur. chaos 15, No. 2, 517-531 (2005) · Zbl 1080.34026 · doi:10.1142/S0218127405012338 [10] Liu, B.; Zhang, Y.; Chen, L.: The dynamical behaviors of a Lotka–Volterra predator–prey model concerning integrated pest management, Nonlinearity anal. 6, 227-243 (2005) · Zbl 1082.34039 · doi:10.1016/j.nonrwa.2004.08.001 [11] Makinde, O. D.: Solving ratio-dependent predator–prey system with constant effort harvesting using Adomian decomposition method, Appl. math. Comput. 186, 17-22 (2007) · Zbl 1114.65081 · doi:10.1016/j.amc.2006.07.083 [12] Xiao, D.; Jennings, L. S.: Bifurcations of a ratio-dependent predator–prey system with constant rate harvesting, SIAM J. Appl. math. 65, No. 3, 737-753 (2005) · Zbl 1094.34024 · doi:10.1137/S0036139903428719 [13] Zhang, S.; Chen, L.: A study of predator–prey models with the beddington–deangelis functional response and impulsive effect, Chaos solitons fractals 27, 237-248 (2006) · Zbl 1102.34032 · doi:10.1016/j.chaos.2005.03.039 [14] Wang, H.; Wang, W.: The dynamical complexity of a ivlev-type prey-predator system with impulsive effect, Chaos solitons fractals (2007) [15] Wang, H.; Wang, W.: The dynamical complexity of a ivlev-type prey-predator system with impulsive effect, Chaos solitons fractals (2007) [16] Wang, W.; Wang, H.; Li, Z.: The dynamic complexity of a three-species beddington-type food chain with impulsive control strategy, Chaos solitons fractals 32, 1772-1785 (2007) · Zbl 1195.92066 · doi:10.1016/j.chaos.2005.12.025 [17] Gakkhar, S.; Negi, K.: Pulse vaccination in SIRS epidemic model with non-monotonic incidence rate, Chaos solitons fractals 35, 626-638 (2008) · Zbl 1131.92052 · doi:10.1016/j.chaos.2006.05.054 [18] Lakmeche, A.; Arino, O.: Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment, Dyn. contin. Discrete impuls. Syst. 7, 265-287 (2000) · Zbl 1011.34031