Modelling the Aggregation Process of Cellular Slime Mold by the Chemical Attraction

Abstract

We put into exercise a comparatively innovative analytical modus operandi, the homotopy decomposition method (HDM), for solving a system of nonlinear partial differential equations arising in an attractor one-dimensional Keller-Segel dynamics system. Numerical solutions are given and some properties show evidence of biologically practical reliance on the parameter values. The reliability of HDM and the reduction in computations give HDM a wider applicability.

1. Introduction

In 1970, Keller and Segel have offered parabolic systems to illustrate the aggregation process of cellular slime mold by the chemical attraction [1]. The system of a simplified form in the one-dimensional case is written as 𝜕𝑢(𝑥,𝑡)𝜕𝜕𝑡=𝑎²𝑢(𝑥,𝑡)𝜕𝑥²−𝜕𝑢()𝜌𝜕𝑥𝑥,𝑡𝜕𝜒,()𝜕𝑥𝜕𝜌𝑥,𝑡𝜕𝜕𝑡=𝑏²𝜌(𝑥,𝑡)𝜕𝑥²+𝑐𝑢(𝑥,𝑡)−𝑑𝜌(𝑥,𝑡), (1) subject to the boundaries conditions 𝜕𝑢(𝛼,𝑡)=𝜕𝑥𝜕𝑢𝛽,𝑡=()𝜕𝑥𝜕𝜌𝛼,𝑡=𝜕𝑥𝜕𝜌𝛽,𝑡𝜕𝑥=0 (2) and initial conditions 𝑢(𝑥,0)=𝑢₀(𝑥),𝜌(𝑥,0)=𝜌₀(𝑥),𝑥∈𝐼, (3) where  𝐼=(𝛼,𝛽) is a bounded open interval and 𝑎, 𝑏, 𝑐, and 𝑑 are positive constants. The unknown functions 𝑢(𝑥,𝑡) and 𝜌(𝑥,𝑡) denote the concentration of amoebae and the concentration of chemical substance, respectively, in 𝐼×(0,∞). The chemotactic term (𝜕/𝜕𝑥)(𝑢(𝑥,𝑡)(𝜕𝜒(𝜌)/𝜕𝑥)) indicates that the cells are sensitive to the chemicals and are attracted by them. 𝜒(𝜌) called the sensitivity function is a smooth function of 𝜌∈(0,∞) which describes cell’s perception and response to the chemical stimulus 𝜌. Several normalized forms have been suggested 𝜌, 𝜌², log(𝜌), 𝜌/(𝜌+1) and 𝜌²/(𝜌²+1), and so forth (see [2, 3]). Recently, the Keller-Segel (KS) equations attracted interests of many mathematicians. Since the modelling of chemotaxis has developed into a large and diverse discipline, one model which is widely used is the Keller-Segel model of chemotaxis; it is important to recall that chemotaxis describes the movement of single or multicellular organisms when they move up or down a chemical gradient [4]. This movement allows the organism to explore its extracellular environment. Organisms move randomly, away from repellents and towards attractants. Questions have arisen on how organisms can detect small changes in their extracellular environment [5]. Usually the organism will undergo a random walk, consisting of smooth swimming and brief direction changes (tumbles). By increasing the attractant, the tumbling is suppressed, which leads to a biased random walk [4]. The organism will then accumulate in areas of high attractant concentration. This type of movement is referred to as runs [5]. A combination of tumbles and runs allows the organism to explore and respond to changes in its extracellular environment [4].

The local solutions were studied by the second author [6]. It was also suggested in [6] that, in the one-dimensional case, (KS) possesses a global solution and that, in the two-dimensional case, when 𝜒(𝜌)=𝑘𝜌 (𝑘 being a positive constant) is a linear function, (KS) possesses a global solution for any sufficiently small initial function 𝑢₀. Horstmann and Wang [7] showed more strongly that the global solution exists if the norm ‖𝑢₀‖_𝐿1 is smaller than a specific number, which is given from the coefficients of the equations. Recently, in the same case, the asymptotic behaviour of the global solutions was studied in [8]. On the other hand, Herrero and Velázquez [9] showed that when 𝜒(𝜌) is linear and the domain is a circular disc, there exist radial local solutions which blow up in a finite time. The blowup of nonradial local solutions was shown recently by [10, 11]. For the study of stationary solutions, we refer to [12–14]. In the field of dynamical systems theory some work has been carried out although the suggested models and algorithms are still in an introductory platform of establishing. It is perhaps important to notice that several analytical methods have been proposed to deal with nonlinear equations, but there exist a lot of nonlinear ordinary differential equations and nonlinear partial differential equations for which exact analytical solution cannot be found. There is no exact solution of (4) in the literature. To solve these problems, some eminent scholars have proposed some powerful iteration methods to deal with this class of nonlinear equation.

