Advances in Mathematical Physics

Volume 2018, Article ID 6152961, 24 pages

https://doi.org/10.1155/2018/6152961

## Motion of a Spot in a Reaction Diffusion System under the Influence of Chemotaxis

Department of Complex and Intelligent Systems, School of Systems Information Science, Future University Hakodate, Hakodate 041-8655, Japan

Correspondence should be addressed to Satoshi Kawaguchi; pj.ca.nuf@ihsotas

Received 7 December 2017; Accepted 29 March 2018; Published 21 May 2018

Academic Editor: Christos Volos

Copyright © 2018 Satoshi Kawaguchi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider the motion of a spot under the influence of chemotaxis. We propose a two-component reaction diffusion system with a global coupling term and a Keller-Segel type chemotaxis term. For the system, we derive the equation of motion of the spot and the time evolution equation of the tensors. We show the existence of an upper limit for the velocity and a critical intensity for the chemotaxis, over which there is no circular motion. The chemotaxis suppresses the range of velocity for the circular motion. This braking effect on velocity originates from the refractory period behind the rear interface of the spot and the negative chemotactic velocity. The physical interpretation of the results and its plausibility are discussed.

#### 1. Introduction

The behaviors of artificial and biological microswimmers such as oil droplets, bimetallic nanorods, catalytic Janus colloids, liposomes, flagellated bacteria, and* Volvox* have attracted widespread attention [1]. Under certain circumstances, some of these microswimmers are self-propelled particles, the mobility mechanism of which has been intensively studied [2, 3]. The motion of oil droplets, especially, has been studied in well-controlled experimental facilities with sufficient reproducibility. Although symmetric droplets cannot move in the absence of external force, the Marangoni effect can cause motion in the presence of an inhomogeneous chemical substance outside the droplet or a temperature gradient along the surface [4–6]. Numerical simulations and theoretical results support this mechanism and the existence of straight, circular, and complicated motions of droplets [7, 8], and experimental results qualitatively agree with the numerical results [9–12]. Droplet motion has also been the subject of a review article [13, 14].

In a two-dimensional reaction diffusion (RD) system, the droplet is often referred to as a spot solution. In order to systematically describe the motion of spots in an RD system, the time evolution equation of the spot was derived and the mechanism of elastic collision of moving spots was clarified in a previous study [15]. This study was extended by studies on the drift and rotation bifurcations of spot solutions in RD systems [16, 17]. In order to describe the deformations of the spot, tensors were introduced. The bifurcation diagram of the spot suggested that, with increasing velocity of the spot, rotation bifurcation occurred causing the straight motion to become destabilized into circular motion.

In addition to the Marangoni effect, which plays an important role in the motion of oil droplets, chemotaxis is an important property of cell migration; it is important in mass transfer and immunological response in biology. In inflammatory response, the neutrophils among blood cells have a remarkable migration potency (chemotaxis) and can change their form by generating pseudopods toward the antigen. In biophylaxis, several chemokines (chemoattractants) are released from the macrophages and mast cells. Then other immunocompetent cells (neutrophils) respond to the gradient of the chemoattractant. Consequently, the immunocompetent cells move unidirectionally to the source point of the antigen [18].

The mathematical model for chemotaxis was first proposed by Keller and Segel [19], wherein the gradient of the chemoattractant was taken into consideration for the flow of amoeba. Neutrophil migration was considered with a Keller-Segel type chemotaxis term in [20, 21]. In these studies, the Cahn-Hilliard (CH) equation was employed, and the kinematic properties and morphological changes of the crawling cell distribution were shown. In addition to the chemotaxis of the neutrophil, cancer cell invasion under haptotaxis was modeled by the CH equation [22, 23]. The haptotactic response of cancer cells is described by the gradient of the haptoattractant. However, in the above studies, the gradients of chemoattractant and haptoattractant are assumed to be constant; there is no feedback between the cells and these chemical substances.

