# zbMATH — the first resource for mathematics

Seasonal effects on a Beddington-DeAngelis type predator-prey system with impulsive perturbations. (English) Zbl 1187.34058
The authors consider the following periodical impulsive differential equations which represent a predator-prey system
$\begin{gathered} x'(t)= rx(t)\Biggl(1- {x(t)\over k}\Biggr)- {ax(t) y(t)\over by(t)+ x(t)+c}+\lambda x(t)\sin(\omega t),\\ y'(t)= -dy(t)+ {eax(t) y(t)\over by(t)+ x(t)+c},\quad t\neq n\tau,\\ x(t^+)= (1- p_1) x(t),\;t= n\tau,\;y(t^+)= (1- p_2) y(t)+ q,\;(x(0^+), y(0^+))= (x_0, y_0),\end{gathered}\tag{1}$
where $$\tau$$ is the period of impulsive immigration or stock of the predator, $$0\leq p_1$$, $$p_2< 1$$ and $$x(t)$$, $$y(t)$$ represent the population densities of prey and predator, respectively.
The authors perform a numerical analysis of (1) for the case without of impulses, i.e. $$p_1= p_2= 0$$. Sufficient conditions for the local asymptotic stability of (1) are derived.
For the periodic solution $$(0,y^*(t))$$ sufficient conditions for its local asymptotic stability are found. A numerical analysis of seasonal effect and impulsive perturbations is performed.

##### MSC:
 34C60 Qualitative investigation and simulation of ordinary differential equation models 34A37 Ordinary differential equations with impulses 92D25 Population dynamics (general) 34C25 Periodic solutions to ordinary differential equations
Full Text:
##### References:
 [1] J. R. Beddington, “Mutual interference between parasites or predator and its effect on searching efficiency,” Journal of Animal Ecology, vol. 44, pp. 331-340, 1975. [2] D. L. DeAngelis, R. A. Goldstein, and R. V. O/Neill, “A model for trophic interaction,” Ecology, vol. 56, pp. 881-892, 1975. [3] G. T. Skalski and J. F. Gilliam, “Functional responses with predator interference: viable alternatives to the Holling type II mode,” Ecology, vol. 82, pp. 3083-3092, 2001. [4] M. Fan and Y. Kuang, “Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response,” Journal of Mathematical Analysis and Applications, vol. 295, no. 1, pp. 15-39, 2004. · Zbl 1051.34033 [5] T. W. Hwang, “Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response,” Journal of Mathematical Analysis and Applications, vol. 290, pp. 113-122, 2004. · Zbl 1086.34028 [6] J. M. Cushing, “Periodic time-dependent predator-prey systems,” SIAM Journal on Applied Mathematics, vol. 32, no. 1, pp. 82-95, 1977. · Zbl 0348.34031 [7] S. M. Moghadas and M. E. Alexander, “Dynamics of a generalized Gause-type predator-prey model with a seasonal functional response,” Chaos, Solitons & Fractals, vol. 23, pp. 55-65, 2005. · Zbl 1058.92049 [8] Yu. A. Kuznetsov, S. Muratori, and S. Rinaldi, “Bifurcations and chaos in a periodic predator-prey model,” International Journal of Bifurcation and Chaos, vol. 2, no. 1, pp. 117-128, 1992. · Zbl 1126.92316 [9] Z. Li, W. Wang, and H. Wang, “The dynamics of a Beddington-type system with impulsive control strategy,” Chaos, Solitons & Fractals, vol. 29, pp. 1229-1239, 2006. · Zbl 1142.34305 [10] X. Liu and L. Chen, “Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator,” Chaos, Solitons & Fractals, vol. 16, pp. 311-320, 2003. · Zbl 1085.34529 [11] B. Liu, Y. Zhang, and L. Chen, “Dynamic complexities in a Lotka-Volterra predator-prey model concerning impulsive control strategy,” International Journal of Bifurcation and Chaos, vol. 15, no. 2, pp. 517-531, 2005. · Zbl 1080.34026 [12] S. Rinaldi, S. Muratori, and Y. Kuznetsov, “Multiple attractors, catastrophes and chaos in seasonally perturbed predator-prey communities,” Bulletin of Mathematical Biology, vol. 55, pp. 15-35, 1993. · Zbl 0756.92026 [13] W. Wang, H. Wang, and Z. Li, “The dynamic complexity of a three-species Beddington-type food chain with impulsive control strategy,” Chaos, Solitons & Fractals, vol. 32, pp. 1772-1785, 2007. · Zbl 1195.92066 [14] S. Zhang, D. Tan, and L. Chen, “Chaos in periodically forced Holling type II predator-prey system with impulsive perturbations,” Chaos, Solitons & Fractals, vol. 28, no. 2, pp. 367-376, 2006. · Zbl 1083.37537 [15] S. Zhang and L. Chen, “A study of predator-prey models with the Beddington-DeAnglis functional response and impulsive effect,” Chaos, Solitons & Fractals, vol. 27, pp. 237-248, 2006. · Zbl 1102.34032 [16] S. Gakkhar and R. K. Naji, “Chaos in seasonally perturbed ratio-dependent prey-predator system,” Chaos, Solitons & Fractals, vol. 15, pp. 107-118, 2003. · Zbl 1033.92026 [17] G. C. W. Sabin and D. Summers, “Chaos in a periodically forced predator-prey ecosystem model,” Mathematical Biosciences, vol. 113, pp. 91-113, 1993. · Zbl 0767.92028 [18] H. Baek, “Dynamic complexities of a three-species beddington-DeAngelis system with impulsive control strategy,” Acta Applicandae Mathematicae. · Zbl 1194.34087 [19] H. Baek and Y. Do, “Stability for a holling type IV food chain system with impulsive perturbations,” Kyungpook Mathematical Journal, vol. 48, no. 3, pp. 515-527, 2008. · Zbl 1165.34307 [20] H. Wang and W. Wang, “The dynamical complexity of a Ivlev-type prey-predator system with impulsive effect,” Chaos, Solitons & Fractals, vol. 38, pp. 1168-1176, 2007. · Zbl 1152.34310 [21] S. Zhang, L. Dong, and L. Chen, “The study of predator-prey system with defensive ability of prey and impulsive perturbations on the predator,” Chaos, Solitons & Fractals, vol. 23, pp. 631-643, 2005. · Zbl 1081.34041 [22] V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. · Zbl 0719.34002
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.