Study on Space-Time Fractional Nonlinear Biological Equation in Radial Symmetry

Abstract

We consider the initial stage of space-time fractional generalized biological equation in radial symmetry. Dimensionless multiorder fractional nonlinear equation was first given, and approximate solutions were derived in the form of series using the homotopy perturbation method with a new modification. And the influence of fractional derivative is also discussed.

1. Introduction

The problem of biological diffusion is an issue of increasing significance in contemporary ecology [1, 2]. In case of favourable environmental conditions, the alien population may begin to grow and spread over the area and thus the local initial structural perturbation of the native biological community may lead to large-scale dramatic changes in the community structure. Recently, it has turned out that many phenomena in engineering, physics, chemistry, and other sciences [3–5] can be described very successfully by models using mathematical tools fractional calculus [6, 7], such as anomalous transport in disordered systems [8–10], some percolations in porous media, and the diffusion biological population. Mathematical aspects of the biological problem have been considered in many papers [11–13]. El-Sayed et al. [14] studied the fractional-order biological population model in the form 𝜕^𝛼𝑢/𝜕𝑡^𝛼=𝜕²(𝑢²)/𝜕𝑥²+𝜕²(𝑢²)/𝜕𝑦²+𝑓(𝑢) using the Adomian decomposition method. Wazwaz and Gorguis [15] gave a detailed study of integer Fisher’s diffusion equation by using Adomian decomposition method. Najeeb et al. [16] studied the time fractional Fisher’s equation and approximate analytical solutions were obtained by using homotopy analysis method. Petrovskii et al. [17] obtained an exact solution of the spatiotemporal dynamics of a predator-prey community by using an approximate change of variables, and the properties of the solution exhibit biologically reasonable dependence on the parameter values. Liu and Xin [18] studied the fractional Lotka-Volterra equations using the homotopy perturbation method.

This paper is devoted to investigating approximate solutions of a generalized fractional nonlinear population diffusion equation in radical symmetry. The structure of the paper is as follows. In Section 2, a brief review of the theory of fractional calculus will be given to fix notation and provide a convenient reference. In Section 3, a mathematical formulation of the generalized multifractional population diffusion model in radical symmetry is given. In Section 4, we extend the homotopy perturbation method and a new reliable modification to the fractional nonlinear population diffusion system and give some properties of this model. Conclusions and prospects will be presented in Section 5.

2. Fractional Calculus

There are several approaches to define the fractional calculus; the Riemann-Liouville and Caputo fractional operators are defined as follows.

Definition 1. The Riemann-Liouville fractional integral operator 𝐽^𝛼(𝛼≥0) of a function 𝑓(𝑡) is defined as 𝐽^𝛼𝑓(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝑓(𝜏)𝑑𝜏,(𝛼≥0), (1) where Γ(⋅) is the well-known gamma function, and some properties of the operator 𝐽^𝛼 are as follows: 𝐽^𝛼𝐽^𝛽𝑓(𝑡)=𝐽^𝛼+𝛽𝑓(𝑡),,𝐽𝛼≥0,𝛽≥0_𝛼𝑡^𝛾=Γ1+𝛾Γ𝑡1+𝛾+𝛼^𝛼+𝛾,𝛾≥−1. (2)

Definition 2. The Caputo fractional derivative 𝐷^𝛼 of a function 𝑓(𝑡) is defined as ₀𝐷^𝛼_𝑡𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡₀𝑓^(𝑛)(𝑡)𝑑𝜏(𝑡−𝜏)^{𝛼+1−𝑛},(𝑛−1<Re(𝛼)≤𝑛,𝑛∈𝑁). (3) The following are two basic properties of the Caputo fractional derivative: ₀𝐷^𝛼_𝑡𝑡^𝛽=Γ1+𝛽Γ𝑡1+𝛽−𝛼^𝛽−𝛼₀,𝐷^𝛼_𝑡0𝐷^𝑚_𝑡𝑓(𝑡)=₀𝐷_𝑡^𝛼+𝑚𝑓(𝑡),(),𝐽𝑚=0,1,2,…;𝑛−1<𝛼≤𝑛_𝛼𝐷^𝛼𝑓(𝑡)=𝑓(𝑡)−^𝑛−1_𝑘=0𝑓^(𝑘)0⁺𝑡^𝑘.𝑘! (4) We have chosen the Caputo fractional derivative because it allows traditional initial and boundary conditions to be included in the formulation of the problem. And some other properties of fractional derivative can be found in [3, 5].