As V.M. Alexandrov wrote in the introduction of a well-liked science book Asymtotology: Ideas, Methods, and Applications [15, 16], asymptotic methods belong to the, perhaps, most romantic area of modern mathematics [15–19]. Though computer science is growing very fast and numerical simulation is applied everywhere, nonnumerical issues will still play a large role [16, 20–22]. There exist some alternative analytical asymptotic approaches such as the nonperturbative method, modified Lindstedt-Poincare method [21], variational iteration method [22], Adomian decomposition method [23], homotopy perturbation method [17, 24], and bookkeeping artificial parameter perturbation method [18].

The purpose of this paper is to derive analytical solutions of attractor one-dimensional Keller-Segel equations (1) via the relatively new analytical method the modified homotopy perturbation method. The HDM was recently used in [19, 25–27]. This method displays some advantages over existing methods.

The paper is prearranged as follows: in Section 2, we present the basic idea of the HDM for solving high orders differential equations. We present the application of the HDM for attractor one-dimensional Keller-Segel equations and numerical results in Section 3. In Section 4 we present the discussions. The conclusions are then given in Section 5.

2. Basic Properties of Homotopy Decomposition Method

With the purpose of making the fundamental possessions of the homotopy decomposition method [28] clear, we think about a universal nonlinear nonhomogeneous partial differential equation with initial conditions of the following form: 𝜕^𝑚𝑈(𝑥,𝑡)𝜕𝑡^𝑚=𝐿(𝑈(𝑥,𝑡))+𝑁(𝑈(𝑥,𝑡))+𝑓(𝑥,𝑡),𝑚=1,2,3…. (4) Subject to the initial condition 𝜕^𝑖𝑈(𝑥,0)𝜕𝑡^𝑖=𝑦_𝑖(𝑥),𝜕^𝑚−1𝑈(𝑥,0)𝜕𝑡^𝑚−1=0,𝑖=0,1,2…𝑚−2, (5)𝑚 is the order of the derivative.

