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A discrete predator-prey system with age-structure for predator and naturalbarriers for prey

Published online by Cambridge University Press:  15 April 2002

Sanyi Tang  and
Lansun Chen
Sanyi Tang
Affiliation:
Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, P. R. China. (tsy@math08.math.ac.cn)
Lansun Chen
Affiliation:
Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, P. R. China. (tsy@math08.math.ac.cn)
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Abstract

We analyze a two species discrete predator-prey model in which the prey disperses between two patches of a heterogeneous environment with barriers and the mature predator disperses between the patches with no barrier. By using the discrete dynamical system generated by a monotone, concave maps for subcommunity of prey, we obtain the subcommunity of prey exists an equilibrium which attracts all positive solutions, and using the stability trichotomy results on the monotone and continuous operator, we obtain some sufficient conditions for the permanence of species. These results are applied to the models with rational growth functions and exponential growth functions. We also present numerical examples to illustrate the dynamic complexity of systems.

Keywords

Age-structurenatural barriersubcommunitypersistence.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 4 , July 2001 , pp. 675 - 690
Copyright
© EDP Sciences, SMAI, 2001

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References

Allen, J.S., Moulton, M.P. and Rose, F.L., Persistence in an age-structure population for a patch-type environment. Nat. Resour. Model. 4 (1990) 197-214. CrossRef
Beddington, J.R., Free, C.A. and Lawton, J.H., Dynamical complexity in predator-prey models framed in difference equations. Nature 255 (1975) 58-60. CrossRef
Chen, J.F., Influence of high dimension terms for qualitative structure of solutions of a second order linear difference system with ordinary coefficient in the neighborhood of a singular point. Acta Math. Appl. Sinica (China) 11 (1988) 299-311.
Clark, M.E. and Gross, L.J., Periodic solutions to nonautonomous difference equations. Math. Biosci. 102 (1990) 105-119. CrossRef
J.M. Cushing, An introduction to structured population dynamics. SIAM Soc. Indus. Appl. Math., Philadelphia (1998).
Freedman, H.I. and Persistence, J.W.-H. So in discrete models of a population which may be subjected to harvesting. Nat. Resour. Model. 2 (1987) 135-145. CrossRef
Freedman, H.I. and Takeuchi, Y., Global stability and predator dynamics in a model of prey dispersal in a patchy environment. Nonlinear Anal. TMA 13 (1989) 993-1002. CrossRef
Freedmen, H.I. and Josephso, W.H., Persistence in discrete semidynamical systems. SIAM J. Math. Anal. 20 (1989) 930-938. CrossRef
Gumowski, I. and C.Mira, Recurrences and discrete dynamics systems. Lect. Notes Math. 809 (1980) 61-96. CrossRef
Hastings, A., Complex interactions between dispersal and dynamics: Lessons from coupled logistic equations. Ecology 74 (1993) 1362-1372. CrossRef
Hutson, V. and Schmitt, K., Permanence and the dynamics of biological systems. Math. Biosci. 111 (1992) 1-71. CrossRef
Krause, U. and Ranet, P., A limit set trichotomy for monotone nonlinear dynamical systems. Nonlinear Anal. TMA 19 (1992) 375-392. CrossRef
Kuang, Y. and Takeuchi, Y., Predator-prey dynamics in models of prey dispersal in two patch environments. Math. Biosci. 120 (1994) 77-98. CrossRef
N.R. Leblond, Porcupine caribou herd. Canadian Arctic Resources Comn., Offuwa (1979).
Levin, S.A., Dispersion and population interactions. Amer. Natur. 108 (1974) 207-228. CrossRef
Persistence, J. Li in discrete age-structure population models. Bull. Math. Biol. 50 (1992) 351-366.
A. Okubo, Diffusion and ecological problems, math. models. Springer, Berlin (1980).
Schumacher, K., Regions and oscillations in second order predator-prey recurrences. J. Math. Biol. 16 (1983) 221-231. CrossRef
Selgrade, J.F. and Namkoong, G., Stable periodic behavior in a pioneer-climax model. Nat. Resour. Model. 4 (1990) 215-227. CrossRef
Skellam, J.G., Random dispersal in theoretical populations. Biometrika 38 (1951) 196-218. CrossRef
Smith, H.L., Cooperative systems of differential equations with concave nonlinearities. Nonlinear Anal. TMA 10 (1986) 1037-1052. CrossRef
Takeuchi, Y., Cooperative systems theory and global stability of diffusion models. Acta Appl. Math. 14 (1989) 49-57. CrossRef
Wang, W.D. and Chen, L.S., A predator-prey system with stage-structure for predator. Comput. Math. Appl. 33 (1997) 83-91. CrossRef
Yakubu, A.-A., Prey dominance in discrete predator-prey system with a prey refuge. Math. Biosci. 144 (1997) 155-178. CrossRef