Top

Published in:

2020 | OriginalPaper | Chapter

Stability Investigation of Biosensor Model Based on Finite Lattice Difference Equations

Authors : Vasyl Martsenyuk, Aleksandra Klos-Witkowska, Andriy Sverstiuk

Published in: Difference Equations and Discrete Dynamical Systems with Applications

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

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.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter On the Boundedness Character of a Rational System of Difference Equations with Non-constant Coefficients

next chapter Discrete Reaction-Dispersion Equation

Diffusion term is considered be additive in order to get clear permanence and stability results. Actually the diffusion on discrete space may be represented by a matrix multiplication also [4].

Lemma 4 offers necessary condition (12).

In order to substantiate it, we assume the contrary, namely, there are $\epsilon _1>0$ and $i^\star ,j^\star \in \overline{1,N}$ such that $x_{i^\star ,j^\star }(n)>M_{x,i^\star ,j^\star }(r)\frac{\exp (\beta - 1)}{\delta _x} + \epsilon _1$, for all $n>0$. Then

$$\begin{aligned} \begin{aligned} \lim _{n\rightarrow \infty } \sup \sum _{i,j=1}^N{x_{i,j}(n)}&\le \frac{\exp (\beta -1)}{\delta _x}\sum _{i,j=1}^N{M_{x,i,j}(r)} < \frac{\exp (\beta -1)}{\delta _x}\sum _{i,j=1,i\ne i^\star , j\ne j^\star }^N{M_{x,i,j}(r)}\\ + x_{i^\star ,j^\star } - \epsilon _1&\le \frac{\exp (\beta -1)}{\delta _x}\sum _{i,j=1}^N{M_{x,i,j}(r)} + \hat{S}\left\{ x_{i^\star ,j^\star }(n-1) \right\} - \epsilon _1, \end{aligned} \end{aligned}$$

which is a contradiction at $n\rightarrow \infty $.

Here we use that $\frac{1}{x} \exp (x-1)>1$ for $x>0$.

Here we use epidemiological term “endemic” meaning the state when the “infection” (in this context, antigen) is constantly maintained at a baseline level in an area without external inputs.

Hereinafter we omit units of dimensions of parameters.

After scaling of $\gamma $ the value $\eta = 0.8/\gamma $ may not be applicable for Nicholson-type difference system (it causes number overflow). So, we have decreased it to 0.01184 experimentally.

Allen, L.J.S.: Persistence, extinction, and critical patch number for island populations. J. Math. Biol. 24(6), 617–625 (1987)MathSciNetCrossRef

Berger, C., Hees, A., Braunreuther, S., Reinhart, G.: Characterization of cyber-physical sensor systems. Proc. CIRP 41, 638–643 (2016)CrossRef

Bevers, M., Flather, C.H.: Numerically exploring habitat fragmentation effects on populations using cell-based coupled map lattices. Theor. Popul. Biol. 55(1), 61–76 (1999)CrossRef

Caswell, H.: Matrix Population Models: Construction, Analysis, and Interpretation. Sinauer Associates, Massachusetts (1989)

Chow, S.-N., Mallet-Paret, J., Shen, W.: Traveling waves in lattice dynamical systems. J. Differ. Equ. 149(2), 248–291 (1998)MathSciNetCrossRef

Chow, S.-N., Shen, W.: Dynamics in a discrete nagumo equation: spatial topological chaos. SIAM J. Appl. Math. 55(6), 1764–1781 (1995)MathSciNetCrossRef

Faria, T., Röst, G.: Persistece, permaece ad global stability for a $n$-dimesional nicholson system. J. Dyn. Differ. Equ. 26(3), 723–744 (2014)CrossRef

Hofbauer, J., Iooss, G.: A Hopf bifurcation theorem for difference equations approximating a differential equation. Monatshefte für Mathematik 98(2), 99–113 (1984)MathSciNetCrossRef

Hupkes, H.J., Van Vleck, E.S.: Negative diffusion and traveling waves in high dimensional lattice systems. SIAM J. Math. Anal. 45(3), 1068–1135 (2013)MathSciNetCrossRef

10.

Hupkes, H.J., Van Vleck, E.S.: Travelling waves for complete discretizations of reaction diffusion systems. J. Dyn. Differ. Equ. 28(3–4), 955–1006 (2016)MathSciNetCrossRef

11.

