On non-selective harvesting of a multispecies fishery incorporating partial closure for the populations. (English) Zbl 1329.92100
Summary: A prey-predator type fishery model incorporating partial closure for the populations is described in this paper. The proposed model deals with a problem of non-selective harvesting of a prey-predator system in which both the prey and the predator species obey logistic law of growth. The predator dependent Beddington DeAngelis type functional response is taken into consideration. Dynamic behavior of the system is analyzed. Partial closure for the populations is considered as a controlling instrument to regulate the harvesting of the populations. A dynamic framework towards the optimal utilization of the resource is developed using Pontryagin’s maximum principle. The optimal system is numerically solved using an iterative method with Runge-Kutta fourth order scheme. Simulation results show that the optimal control scheme can achieve sustainable ecosystem. Results are analyzed with the help of graphical illustrations.
MSC:
| 92D25 | Population dynamics (general) |
| 91B76 | Environmental economics (natural resource models, harvesting, pollution, etc.) |
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
Keywords:
prey-predator; beddington deangelis functional response; partial closure; global stability; optimal controlReferences:
| [1] | Clark, C. W., Bioeconomic Modelling and Fisheries Management (1985), Wiley Series: Wiley Series New York |
| [2] | Clark, C. W., Mathematical Bioeconomics: The Optimal Management of Renewable Resources (1990), Wiley Series: Wiley Series New York · Zbl 0712.90018 |
| [3] | Kot, M., Elements of Mathematical Ecology (2001), Cambridge University Press: Cambridge University Press Cambridge, UK |
| [4] | Chaudhuri, K. S., A bioeconomic model of harvesting a multispecies fishery, Ecol. Modell., 32, 267 (1986) |
| [5] | Kar, T. K.; Chaudhuri, K. S., On non-selective harvesting of a multispecies fishery, Int. J. Math. Educ. Sci. Technol., 33, 4, 543-556 (2002) |
| [6] | 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) |
| [7] | Mesterton-Gibbons, M., On the optimal policy for the combined harvesting of predator and prey, Nat. Resour. Model., 3, 63-90 (1988) · Zbl 0850.92067 |
| [8] | Mesterton-Gibbons, M., A technique for finding optimal two species harvesting policies, Ecol. Modell., 92, 235-244 (1996) |
| [9] | Leard, B.; Rebaza, J., Analysis of predator-prey models with continuous threshold harvesting, Appl. Math. Comput., 217, 12, 5265-5278 (2011) · Zbl 1207.92046 |
| [10] | Chaudhuri, K. S.; Saharay, S., On the combined harvesting of a prey-predator system, J. Biol. Syst., 4, 373-389 (1996) |
| [11] | 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 |
| [12] | Brauer, F.; Soudack, A. C., Constant-rate stocking of predator-prey systems, J. Math. Biol., 11, 1-14 (1981) · Zbl 0448.92020 |
| [13] | Dai, G.; Tang, M., Coexistence region and global dynamics of a harvested predator-prey systems, SIAM J. Appl. Math., 58, 193-210 (1998) · Zbl 0916.34034 |
| [14] | Myerscough, M. R.; Gray, B. F.; Hogarth, W. L.; Norbury, J., An analysis of an ordinary differential equations model for a two species predator-prey system with harvesting and stocking, J. Math. Biol., 30, 389-411 (1992) · Zbl 0749.92022 |
| [15] | Xiao, D.; Ruan, S., Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, Fields Inst. Commun., 21, 493-506 (1999) · Zbl 0917.34029 |
| [16] | Jia, Y.; Xu, H.; Agarwal, R. P., Existence of positive solutions for a prey-predator model with refuge and diffusion, Appl. Math. Comput., 217, 21, 8264-8276 (2011) · Zbl 1215.92064 |
| [17] | Liu, S.; Beretta, E.; Breda, D., Predator-prey model of Beddington-DeAngelis type with maturation and gestation delays, Nonlinear Anal. Real World Appl., 11, 5, 4072-4091 (2010) · Zbl 1208.34124 |
| [18] | Lin, G.; Hong, Y., Delay induced oscillation in predator-prey system with Beddington-DeAngelis functional response, Appl. Math. Comput., 190, 2, 1296-1311 (2007) · Zbl 1117.92055 |
| [19] | Chakraborty, K.; Jana, S.; Kar, T. K., Global dynamics and bifurcation in a stage structured prey-predator fishery model with harvesting, Appl. Math. Comput., 218, 18, 9271-9290 (2012) · Zbl 1246.92026 |
| [20] | Chakraborty, K.; Das, K.; Kar, T. K., Combined harvesting of a stage structured prey-predator model incorporating cannibalism in competitive environment, C. R. Biol., 336, 34-45 (2013) |
| [21] | Dimitrov, D. T.; Kojouharov, H. V., Complete mathematical analysis of predator-prey models with linear prey growth and Beddington-DeAngelis functional response, Appl. Math. Comput., 162, 2, 523-538 (2005) · Zbl 1057.92050 |
| [22] | Xu, R.; Ma, Z., Stability and Hopf bifurcation in a ratio-dependent predator prey system with stage-structure, Chaos, Solitons Fractals, 38, 669-684 (2008) · Zbl 1146.34323 |
| [23] | Gao, S. J.; Chen, L. S.; Teng, Z. D., Hopf bifurcation and global stability for a delayed predator-prey system with stage structure for predator, Appl. Math. Comput., 202, 721-729 (2008) · Zbl 1151.34067 |
| [24] | Kuang, Y.; Takeuchi, Y., Predator-prey dynamics in models of prey dispersal in two-patch environments, Math. Biosci., 120, 77-98 (1994) · Zbl 0793.92014 |
| [25] | Song, X.; Guo, H., Global stability of a stage-structured predator-prey system, Int. J. Biomath., 1, 3, 313-326 (2008) · Zbl 1173.34043 |
| [26] | Beddington, J. R., Mutual interference between parasites or predator and its effect on searching efficiency, J. Anim. Ecol., 44, 331-340 (1975) |
| [27] | DeAngelis, D. L.; Goldstain, R. A.; O’Neil, R. V., A model for tropic interaction, Ecology, 56, 881-892 (1975) |
| [28] | Huisman, G.; De Boer, R. J., A formal derivation of the “Beddington” functional response, J. Theor. Biol., 185, 389-400 (1997) |
| [29] | Skalski, G. T.; Gilliam, J. F., Functional responses with predator interference: viable alternatives to the Holling type II model, Ecology, 82, 3083-3092 (2001) |
| [31] | Workman, J. T.; Lenhart, S., Optimal Control Applied to Biological Models (2007), Chapman and Hall/CRC · Zbl 1291.92010 |
| [32] | Hackbush, W., A numerical method for solving parabolic equations with opposite orientations, Computing, 20, 3, 229-240 (1978) · Zbl 0391.65044 |
| [33] | Jakobsson, J., Recent variability in the fisheries of the North Atlantic, ICES Mar. Sci. Symp., 195, 291-315 (1992) |
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