Stability, bifurcation and transition to chaos in a model of immunosensor based on lattice differential equations with delay

• Main Authors: Vasyl Martsenyuk, Aleksandra Kłos-Witkowska, Andriy Sverstiuk
• Format: Article
• Language: English
• Published: University of Szeged 2018-05-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: biosensor; immunosensor; lattice differential equations; differential equations with delay; asymptotic stability; lyapunov functional; bifurcation; chaos
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=6354

In the work we proposed the model of immunosensor, which is based on the system of lattice differential equations with delay. The conditions of local asymptotic stability for endemic state are gotten. For this purpose we have used method of Lyapunov functionals. It combines general approach to construction of Lyapunov functionals of the predator–prey models with lattice differential equations. Numerical examples have showed the influence on stability of model parameters. From our numerical simulations, we have found evidence that chaos can occur through variation in the time delay. Namely, as the time delay was increased, the stable endemic solution changed at a critical value of $\tau$ to a stable limit cycle. Further, when increasing the time delay, the behavior changed from convergence to simple limit cycle to convergence to complicated limit cycles with an increasing number of local maxima and minima per cycle until at sufficiently high time delay the behavior became chaotic.