Persistence and global stability in discrete models of pure-delay nonautonomous Lotka–Volterra type

Consider the persistence and the global asymptotic stability of the following discrete model of pure-delay nonautonomous Lotka–Volterra type: <fen><cp type="lcub" STYLE="S"><ar><r><c CSPAN="1" RSPAN="1" CA="L" RA="T">Ni(p+1)=Ni(p)exp<fen><cp type="lcub" STYLE="S">ci(p)−∑lower limit j=1, upper limit n ∑lower limit l=0, upper limit m aijl(p)Nj(p−kl)<cp type="rcub" STYLE="S"></fen>,</c><c CSPAN="1" RSPAN="1" CA="L" RA="T">p=0,1,2,…, 1&les;i&les;n,</c></r><r><c CSPAN="1" RSPAN="1" CA="L" RA="T">Ni(p)=Ni,p&ges;0, p&les;0, and Ni,0>0, 1&les;i&les;n,</c><c CSPAN="1" RSPAN="1" CA="L" RA="T"></c></r></ar></fen> where each <f>ci(p)</f> and <f>aijl(p)</f> are bounded for <f>p&ges;0</f> and inflower limit p&ges;0 <fen><cp type="lpar" STYLE="S">∑lower limit l=0, upper limit m aiil(p)<cp type="rpar" STYLE="S"></fen>>0, aijl(p)&ges;0, i&les;j&les;n, 1&les;i&les;n, and kl&ges;0, 1&les;l&les;m. In this paper, for the above discrete system of pure-delay type, by improving the former work [J. Math. Anal. Appl. 273 (2002) 492–511] which extended the averaged condition offered by S. Ahmad and A.C. Lazer [Nonlinear Anal. 40 (2000) 37–49], we offer conditions of persistence, and considering a Lyapunov-like discrete function to the above discrete system, we establish sufficient conditions of global asymptotic stability. [Copyright &y& Elsevier]