Where 𝑓 is a known function, 𝑁 is the general nonlinear differential operator and 𝐿 represents a linear differential operator, and 𝑚 is the order of the derivative. The method’s first step is to apply the inverse operator of 𝜕^𝑚/𝜕𝑡^𝑚 on both sides of (4) to obtain 𝑈(𝑥,𝑡)=^𝑚−1_𝑘=0𝑡^𝑘𝑑𝑘!^𝑘𝑢(𝑥,0)𝑑𝑡^𝑘+^𝑡₀^𝑡1₀⋯^𝑡𝑚−1₀𝐿(𝑈((𝑈(()𝑥,𝜏))+𝑁𝑥,𝜏))+𝑓𝑥,𝜏𝑑𝜏⋯𝑑𝑡. (6) The multi-integral in (4) can be transformed to ^𝑡₀^𝑡1₀⋯^𝑡𝑚−1₀𝐿(𝑈((𝑈(()=1𝑥,𝜏))+𝑁𝑥,𝜏))+𝑓𝑥,𝜏𝑑𝜏⋯𝑑𝑡(𝑚−1)!^𝑡₀(𝑡−𝜏)^𝑚−1𝐿(𝑈((𝑈(()𝑥,𝜏))+𝑁𝑥,𝜏))+𝑓𝑥,𝜏𝑑𝜏, (7) so that (4) can be reformulated as 𝑈(𝑥,𝑡)=^𝑚−1_𝑘=0𝑡^𝑘𝑦𝑘!^𝑖(𝑥)+1(𝑚−1)!×^𝑡₀(𝑡−𝜏)^𝑚−1𝐿(𝑈((𝑈(()𝑥,𝜏))+𝑁𝑥,𝜏))+𝑓𝑥,𝜏𝑑𝜏. (8) Using the homotopy scheme the solution of the above integral equation is given in series form as 𝑈𝑥,𝑡,𝑝=^∞_𝑛=0𝑝^𝑛𝑈_𝑛(),𝑈()𝑥,𝑡𝑥,𝑡=lim_𝑝→1𝑈𝑥,𝑡,𝑝 (9) and the nonlinear term can be decomposed as 𝑁𝑈(𝑟,𝑡)=^∞_𝑛=1𝑝^𝑛ℋ_𝑛(𝑈), (10) where 𝑝∈(0,1] is an embedding parameter. ℋ_𝑛(𝑈) is the He’s polynomials that can be generated by ℋ_𝑛𝑈₀,…,𝑈_𝑛=1𝜕𝑛!𝜕𝑝^𝑛𝑛𝑁^𝑛_𝑗=0𝑝^𝑗𝑈_𝑗(𝑥,𝑡),𝑛=0,1,2…. (11) The modified homotopy perturbation method is obtained by the coupling of decomposition method with Abel integral and is given by ^∞_𝑛=0𝑝^𝑛𝑈_𝑛()()1𝑥,𝑡=𝑇𝑥,𝑡+𝑝(𝑚−1)!×^𝑡₀(𝑡−𝜏)^𝑚−1𝑓(𝑥,𝜏)+𝐿^∞_𝑛=0𝑝^𝑛𝑈_𝑛()+𝑥,𝜏_∞𝑛=0𝑝^𝑛ℋ_𝑛(𝑈)𝑑𝜏 (12) with 𝑇(𝑥,𝑡)=^𝑚−1_𝑘=0𝑡^𝑘𝑦𝑘!^𝑖(𝑥). (13) Comparing the terms of same powers of 𝑝 produces solutions of various orders. The initial guess of the approximation is 𝑇(𝑥,𝑡) [26, 27]. This is actually the Taylor series of the exact solution of order 𝑚. Note that this initial guess insures the uniqueness of the series decompositions [26, 27].

3. Application

In this section we apply this method for solving coupled attractor one-dimensional Keller-Segel equations.