3. Main Equations

It is widely accepted that the spatiotemporal dynamics of a biological community can be qualitatively described by diffusion-reaction equations [2, 19]. A remarkable point is that, in some cases, relatively simple single-species models provide not only qualitative but also quantitative descriptions of the dynamics of a population. In this paper, we first consider a single-species parabolic nonlinear equation arising in the spatial diffusion of biological populations [11, 14] 𝜕𝑢𝑥,𝑦,𝑡𝜕𝜕𝑡=𝐷²𝜕𝑥²𝑢²+𝜕²𝑢𝜕𝑦²𝑢²+𝑓(𝑢), (5) where 𝑢(𝑥,𝑦,𝑡)≥0 is the population density at position 𝑥,𝑦, and time 𝑡, coefficient 𝐷 describes the intensity of mixing due to animals self-motion, the term 𝑓(𝑢) describes multiplication and mortality of a given population, 𝑓(0)=𝑓(𝐾)=0, and parameter 𝐾 is being treated as the carrying capacity for the given population. Let us assume that the initial distribution of the species is symmetrical, with the density of the species depending only on the distance from the origin. Assuming also that the environment is homogeneous, that is, both  𝐷  and  𝑓(𝑢)  not depends explicitly on the position in space, we arrive at the following problem: 𝜕𝑢(𝑟,𝑡)𝜕𝜕𝑡=𝐷²𝜕𝑟²𝑢²+1𝑟𝜕𝑢𝜕𝑟²+𝑓(𝑢),𝑢(𝑟,0)=𝜑(𝑟,𝑙), (6) where 0<𝑟<𝑙 and 𝑙 is the typical size of the domain and initial condition 𝜑(𝑟,𝑙) promptly approaches zero when 𝑟/𝑙≫1. It has turned out that the diffusion of biological population can be described very successfully by fractional calculus. In this paper, we discuss the corresponding fractional equation and the main aim is to solve the nonlinear fractional biological population model in the following form: 𝜕^𝛼𝑢(𝑟,𝑡)𝜕𝑡^𝛼𝜕=𝐷^1+𝛽𝜕𝑟^1+𝛽𝑢²+1𝑟𝜕^𝛽𝜕𝑟^𝛽𝑢²+𝑓(𝑢),𝑢(𝑟,0)=𝜑(𝑟,𝑙), (7) where 0<𝛼≤1,0<𝛽≤1 is the Caputo derivative. A proper choice of the dimensionless variables, that is, in our case, the choice of scales for the variables 𝑢,𝑟, and 𝑡, is an important point. Coming with the property ₀𝐷^𝛼_𝑡𝑓(𝑡)=𝑎₀^−𝛼𝐷^𝛼_𝑡𝑓(𝑎𝑡) of the Caputo derivative and using reduced dimensionless variables defined as 𝑢𝑢=𝐾,̃𝑟𝑟=𝑙,𝑡̃=𝐾𝐷𝑙^1+𝛽1/𝛼𝑡 (8) (8) can be reduced to the respective dimensionless forms (tildes will be omitted hereafter): 𝜕^𝛼𝑢𝜕𝑡^𝛼=𝜕^1+𝛽𝜕𝑟^1+𝛽𝑢²+1𝑟𝜕^𝛽𝜕𝑟^𝛽𝑢²+𝐹(𝑢),𝑢(𝑟,0)=𝜑(𝑟,1), (9) where 𝐹(𝑢)=(𝑙^1+𝛽/𝐾𝐷)𝑓(𝑢).

4. Approximate Solution to the Equation

We now proceed to derive approximate solution to fractional nonlinear population diffusion equation (9).

