Variational iteration method for solving multispecies Lotka Volterra equations

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1 Computers and Mathematics with Applications Variational iteration method for solving multispecies Lotka Volterra equations B. Batiha, M.S.M. Noorani, I. Hashim School of Mathematical Sciences, National University of Malaysia, 436 UKM Bangi Selangor, Malaysia Received 28 September 26; accepted 2 December 26 Abstract This paper applies the variational iteration method to multispecies Lotka Volterra equations. Comparisons with the Adomian decomposition and the fourth-order Runge Kutta methods show that the variational iteration method is a powerful method for nonlinear equations. c 27 Elsevier Ltd. All rights reserved. Keywords: Variational iteration method; Adomian decomposition method; Fourth-order Runge Kutta method; Lotka Volterra equations 1. Introduction The Lotka Volterra equations model the dynamic behaviour of an arbitrary number of competitors [1]. Though originally formulated to describe the time history of a biological system, these equations find their application in a number of engineering fields such as simultaneous chemical and nonlinear control. In fact, the one-predator one-prey Lotka Volterra model is one of the most popular ones to demonstrate a simple nonlinear control system. The accurate solutions of the Lotka Volterra equations may become a difficult task either if the equations are stiff even with a small number of species, or when the number of species is large [2]. Unlike the discrete solutions obtained by the purely numerical methods like the fourth-order Runge Kutta method RK4, approximate analytical solutions can increase our insights into the natural behaviour of complex systems. An analytical method called the Adomian decomposition method ADM proposed by Adomian [3] aims to solve frontier nonlinear physical problems. It has been applied to a wide class of deterministic and stochastic problems, linear and nonlinear, in physics, biology and chemical reactions etc. For nonlinear models, the method has shown reliable results in supplying analytical approximations that converge rapidly [4]. Yet another powerful analytical method for nonlinear equations is called the variational iteration method VIM, which was first envisioned by He [5] modifying the approach by Inokuti et al. [6]. VIM has successfully been applied to many situations. For example, He [7] solved the classical Blasius equation using VIM. In [8], He gave a solution for a seepage flow problem with fractional derivatives in porous media using VIM. He [9] also employed VIM to Corresponding author. Tel.: ; fax: address: msn@ukm.my M.S.M. Noorani /$ - see front matter c 27 Elsevier Ltd. All rights reserved. doi:1.116/j.camwa

2 94 B. Batiha et al. / Computers and Mathematics with Applications give approximate solutions for some well-known nonlinear problems and in [1], a successful application of VIM to autonomous systems of ordinary differential equations is shown. VIM was also demonstrated to be a powerful method for strongly nonlinear equations by He [11 14]. Many other researchers have shown further applications of VIM. For example, Soliman [15] applied VIM to solve the KdV Burger s and Lax s seventh-order KdV equations. For the application of VIM to other Burger s related equations, see [16,17]. VIM has recently been applied to the solution of nonlinear coagulation problem with mass loss by Abulwafa et al. [18]. Momani et al. [19] applied VIM to the Helmholtz equation. VIM has been applied for solving nonlinear fractional differential equations by Odibat et al. [2]. Bildik et al. [21] used VIM for solving different types of nonlinear partial differential equations. In this paper, we apply VIM to the nonlinear multispecies Lotka Volterra equations. Comparisons with ADM and RK4 shall be made to determine the performance of VIM. 2. Variational iteration method The main feature of the method is that the solution of a mathematical problem with linearization assumption is used as initial approximation or trial function, then a more highly precise approximation at some special point can be obtained. This approximation converges rapidly to an accurate solution. To illustrate the basic concepts of VIM, we consider the following nonlinear differential equation: Lu + Nu = gx, where L is a linear operator, N is a nonlinear operator, and gx is an inhomogeneous term. According to VIM [9 14], we can construct a correction functional as follows: u n+1 x = u n x + x λ{lu n τ + Nũ n τ gτ}dτ, 2 where λ is a general Lagrangian multiplier [6], which can be identified optimally via the variational theory, the subscript n denotes the nth-order approximation, ũ n is considered as a restricted variation [9 11], i.e. δũ n =. 3. Analysis of multispecies Lotka Volterra equations Consider the general Lotka Volterra model for an m species system given as dn i m = N i b i + a i j N j, i = 1, 2,..., m. 3 j=1 These equations may represent either predator prey or competition cases One species In the one-species case, Eq. 3 reduces to one species competing for a given finite source of food, dn = Nb + an, b >, a <, N >, 4 where a and b are constants. This equation has an exact solution be bt, for b, b+an Nt = N aebt N, for b =, 1 ant where N is the initial condition. Solving Eq. 4 by ADM [3] yields the following recursive algorithm, N = N, N n+1 t = bnn + a A 1,n, n, 6 1 5

