Qualitative analysis of discrete nonlinear delay survival red blood cells model

The objective of this paper is to systematically study the qualitative properties of the solutions of the discrete nonlinear delay survival red blood cells model x(n+1)-x(n)=-δ(n)x(n)+p(n)e-q(n)x(n-ω),n=1,2,…,where δ(n)δ(n), p(n)p(n) and q(n)q(n) are positive periodic sequences of period ωω. First, by using the continuation theorem in coincidence degree theory, we prove that the equation has a positive periodic solution x¯(n) with strictly positive components. Second, we prove that the solutions are permanent and establish some sufficient conditions for oscillation of the positive solutions about x¯(n). Finally, we give an estimation of the lower and upper bounds of the oscillatory solution and establish some sufficient conditions for global attractivity of x¯(n). From applications point of view permanence guarantees the long term survival of mature cells, oscillation implies the prevalence of the mature cells around the periodic solution and the convergence implies the absence of any dynamical diseases in the population. Our results in the special case when the coefficients are positive constants involve and improve the oscillation and global attractivity results on the literature.