MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 654759 10.1155/2013/654759 654759 Research Article Study on Space-Time Fractional Nonlinear Biological Equation in Radial Symmetry Liu Yanqin 1, 2 Ionescu Clara 1 Department of Mathematics Dezhou University, Dezhou 253023 China dzu.edu.cn 2 Nonlinear Dynamics and Chaos Group School of Management, Tianjin University, Tianjin 30072 China tju.edu.cn 2013 11 2 2013 2013 01 09 2012 24 12 2012 25 12 2012 2013 Copyright © 2013 Yanqin Liu. 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.

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  can be described very successfully by models using mathematical tools fractional calculus [6, 7], such as anomalous transport in disordered systems , some percolations in porous media, and the diffusion biological population. Mathematical aspects of the biological problem have been considered in many papers . El-Sayed et al.  studied the fractional-order biological population model in the form αu/tα=2(u2)/x2+2(u2)/y2+f(u) using the Adomian decomposition method. Wazwaz and Gorguis  gave a detailed study of integer Fisher’s diffusion equation by using Adomian decomposition method. Najeeb et al.  studied the time fractional Fisher’s equation and approximate analytical solutions were obtained by using homotopy analysis method. Petrovskii et al.  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  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 Jα(α0) of a function f(t) is defined as (1)Jαf(t)=1Γ(α)0t(t-τ)α-1f(τ)dτ,(α0), where Γ(·) is the well-known gamma function, and some properties of the operator Jα are as follows: (2)JαJβf(t)=Jα+βf(t),(α0,β0),Jαtγ=Γ(1+γ)Γ(1+γ+α)tα+γ,(γ-1).

Definition 2.

The Caputo fractional derivative Dα of a function f(t) is defined as (3)  0Dtαf(t)=1Γ(n-α)0tf(n)(t)dτ(t-τ)α+1-n,(n-1<Re(α)n,nN). The following are two basic properties of the Caputo fractional derivative: (4)  0Dtαtβ=Γ(1+β)Γ(1+β-α)tβ-α,  0Dtα(D0tmf(t))=D0tα+mf(t),(m=0,1,2,;n-1<αn),(JαDα)f(t)=f(t)-k=0n-1f(k)(0+)tkk!. 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] (5)u(x,y,t)t=D(2x2(u2)+2uy2(u2))+f(u), 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: (6)u(r,t)t=D(2r2(u2)+1rr(u2))+f(u),u(r,0)=φ(r,l), where 0<r<l and l is the typical size of the domain and initial condition φ(r,l) promptly approaches zero when r/l1. 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: (7)αu(r,t)tα=D(1+βr1+β(u2)+1rβrβ(u2))+f(u),u(r,0)=φ(r,l), where 0<α1,0<β1 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   0Dtαf(t)=a0-αDtαf(at) of the Caputo derivative and using reduced dimensionless variables defined as (8)u~=uK,r~=rl,t~=(KDl1+β)1/αt (8) can be reduced to the respective dimensionless forms (tildes will be omitted hereafter): (9)αutα=1+βr1+β(u2)+1rβrβ(u2)+F(u),u(r,0)=φ(r,1), where F(u)=(l1+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: (10)αutα=1+βr1+β(u2)+1rβrβ(u2)+hu,(11)u(r,0)=r, where F(u)=hu,h = constant, corresponding to the Malthusian law. According to the homotopy perturbation method, we construct the following simple homotopy: (12)αutα=p[1+βr1+β(u2)+1rβrβ(u2)+hu], where p[0,1] is an embedding parameter. In case p=0, (12) is a fractional differential equation, Dtαu=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: (13)u=n=0pnun=u0+pu1+p2u2+p3u3+,u0 is an initial approximation of (10). The approximate solutions of the original equations can be obtained by setting p=1, that is, (14)u(x,t)=limp1n=0pnun=u0+u1+u2+u3+. Instituting (13) into (12) and comparing coefficients of terms with identical powers of p then applying Jα on both sides of equations yield (15)u0=r,u1=Jβ[1+βr1+β(u02)+1rβrβ(u02)+hu0]=hrtαΓ(1+α)+2r1-βtαΓ(1+α)Γ(2-β)+2r1-βtαΓ(1+α)Γ(3-β),u2=Jβ[1+βr1+β(2u0u1)+1rβrβ(2u0u1)+hu1]=h2rt2αΓ(1+2α)+8r1-2βt2αΓ(1+2α)Γ(2-2β)-4r1-2βt2αβΓ(1+2α)Γ(2-2β)+4r1-2βt2αΓ(1+2α)Γ(3-2β)+6hr1-βt2αΓ(1+2α)Γ(2-β)+6hr1-βt2αΓ(1+2α)Γ(3-β)+8r1-2βt2αΓ(2-β)Γ(1+2α)Γ(2-2β)Γ(3-β)-4r1-2βt2αβΓ(2-β)Γ(1+2α)Γ(2-2β)Γ(3-β)+4r1-2βt2αΓ(3-β)Γ(1+2α)Γ(3-2β)Γ(2-β),u3=Jβ[1+βr1+β(2u0u2+u12)+1rβrβ(2u0u2+u12)+hu2] 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 (16)u(r,t)=r+hrtαΓ(1+α)+2r1-βtαΓ(1+α)Γ(2-β)+2r1-βtαΓ(1+α)Γ(3-β)+.

