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Bifurcations and Control in a Singular Biological Economic Model

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Complexity, Analysis and Control of Singular Biological Systems

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 421))

Introduction

In the natural world, there are many species whose individuals have a life history that takes them through two stages, juvenile stage and adult stage. Individuals in each stage are identical in biological characteristics, and some vital rates (rates of survival, development, and reproduction) of individuals in a population almost always depend on stage structure. Furthermore, there is a strong interaction relationship between the mature population and the immature population , which is to some extent relevant to the persistence and extinction of the related population. Consequently, it is constructive to investigate the dynamics of such ecosystem without ignorance of stage structure for population.

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References

  1. Aiello, W.G., Freedman, H.I.: A time-delay model of single-species growth with stage structure. Math. Bios. 101, 139–153 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aiello, W.G., Freedman, H.J., Wu, J.: Analysis of a model representing stage-structured population growth with state-dependent time delay. SIAM J. Appl. Math. 52, 855–869 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. Al-Omari, J., Gourley, S.: Stability and traveling fronts in Lotka-Volterra competition models with stage structure. SIAM J. Appl. Math. 63, 2063–2086 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arino, O., Sanchez, E., Fathallah, A.: State-dependent delay differential equations in population dynamics: Modeling and analysis. Fiel. Inst. Commu. Amer. Math. Soci. 29, 19–36 (2001)

    MathSciNet  Google Scholar 

  5. Bandyopadhyay, M., Banerjee, S.: A stage-structured prey-predator model with discrete time delay. Appl. Math. Compu. 182, 1385–1398 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Barclay, H.J., Driessche, V.D.: A model for a single species with two life history stages and added mortality. Ecol. Model 11, 157–166 (1980)

    Article  Google Scholar 

  7. Beardmore, R.E.: The singularity-induced bifurcation and its kronecker normal form. SIAM J. Matr. Ana. Appl. 23(1), 126–137 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, L.S., Ruan, S.G., Zhu, J.: Adv. Top. Biomath. Oscillations in a stage structured predator-prey system with time dependent coefficients. World Scientific, Singapore (1997)

    Google Scholar 

  9. Clark, C.W.: Mathematical Bioeconomics: The Optimal Management of Renewable Resource, 2nd edn. John Wiley and Sons, New York (1990)

    Google Scholar 

  10. Cui, J., Chen, L.S., Wang, W.D.: The effect of dispersal on population growth with stage-structure. Compu. Math. Appl. 39, 91–102 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dai, G.R., Tang, M.X.: Coexistence region and global dynamics of a harvested predator-prey system. SIAM J. Appl. Math. 58(1), 193–210 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dai, L.: Singular Control System. Springer, New York (1989)

    Book  Google Scholar 

  13. Edwards, H.J., Dytham, C., Pitchford, J.W., Righton, D.: Prey selection, vertical migrations and the impacts of harvesting upon the population dynamics of a predator-prey system. Bull. Math. Biol. 69, 1827–1846 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Freedman, H.I., Joseph, W.H.S., Wu, J.H.: A model for the growth of a population exhibiting stage structure: Cannibalism and cooperation. J. Compu. Appl. Math. 52, 177–198 (1994)

    Article  MATH  Google Scholar 

  15. Freedman, H., Rao, V.S.H.: The trade-off between mutual interference and time lags in predator-prey systems. Bull. Math. Biol. 45, 991–1004 (1983)

    MathSciNet  MATH  Google Scholar 

  16. Gakkhar, S., Singh, B.: The dynamics of a food web consisting of two preys and a harvesting predator. Chaos Soli. Frac. 34(4), 1346–1356 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gordon, H.S.: The economic theory of a common property resource: The fishery. J. Poli. Eco. 62(2), 124–142 (1954)

    Article  Google Scholar 

  18. Gourley, S.A., Kuang, Y.: A stage structured predator-prey model and its dependence on maturation delay and death rate. J. Math. Biol. 49, 188–200 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Field. Springer, New York (1983)

    Google Scholar 

  20. Gurney, W.S.C., Blythe, S.P., Nisbet, R.M.: Nicholson’s blowflies revisited. Nature 287, 17–21 (1980)

    Article  Google Scholar 

  21. Gurney, W.S.C., Nisbet, R.M.: Fluctuating periodicity, generation separation, and the expression of larval competition. Theo. Popu. Biol. 28, 150–180 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1997)

