Selective harvesting in a prey-predator fishery with time delay. (English) Zbl 1045.92046

Summary: We have considered a prey-predator fishery model and discussed selective harvesting of fishes above a certain age or size by incorporating a time delay in the harvesting term. It is shown that the time delay can cause a stable equilibrium to become unstable and even a switching of stabilities. Computer simulations are carried out to explain some mathematical conclusions.


92D40 Ecology
34K20 Stability theory of functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations
91B76 Environmental economics (natural resource models, harvesting, pollution, etc.)
Full Text: DOI


[1] Brauer, F.; Soudack, A. C., Stability regions and transition phenomena for harvested predator-prey systems, J. Math. Biol., 7, 319-337 (1979) · Zbl 0397.92019
[2] Brauer, F.; Soudack, A. C., Stability regions in predator-prey systems with constant-rate prey harvesting, J. Math. Biol., 8, 55-71 (1979) · Zbl 0406.92020
[3] Dai, G.; Tang, M., Coexistence region and global dynamics of a harvested predator-prey system, SIAM J Appl. Math., 58, 193-210 (1978) · Zbl 0916.34034
[4] Myerscough, M. R.; Gray, B. E.; Hograth, W. L.; Norbury, J., An analysis of an ordinary differential equation model for a two-species predator-prey system with harvesting and stocking, J. Math. Biol., 30, 389-411 (1992) · Zbl 0749.92022
[5] Chaudhuri, K. S.; Ray, S. S., On the combined harvesting of a prey-predator system, J. Biol. Sys., 4, 373-389 (1996)
[6] Leung, A., Optimal harvesting coefficient control of steady state prey-predator diffusive Volterra-Lotka systems, Appl. Math. Optim., 31, 219 (1995) · Zbl 0820.49011
[7] Clark, C. W., Mathematical Bioeconomics, The Optimal Management of Renewable Resources (1979), John Wiley & Sons: John Wiley & Sons New York · Zbl 0364.90002
[8] Levin, S. A.; Hallam, T. G.; Gross, J. L., Applied Mathematical Ecology (1989), Springer-Verlag
[9] Aiello, W. G.; Freedman, H. I., A time delay model of single species growth with stage structure, Math. Biosci., 101, 139 (1990) · Zbl 0719.92017
[10] Freedman, H. I.; Gopalsammy, K., Global stability in time-delayed single species dynamics, Bull. Math. Biol., 48, 485 (1986) · Zbl 0606.92020
[11] Rosen, G., Time delays produced by essential nonlinearity in population growth models, Bull. Math. Biol., 49, 253 (1987) · Zbl 0614.92015
[12] Fisher, M. E.; Goh, B. S., Stability results for delayed recruitment models in population dynamics, J. Math. Biol., 19, 147 (1984) · Zbl 0533.92017
[13] Cushing, J. M., Stability and maturation periods in age structured populations, (Busenberg, S.; Cooke, K. L., Differential Equations and Applications in Ecology, Epidemics and Population Problems (1981), Academic Press: Academic Press New York) · Zbl 0484.92022
[14] Hale, J. K., Ordinary Differential Equations (1969), Wiley: Wiley New York · Zbl 0186.40901
[15] Cushing, J. M., Integro-Differential Equations and Delay Models in Population Dynamics (1977), Springer-Verlag: Springer-Verlag Heidelberg · Zbl 0363.92014
[16] Cushing, J. M.; Saleem, M., A predator-prey model with age structure, J. Math. Biol., 14, 231-250 (1982) · Zbl 0501.92018
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.