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,and  r.

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

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

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

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: (17)αutα=1+βr1+β(u2)+1rβrβ(u2)+hu(1-gu),(18)u(r,0)=er,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: (19)αutα=p[1+βr1+β(u2)+1rβrβ(u2)+hu(1-gu)]. 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 (20)er=1+r+r22+r36+. Instituting (13) into (19) and comparing coefficients of terms with identical powers of p then applying Jα on both sides of equations yield (21)u0=1,u1=r+Jβ[1+βr1+β(u02)+1rβrβ(u02)+hu0-hgu02]=r+(h-hg)tαΓ(1+α),u2=r22+Jβ[1+βr1+β(2u0u1)+1rβrβ(2u0u1)+hu1-2hgu0u1]=r22+(hr-2ghr)tαΓ(1+α)+h2tαΓ(1+2α)-3gh2t2αΓ(1+2α)+2g2h2t2αΓ(1+2α)+2r-βtαΓ(2-β)Γ(1+α),u3=Jβ[1+βr1+β(2u0u2+u12)+1rβrβ×(2u0u2+u12)+hu2-hg(2u0u2+u12)1+βr1+β]=r36+hrtαΓ(1+α)-2hgr2tαΓ(1+α)-h3gt3αΓ(1+2α)Γ2(1+α)Γ(1+3α)+2h3g2t3αΓ(1+2α)Γ2(1+α)Γ(1+3α)-h3g3t3αΓ(1+2α)Γ2(1+α)Γ(1+3α)+h2t2αΓ(1+2α)-gh2t2αΓ(1+2α)-4h2grt2αΓ(1+2α)+6g2h2t2αΓ(1+2α)-2h3gt3αΓ(1+3α)+6g2h3t3αΓ(1+3α)-4g3h3t3αΓ(1+3α)+4r1-βtαΓ(2-β)Γ(1+α)+4hr-βt2αΓ(2-β)Γ(1+2α)-6ghr-βt2αΓ(2-β)Γ(1+2α)-4hgr-βt2αΓ(2-β)Γ(1+2α)+4r1-2βt2αΓ(1-β)Γ(1-2β)Γ(2-β)Γ(1+2α)+4r1-βtαΓ(3-β)Γ(1+α)-4r-1-2βt2αβΓ(-β)Γ(1+2α)Γ(2-β)Γ(-2β) 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 (22)u(r,t)=er+(h-hg)tαΓ(1+α)+(hr-2ghr)tαΓ(1+α)+h2tαΓ(1+2α)-3gh2t2αΓ(1+2α)+.

Case 3.

We will consider the following initial value problem of time-space fractional nonlinear diffusion equation: (23)αutα=1+βr1+β(u2)+1rβrβ(u2)+F(u),(24)u(r,0)=φ(r). According to the homotopy perturbation method, we construct the following simple homotopy: (25)αutα=p[1+βr1+β(u2)+1rβrβ(u2)+F(u)]. 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 (26)u(r,0)=φ(r)=n=0φn(r). Instituting (12) into (23) and equating coefficients of terms with identical powers of p(27)p0:Dtαu0=0,u0(r,0)=φ0(r),pn:Dtαun=1+βr1+β(vn-1)+1rβrβ(vn-1)+Fn-1,un(r,0)=φn(r), where vn-1 is the coefficient of pn-1 in u2 and Fn-1 is the coefficient of pn-1 in F(u), then applying Jα, the inverse operator of Dtα, on both sides of equations, it is obvious that (27) are easy to solve, the components un,n0 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.)