Example 1. Consider the following Keller Segel equation with the sensitivity function 𝜒(𝜌)=1.
Then the chemotactic term 𝜕𝑢()𝜌𝜕𝑥𝑥,𝑡𝜕𝜒()𝜕𝑥=0,𝜕𝑢𝑥,𝑡𝜕𝜕𝑡=𝑎²𝑢(𝑥,𝑡)𝜕𝑥²,𝜕𝑢(𝑥,𝑡)𝜕𝜕𝑡=𝑏²𝜌(𝑥,𝑡)𝜕𝑥²+𝑐𝑢(𝑥,𝑡)−𝑑𝜌(𝑥,𝑡). (14) Subject to the initial conditions 𝑢(𝑥,0)=𝑚𝑒^−𝑥2,𝜌(𝑥,0)=𝑛𝑒^−𝑥2,𝑥∈𝐼. (15) In the view of the HDM, we obtain the following equation: ^∞_𝑛=0𝑝^𝑛𝑢_𝑛(𝑥,𝑡)−𝑢(𝑥,0)=𝑝^𝑡₀𝑎𝜕²𝜕𝑥^2∞_𝑛=0𝑝^𝑛𝑢_𝑛(𝑥,𝜏)𝑑𝜏,_∞𝑛=0𝑝^𝑛𝜌_𝑛(𝑥,𝑡)+𝜌(𝑥,0)=𝑝^𝑡₀𝑏𝜕²𝜕𝑥^2∞_𝑛=0𝑝^𝑛𝜌_𝑛(𝑥,𝜏)+𝑐^∞_𝑛=0𝑝^𝑛𝑢_𝑛(𝑥,𝜏)−𝑑^∞_𝑛=0𝑝^𝑛𝜌_𝑛(𝑥,𝜏)𝑑𝜏. (16) Now comparing the terms of same power of 𝑝, we obtained the following integral equations: 𝑝⁰∶𝑢₀(𝑥,𝑡)=𝑢(𝑥,0)=𝑚𝑒^−𝑥2,𝑢₀()(),𝑝𝑥,0=𝑢𝑥,0₀∶𝜌₀(𝑥,𝑡)=𝜌(𝑥,0)=𝑛𝑒^−𝑥2,𝜌₀()(),𝑝𝑥,0=𝜌𝑥,0₁∶𝑢₁(𝑥,𝑡)=𝑎^𝑡₀𝜕²𝑢₀𝜕𝑥²𝑑𝜏,𝑢₁()𝑝𝑥,0=0,₁∶𝜌₁(𝑥,𝑡)=^𝑡₀𝑏𝜕²𝜌₀𝜕𝑥²+𝑐𝑢₀−𝑑𝜌₀𝑑𝜏,𝜌₁()⋮𝑝𝑥,0=0,_𝑛∶𝑢_𝑛(𝑥,𝑡)=𝑎^𝑡₀𝜕²𝑢_𝑛−1𝜕𝑥²𝑑𝜏,𝑢_𝑛()𝑝𝑥,0=0,_𝑛∶𝜌_𝑛(𝑥,𝑡)=^𝑡₀𝑏𝜕²𝜌_𝑛−1𝜕𝑥²+𝑐𝑢_𝑛−1−𝑑𝜌_𝑛−1𝜌𝑑𝜏,_𝑛(𝑥,0)=0. (17) The following solutions are obtained straightforwardly: 𝑢(𝑥,0)=𝑚𝑒^−𝑥2,𝑛𝑒^−𝑥2,𝑢₁(𝑥,𝑡)=2𝑎𝑒^−𝑥2𝑚𝑡−1+2𝑥²,𝜌₁(𝑥,𝑡)=𝑒^−𝑥2𝑡𝑐𝑚−𝑛𝑑+𝑏2−4𝑥²,𝑢₂(𝑥,𝑡)=2𝑎²𝑒^−𝑥2𝑚𝑡²3−12𝑥²+4𝑥⁴,𝜌₂()=1𝑥,𝑡2𝑒^−𝑥2𝑡²𝑑(−𝑐𝑚+𝑛𝑑)+2𝑎𝑐𝑚−1+2𝑥²+2𝑏(𝑐𝑚−2𝑑𝑛)−1+2𝑥²+4𝑏²𝑛3−12𝑥²+4𝑥⁴,𝑢₃()=4𝑥,𝑡3𝑎³𝑒^−𝑥2𝑚𝑡³−15+90𝑥²−60𝑥⁴+8𝑥⁶𝜌₃()=1𝑥,𝑡6𝑒^−𝑥2𝑡³𝑑²()()×𝑐𝑚−𝑑𝑛+2𝑏𝑑−2𝑐𝑚+3𝑑𝑛−1+2𝑥²+4𝑎²𝑐𝑚3−12𝑥²+4𝑥²+4𝑏²(𝑐𝑚−3𝑑𝑛)3−12𝑥²+4𝑥⁴+8𝑏³𝑛−15+90𝑥²−60𝑥⁴+8𝑥⁶+2𝑎𝑐𝑚𝑑−2𝑑𝑥²+𝑏6−24𝑥²+8𝑥⁴. (18) Using the iterative formula, the remaining terms can be obtained. But here, only few terms of the series solutions are considered and the asymptotic solution is given as 𝑢(𝑥,𝑡)=𝑢₀(𝑥,𝑡)+𝑢₁(𝑥,𝑡)+𝑢₂(𝑥,𝑡)+𝑢₃()𝜌()𝑥,𝑡+⋯,𝑥,𝑡=𝜌₀(𝑥,𝑡)+𝜌₁(𝑥,𝑡)+𝜌₂(𝑥,𝑡)+𝜌₃(𝑥,𝑡)+⋯. (19) The following figures show the biological behaviour of the coupled solutions for the following set of theoretical parameters: 𝑚=120, 𝑛=160, 𝑎=0.5, 𝑏=3, 𝑐=1, and 𝑑=2, first for a fixed time 𝑡=5 and secondly for a fixed distance 𝑥=1.

Figures 1, 2, and 3 show the behaviour of the solution of the system of (15) describing the concentrations of the chemical substance and the amoebae in the human body. While on one hand, Figure 1 shows the behaviour as function of space, Figures 2 and 3 show the behaviour of these solutions as function of time. From the above figures, one can see that the concentration of amoebae reduces in space as the concentration of the chemical substance reduces. This simply implies that if the concentration of the chemical substance introduced in the human system to combat the spread of the disease is not sufficient enough, the amoebae will spread all over and the patient will certainly die. However, if this concentration is sufficient enough, the amoebae will decrease in space. It is observed from the graphical representation that the approximate solutions obtained here display the behaviour of the real world problem.

