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Harvesting of a prey-predator fishery in the presence of toxicity. (English) Zbl 1185.91120
Summary: We discuss the bioeconomic harvesting of a prey-predator fishery in which both the species are infected by some toxicants released by some other species. Here both the species are harvested where we use the usual catch-per-unit-effort hypothesis. The dynamical behaviour of the exploited system is examined. The possibility of existence of a bionomic equilibrium is considered. The optimal harvesting policy is studied by using Pontryagin’s maximal principle. Some numerical examples and the corresponding solution curves are studied to illustrate the results of the model. Finally, the existence of limit cycle is discussed.

MSC:
91B76 Environmental economics (natural resource models, harvesting, pollution, etc.)
92D25 Population dynamics (general)
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[1] Clark, C.W., Mathematical bioeconomics: the optimal management of renewable resources, (1976), Wiley New York · Zbl 0364.90002
[2] Clark, C.W., Bioeconomic modeling and fisheries management, (1985), Wiley New York
[3] Mesterton-Gibbons, M., On the optimal policy for the combined harvesting of predator and prey, Nat. resour. model, 3, 63-90, (1988)
[4] Mesterton-Gibbons, M., A technique for finding optimal two species harvesting policies, Ecol. modell., 92, 235-244, (1996)
[5] Chaudhuri, K.S.; Saha Ray, S., Bionomic exploitation of a lotka – volterra prey predator system, Bull. cal. math. soc., 83, 175-186, (1991) · Zbl 0744.34046
[6] Chaudhuri, K.S.; Saha ray, S., On the combined harvesting of a prey – predator system, J. biol. syst., 4, 373-389, (1996)
[7] Hallam, T.G.; Clark, C.W., Non-autonomous logistic equations as models of populations in a deteriorating environment, J. theor. biol., 93, 303-311, (1982)
[8] Hallam, T.G.; De Luna, T.J., Effects of toxicants on populations: a qualitative approach III, environmental and food chain pathways, Theor. biol., 109, 411-429, (1984)
[9] Chattopadhyay, J., Effect of toxic substances on a two-species competitive system, Ecol. modell., 84, 287, (1996)
[10] Mukhopadhyay, A.; Chattopadhyay, J.; Tapaswi, P.K., A delay differential equations model of plankton allelopathy, Math. biosci., 149, 167-189, (1998) · Zbl 0946.92031
[11] Dubey, B.; Hussain, J., A model for the allelopathic effect on two competing species, Ecol. modell., 129, 195-207, (2000)
[12] Kar, T.K.; Chaudhuri, K.S., On non-selective harvesting of two competing fish species in the presence of toxicity, Ecol. modell., 161, 125-137, (2003)
[13] Kar, T.K.; Pahari, U.K.; Chaudhuri, K.S., Management of a prey – predator fishery based on continuous fishing effort, J. biol. syst., 12, 3, 301-313, (2004) · Zbl 1124.91355
[14] Maynard Smith, J., Models in ecology, (1974), Cambridge University Press, p. 146 · Zbl 0312.92001
[15] Pontryagin, L.S.; Boltyanski, V.S.; Gamkrelidze, R.V.; Mishchenco, E.F., The mathematical theory of optimal processes, (1962), Wiley New York
[16] Arrow, K.J.; Kurz, M., Public investment, The rate of return and optimal fiscal policy, (1970), John Hopkins Baltimore
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