Case 1. In this case, we will examine the following time-space fractional nonlinear population model: 𝜕^𝛼𝑢𝜕𝑡^𝛼=𝜕^1+𝛽𝜕𝑟^1+𝛽𝑢²+1𝑟𝜕^𝛽𝜕𝑟^𝛽𝑢²𝑢()+ℎ𝑢,𝑟,0=𝑟, (10)(11) where 𝐹(𝑢)=ℎ𝑢,ℎ = constant, corresponding to the Malthusian law. According to the homotopy perturbation method, we construct the following simple homotopy: 𝜕^𝛼𝑢𝜕𝑡^𝛼𝜕=𝑝^1+𝛽𝜕𝑟^1+𝛽𝑢²+1𝑟𝜕^𝛽𝜕𝑟^𝛽𝑢²+ℎ𝑢, (12) where 𝑝∈[0,1] is an embedding parameter. In case 𝑝=0, (12) is a fractional differential equation, 𝐷^𝛼_𝑡𝑢=0, which is easy to solve, and when 𝑝=1, (12) turns out to be the original one (10). The basic assumption is that the solutions can be written as a power series in 𝑝: 𝑢=^∞_𝑛=0𝑝^𝑛𝑢_𝑛=𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+𝑝³𝑢₃+⋯, (13)𝑢₀ is an initial approximation of (10). The approximate solutions of the original equations can be obtained by setting 𝑝=1, that is, 𝑢(𝑥,𝑡)=lim^∞_{𝑝→1𝑛=0}𝑝^𝑛𝑢_𝑛=𝑢₀+𝑢₁+𝑢₂+𝑢₃+⋯. (14) Instituting (13) into (12) and comparing coefficients of terms with identical powers of 𝑝 then applying 𝐽^𝛼 on both sides of equations yield 𝑢₀𝑢=𝑟,₁=𝐽^𝛽𝜕^1+𝛽𝜕𝑟^1+𝛽𝑢²₀+1𝑟𝜕^𝛽𝜕𝑟^𝛽𝑢²₀+ℎ𝑢₀=ℎ𝑟𝑡^𝛼Γ()+1+𝛼2𝑟^1−𝛽𝑡^𝛼Γ()Γ+1+𝛼2−𝛽2𝑟^1−𝛽𝑡^𝛼Γ()Γ,𝑢1+𝛼3−𝛽₂=𝐽^𝛽𝜕^1+𝛽𝜕𝑟^1+𝛽2𝑢₀𝑢₁+1𝑟𝜕^𝛽𝜕𝑟^𝛽2𝑢₀𝑢₁+ℎ𝑢₁=ℎ²𝑟𝑡^2𝛼Γ()+1+2𝛼8𝑟^1−2𝛽𝑡^2𝛼Γ()Γ−1+2𝛼2−2𝛽4𝑟^1−2𝛽𝑡^2𝛼𝛽Γ()Γ+1+2𝛼2−2𝛽4𝑟^1−2𝛽𝑡^2𝛼Γ()Γ+1+2𝛼3−2𝛽6ℎ𝑟^1−𝛽𝑡^2𝛼Γ()Γ+1+2𝛼2−𝛽6ℎ𝑟^1−𝛽𝑡^2𝛼Γ()Γ+1+2𝛼3−𝛽8𝑟^1−2𝛽𝑡^2𝛼Γ2−𝛽Γ()ΓΓ−1+2𝛼2−2𝛽3−𝛽4𝑟^1−2𝛽𝑡^2𝛼𝛽Γ2−𝛽Γ()ΓΓ+1+2𝛼2−2𝛽3−𝛽4𝑟^1−2𝛽𝑡^2𝛼Γ3−𝛽Γ()ΓΓ,𝑢1+2𝛼3−2𝛽2−𝛽₃=𝐽^𝛽𝜕^1+𝛽𝜕𝑟^1+𝛽2𝑢₀𝑢₂+𝑢²₁+1𝑟𝜕^𝛽𝜕𝑟^𝛽2𝑢₀𝑢₂+𝑢²₁+ℎ𝑢₂⋮ (15) and so on, in this manner the rest of components of the solution can be obtained. The solution of (10) in series form is given by 𝑢(𝑟,𝑡)=𝑟+ℎ𝑟𝑡^𝛼Γ()+1+𝛼2𝑟^1−𝛽𝑡^𝛼Γ()Γ+1+𝛼2−𝛽2𝑟^1−𝛽𝑡^𝛼Γ(1+𝛼)Γ3−𝛽+⋯. (16)

Figure 1 shows the approximate solution for (10) and (11) by using the homotopy perturbation method when choosing 𝑟=0.6,𝛽=1. From the figure, it is clear to see the time evolution of nonlinear population diffusion density and we also know that the approximate solution of fractional population model is continuous with the fractional parameter 𝛼. Figure 2 shows the approximate solution for (10) and (11) when 𝑟=0.6,𝛼=1, and the approximate solution of fractional population model is continuous with the fractional parameter 𝛽. Figures 3 and 4 show the approximate solution for (10) and (11) when the time 𝑡=10, from the figures, we also know that the population density changes with the parameters 𝛼,𝛽,and𝑟.

Figure 1 

The surface of second-order approximate solution of (10) when 𝑟=0.6,𝛽=1.