(a)

(b)

Figure 1 

Biological behaviour of concentrations of the chemical substance and amoebae as function of space.

(a)

(b)

Figure 2 

Biological behaviour of concentrations as function of time.

Figure 3 

The behaviour of the coupled solutions.

Example 2. Consider the following Keller-Segel equation with the sensitivity function 𝜒(𝜌)=𝜌.
With the chemotactic term (𝜕/𝜕𝑥)(𝑢(𝑥,𝑡)(𝜕𝜒(𝜌)/𝜕𝑥)) = (𝜕𝑢(𝑥,𝑡)/𝜕𝑥)(𝜕𝜌(𝑥,𝑡)/𝜕𝑥) + 𝑢(𝑥,𝑡)(𝜕²𝜌(𝑥)/𝜕𝑥²), 𝜕𝑢(𝑥,𝑡)𝜕𝜕𝑡=𝑎²𝑢(𝑥,𝑡)𝜕𝑥²−𝜕𝑢(𝑥,𝑡)()𝜕𝑥𝜕𝜌𝑥,𝑡()𝜕𝜕𝑥+𝑢𝑥,𝑡²𝜌(𝑥)𝜕𝑥²,𝜕𝑢(𝑥,𝑡)𝜕𝜕𝑡=𝑏²𝜌(𝑥,𝑡)𝜕𝑥²+𝑐𝑢(𝑥,𝑡)+𝑐𝑢(𝑥,𝑡)−𝑑𝜌(𝑥,𝑡), (20) subject to the initial conditions 𝑢(𝑥,0)=𝑢₀(𝑥),𝜌(𝑥,0)=𝜌₀(𝑥),𝑥∈𝐼. (21) In the view of the homotopy decomposition method, we arrive at the following set of integral equations that are very easy to handle: 𝑝⁰∶𝑢₀(𝑥,𝑡)=𝑢₀(𝑥),𝑢₀()(),𝑝𝑥,0=𝑢𝑥,0₀∶𝜌₀(𝑥,𝑡)=𝜌₀(𝑥),𝜌₀(𝑥,0)=𝜌₀(𝑥),𝑝¹∶𝑢₁(𝑥,𝑡)=^𝑡₀𝑎𝜕²𝑢₀𝜕𝑥²−𝜕𝑢₀𝜕𝑥𝜕𝜌₀𝜕𝑥+𝑢₀𝜕²𝜌₀𝜕𝑥²𝑝𝑑𝜏,₁∶𝜌₁(𝑥,𝑡)=^𝑡₀𝑏𝜕²𝜌₀𝜕𝑥²+𝑐𝑢₀−𝑑𝜌₀𝑑𝜏,𝜌₁()⋮𝑝𝑥,0=0,_𝑛∶𝑢_𝑛(𝑥,𝑡)=^𝑡₀𝑎𝜕²𝑢_𝑛−1𝜕𝑥²−^𝑛−1_𝑗=0𝜕𝑢_𝑗𝜕𝑥𝜕𝜌_{𝑛−𝑗−1}+𝜕𝑥_{𝑛−1𝑗=0}𝑢_𝑗𝜕²𝜌_{𝑛−𝑗−1}𝜕𝑥²𝑢𝑑𝜏,_𝑛()𝑝𝑥,0=0,_𝑛∶𝜌_𝑛(𝑥,𝑡)=^𝑡₀𝑏𝜕²𝜌_𝑛−1𝜕𝑥²+𝑐𝑢_𝑛−1−𝑑𝜌_𝑛−1𝜌𝑑𝜏,_𝑛(𝑥,0)=0. (22) Here we will consider two cases. Case one: we suppose that 𝑢(𝑥,0)=𝑚𝑒^−𝑥 and 𝜌(𝑥,0)=𝑛𝑒^−𝑥. The following series solutions are obtained: 𝑢₀(𝑥,𝑡)=𝑚𝑒^−𝑥,𝜌₀(𝑥,𝑡)=𝑛𝑒^−𝑥,𝑢₁(𝑥,𝑡)=𝑎𝑒^−𝑥𝑚𝑡,𝜌₁(𝑥,𝑡)=𝑒^−𝑥(()𝑛)𝑢𝑐𝑚+𝑏−𝑑𝑡,₂()=1𝑥,𝑡2𝑎²𝑒^−𝑥𝑚𝑡²,𝜌₂()=1𝑥,𝑡2𝑒^−𝑥(𝑎𝑐𝑚+(𝑏−𝑑)(𝑐𝑚+(𝑏−𝑑)𝑛))𝑡²,𝑢₃()=1𝑥,𝑡𝑒3!^−𝑥𝑚(𝑎𝑡)³,𝜌₃()=1𝑥,𝑡𝑒3!^−𝑥𝑡³𝑎²()𝑚+()𝑐𝑚+𝑎𝑐𝑏−𝑑𝑏−𝑑²(()𝑛,𝑢𝑐𝑚+𝑏−𝑑)₄()=1𝑥,𝑡𝑒4!^−𝑥𝑚(𝑎𝑡)⁴,𝜌₄()=1𝑥,𝑡𝑒4!