Stability Investigation of Biosensor Model Based on Finite Lattice Difference Equations

Abstract

We consider the delayed antibody-antigen competition model for two-dimensional array of biopixels

$$\begin{aligned} \begin{aligned} x_{i,j}(n+1)&=x_{i,j}(n)\exp \big \{\beta - \gamma y_{i,j}(n-r) - \delta _x x_{i,j}(n-r) \big \} + \hat{S}\left\{ x_{i,j}(n) \right\} ,\\ y_{i,j}(n+1)&=y_{i,j}(n)\exp \big \{-\mu _y + \eta \gamma x_{i,j}(n-r) - \delta _y y_{i,j}(n) \big \}, i,j=\overline{1,N}, \end{aligned} \end{aligned}$$

$n, r\in \mathbb {N}$. Here $x_{i,j}(t)$ is the concentration of antigens, $y_{i,j}(t)$ is the concentration of antibodies in biopixel (i, j), $i,j=\overline{1,N}$. $\hat{S} \{ x_{i,j}(n)\} = (D/\varDelta ^2)\{x_{i-1,j}(n)+x_{i+1,j} (n)+x_{i,j-1}(n)+x_{i,j+1}(n) - 4x_{i,j}(n)\}$ is spatial diffusion-like operator. Permanence of the system is investigated. Stability research uses approach of Lyapunov functions. Numerical simulations are used in order to investigate qualitative behavior when changing the value of time delay $r \in \mathbb {N}$ and diffusion $D/\varDelta ^2$. It was shown that when increasing the value of time delay r, we transit from steady state through Hopf bifurcation, increasing period and finally to chaotic behavior. The increase of diffusion causes an appearance of chaotic solutions also.