    Google Scholar 

  23. Hastings, A.: Age-dependent predation is not a simple process, I, continuous time models. Theo. Popu. Biol. 23, 347–362 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hastings, A.: Delay in recruitment at different tropic levels: Effects on stability. J. Math. Biol. 21, 35–44 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hofbauer, J., Sigmund, K.: Evolutionary Games and Population Dynamics and Dynamical Systems. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  26. Huo, H., Li, W., Agarwal, R.P.: Optimal harvesting and stability for two species stage-structured system with cannibalism. Int. J. Appl. Math. 6, 59–79 (2001)

    MathSciNet  MATH  Google Scholar 

  27. Jiao, J.J., Chen, L.S.: Global attractivity of a stage-structure variable coefficients predator-prey system with time delay and impulsive perturbations on predators. Int. J. Biomath. 1(2), 197–208 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kar, T.K., Matsuda, H.: Controllability of a harvested prey-predator with time delay. J. Biol. Syst. 14(2), 243–254 (2006)

    Article  MATH  Google Scholar 

  29. Kolesov, Y.S.: Properties of solutions of a class of equations with lag which describe the dynamics of change in the population of a species with the age structure taken into account. Math. USSR Sbo. Tom. 45, 91–100 (1983)

    Article  MATH  Google Scholar 

  30. Kot, M.: Elements of Mathematical Biology. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  31. Landahl, H.D., Hanson, B.D.: A three stage population model with cannibalism. Bull. Math. Biol. 37, 11–17 (1995)

    Google Scholar 

  32. Liu, B., Zhang, Y.J., Chen, L.S.: Dynamic complexities in a Lotka-Volterra predator-prey model concerning impulsive control strategy. Int. J. Bifur. Chaos. 15(2), 517–531 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Liu, C., Zhang, Q.L., Zhang, Y., Duan, X.D.: Bifurcation and control in a differential-algebraic harvested prey-predator model with stage structure for predator. Int. J. Bifur. Chaos. 18(10), 3159–3168 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Liu, S.Q., Chen, L.S.: Extinction and permanence in competitive stage-structured system with time delay. Non. Anal. 51, 1347–1361 (2002)

    Article  MATH  Google Scholar 

  35. Liu, S.Q., Chen, L.S., Agarwal, R.: Recent progress on stage-structured population dynamics. Math. Compu. Model 36, 1319–1360 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  36. Liu, S.Q., Chen, L.S., Liu, Z.: Extinction and permanence in nonautonomous competitive system with stage structure. J. Math. Anal. Appl. 274, 667–684 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  37. Liu, S.Q., Chen, L.S., Luo, G.L., Jiang, Y.L.: Asymptotic behavior of competive Lotka-Volterra system with stage structure. J. Math. Anal. Appl. 271, 124–138 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  38. Liu, S.Q., Kouche, M., Tatar, N.: Permanence and global asymptotic stability in a stage structured system with distributed delays. J. Math. Anal. Appl. 301, 187–207 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  39. Magnusson, J.G.: Destabilizing effect of cannibalism on age structured predator-prey system. Math. Bios. 155, 195–200 (1999)

    Article  Google Scholar 

  40. Ou, L., Luo, G.L., Jiang, Y.L., Li, Y.P.: The asymptotic behavior of a stage-structured autonomous predator-prey system with time delay. J. Math. Anal. Appl. 283, 534–548 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  41. Song, X.Y., Chen, L.S.: Optimal harvesting and stability for a two species competitive system with stage structure. Math. Bios. 170, 173–186 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  42. Song, X.Y., Chen, L.S.: A predator-prey system with stage structure and harvesting for predator. Anna. Diff. Equa. 18, 264–277 (2002)

    MathSciNet  MATH  Google Scholar 

  43. Song, X.Y., Chen, L.S.: Modelling and analysis of a single species system with stage structure and harvesting. Math. Compu. Model 36, 67–82 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  44. Song, X.Y., Chen, L.S., Neumann, A.U.: Ratio-dependent predator-prey system with stage structure for prey. Disc. Conti. Dyna. Syst. B 4, 747–758 (2004)