^−𝑥𝑡⁴𝑎³𝑐𝑚+𝑎²𝑐(𝑏−𝑑)𝑚+𝑎𝑐(𝑏−𝑑)²𝑚+(𝑏−𝑑)³(()𝑛)⋮𝑐𝑚+𝑏−𝑑 (23) The remaining terms can be obtained by using the iteration formula. But here, only few terms of the series solutions are considered and the asymptotic solution is given as 𝑢_𝑁(𝑥,𝑡)=^𝑁_𝑛=0𝑚𝑒^−𝑥(𝑎𝑡)^𝑛,𝜌()𝑛!𝑥,𝑡=𝜌₀(𝑥,𝑡)+𝜌₁(𝑥,𝑡)+𝜌₂(𝑥,𝑡)+𝜌₃(𝑥,𝑡)+⋯. (24) Therefore when 𝑁 tends to infinity the concentration of amoebae converges to 𝑢(𝑥,𝑡)=𝑚𝑒^{(𝑎𝑡−𝑥)}. (25) The following figures show the biological behaviour of the coupled solution for the following set of theoretical parameters: 𝑚=120, 𝑛=160, 𝑎=0.5, 𝑏=0.001, 𝑐=1, and 𝑑=2, first for a fixed time 𝑡=5 and secondly for a fixed distance 𝑥=5.
Second case, we suppose that 𝑢(𝑥,0)=𝑚𝑒^−𝑥2,𝜌(𝑥,0)=𝑛𝑒^−𝑥2. (26) Following the homotopy decomposition steps, we arrived at the following series solutions: 𝑢(𝑥,0)=𝑚𝑒^−𝑥2,𝜌(𝑥,0)=𝑛𝑒^−𝑥2,𝑢₁(𝑥,𝑡)=2𝑒^−2𝑥2𝑚𝑡−𝑛+𝑎𝑒^𝑥2−1+2𝑥²,𝜌₁(𝑥,𝑡)=𝑒^−𝑥2𝑡𝑐𝑚−𝑛𝑑+𝑏2−4𝑥²,𝑢₂(𝑥,𝑡)=𝑒^−3𝑥2𝑚𝑡²×−𝑐𝑒^𝑥2𝑚−6𝑎𝑒^𝑥2𝑛−1+2𝑥²+2𝑎²𝑒^2𝑥23−12𝑥²+4𝑥⁴+𝑛𝑑𝑒^𝑥2+2𝑛+4𝑛𝑥²−6𝑏𝑒^𝑥2−1+2𝑥²,𝜌₂()=1𝑥,𝑡2𝑒^−2𝑥2𝑡²×−𝑐𝑑𝑒^𝑥2𝑚+𝑑²𝑒^𝑥2𝑛−2𝑐𝑚𝑛+2𝑎𝑐𝑒^𝑥2𝑚−1+2𝑥²+2𝑏𝑒^𝑥2(𝑐𝑚−2𝑑𝑛)−1+2𝑥²+4𝑏²𝑒^𝑥2𝑛3−12𝑥²+4𝑥⁴. (27) Using the iteration formulas, the remaining terms can be obtained. But here, only few terms of the series solutions are considered and the asymptotic solution is given as 𝑢(𝑥,𝑡)=𝑢₀(𝑥,𝑡)+𝑢₁(𝑥,𝑡)+𝑢₂(𝑥,𝑡)+𝑢₃()𝜌()𝑥,𝑡+⋯,𝑥,𝑡=𝜌₀(𝑥,𝑡)+𝜌₁(𝑥,𝑡)+𝜌₂(𝑥,𝑡)+𝜌₃(𝑥,𝑡)+⋯. (28) The following figures show the biological behaviour of the coupled solutions for the following set of theoretical parameters: 𝑚=120, 𝑛=160, 𝑎=0.5, 𝑏=0.001, 𝑐=1, and 𝑑=2, first for a fixed time 𝑡=5 and secondly for a fixed distance 𝑥=5.