Figure 2 

The surface of second-order approximate solution of (10) when 𝑟=0.6,𝛼=1.

Figure 3 

The surface of second-order approximate solution of (10) when 𝑡=10,𝛽=1.

Figure 4 

The surface of second-order approximate solution of (10) when 𝑡=10,𝛼=1.

Case 2. In this case, we will examine the following time-space fractional nonlinear population model: 𝜕^𝛼𝑢𝜕𝑡^𝛼=𝜕^1+𝛽𝜕𝑟^1+𝛽𝑢²+1𝑟𝜕^𝛽𝜕𝑟^𝛽𝑢²,𝑢()+ℎ𝑢1−𝑔𝑢𝑟,0=𝑒^𝑟, (17)(18)𝐹(𝑢)=ℎ𝑢(1−𝑔𝑢),ℎ,𝑔 are constant, corresponding to the Verhulst law. According to the homotopy perturbation method, we construct the following simple homotopy: 𝜕^𝛼𝑢𝜕𝑡^𝛼𝜕=𝑝^1+𝛽𝜕𝑟^1+𝛽𝑢²+1𝑟𝜕^𝛽𝜕𝑟^𝛽𝑢²+ℎ𝑢1−𝑔𝑢. (19) For this case, it is difficult to solve the multifractional equation. We give a new modification of the homotopy perturbation method; the modified form of the homotopy perturbation method can be established based on the initial condition expressed in the Taylor series. We suggest that 𝑢(𝑟,0) be expressed in the Taylor series 𝑒^𝑟𝑟=1+𝑟+²2+𝑟³6+⋯. (20) Instituting (13) into (19) and comparing coefficients of terms with identical powers of 𝑝 then applying 𝐽^𝛼 on both sides of equations yield 𝑢₀𝑢=1,₁=𝑟+𝐽^𝛽𝜕^1+𝛽𝜕𝑟^1+𝛽𝑢²₀+1𝑟𝜕^𝛽𝜕𝑟^𝛽𝑢²₀+ℎ𝑢₀−ℎ𝑔𝑢²₀𝑡=𝑟+ℎ−ℎ𝑔^𝛼Γ(),𝑢1+𝛼₂=𝑟²2+𝐽^𝛽𝜕^1+𝛽𝜕𝑟^1+𝛽2𝑢₀𝑢₁+1𝑟𝜕^𝛽𝜕𝑟^𝛽2𝑢₀𝑢₁+ℎ𝑢₁−2ℎ𝑔𝑢₀𝑢₁=𝑟²2+ℎ𝑟−2𝑔ℎ𝑟𝑡^𝛼Γ()+ℎ1+𝛼²𝑡^𝛼Γ()−1+2𝛼3𝑔ℎ²𝑡^2𝛼Γ()+1+2𝛼2𝑔²ℎ²𝑡^2𝛼Γ()+1+2𝛼2𝑟^−𝛽𝑡^𝛼ΓΓ(),𝑢2−𝛽1+𝛼₃=𝐽^𝛽𝜕^1+𝛽𝜕𝑟^1+𝛽2𝑢₀𝑢₂+𝑢²₁+1𝑟𝜕^𝛽𝜕𝑟^𝛽×2𝑢₀𝑢₂+𝑢²₁+ℎ𝑢₂−ℎ𝑔2𝑢₀𝑢₂+𝑢²₁=𝑟³6+ℎ𝑟𝑡^𝛼Γ()−1+𝛼2ℎ𝑔𝑟²𝑡^𝛼Γ()−ℎ1+𝛼₃𝑔𝑡^3𝛼Γ(1+2𝛼)Γ²()Γ()+1+𝛼1+3𝛼2ℎ³𝑔²𝑡^3𝛼Γ(1+2𝛼)Γ²()Γ()−ℎ1+𝛼1+3𝛼₃𝑔³𝑡^3𝛼Γ(1+2𝛼)Γ²()Γ()+ℎ1+𝛼1+3𝛼²𝑡^2𝛼Γ()−1+2𝛼𝑔ℎ²𝑡^2𝛼Γ()−1+2𝛼4ℎ²𝑔𝑟𝑡^2𝛼Γ()+1+2𝛼6𝑔²ℎ²𝑡^2𝛼Γ()−1+2𝛼2ℎ³𝑔𝑡^3𝛼Γ()+1+3𝛼6𝑔²ℎ³𝑡^3𝛼Γ()−1+3𝛼4𝑔³ℎ³𝑡^3𝛼Γ()+1+3𝛼4𝑟^1−𝛽𝑡^𝛼ΓΓ()+2−𝛽1+𝛼4ℎ𝑟^−𝛽𝑡^2𝛼ΓΓ()−2−𝛽1+2𝛼6𝑔ℎ𝑟^−𝛽𝑡^2𝛼ΓΓ()−2−𝛽1+2𝛼4ℎ𝑔𝑟^−𝛽𝑡^2𝛼ΓΓ()+2−𝛽1+2𝛼4𝑟^1−2𝛽𝑡^2𝛼Γ1−𝛽ΓΓΓ()+1−2𝛽2−𝛽1+2𝛼4𝑟^1−𝛽𝑡^𝛼ΓΓ()−3−𝛽1+𝛼4𝑟^−1−2𝛽𝑡^2𝛼𝛽Γ−𝛽Γ()ΓΓ⋮1+2𝛼2−𝛽−2𝛽 (21) and so on, in this manner the rest of components of the solution can be obtained. The solution of (17) in series form is given by 𝑢(𝑟,𝑡)=𝑒^𝑟+ℎ−ℎ𝑔𝑡^𝛼Γ()+𝑡1+𝛼ℎ𝑟−2𝑔ℎ𝑟^𝛼Γ()+ℎ1+𝛼₂𝑡^𝛼Γ()−1+2𝛼3𝑔ℎ²𝑡^2𝛼Γ(1+2𝛼)+⋯. (22)

