Periodic solutions of a discrete-time diffusive system governed by backward difference equations
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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:
a13 (t)x3 (t) + D1 (t) x2 (t) − x1 (t) , x˙1 (t) = x1 (t) a1 (t) − a11 (t)x1 (t) − m(t)x3 (t) + x1 (t)
x˙2 (t) = x2 (t) a2 (t) − a22 (t)x2 (t) + D2 (t) x1 (t) − x2 (t) ,
x˙3 (t) = x3 (t) − a3 (t) +
(1.1)
a31 (t)x1 (t − τ) , m(t)x3 (t − τ) + x1 (t − τ)
where xi (t) represents the prey population in the ith patch (i = 1,2), and x3 (t) represents the predator population, τ > 0 is a constant delay due to gestation, and Di (t) denotes the dispersal rate of the prey in the ith patch (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 ω-periodic functions. They proved that system (1.1) has at least one positive ω-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
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Periodic solutions of a discrete-time diffusive system
One question arises naturally. Does the discrete analog of system (1.1) have a positive periodic solution? The purpose of this paper is to answer this question to some extent. More precisely, we consider the following discrete-time diffusion system governed by backward difference equations:
a13 (k)x3 (k) x (k) − x1 (k) x1 (k) = x1 (k − 1)exp a1 (k) − a11 (k)x1 (k) − + D1 (k) 2 , m(k)x3 (k) + x1 (k) x1 (k)
x (k) − x2 (k) x2 (k) = x2 (k − 1)exp a2 (k) − a22 (k)x2 (k) + D2 (k) 1 , x2 (k)
a31 (k)x1 (k − l) x3 (k) = x3 (k − 1)exp − a3 (k) + m(k)x3 (k − l) + x1 (k − l)
(1.2) with initial condition xi (−m) ≥ 0,
m = 1,2,...,l;
xi (0) > 0
(i = 1,2,3),
(1.3)
where Di (k) (i = 1,2), ai (k) (i = 1,2,3), a11 (k), a13 (k), a22 (k), a31 (k), m(k) are strictly positive ω-periodic sequence, that is, Di (k + ω) = Di (k), ai (k + ω) = ai (k),
i = 1,2, i = 1,2,3,
a11 (k + ω) = a11 (k),
a13 (k + ω) = a13 (k),
a22 (k + ω) = a22 (k),
a31 (k + ω) = a31 (k),
(1.4)
m(k + ω) = m(k) for arbitrary integer k, where ω, a fixed positive integer, denotes the pr
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