Positive Periodic Solution and Numerical Optimization in the Harvesting Effort for a Single-Species Stage-Structured System with Birth Pulses
In this paper, we consider a single-species stage-structured model with birth pulses and the harvesting of the species. Especially, we assume that the species can be divided into the immature and the mature, which exhibit different death rates. The mature species reproduces at fixed moments each year because the birth of many species is seasonal or occurs in a regular pulse, and the species is harvested not during the whole year but during a single period of the year. For such a system, we obtain conditions which guarantee the existence of a stable positive periodic solution. This implies that sustainable exploitation of the species can be maintained if we use the proper strategy in the harvesting effort and timing. Further, in order to get the maximum annual sustainable yield, we optimize the harvesting using numerical analysis; in addition, we find that the harvesting timing affects the maximum annual sustainable yield. Lastly, we show the effects of birth rate and harvesting effort on the dynamical complexity of the system with the help of a bifurcation graph.
KeywordsDeath Rate Mathematical Modeling Birth Rate Computational Mathematic Industrial Mathematic
Unable to display preview. Download preview PDF.
- 2.J. Cui and L. Chen, “The effect of dispersal on population growth with stage-structure,” Comput. Math. Appl., 39, No.1/2, 91–102 (2000).Google Scholar
- 4.H. I. Freedman, J. W.-H., So and J. Wu, “A model for the growth of a population exhibiting stage structure: Cannibalism and cooperation,” J. Comp. Appl. Math., 52, 177–198 (1994).Google Scholar
- 5.W. G. Aiello, H. I. Freedman, and J. Wu, “Analysis of a model representing stage-structured population growth with state-dependent time delay,” SIAM J. Appl. Math., 52, 855–869 (1992).Google Scholar
- 6.S. Liu, L. Chen, and G. Luo, “Extinction and permanence in competitive stage structured system with time-delays,” Nonlinear Anal., 51, 1347–1361 (2002).Google Scholar
- 7.S. Liu, L. Chen, and R. Agarwal, “Recent progress on stage-structured population dynamics,” Math. Comput. Model., 36, 1319–1360 (2002).Google Scholar
- 8.C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Wiley, New York, 1990.Google Scholar
- 9.P. H. Spencer and J. S. Collie, “A simple predator-prey model of exploited marine fish population incorporating alternative prey,” ICES J. Mar. Sci., 53, 615 (1994).Google Scholar
- 10.A. W. Leung, “Optimal harvesting-coefficients control of steady-state prey-predator diffusive Volterra-Lotka system,” Appl. Math. Optim., 31, No.2, 219–241 (1995).Google Scholar
- 11.X. Song and L. Chen, “Optimal harvesting policy and stability for single-species growth model with stage structure,” J. Syst. Sci. Complex., 15, No.2, 194–201 (2002).Google Scholar
- 12.X. Song and L. Chen, “Optimal harvesting and stability for a predator-prey system with stage-structure,” Acta Math. Appl. Sini., 18, 423–430 (2002).Google Scholar