Shigesada N. Kawasaki K. Biological Invasions: Theory and Practice 1997 Oxford, UK Oxford University Petrovskii S. Shigesada N. Some exact solutions of a generalized Fisher equation related to the problem of biological invasion Mathematical Biosciences 2001 172 2 73 94 10.1016/S0025-5564(01)00068-2 MR1853470 Podlubny I. Fractional Differential Equations 1999 New York, NY, USA Academic Press xxiv+340 MR1658022 Sabatier J. Agrawal O. P. Machado J. A. T. Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering 2007 Dordrecht, The Netherlands Springer Hilfer R. Applications of Fractional Calculus in Physics 2000 Singapore World Scientific Machado J. T. Kiryakova V. Mainardi F. Recent history of fractional calculus Communications in Nonlinear Science and Numerical Simulation 2011 16 3 1140 1153 10.1016/j.cnsns.2010.05.027 MR2736622 ZBL1221.26002 Lenzi E. K. Mendes G. A. Mendes R. S. Da Silva L. R. Lucena . L. S. Exact solutions to nonlinear nonautonomous space-fractional diffusion equations with absorption Physical Review E 2003 67 51 051109 Lenzi E. K. Malacarne L. C. Mendes R. S. Pedron I. T. Anomalous diffsion, nonlinear fractional Fokker-Planck equation and solutions Physica A 2003 319 245 252 Bologna M. Tsallis C. Grigolini P. Anomalous diffusion associated with nonlinear fractional derivative Fokker-Planck-like equation: exact time-dependent solutions Physical Review E 2000 62 2 2213 2218 Tsallis C. Lenzi E. K. Anomalous diffusion: nonlinear fractional Fokker-Planck equation Chemical Physics 2002 284 341 347 Shakeri F. Dehghan M. Numerical solution of a biological population model using He's variational iteration method Computers & Mathematics with Applications 2007 54 7-8 1197 1209 10.1016/j.camwa.2006.12.076 MR2398143 ZBL1137.92033 Tan Y. Xu H. Liao S.-J. Explicit series solution of travelling waves with a front of Fisher equation Chaos, Solitons and Fractals 2007 31 2 462 472 10.1016/j.chaos.2005.10.001 MR2259770 ZBL1143.35313 Kadem A. Baleanu D. Homotopy perturbation method for the coupled fractional Lotka-Volterra equations Romanian Journal of Physics 2011 56 3-4 332 338 MR2803287 ZBL1231.65134 El-Sayed A. M. A. Rida S. Z. Arafa A. A. M. Exact solutions of fractional-order biological population model Communications in Theoretical Physics 2009 52 6 992 996 10.1088/0253-6102/52/6/04 MR2683018 ZBL1184.92038 Wazwaz A.-M. Gorguis A. An analytic study of Fisher's equation by using Adomian decomposition method Applied Mathematics and Computation 2004 154 3 609 620 10.1016/S0096-3003(03)00738-0 MR2072808 ZBL1054.65107 Najeeb A. K. Nasir-Uddin K. Asmat A. Muhammad J. Approximate analytical solutions of fractional reaction-diffusion equations Journal of King Saud University 2012 24 2 111 118 Petrovskii S. Malchow H. Li B.-L. An exact solution of a diffusive predator-prey system Proceedings of The Royal Society of London A 2005 461 2056 1029 1053 10.1098/rspa.2004.1404 MR2144614 ZBL1145.92341 Liu Y. Xin B. Numerical solutions of a fractional predator-prey system Advances in Difference Equations 2011 2011 11 190475 MR2780665 ZBL1217.35205 Khan N. A. Ayaz M. Jin L. Yildirim A. On the approximate solutions for the time-fractional reaction-diffusion equation of Fisher type International Journal of the Physical Science 2011 6 2483 2496