The above figures show the behaviour of the solution of the system of (20) with initial conditions in (21) and (26). These solutions are describing the concentrations of the chemical substance and the amoebae in the human body for a given set of theoretical parameters chosen according to the literatures. While on one hand, Figure 4 shows the behaviour as function of space, Figures 5, 6, and 7 show the behaviour of these solutions as function of time. From the above figures, one can deduce that the concentration of amoebae reduces in space as the concentration of the chemical substance reduces. It is observed from the graphical representation that the approximate solutions obtained display the behaviour of the real world problem.

Figure 4 

Coupled solutions.

Figure 5 

Coupled solutions.

Figure 6 

Coupled solutions.

Figure 7 

Coupled solutions.

Example 3. Consider the following Keller-Segel equation with the sensitivity function 𝜒(𝜌)=𝜌².
With the chemotactic term (𝜕/𝜕𝑥)(𝑢(𝑥,𝑡)(𝜕𝜒(𝜌)/𝜕𝑥)) = (𝜕𝑢(𝑥,𝑡)/𝜕𝑥)(𝜕𝜌²(𝑥,𝑡)/𝜕𝑥) + 𝑢(𝑥,𝑡)(𝜕²𝜌²(𝑥,𝑡)/𝜕𝑥²), 𝜕𝑢(𝑥,𝑡)𝜕𝜕𝑡=𝑎²𝑢(𝑥,𝑡)𝜕𝑥²−𝜕𝑢(𝑥,𝑡)𝜕𝑥𝜕𝜌²(𝑥,𝑡)()𝜕𝜕𝑥+𝑢𝑥,𝑡²𝜌²(𝑥)𝜕𝑥²,𝜕𝑢(𝑥,𝑡)𝜕𝜕𝑡=𝑏²𝜌(𝑥,𝑡)𝜕𝑥²+𝑐𝑢(𝑥,𝑡)+𝑐𝑢(𝑥,𝑡)−𝑑𝜌(𝑥,𝑡) (29) subject to the initial conditions 𝑢(𝑥,0)=𝑢₀(𝑥),𝜌(𝑥,0)=𝜌₀(𝑥),𝑥∈𝐼. (30) Following the homotopy decomposition steps, we arrive at the following integral equations that are very easy to solve: 𝑝⁰∶𝑢₀(𝑥,𝑡)=𝑢₀(𝑥),𝑢₀()(),𝑝𝑥,0=𝑢𝑥,0₀∶𝜌₀(𝑥,𝑡)=𝜌₀(𝑥),𝜌₀(𝑥,0)=𝜌₀(𝑥),𝑝¹∶𝑢₁(𝑥,𝑡)=^𝑡₀𝑎𝜕²𝑢₀𝜕𝑥²−2𝜕𝑢₀𝜕𝑥𝜕𝜌₀𝜌𝜕𝑥⁰+2𝑢₀𝜌₀𝜕²𝜌₀𝜕𝑥²+2𝑢₀𝜕𝜌₀𝜕𝑥²𝑝𝑑𝜏,₁∶𝜌₁(𝑥,𝑡)=^𝑡₀𝑏𝜕²𝜌₀𝜕𝑥²+𝑐𝑢₀−𝑑𝜌₀𝜌𝑑𝜏,₁()𝑝𝑥,0=0,_𝑛∶𝑢_𝑛()=𝑥,𝑡_𝑡0𝑎𝜕²𝑢_𝑛−1𝜕𝑥²−2^𝑛−1𝑗_𝑗=0𝑘=0𝜌_𝑘𝜕𝑢_𝑗−𝑘𝜕𝑥𝜕𝜌_{𝑛−𝑗−1}𝜕𝑥+2^𝑛−1𝑗_𝑗=0𝑘=0𝑢_𝑘𝜌_𝑗−𝑘𝜕²𝜌_{𝑛−𝑗−1}𝜕𝑥²+2^𝑛−1𝑗_𝑗=0𝑘=0𝑢_𝑘𝜕𝜌_𝑗−𝑘𝜕𝑥𝜕𝜌_{𝑛−𝑗−1}𝑢𝜕𝑥𝑑𝜏,_𝑛()𝑝𝑥,0=0,_𝑛∶𝜌_𝑛(𝑥,𝑡)=^𝑡₀𝑏𝜕²𝜌_𝑛−1𝜕𝑥²+𝑐𝑢_𝑛−1−𝑑𝜌_𝑛−1𝜌𝑑𝜏,_𝑛(𝑥,0)=0. (31) We will suppose that 𝑢₀(𝑥,0)=𝑚sin(𝑥) and 𝜌(𝑥,0)=𝑛sin(𝑥). The following series solutions are obtained: 𝑢₀(𝑥,𝑡)=𝑚sin(𝑥),𝜌₀()(𝑥),𝑢𝑥,𝑡=𝑛sin₁(𝑥,𝑡)=−𝑚𝑡𝑎+2𝑛²(cos(𝑥))²(𝑥),𝜌sin₁()(𝑥()𝑛),𝑢𝑥,𝑡=𝑡sin)(𝑐𝑚−𝑏+𝑑₂()(𝑥)(𝑥,𝑡=−𝑚𝑡sin𝑎+𝑛𝑡2𝑐𝑚−𝑎𝑛−2𝑏𝑛−2𝑑𝑛+𝑛³+𝑛³cos(2𝑥)(sin(𝑥))²,𝜌₂()1𝑥,𝑡=−2𝑡²𝑎𝑐𝑚+𝑏𝑐𝑚+𝑐𝑑𝑚−𝑏²𝑛−2𝑑𝑏𝑛−𝑑²𝑛+𝑐𝑛𝑚²+𝑐𝑚𝑛²cos(2𝑥)sin(𝑥). (32) Using the iteration formulas, the remaining terms can be obtained. But here, only few terms of the series solutions are considered and the asymptotic solution is given as 𝑢(𝑥,𝑡)=𝑢₀(𝑥,𝑡)+𝑢₁(𝑥,𝑡)+𝑢₂(𝑥,𝑡)+𝑢₃()𝜌()𝑥,𝑡+⋯𝑥,𝑡=𝜌₀(𝑥,𝑡)+𝜌₁(𝑥,𝑡)+𝜌₂(𝑥,𝑡)+𝜌₃(𝑥,𝑡)+⋯. (33)