As described above, with the recent increase in importance of chemotaxis in biology, medicine, and cytoengineering [24–27], many experimental and theoretical studies have been performed. Although there are model systems for the cell density and concentration of chemotactic substances, no mathematical analysis has been reported on the motion of the cell. Inspired by these points, we first propose an RD system including a naive Keller-Segel type chemotaxis term. The system is autonomous, the spot secretes a chemotactic substance, and the motion of the spot is influenced by it. For the proposed RD system, we apply the method reported in [16] to derive the equation of motion of the spot and time evolution equation of the tensors. Based on these equations, we study the bifurcation from straight motion to circular motion as well as the upper limit of the velocity of circular motion. In order to verify the theoretical result, we perform numerical simulations for the tensor model. The physical meaning and validity of the results are discussed.

#### 2. Model Equation

We first consider the following three-component RD system with an activator , a chemotactic substance , and an inhibitor :where , , , , , , , , and are positive constants, and is a step function satisfying for and for . Throughout this study, we consider the system in a two-dimensional space, with and . We choose such that the system is monostable. Here, we fix . In the above excitable system, there are two stationary states: a rest state and an excited state. The rest state is , and the excited state has spatially nonuniform values of , , and . Between the rest and excited states, there appear boundary layers with thickness , connecting the two different states.

When the second term on the right hand side of (1) is absent, (1)–(3) describe an RD system with one activator and two inhibitors, which was studied in [28]. In that system, when is large, the localized domain (motionless spot solution) of an activator appears. With decreasing , the motionless spot is destabilized through static bifurcation or oscillatory bifurcation; however, when is small and and are large, these bifurcations are suppressed by . When is small, the motionless spot is primarily destabilized through translational bifurcation, causing the spot to move.

In the presence of the second term on the right hand side of (1), the moving spot is influenced by the chemotaxis. A system similar to that described by (1)–(3), but with bistability, was studied in [29]. In that system, the nonlinear term in (1) was replaced by , and a front solution was obtained. Furthermore, maze patterns and branching from a front solution were observed. The stability analyses of the spot and front solutions were conducted by applying the singular perturbation method [30].

The time evolution equation for is obtained using the conservation equation. The diffusion term is derived from , where the flux is the sum of the normal diffusion (random motility) term and the chemotaxis term . That is, , where and . It should be noted that the signs of these fluxes are different. The sign of suggests that the chemotaxis term provides a negative diffusion effect, which suppresses the expansion of . The second term on the right hand side of (1) is the Keller-Segel type chemotaxis term; we express the chemotactic sensitivity function as , where with . In order to satisfy the condition , we choose . We call the intensity of chemotaxis [31].

In (3), and represent the relaxation time and diffusion constant of , respectively. Let us consider a situation in which plays the role of feedback to suppress the static bifurcation and oscillatory bifurcation. For the rapid feedback mechanism, and must be small and large, respectively. In the limits and , becomes a time-dependent but spatially independent variable, which is denoted by :where is the area of the entire system. Replacing in by , becomes a global coupling term. In the case where is very small and is very large, we reduce the three-component RD system to the following two-component RD system with a global coupling term:whereThe functional represents a global coupling term given bywhere the integral is over the entire domain, and and are rescaled to absorb and , and the primes are dropped. corresponds to the intensity of the global coupling, and the value of is chosen as . Hereinafter, we consider the above two-component RD system to be described by (5) and (6).

In the absence of chemotaxis, (5) and (6) describe the same system proposed by Krischer and Mikhailov [32]. This system had an activator and an inhibitor, and, for large , the motionless spot (localized particle-like structure) in two dimensions was stable. With decreasing under a large , it was shown that the system had a stable moving spot. In order to understand intuitively the bifurcation from the motionless spot to the moving spot, we consider the limit . In the limit , the boundary layer of becomes an interface, as shown in Figure 1. The location of the interface is defined by the condition . In this limit, and satisfy the relation (inside the domain) and 0 (outside the domain), which is obtained from (5). In the limits of and , the area of the spot is conserved; using (8), we obtainThis restriction prohibits the expansion and oscillation of the spot; the translational bifurcation firstly occurs with decreasing . Even for finite values of a large and small , the area of the spot is approximately conserved by the feedback mechanism. This supports the existence of a moving spot for small with large . In contrast, for small , the static bifurcation or oscillatory bifurcation firstly occurs with decreasing , and the motionless spot is destabilized to form an expanding wave or is disintegrated by unstable oscillation.