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.
@article{M2AN_2001__35_4_675_0, author = {Tang, Sanyi and Chen, Lansun}, title = {A discrete predator-prey system with age-structure for predator and natural barriers for prey}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, volume = {35}, year = {2001}, pages = {675-690}, mrnumber = {1862874}, zbl = {0993.39009}, language = {en}, url = {http://dml.mathdoc.fr/item/M2AN_2001__35_4_675_0} }
Tang, Sanyi; Chen, Lansun. A discrete predator-prey system with age-structure for predator and natural barriers for prey. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 35 (2001) pp. 675-690. http://gdmltest.u-ga.fr/item/M2AN_2001__35_4_675_0/
[1] Persistence in an age-structure population for a patch-type environment. Nat. Resour. Model. 4 (1990) 197-214. | Zbl 0850.92063
, and ,[2] Dynamical complexity in predator-prey models framed in difference equations. Nature 255 (1975) 58-60.
, and ,[3] 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. | Zbl 0673.39002
,[4] Periodic solutions to nonautonomous difference equations. Math. Biosci. 102 (1990) 105-119. | Zbl 0712.39014
and ,[5] An introduction to structured population dynamics. SIAM Soc. Indus. Appl. Math., Philadelphia (1998). | MR 1636703 | Zbl 0939.92026
,[6] Persistence in discrete models of a population which may be subjected to harvesting. Nat. Resour. Model. 2 (1987) 135-145. | Zbl 0850.92074
and ,[7] Global stability and predator dynamics in a model of prey dispersal in a patchy environment. Nonlinear Anal. TMA 13 (1989) 993-1002. | Zbl 0685.92018
and ,[8] Persistence in discrete semidynamical systems. SIAM J. Math. Anal. 20 (1989) 930-938. | Zbl 0676.92011
and ,[9] Recurrences and discrete dynamics systems. Lect. Notes Math. 809 (1980) 61-96. | Zbl 0449.58003
and ,[10] Complex interactions between dispersal and dynamics: Lessons from coupled logistic equations. Ecology 74 (1993) 1362-1372.
,[11] Permanence and the dynamics of biological systems. Math. Biosci. 111 (1992) 1-71. | Zbl 0783.92002
and ,[12] A limit set trichotomy for monotone nonlinear dynamical systems. Nonlinear Anal. TMA 19 (1992) 375-392. | Zbl 0779.34038
and ,[13] Predator-prey dynamics in models of prey dispersal in two patch environments. Math. Biosci. 120 (1994) 77-98. | Zbl 0793.92014
and ,[14] Porcupine caribou herd. Canadian Arctic Resources Comn., Offuwa (1979).
,[15] Dispersion and population interactions. Amer. Natur. 108 (1974) 207-228.
,[16] Persistence in discrete age-structure population models. Bull. Math. Biol. 50 (1992) 351-366. | Zbl 0659.92019
,[17] Diffusion and ecological problems, math. models. Springer, Berlin (1980). | MR 572962 | Zbl 0422.92025
,[18] Regions and oscillations in second order predator-prey recurrences. J. Math. Biol. 16 (1983) 221-231. | Zbl 0513.92018
,[19] Stable periodic behavior in a pioneer-climax model. Nat. Resour. Model. 4 (1990) 215-227. | Zbl 0850.92060
and ,[20] Random dispersal in theoretical populations. Biometrika 38 (1951) 196-218. | Zbl 0043.14401
,[21] Cooperative systems of differential equations with concave nonlinearities. Nonlinear Anal. TMA 10 (1986) 1037-1052. | Zbl 0612.34035
,[22] Cooperative systems theory and global stability of diffusion models. Acta Appl. Math. 14 (1989) 49-57. | Zbl 0665.92017
,[23] A predator-prey system with stage-structure for predator. Comput. Math. Appl. 33 (1997) 83-91.
and ,[24] Prey dominance in discrete predator-prey system with a prey refuge. Math. Biosci. 144 (1997) 155-178. | Zbl 0896.92031
,