4. Discussion

The homotopy decomposition method is chosen to solve this kind of nonlinear problem. Because of the following advantages that, the HDM has over the exiting methods. The method does not require the linearization or assumptions of weak nonlinearity [29, 30]. The solutions are not generated in the form of general solution as in Adomian decomposition method [29, 31, 32]. No Lagrange multiplier and correction functional are required as in the case of the variational iteration method [22, 30, 31, 34]. It is more realistic compared to the method of simplifying the physical problems. If the exact solution of the partial differential equation exists, the approximated solution via the method converges to the exact solution [26]. A construction of a homotopy 𝑣(𝑟,𝑝)∶Ω×[0,1] is not needed as in the case of the homotopy perturbation method, because in this case one needs first to continuously deform a difficult problem into another one, which is easy to solve [17, 18, 24, 33]. HDM provides us with a convenient way to control the convergence of approximation series without adapting ℎ, as in the case of [24] which is a fundamental qualitative difference in analysis between HDM and other methods [29–32, 34–38].

5. Conclusion

An interesting biological problem describing theaggregation process of cellular slime mold by the chemical attraction was investigated in this paper. We made use of the efficient method called homotopy decomposition method to derive the solution of the mathematical equation underpinning this problem. Analysis and results of nonlinear system of attractor one-dimensional Keller-Segel equation indicate that the model matches the regular biological diffusion behaviour observed in the field.

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors’ Contribution

Abdon Atangana wrote the first draft, P. D. Vermeulen revised the paper, and all the authors corrected the last version.