    Article  MATH  Google Scholar 

  45. Song, X.Y., Cui, J.: The stage-structured predator-prey system with delay and harvesting. Appl. Anal. 81, 1127–1142 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  46. Song, X.Y., Guo, H.J.: Global stability of a stage-structured predator-prey system. Int. J. Biomath. 1(3), 313–326 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  47. Tang, S.Y., Chen, L.S.: Density-dependent birth rate, birth pulses and their population dynamic consequences. J. Math. Biol. 44, 185–199 (2002)

    Article  MathSciNet  Google Scholar 

  48. Tognetti, K.: The two stage stochastic model. Math. Bios. 25, 195–204 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  49. Venkatasubramanian, V., Schaettler, H., Zaborszky, J.: Local bifurcations and feasibility regions in differential-algebraic systems. IEEE Trans. Auto. Contr. 40(12), 1992–2013 (1995)

    Article  MATH  Google Scholar 

  50. Venkatasubramanian, V.: Singularity induced bifurcation and the van den Pol oscillator. IEEE Trans. Cir. Syst. I 41, 765–769 (1994)

    Article  MATH  Google Scholar 

  51. Wang, W.D.: Global dynamics of a population model with stage structure for predator. Advanced Topics in Biomathmatics, pp. 253–257. World Scientific, Singapore (1998)

    Google Scholar 

  52. Wang, W.D., Chen, L.S.: A predator-prey system with stage-structure for predator. Compu. Math. Appl. 33, 83–91 (1997)

    Article  Google Scholar 

  53. Wood, S.N., Blythe, S.P., Gurney, W.S.C., Nisbet, R.M.: Instability in mortality estimation schemes related to stage-structure population models. IMA J. Math. Appl. Medi. Biol. 6, 47–68 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  54. Xu, D., Zhao, X.: Dynamics in a periodic competitive model with stage structure. J. Math. Anal. Appl. 311(2), 417–438 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  55. Xu, R., Chaplin, M.A., Davidson, F.A.: Persistence and stability of a stage-structured predator-prey model with time delays. Appl. Math. Comp. 150, 259–277 (2004)

    Article  MATH  Google Scholar 

  56. Yang, L.J., Tang, Y.: An improved version of the singularity induced bifurcation theorem. IEEE Tran. Auto. Contr. 49(6), 1483–1486 (2001)

    Article  MathSciNet  Google Scholar 

  57. Zhang, H., Georgescu, P., Chen, L.S.: An impulsive predator-prey system with Beddington-DeAngelis functional response and time delays. Int. J. Biomath. 1(1), 1–17 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  58. Zhang, X., Chen, L., Neumann, U.A.: The stage-structured predator-prey model and optimal harvesting policy. Math. Bios. 168, 201–210 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  59. Zhang, X., Zhang, Q.L., Zhang, Y.: Bifurcations of a class of singular biological economic models. Chaos Soli. Frac. 42(3), 1485–1494 (2009)

    Article  MATH  Google Scholar 

  60. Zhang, Y., Zhang, Q.L.: Chaotic control based on descriptor bioeconomic systems. Contr. Deci. 22(4), 445–452 (2007)

    Google Scholar 

  61. Zhang, Y., Zhang, Q.L., Zhao, L.C.: Bifurcations and control in singular biological economical model with stage structure. J. Syst. Engin. 22(3), 232–238 (2007)

    Google Scholar 

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Authors and Affiliations

  1. College of Sciences, Northeastern University, WenHua Road, 110819, Shenyang, China, People’s Republic

    Qingling Zhang, Chao Liu & Xue Zhang

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  1. Qingling Zhang
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  2. Chao Liu
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Correspondence to Qingling Zhang .

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Zhang, Q., Liu, C., Zhang, X. (2012). Bifurcations and Control in a Singular Biological Economic Model. In: Complexity, Analysis and Control of Singular Biological Systems. Lecture Notes in Control and Information Sciences, vol 421. Springer, London. https://doi.org/10.1007/978-1-4471-2303-3_4

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  • DOI: https://doi.org/10.1007/978-1-4471-2303-3_4

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