×

zbMATH — the first resource for mathematics

Harvesting in a two-prey one-predator fishery: A bioeconomic model. (English) Zbl 1052.92052
Summary: A multispecies harvesting model with interference is proposed. The model is based on Lotka-Volterra dynamics with two competing species which are affected not only by harvesting but also by the presence of a predator, the third species. In order to understand the dynamics of this complicated system, we choose to model the simplest possible predator response function in which the feeding rate of the predator increases linearly with prey density. We derive the conditions for global stability of the system using a Lyapunov function. The possibility of existence of a bioeconomic equilibrium is discussed. The optimal harvest policy is studied and the solution is derived in the equilibrium case using Pontryagin’s maximum principle. Finally, some numerical examples are discussed.

MSC:
92D40 Ecology
34D23 Global stability of solutions to ordinary differential equations
49N90 Applications of optimal control and differential games
34D20 Stability of solutions to ordinary differential equations
91B76 Environmental economics (natural resource models, harvesting, pollution, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1142/S0218339096000259 · doi:10.1142/S0218339096000259
[2] DOI: 10.1016/0025-5564(90)90006-K · Zbl 0699.92024 · doi:10.1016/0025-5564(90)90006-K
[3] DOI: 10.1016/0304-3800(88)90041-5 · doi:10.1016/0304-3800(88)90041-5
[4] DOI: 10.1016/0304-3800(86)90091-8 · doi:10.1016/0304-3800(86)90091-8
[5] Clark, Mathematical bioeconomics: the optimal management of renewable resources (1976) · Zbl 0364.90002
[6] DOI: 10.1016/0095-0696(85)90025-7 · doi:10.1016/0095-0696(85)90025-7
[7] Murray, Mathematical Biology (1993) · Zbl 0779.92001
[8] Mesterton-Gibbons, Nat. Resour. Model. 3 pp 63– (1988)
[9] Clark, Bioeconomic modelling and fisheries management (1985)
[10] Pontryagin, The mathematical theory of optimal processes (1962) · Zbl 0112.05502
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.