Lancaster, P., Tismenetsky, M.: The Theory of Matrices: with Applications. Elsevier, Amsterdam (1985)

12.

Letellier, C., Elaydi, S., Aguirre, L.A., Alaoui, A.: Difference equations versus differential equations, a possible equivalence for the Rössler system? Phys. D: Nonlinear Phenom. 195(1–2), 29–49 (2004)

13.

Liu, L., Liu, Z.: Asymptotic behaviors of a delayed nonautonomous predator-prey system governed by difference equations. Discret. Dyn. Nat. Soc. 1–15, 2011 (2011)MathSciNetMATH

14.

Ma, S., Weng, P., Zou, X.: Asymptotic speed of propagation and traveling wavefronts in a non-local delayed lattice differential equation. Nonlinear Anal.: Theory, Methods Appl. 65(10), 1858–1890 (2006)MathSciNetCrossRef

15.

Martsenyuk, V., Kłos-Witkowska, A., Sverstiuk, A.: Stability, bifurcation and transition to chaos in a model of immunosensor based on lattice differential equations with delay. Electron. J. Qual. Theory Differ. Equ. 27, 1–31 (2018)MathSciNetCrossRef

16.

Mickens, R.E.: Nonstandard Finite Difference Models of Differential Equations. World Scientific, Singapore (1994)

17.

Mickens, R.E.: Advances in the Applications of Nonstandard Finite Difference Schemes. World Scientific, Singapore (2005)CrossRef

18.

Nicholson, A.J., Bailey, V.A.: The balance of animal populations. part i. In: Proceedings of the Zoological Society of London, vol. 105, pp. 551–598. Wiley Online Library (1935)

19.

Prindle, A., Samayoa, P., Razinkov, I., Danino, T., Tsimring, L.S., Hasty, J.: A sensing array of radically coupled genetic ‘biopixels’. Nature 481(7379), 39–44 (2012)CrossRef

20.

Schütze, A., Helwig, N., Schneider, T.: Sensors 4.0 – smart sensors and measurement technology enable industry 4.0. J. Sens. Sens. Syst. 7(1), 359–371 (2018)CrossRef

21.

So, J.O.S.E.P.H.W.-H., Yu, J.S.: On the stability and uniform persistence of a discrete model of nicholson’s blowflies. J. Math. Anal. Appl. 193(1), 233–244 (1995)MathSciNetCrossRef

22.

Stehlík, P.: Exponential number of stationary solutions for nagumo equations on graphs. J. Math. Anal. Appl. 455(2), 1749–1764 (2017)MathSciNetCrossRef

23.

Stehlík, P., Volek, J.: Maximum principles for discrete and semidiscrete reaction-diffusion equation. Discret. Dyn. Nat. Soc. 2015, (2015)

24.

Wang, L., Wang, M.Q.: Ordinary Difference Equation. Xinjiang University Press, Xinjiang, China (1989)

25.

Wang, Z.-C., Li, W.-T., Jianhong, W.: Entire solutions in delayed lattice differential equations with monostable nonlinearity. SIAM J. Math. Anal. 40(6), 2392–2420 (2009)MathSciNetCrossRef

26.

Yang, X.: Uniform persistence and periodic solutions for a discrete predator-prey system with delays. J. Math. Anal. Appl. 316(1), 161–177 (2006)MathSciNetCrossRef

27.

Yi, T., Zou, X.: Global attractivity of the diffusive nicholson blowflies equation with neumann boundary condition: a non-monotone case. J. Differ. Equ. 245(11), 3376–3388 (2008)MathSciNetCrossRef

28.

Zhang, B.G., Xu, H.X.: A note on the global attractivity of a discrete model of nicholsonś blowflies. Discret. Dyn. Nat. Soc. 3(1), 51–55 (1999)CrossRef

Title: Stability Investigation of Biosensor Model Based on Finite Lattice Difference Equations
Authors: Vasyl Martsenyuk
Aleksandra Klos-Witkowska
Andriy Sverstiuk
Publisher: Springer International Publishing
Book: Difference Equations and Discrete Dynamical Systems with Applications
Print ISBN: 978-3-030-35501-2

Electronic ISBN: 978-3-030-35502-9

Copyright Year: 2020
DOI: https://doi.org/10.1007/978-3-030-35502-9_13

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Premium Partner