Complex dynamics of Holling type II Lotka–Volterra predator–prey system with impulsive perturbations on the predator. (English) Zbl 1085.34529

Liu, Xianning; Chen, Lansun

Chaos Solitons Fractals 16, No. 2, 311-320 (2003).

The authors consider the so called Holling-type II predator-prey model with logistic term, subjected to an impulsive periodic perturbation that corresponds to punctuated predator in-migration. In the first part of the paper, the authors give conditions for the existence of bounded nontrivial soutions of the perturbed system. In the second part of the paper, a numerical exploration of the system is given, with the forcing amplitude as a parameter. The bifurcation diagram exhibits complex dynamical behaviour characteristic for periodically forced nonlinear oscillators.

Reviewer: Ana Nunes (Lisboa)

MSC:

34C60	Qualitative investigation and simulation of ordinary differential equation models
34C15	Nonlinear oscillations and coupled oscillators for ordinary differential equations
37N25	Dynamical systems in biology
34A37	Ordinary differential equations with impulses
34C23	Bifurcation theory for ordinary differential equations
34C28	Complex behavior and chaotic systems of ordinary differential equations

Keywords:

predator-prey; impulsive perturbations

References:

[1]	Holling, C.S., The functional response of predator to prey density and its role in mimicry and population regulation, Mem ent sec can, 45, 1-60, (1965)
[2]	Komogorov, A., Sulla teoria di Volterra lotta per l’esistenza, Gi inst ital attuari, 7, 74-80, (1936) · JFM 62.1263.01
[3]	Alberecht, F.; Gatzke, H.; Haddad, A.; Wax, N., The dynamics of two interacting populations, J math anal appl, 46, 658-670, (1974) · Zbl 0281.92012
[4]	Chen, K., Uniqueness of a limit cycle for a predator – prey system, SIAM J math anal, 12, 541-548, (1981) · Zbl 0471.92021
[5]	Cushing, J.M., Periodic Kolmogorov systems, SIAM J math anal, 13, 811-827, (1982) · Zbl 0506.34039
[6]	Chen, L.; Jing, Z., The existence and uniqueness of predator – prey differential equations, Chin sci bull, 9, 521-523, (1984)
[7]	Huang, X.; Merriu, S.J., Conditions for uniqueness of limit cycles in general predator – prey systems, Math biosci, 96, 47-60, (1989) · Zbl 0676.92008
[8]	Cushing, J.M., Periodic time-dependent predator – prey system, SIAM J appl math, 32, 82-95, (1977) · Zbl 0348.34031
[9]	Bardi, M., Predator – prey models in periodically fluctuating environments, J math biol, 12, 127-140, (1981) · Zbl 0466.92019
[10]	Inoue, M.; Kamifukumoto, H., Scenarios leading to chaos in forced lotka – volterra model, Progr theor phys, 71, 930-937, (1984) · Zbl 1074.37522
[11]	Levin, R.W.; Cock, B.P.; Markman, G.S., Periodic, quasiperiodic, and chaotic motion in a forced predator – prey ecosystem. dynamical systems and environmental models, (1987), Akademie-Verlag Berlin, pp. 95-104
[12]	Schaffer, W.M., Perceiving order in the chaos of nature. evolution of life histories of mammals, (1988), Yale Univ Press New Haven, pp. 313-350
[13]	Allen, J.C., Chaos and phase-locking in predator – prey models in relation to functional response, Fla entomol, 73, 100-110, (1990)
[14]	Doveri, F.; Kuznetsov, Y.; Muratori, S.; Rinaldi, S.; Scheffer, M., Seasonality and chaos in a plankton-fish model, Theor popul biol, 43, 159-183, (1993) · Zbl 0825.92135
[15]	Kot, M.; Sayler, G.S.; Schultz, T.W., Complex dynamics in a model microbial system, Bull math biol, 54, 619-648, (1992) · Zbl 0761.92041
