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PERIODIC SOLUTIONS OF A DISCRETE-TIME DIFFUSIVE SYSTEM GOVERNED BY BACKWARD DIFFERENCE

PERIODIC SOLUTIONS OF A DISCRETE-TIME DIFFUSIVE SYSTEM GOVERNED BY BACKWARD DIFFERENCE EQUATIONS BINXIANG DAI AND JIEZHONG ZOU Received 22 November 2004 and in revised form 16 January 2005 A discrete-time delayed diffusion model governed by backward difference equations is investigated. By using the coincidence degree and the related continuation theorem as well as some priori estimates, easily verifiable sufficient criteria are established for the existence of positive periodic solutions. 1. Introduction Recently, some biologists have argued that the ratio-dependent predator-prey model is more appropriate than the Gauss-type models for modelling predator-prey interactions where predation involves searching processes. This is strongly supported by. | PERIODIC SOLUTIONS OF A DISCRETE-TIME DIFFUSIVE SYSTEM GOVERNED BY BACKWARD DIFFERENCE EQUATIONS BINXIANG DAI AND JIEZHONG ZOU Received 22 November 2004 and in revised form 16 January 2005 A discrete-time delayed diffusion model governed by backward difference equations is investigated. By using the coincidence degree and the related continuation theorem as well as some priori estimates easily verifiable sufficient criteria are established for the existence of positive periodic solutions. 1. Introduction Recently some biologists have argued that the ratio-dependent predator-prey model is more appropriate than the Gauss-type models for modelling predator-prey interactions where predation involves searching processes. This is strongly supported by numerous laboratory experiments and observations 1 2 3 4 10 11 12 . Many authors 1 5 7 13 14 have observed that the ratio-dependent predator-prey systems exhibit much richer more complicated and more reasonable or acceptable dynamics. In view of periodicity of the actual environment Chen et al. 6 considered the following two-species ratiodependent predator-prey nonautonomous diffusion system with time delay x1 t - x1 t i a1 t - a11 t x1 t - si D1 t x2 t - x1 t m t x3 t X1 t x2 t - x2 t a2 t - a22 t x2 t D2 t x1 t - x2 t 1.1 31 t X1 t - T x3 t - x3 t - a3 t m t x3 t - T x1 t - T where xi t represents the prey population in the ith patch i 1 2 and x3 t represents the predator population T 0 is a constant delay due to gestation and Di t denotes the dispersal rate of the prey in the ithpatch i 1 2 . Di t i 1 2 ai t i 1 2 3 a11 t a13 t a22 t a31 t and m t are strictly positive continuous w-periodic functions. They proved that system 1.1 has at least one positive w-periodic solution if the conditions a31 t a3 t and m t a1 t a13 t are satisfied. Copyright 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005 3 2005 263-274 DOI 10.1155 ADE.2005.263 264 Periodic solutions of a discrete-time diffusive system One .