Abstract
In this article, by a nonstandard finite-difference-θ (NSFD-θ) method we study the dynamics of a discrete red blood cell survival model. Firstly, the linear stability of the model is discussed. It is found that the Neimark-Sacker bifurcation exists when the delay passes a sequence of critical values. Then the explicit algorithm for determining the direction of the Neimark-Sacker bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form method and center manifold theorem. Our results show the NSFD-θ method could inherit the Hopf bifurcation and the asymptotically stability for sufficiently small step-size $h=1/m$ , where m is a positive integer. In particular, for $\theta=0,1/2,1$ the results hold for any step-size $h=1/m$ . Finally, numerical examples are provided to illustrate the theoretical results.
Highlights
1 Introduction In order to describe the survival of red blood cells in animal, Wazewska-Czyzewska and Lasota [ ] proposed the following autonomous functional differential equation: dx = –ax(t) + be–cx(t–τ), ( . )
It is desired that the discrete-time model is ‘dynamically consistent’ with the continuous-time model
In Section, we analyze the distribution of the characteristic equation associated with the discrete red blood cell survival model, and we obtain the existence of the Neimark-Sacker bifurcation
Summary
In [ , ], Wulf and Ford showed that, if applying the Euler forward method to solve the delay differential equation, the discrete scheme is ‘dynamically consistent.’ It means that for all sufficiently small step-sizes the discrete model undergoes a Hopf bifurcation of the same type as the original model. ). We obtain the consistent dynamical results of the corresponding continuous-time model by the NSFD-θ method for sufficiently small step-size. In Section , we analyze the distribution of the characteristic equation associated with the discrete red blood cell survival model, and we obtain the existence of the Neimark-Sacker bifurcation. (ii) If cu∗ > , applying Lemma , we know that all roots of Remark According to the conclusions of Theorem , we have results that are consistent with those for the corresponding continuous-time model (see [ ])
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