[16]	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
[17]	Rinaldi, S.; Muratori, S., Conditioned chaos in seasonally perturbed predator – prey models, Ecol model, 69, 79-97, (1993)
[18]	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
[19]	Sabin, G.C.W.; Summers, D., Chaos in a periodically forced predator – prey ecosystem model, Math biosci, 113, 91-114, (1993) · Zbl 0767.92028
[20]	Vandermeer, J.; Stone, L.; Blasius, B., Categeories of chaos and fractal basin boundaries in forced predator – prey models, Chaos, solitons & fractals, 12, 265-276, (2001) · Zbl 0976.92033
[21]	Pandit, S.G., On the stability of impulsively perturbed differential systems, Bull austral math soc, 17, 423-432, (1977) · Zbl 0367.34038
[22]	Pavidis, T., Stabling of systems described by differential equations containing impulses, IEEE trans, AC-2, 43-45, (1967)
[23]	Simeonov, P.S.; Bainov, D.D., The second method of Liapunov for systems with impulsive effect, Tamkang J math, 16, 19-40, (1985) · Zbl 0641.34051
[24]	Simeonov, P.S.; Bainov, D.D., Stability with respect to part of the variables in systems with impulsive effect, J math anal appl, 117, 247-263, (1986) · Zbl 0588.34044
[25]	Kulev, G.K.; Bainov, D.D., On the asymptotic stability of systems with impulses by direct method of Lyapunov, J math anal appl, 140, 324-340, (1989) · Zbl 0681.34042
[26]	Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002
[27]	Bainov, D.D.; Simeonov, P.S., Impulsive differential equations: periodic solutions and applications, (1993), Longman England · Zbl 0793.34011
[28]	Roberts, M.G.; Kao, R.R., The dynamics of an infectious disease in a population with birth pulses, Math. biosci, 149, 23-36, (1998) · Zbl 0928.92027
[29]	Lakmeche, A.; Arino, O., Bifurcation of non-trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment, Dyn cont discr impulsive syst, 7, 265-287, (2000) · Zbl 1011.34031
[30]	Cushing, J.M.; Henson, S.M.; Desharnais, R.A.; Dennis, B.; Costantino, R.F.; King, A., A chaotic attractor in ecology: theory and experimental data, Chaos, solitons & fractals, 12, 219-234, (2001) · Zbl 0976.92022
[31]	Kendall, B.E., Cycles, chaos and noise in predator – prey dynamics, Chaos, solitons & fractals, 12, 321-332, (2001) · Zbl 0977.92028
[32]	Upadhyay, R.K.; Rai, V., Crisis-limited chaotics in ecological systems, Chaos, solitons & fractals, 12, 205-218, (2001) · Zbl 0977.92033
[33]	Davies, B., Exploring chaos, theory and experiment, (1999), Perseus Books Reading, MA · Zbl 0959.37001
[34]	May, R.M., Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos, Science, 186, 645-647, (1974)
[35]	May, R.M.; Oster, G.F., Bifurcations and dynamic complexity in simple ecological models, Am nature, 110, 573-599, (1976)
[36]	Collet, P.; Eeckmann, J.P., Iterated maps of the interval as dynamical systems, (1980), Birkhauser Boston
[37]	Eckmann, J.P., Routes to chaos with special emphasis on period doubling, () · Zbl 0616.58032
[38]	Neubert, M.G.; Caswell, H., Density-dependent vital rates and their population dynamic consequences, J math biol, 41, 103-121, (2000) · Zbl 0956.92029
[39]	Wikan, A., From chaos to chaos. an analysis of a discrete age-structured prey-predator model, J math biol, 43, 471-500, (2001) · Zbl 0996.92031
[40]	Kaitala, V.; Ylikarjula, J.; Heino, M., Dynamic complexities in host-parasitioid interaction, J theor biol, 197, 331-341, (1999)