The stage-structured predator-prey model and optimal harvesting policy. (English) Zbl 0961.92037

Summary: We establish a mathematical model of two species with stage structure and the relation of predator-prey, to obtain the necessary and sufficient condition for the permanence of two species and the extinction of one species or two species. We also obtain the optimal harvesting policy and the threshold of the harvesting for sustainable development.


92D40 Ecology
34D05 Asymptotic properties of solutions to ordinary differential equations
37N25 Dynamical systems in biology
34D20 Stability of solutions to ordinary differential equations
Full Text: DOI


[1] L. Chen, J. Chen, Nonlinear Biological Dynamic Systems, Science, Beijing, 1993 (in Chinese)
[2] L. Chen, Mathematical Models and Methods in Ecology, Science, Beijing, 1988 (in Chinese)
[3] J. Hofbauer, K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University, Cambridge, 1998 · Zbl 0914.90287
[4] H.I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York, 1980 · Zbl 0448.92023
[5] J.D. Murray, Mathematical Biology, 2nd, corrected Ed., Springer, Heidelberg, 1993 · Zbl 0779.92001
[6] Y. Takeuchi, Global Dynamical Properties of Lotka-Volterra Systems, World Scientific, Singapore, 1996 · Zbl 0844.34006
[7] R.M. May, Stability and Complexity in Model Ecosystems, 2nd Ed., Princeton University, Princeton, NJ, 1975
[8] R.M. May, Theoretical Ecology, Principles and Applications, 2nd Ed., Blackwell, Oxford, 1981
[9] Takeuchi, Y; Oshime, Y; Matsuda, H, Persistence and periodic orbits of a three-competitor model with refuges, Math. biosci., 108, 1, 105, (1992) · Zbl 0748.92017
[10] Matsuda, H; Namba, Toshiyuki co-evolutionarily stable community structure in a patchy environment, J. theoret. biol., 136, 2, 229, (1989)
[11] 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
[12] Aiello, W.G; Freedman, H.I; Wu, J, Analysis of a model representing stage-structured population growth with stage-dependent time delay, SIAM appl. math., 52, 855, (1992) · Zbl 0760.92018
[13] Wang, W; Chen, L, A predator – prey system with stage structure for predator, Comput. math. appl., 33, 8, 207, (1997)
[14] C.W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, 2nd Ed., Wiley, New York, 1990 · Zbl 0712.90018
[15] Leng, A.W, Optimal harvesting-coefficient control of steady-state prey – predator diffusive volterra – lotka systems, Appl. math. optim., 31, 2, 219, (1995) · Zbl 0820.49011
[16] Eiolko; Mariusz; Kozlowski, Some optimal models of growth in biology, IEEE trans. automat. control, 40, 10, 1779, (1995) · Zbl 0833.92009
[17] Bhattacharya, D.K; Begun, S, Bioeconomic equilibrium of two-species systems I, Math. biosci., 135, 2, 111, (1996) · Zbl 0856.92018
[18] J.K. Hale, Ordinary Differential Equations, Wiley, New York, 1969 · Zbl 0186.40901
[19] B. Chen, The Chinese Alligator or Yangtzi Alligator, in: S. Wang, E. Zhao (Eds.), China Red Data Book of Endangered Animals - Amphibia and Raptilia, Science, Beijing, 1998, p. 311
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