# On Nonlinear Reaction-Diffusion Model with Time Delay on Hexagonal Lattice

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## Abstract

**:**

## 1. Introduction

## 2. Lattice Reaction-Diffusion Model on a Hexagonal Biopixel Array

- Antigens are born with the constant birthrate $\beta >0$.
- We have some probability rate $\gamma >0$ for the binding of antigens with antibodies.
- The antigen population tends to some carrying capacity with the rate ${\delta}_{v}>0$.
- Antibodies die with the constant birthrate ${\mu}_{f}>0$.
- As a result of immune response, we have the increase of the density of antibodies with probability rate $\eta \gamma $.
- The antibody population tends to some carrying capacity with rate ${\delta}_{f}>0$.
- The immune response appears with some constant time delay $\tau >0$.

## 3. Persistence of the Solutions

**Definition**

**1.**

**Lemma**

**1.**

**Proof.**

**Remark**

**1.**

## 4. Stability Problem in Immunosensors

#### 4.1. Steady States

**Prey-free steady state:**In case of the absence of antigens (preys), ${V}_{i,j,k}\left(t\right)\equiv 0$, we have two possible opportunities, namely, either ${\mathcal{E}}_{i,j,k}^{0}\equiv \left(0,0\right)$, $i,j,k=\overline{-N,N}$ or ${\mathcal{E}}_{i,j,k}^{0}\equiv \left(0,-\frac{{\mu}_{f}}{{\delta}_{f}}\right)$, $i,j,k=\overline{-N,N}$, $i+j+k=0$. The last one is of biological meaning and is not reachable for positive initial values (3).

**Pixel-independent endemic state:**In the case of the existence of solutions of (10), which are identical for all pixels, i.e., ${V}_{i,j,k}^{*}\equiv {V}^{*}$, ${F}_{i,j,k}^{*}\equiv {F}^{*}$, $i,j,k=\overline{-N,N}$, $\widehat{S}\left\{{V}_{i,j,k}^{*}\right\}\equiv 0$, the endemic steady states ${\mathcal{E}}_{i,j,k}^{*}=\left({V}^{*},{F}^{*}\right)$, $i,j,k=\overline{-N,N}$ are presented as:

**Pixel-dependent endemic state:**Because of the diffusion between pixels $D/{\Delta}^{-2}$, the endemic steady states are not equal to (11), and in Section 5, it is evidenced numerically.

#### 4.2. Global Asymptotic Stability

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Remark**

**2.**

## 5. Numerical Study

^{−1}, $\gamma $ = 2 $\frac{\mathrm{mL}}{\mathrm{min}\xb7\mathsf{\mu}\mathrm{g}}$, ${\mu}_{f}$ = 1 min

^{−1}, $\eta =0.8/\gamma $, ${\delta}_{v}$ = 0.5 $\frac{\mathrm{mL}}{\mathrm{min}\xb7\mathsf{\mu}\mathrm{g}}$, ${\delta}_{f}$ = 0.5 $\frac{\mathrm{mL}}{\mathrm{min}\xb7\mathsf{\mu}\mathrm{g}}$, D = 0.2 $\frac{{\mathrm{nm}}^{2}}{\mathrm{min}}$, $\Delta =0.3$ nm. Numerical modeling was implemented at different values of $n\in (0,1]$. For this purpose, we used the RStudio environment.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Diffusion of antigens for the hexagonal lattice model. Antigens from six neighboring pixels interact. $n>0$ is the constant of disbalance. Here, ‘1’, ‘3’, ‘5’, ‘8’, ‘9’, ‘11’ have to be replaced with $D{\Delta}^{-2}{V}_{i,j,k}\left(t\right)$, ‘2’ with $D{\Delta}^{-2}{V}_{i+1,j,k-1}\left(t\right)$, ‘4’ with $D{\Delta}^{-2}{V}_{i+1,j-1,k}\left(t\right)$, ‘6’ with $D{\Delta}^{-2}{V}_{i,j-1,k+1}\left(t\right)$, ‘7’ with $D{\Delta}^{-2}{V}_{i-1,j,k+1}\left(t\right)$, ‘10’ with $D{\Delta}^{-2}{V}_{i-1,j+1,k}\left(t\right)$, and ‘12’ with $D{\Delta}^{-2}{V}_{i,j+1,k-1}\left(t\right)$.

**Figure 3.**Phase plots of the system (2) at $\tau =0.05$, which correspond to stable focus. Here, ● indicates the initial state, ● the pixel-independent endemic state, and ● the pixel-dependent endemic state.

**Figure 4.**Phase plots of the system (2) at $\tau =0.25$. The solution tends to a stable limit cycle. Here, ● indicates the initial state, ● the pixel-independent endemic state, and ● the pixel-dependent endemic state.

**Figure 5.**Phase plots of the system (2) at $\tau =0.287$. Here, ● indicates the initial state, ● the pixel-independent endemic state, and ● the pixel-dependent endemic state. The solution tends to a stable limit cycle with six local extrema per cycle.

**Figure 8.**Example of the hexagonal tiling plot for the probabilities of binding antigens by antibodies, i.e., $V\times F$. In the case of the optical immunosensor, it is fluorescence intensity.

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Martsenyuk, V.; Veselska, O.
On Nonlinear Reaction-Diffusion Model with Time Delay on Hexagonal Lattice. *Symmetry* **2019**, *11*, 758.
https://doi.org/10.3390/sym11060758

**AMA Style**

Martsenyuk V, Veselska O.
On Nonlinear Reaction-Diffusion Model with Time Delay on Hexagonal Lattice. *Symmetry*. 2019; 11(6):758.
https://doi.org/10.3390/sym11060758

**Chicago/Turabian Style**

Martsenyuk, Vasyl, and Olga Veselska.
2019. "On Nonlinear Reaction-Diffusion Model with Time Delay on Hexagonal Lattice" *Symmetry* 11, no. 6: 758.
https://doi.org/10.3390/sym11060758