Chaos in a Lotka-Volterra predator-prey system with periodically impulsive ratio-harvesting the prey and time delays. (English) Zbl 1130.37042

Wang, Fengyan; Zeng, Guangzhao

Chaos Solitons Fractals 32, No. 4, 1499-1512 (2007).

Summary: We introduce and study a Lotka-Volterra predator-prey system with impulsive ratio-harvesting the prey and time delays. By using Floquet theory and small amplitude perturbation skills, we discuss the boundary periodic solutions for predator-prey system under periodic pulsed conditions. The stability analysis of the boundary periodic solution yields an invasion threshold of the predator. Further, by use of the coincidence degree theorem and its related continuous theorem we prove the existence of the positive periodic solutions of the system when the value of the coefficient is large than the threshold. Finally, by comparing bifurcation diagrams with different bifurcation parameters, we show that the impulsive effect and the time delays bring to the system to be more complex, which experiences a complex process of cycles \(\rightarrow \) quasi-periodic oscillation \(\rightarrow \) periodic doubling cascade \(\rightarrow \) chaos.

MSC:

37N25	Dynamical systems in biology
92D25	Population dynamics (general)
37D45	Strange attractors, chaotic dynamics of systems with hyperbolic behavior
34C28	Complex behavior and chaotic systems of ordinary differential equations

Keywords:

Floquet theory; amplitude perturbation; stability analysis; boundary periodic solution; coincidence degree theorem; bifurcation diagrams; quasi-periodic oscillation; periodic doubling cascade

References:

[1]	Rinaldi, S.; Muratori, S.; Kuznetsov, Y.A., Multiple, attractors, catastrophes, and chaos in seasonally perturbed predator-prey communities, Bull math biol, 55, 15-36, (1993) · Zbl 0756.92026
[2]	Pavlou, S.; Kevrekidis, I.G., Microbial predation in a periodically operated chemostat: a global study of the interaction between natural and externally imposed frequencies, Math biosci, 108, 1-55, (1992) · Zbl 0729.92522
[3]	Wang, F.; Zhang, S.; Chen, L.; Sun, L., Bifurcation and complexity of monod type predator-prey system in a pulsed chemostat, Chaos, solitons & fractals, 27, 447-458, (2006) · Zbl 1096.34029
[4]	Kor, M.; Sayler, G.S.; Waltman, T.W., Complex dynamics in a model microbial system, Bull math biol, 54, 619-648, (1992) · Zbl 0761.92041
[5]	Cushing, J.M., Two species competition in a periodic environment, J math biol, 10, 348-400, (1980) · Zbl 0455.92012
[6]	Schaffer, W.M., Can nonlinear dynamics elucidate mechanisms in ecology and epidemiology?, IMA J math appl med biol, 2, 221-252, (1985) · Zbl 0609.92034
[7]	Bainov D, Simeonor P. Impulsive differential equations: periodic solutions and applications. Pitman monographs and surreys in pure and applied mathematicsm, 1993. p. 66.
[8]	Laksmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore
[9]	Panetta, J.C., A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competition environment, Bull math biol, 58, 425-447, (1996) · Zbl 0859.92014
[10]	Zeng, G.Z.; Chen, L.; Sun, L., Complexity of an SIR epidemic dynamics model with impulsive vaccination control, Chaos, solitons & fractals, 26, 495-505, (2005) · Zbl 1065.92050
[11]	D’Onofrio, A., Stability properties of pulse vaccination strategy in the SIR epidemic model, Bull math biol, 60, 1-26, (2002)
[12]	Liu, X.N.; Chen, L.S., Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator, Chaos, solitons & fractals, 16, 311-320, (2003) · Zbl 1085.34529
[13]	Zhang, S.; Dong, L.; Chen, L., The study of predator-prey system with defensive ability of prey and impulsive perturbations on the predator, Chaos, solitons & fractals, 23, 631-643, (2005) · Zbl 1081.34041
[14]	Wang, F.; Zhang, S.; Chen, L.; Sun, L., Permanence and complexity of a three species food chain with impulsive effect on the top predator, Int J nonlinear sci numer simulat, 6, 169-180, (2005) · Zbl 1401.92167
[15]	Tang, S.Y.; Chen, L.S., Density-dependent birth rate, birth pulse and their population dynamic consequences, J math biol, 44, 185-199, (2002) · Zbl 0990.92033
[16]	Ballinger, G.; Liu, X., Permanence of population growth models with impulsive effects, Math comput model, 26, 59-72, (1997) · Zbl 1185.34014
[17]	Ghaines, R.E.; Mawhin, J.L., Coincidence degree, and nonlinear differential equations, (1977), Springer-Verlag New York
[18]	Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press Boston, MA · Zbl 0777.34002
[19]	May, R.M., Biological populations with non-overlapping generations: stable points, stable cycles, and chaos, Science, 186, 645-647, (1974)
[20]	Gakkhar, S.; Naji, M.A., Order and chaos in predator to prey ratio-dependent food chain, Chaos, solitons & fractals, 18, 229-239, (2003) · Zbl 1068.92044
[21]	Klebanoff, A.; Hastings, A., Chaos in three species food chains, J math biol, 32, 427-451, (1994) · Zbl 0823.92030