×

zbMATH — the first resource for mathematics

Impulsive harvesting and stocking in a Monod-Haldane functional response predator-prey system. (English) Zbl 1127.92045
Summary: A Monod-Haldane functional response predator-prey system with impulsive harvesting and stocking is proposed, where the Monod-Haldane functional response involves group defense theory. Conditions for the system to be extinct are given and permanence conditions are established via the method of comparison involving multiple Lyapunov functions. Further influences of the impulsive harvesting and stocking on the system are studied, and numerical simulations show that the system has rich dynamical behavior.

MSC:
92D40 Ecology
34A37 Ordinary differential equations with impulses
37N25 Dynamical systems in biology
91B76 Environmental economics (natural resource models, harvesting, pollution, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Clark, C.W., Mathematical bioeconomics: the optimal management of renewable resources, (1976), Wiley New York · Zbl 0364.90002
[2] Clark, C.W., Bioeconomic modelling and fisheries management, (1985), Wiley New York
[3] Kar, T.K.; Chaudhuri, K.S., Harvesting in a two-prey one-predator fishery: a bioeconomic model, Anz-iam j, 45, 443-456, (2004) · Zbl 1052.92052
[4] Freedman, H.I., Deterministic mathematical models in population ecology, Monogr textbooks pure appl math, vol. 57, (1980), Marcel Dekker New York · Zbl 0448.92023
[5] Hsu, S.B., On global stability of a predator – prey system, Math biosci, 39, 1-10, (1978) · Zbl 0383.92014
[6] Kooij, R.E.; Zegeling, A., Qualitative properties of two-dimensional predator – prey systems, Nonlinear anal, 29, 693-715, (1997) · Zbl 0883.34040
[7] Kuang, Y.; Freedman, H.I., Uniqueness of limit cycles in gause-type models of predator – prey systems, Math biosci, 88, 67-84, (1988) · Zbl 0642.92016
[8] May, R.M., Limit cycles in predator – prey communities, Science, 177, 900-902, (1972)
[9] Mischaikow, K.; Wolkowicz, G.S., A predator – prey system involving group defense: a connection matrix approach, Nonlinear anal, 14, 955-969, (1990) · Zbl 0724.34015
[10] Sugie, J.; Kohno, R.; Miyazaki, R., On a predator – prey system of Holling type, Proc am math soc, 125, 2041-2050, (1997) · Zbl 0868.34023
[11] Holmes, J.C.; Bethel, W.M., Modification of intermediate host behavior parasites, Zool J Linnean soc, 51, Suppl. 1, 123-149, (1972)
[12] Tener, J.S., Muskoxen, (1965), Queen’s Printer Ottawa, Canada
[13] Andrews, J.F., A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnol bioeng, 10, 707-723, (1968)
[14] Lakshmikantham, V.; Bainov, D.; Simeonov, P., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002
[15] Bainov, D.; Simeonov, P., Impulsive differential equations: periodic solutions and applications, Ptiman monogr surv pure appl math, 66, 54-121, (1993) · Zbl 0815.34001
[16] Lakmeche, A.; Arino, O., Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment, Dynam contin discrete impuls syst, 7, 265-287, (2000) · Zbl 1011.34031
[17] Zhang, S.W.; Tan, D.J.; Chen, L.S., Chaos in periodically forced Holling type IV predator – prey system with impulsive perturbations, Chaos, solitons & fractals, 27, 980-990, (2006) · Zbl 1097.34038
[18] Zhang, S.W.; Wang, F.Y.; Chen, L.S., A food chain model with impulsive perturbations and Holling IV functional response, Chaos, solitons & fractals, 26, 855-866, (2005) · Zbl 1066.92061
[19] 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
[20] Zeng, G.Z.; Chen, L.S.; Sun, L.H., Complexity of an SIR epidemic dynamics model with impulsive vaccination control, Chaos, solitons & fractals, 26, 495-505, (2005) · Zbl 1065.92050
[21] Hui, J.; Zhu, D.M., Dynamic complexities for prey-dependent consumption integrated pest management models with impulsive effects, Chaos, solitons & fractals, 29, 233-251, (2006) · Zbl 1095.92067
[22] Zhang, Y.J.; Xiu, Z.L.; Chen, L.S., Dynamic complexity of a two-prey one-predator system with impulsive effect, Chaos, solitons & fractals, 26, 131-139, (2005) · Zbl 1076.34055
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.