Case 3. We will consider the following initial value problem of time-space fractional nonlinear diffusion equation: 𝜕^𝛼𝑢𝜕𝑡^𝛼=𝜕^1+𝛽𝜕𝑟^1+𝛽𝑢²+1𝑟𝜕^𝛽𝜕𝑟^𝛽𝑢²(𝑢),𝑢()(𝑟).+𝐹𝑟,0=𝜑 (23)(24) According to the homotopy perturbation method, we construct the following simple homotopy: 𝜕^𝛼𝑢𝜕𝑡^𝛼𝜕=𝑝^1+𝛽𝜕𝑟^1+𝛽𝑢²+1𝑟𝜕^𝛽𝜕𝑟^𝛽𝑢²+𝐹(𝑢). (25) The modified form of the homotopy perturbation method can be established based on the initial condition expressed in the Taylor series; the initial condition  𝑢(𝑟,0)  is expressed in the Taylor series 𝑢(𝑟,0)=𝜑(𝑟)=^∞_𝑛=0𝜑_𝑛(𝑟). (26) Instituting (12) into (23) and equating coefficients of terms with identical powers of 𝑝𝑝⁰∶𝐷^𝛼_𝑡𝑢₀=0,𝑢₀(𝑟,0)=𝜑₀(𝑟),⋮𝑝^𝑛∶𝐷^𝛼_𝑡𝑢_𝑛=𝜕^1+𝛽𝜕𝑟^1+𝛽𝑣_𝑛−1+1𝑟𝜕^𝛽𝜕𝑟^𝛽𝑣_𝑛−1+𝐹_𝑛−1,𝑢_𝑛(𝑟,0)=𝜑_𝑛(𝑟), (27) where 𝑣_𝑛−1 is the coefficient of 𝑝^𝑛−1 in 𝑢² and 𝐹_𝑛−1 is the coefficient of 𝑝^𝑛−1 in 𝐹(𝑢), then applying 𝐽^𝛼, the inverse operator of 𝐷^𝛼_𝑡, on both sides of equations, it is obvious that (27) are easy to solve, the components 𝑢_𝑛,𝑛≥0 of the homotopy perturbation method can be completely determined, and series solutions are thus entirely determined.

5. Conclusion

Approximate solutions of the multifractional nonlinear diffusion population equations in radial symmetry were derived using the homotopy perturbation method and the new modification of homotopy perturbation method. The solutions are given in the form of series with easily computable terms. The results reveal that the new modified method is very effective for solving nonlinear diffusion equation of multifractional order. This is the first step to study the multifractional nonlinear population diffusion in radical symmetry, and we will make subsequent research, for example, exact solution and self-similar exact solution of these fractional nonlinear system. And we hope that this work is a step in this direction.

Acknowledgments

The author expresses thanks to the referees for their fruitful advices and comments. This work was supported by the National Science Foundation of Shandong Province (Grants no. Y2007A06 and ZR2010Al019) and the China Postdoctoral Science Foundation (Grant no. 20100470783.)