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Qualitative analysis of discrete nonlinear delay survival red blood cells model. (English) Zbl 1136.92009

Summary: 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
\[ x(n+1)- x(n)= -\delta(n)x(n)+ p(n) e^{-q(n)x(n-\omega)}, \quad n=1,2,\dots, \]
where \(\delta(n)\), \(p(n)\) and \(q(n)\) are positive periodic sequences of period \(\omega\). First, by using the continuation theorem in coincidence degree theory, we prove that the equation has a positive periodic solution \(\overline{x}(n)\) with strictly positive components. Second, we prove that the solutions are permanent and establish some sufficient conditions for oscillation of the positive solutions about \(\overline{x}(n)\). Finally, we give an estimation of the lower and upper bounds of the oscillatory solution and establish some sufficient conditions for global attractivity of \(\overline{x}(n)\). From the 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 in the literature.

MSC:

92C30 Physiology (general)
39A11 Stability of difference equations (MSC2000)
34K60 Qualitative investigation and simulation of models involving functional-differential equations
46N60 Applications of functional analysis in biology and other sciences
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