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The effect of seasonal harvesting on a single-species discrete population model with stage structure and birth pulses

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  • Gao, Shujing
  • Chen, Lansun

Abstract

In this paper, we propose an exploited single-species discrete model with stage structure for the dynamics in a fish population for which births occur in a single pulse once per time period. Using the stroboscopic map, we obtain an exact cycle of the system, and obtain the threshold conditions for its stability. Bifurcation diagrams are constructed with the birth rate as the bifurcation parameter, and these are observed to display complex dynamic behaviors, including chaotic bands with period windows, pitch-fork and tangent bifurcation. This suggests that birth pulse provides a natural period or cyclicity that makes the dynamical behavior more complex. Moreover, we show that the timing of harvesting has a strong impact on the persistence of the fish population, on the volume of mature fish stock and on the maximum annual-sustainable yield. An interesting result is obtained that, after the birth pulse, the earlier culling the mature fish, the larger harvest can tolerate.

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  • Gao, Shujing & Chen, Lansun, 2005. "The effect of seasonal harvesting on a single-species discrete population model with stage structure and birth pulses," Chaos, Solitons & Fractals, Elsevier, vol. 24(4), pages 1013-1023.
    • Handle: RePEc:eee:chsofr:v:24:y:2005:i:4:p:1013-1023
    DOI: 10.1016/j.chaos.2004.09.091
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    Cited by:

    1. Jiao, Jianjun & Chen, Lansun & Cai, Shaohong, 2009. "A delayed stage-structured Holling II predator–prey model with mutual interference and impulsive perturbations on predator," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1946-1955.
    2. Liu, Bing & Teng, Zhidong & Liu, Wanbo, 2007. "Dynamic behaviors of the periodic Lotka–Volterra competing system with impulsive perturbations," Chaos, Solitons & Fractals, Elsevier, vol. 31(2), pages 356-370.
    3. Zhang, Xue & Zhang, Qing-Ling & Liu, Chao & Xiang, Zhong-Yi, 2009. "Bifurcations of a singular prey–predator economic model with time delay and stage structure," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1485-1494.
    4. Wang, Fengyan & Pang, Guoping, 2009. "The global stability of a delayed predator–prey system with two stage-structure," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 778-785.
    5. Jiang, Zhichao & Wei, Junjie, 2008. "Stability and bifurcation analysis in a delayed SIR model," Chaos, Solitons & Fractals, Elsevier, vol. 35(3), pages 609-619.
    6. Zhao, Hongyong & Zhang, Xuebing & Huang, Xuanxuan, 2015. "Hopf bifurcation and spatial patterns of a delayed biological economic system with diffusion," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 462-480.
    7. Yang, Xiaofeng & Jin, Zhen & Xue, Yakui, 2007. "Weak average persistence and extinction of a predator–prey system in a polluted environment with impulsive toxicant input," Chaos, Solitons & Fractals, Elsevier, vol. 31(3), pages 726-735.
    8. Yun Liu & Lifeng Guo & Xijuan Liu, 2023. "Dynamical Behaviors in a Stage-Structured Model with a Birth Pulse," Mathematics, MDPI, vol. 11(15), pages 1-13, July.
    9. Xiang, Zhongyi & Song, Xinyu, 2007. "A model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with periodic input," Chaos, Solitons & Fractals, Elsevier, vol. 32(4), pages 1419-1428.
    10. Gao, Shujing & Chen, Lansun & Sun, Lihua, 2005. "Optimal pulse fishing policy in stage-structured models with birth pulses," Chaos, Solitons & Fractals, Elsevier, vol. 25(5), pages 1209-1219.
    11. Banerjee, Ritwick & Das, Pritha & Mukherjee, Debasis, 2018. "Stability and permanence of a discrete-time two-prey one-predator system with Holling Type-III functional response," Chaos, Solitons & Fractals, Elsevier, vol. 117(C), pages 240-248.
    12. Çelik, C. & Merdan, H. & Duman, O. & Akın, Ö., 2008. "Allee effects on population dynamics with delay," Chaos, Solitons & Fractals, Elsevier, vol. 37(1), pages 65-74.
    13. Cui, Qianqian & Zhang, Qiang & Qiu, Zhipeng & Hu, Zengyun, 2016. "Complex dynamics of a discrete-time predator-prey system with Holling IV functional response," Chaos, Solitons & Fractals, Elsevier, vol. 87(C), pages 158-171.
    14. Merdan, H. & Duman, O. & Akın, Ö. & Çelik, C., 2009. "Allee effects on population dynamics in continuous (overlapping) case," Chaos, Solitons & Fractals, Elsevier, vol. 39(4), pages 1994-2001.
    15. Gu, En-Guo & Hao, Yu-Dong, 2007. "On the global analysis of dynamics in a delayed regulation model with an external interference," Chaos, Solitons & Fractals, Elsevier, vol. 34(4), pages 1272-1284.
    16. Zhao, Hongyong & Huang, Xuanxuan & Zhang, Xuebing, 2015. "Hopf bifurcation and harvesting control of a bioeconomic plankton model with delay and diffusion terms," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 421(C), pages 300-315.
    17. Gao, Shujing & Teng, Zhidong & Xie, Dehui, 2009. "Analysis of a delayed SIR epidemic model with pulse vaccination," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 1004-1011.
    18. Gao, Shujing & Chen, Lansun & Sun, Lihua, 2005. "Dynamic complexities in a seasonal prevention epidemic model with birth pulses," Chaos, Solitons & Fractals, Elsevier, vol. 26(4), pages 1171-1181.
    19. Ahjond S. Garmestani & Craig R. Allen & Colin M. Gallagher & John D. Mittelstaedt, 2007. "Departures from Gibrat's Law, Discontinuities and City Size Distributions," Urban Studies, Urban Studies Journal Limited, vol. 44(10), pages 1997-2007, September.

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