3 B. Batiha et al. / Computers and Mathematics with Applications where the Adomian polynomials A 1,n are given by A 1,n = N k N n k. Now, formally in VIM [5], we construct the correction functional, [ ] dnn s N n+1 t = N n t + λs bn n s añn 2 ds s ds, 8 where Ñ n is considered as restricted variations, i.e. δñ n =. Its stationary conditions can be obtained as 1 + λt =, λ s + bλs s=t =. The Lagrange multiplier λ can therefore be identified as λs = e bs t and the following variational iteration formula is obtained, e bs [ ] dnn s N n+1 t = N n t e bt bn n s ann 2 ds s ds, n. 1 The solution of the linearized version of Eq. 4, where a =, is Nt = Ce bt. Taking this as the initial approximation N gives 7 9 N 1 t = Ce bt + ac2 e bt e bt 1. b The condition N =.1 gives us C =.1. Thus 11 N 1 t =.1e bt +.1aebt e bt 1. b In the same manner, the rest of the components of the iteration formula 1 can be obtained using the computer algebra package Maple Two species The Lotka Volterra equations modelling two species competing for a common ecological niche are dn 1 = N 1 b 1 + a 11 N 1 + a 12 N 2, dn 2 = N 2 b 2 + a 21 N 1 + a 22 N 2, where a 11, a 12, a 21, a 22, b 1 and b 2 are constants. The Adomian recursive algorithms for solving 13 and 14 are, cf. [3], N 1, = N 1, N 2, = N 2, N 1,n+1 = N 2,n+1 = b1 N 1,n + a 11 A 1,n + a 12 A 1,2,n, n, 16 b2 N 2,n + a 21 A 2,1,n + a 22 A 2,n, n, 17 where N 1 and N 2 are the initial conditions and the Adomian polynomials are given by A 1,n = N 1,k N 1,n k, A 1,2,n = N 1,k N 2,n k, 18

4 96 B. Batiha et al. / Computers and Mathematics with Applications A 2,n = N 2,k N 2,n k, A 2,1,n = N 2,k N 1,n k. 19 The correction functionals for system 13 and 14 are [ ] dn1,n s N 1,n+1 t = N 1,n t + λ 1 s b 1 N 1,n s a 11 Ñ1,n 2 ds s a 12Ñ1,nsÑ 2,n s ds, 2 [ ] dn2,n s N 2,n+1 t = N 2,n t + λ 2 s b 2 N 2,n s a 22 Ñ2,n 2 ds s a 21Ñ2,nsÑ 1,n s ds, 21 where Ñ i,n are considered as restricted variations, i.e. δñ i,n =. Its stationary conditions can be obtained as 1 + λ 1 t =, λ 1 s + λ 1s s=t =, λ 2 t =, λ 2 s + λ 2s s=t =. 23 The Lagrange multipliers, therefore, can be identified as λ 1 s = e b 1s t and λ 2 s = e b 2s t. The solutions of the linearized versions of 13 and 14, where a 11 =, a 12 =, a 21 = and a 22 =, are N 1 t = C 1 e b 1t and N 2 t = C 2 e b 2t. Now taking these as initial estimates and imposing the conditions N 1 = 4 and N 2 = 1, for example, give C 1 = 4 and C 2 = 1. Thus the first iteration solutions are given by N 1,1 t = 4e b 1t + 4eb 1t 4a 11 b 2 1a 12 b 1 + 4a 11 b 2 e b 1t + 1a 12 b 1 e b 2t b 1 b 2, 24 N 2,1 t = 1e b 2t + 1eb 2t 4a 21 b 2 1a 22 b 1 + 4a 21 b 2 e b 1t + 1a 22 b 1 e b 2t b 1 b Again, the rest of the components of the iteration formulas 2 and 21 can be obtained using the computer algebra package Maple Three species The following version of the Lotka Volterra equations modelling three species shall be used [22]: dn 1 = N 1 1 N 1 αn 2 β N 3, dn 2 = N 2 1 β N 1 N 2 αn 3, dn 3 = N 3 1 αn 1 β N 2 N 3, where α and β are constants. The Adomian recursive algorithms for solving are, cf. [3], N 1, = N 1, N 2, = N 2, N 3, = N 3, 29 N 1,n+1 = N 2,n+1 = N 3,n+1 = N1,n A 1,n α A 1,2,n β A 1,3,n, n, 3 N2,n β A 2,1,n A 2,n α A 2,3,n, n, 31 N3,n α A 3,1,n β A 3,2,n A 3,n, n, 32 where N 1, N 2 and N 3 are the initial conditions and the Adomian polynomials are A 1,n = N 1,k N 1,n k, A 1,2,n = N 1,k N 2,n k, A 1,3,n = N 1,k N 3,n k, 33

5 B. Batiha et al. / Computers and Mathematics with Applications Table 1 Numerical comparisons when b = 1, a = 3, N =.1 t Exact solution ADM, φ 3 2-iteration VIM A 2,n = A 3,n = N 2,k N 2,n k, A 2,1,n = N 3,k N 3,n k, A 3,1,n = In VIM, the correction functionals are N 1,n+1 t = N 1,n t + N 2,n+1 t = N 2,n t + N 3,n+1 t = N 3,n t + λ 1 s N 2,k N 1,n k, A 2,3,n = N 3,k N 1,n k, A 3,2,n = [ dn1,n ds [ dn2,n λ 2 s ds N 2,k N 3,n k, 34 N 3,k N 2,n k. 35 ] N 1,n + Ñ1,n 2 + αñ 1,n Ñ 2,n + β Ñ 1,n Ñ 3,n ds, 36 ] N 2,n + β Ñ 2,n Ñ 1,n + Ñ2,n 2 + αñ 2,n Ñ 3,n ds, 37 [ dn3,n λ 3 s N 3,n + αñ 3,n Ñ 1,n + β Ñ 3,n Ñ 2,n + Ñ3,n 2 ds ] ds, 38 where Ñ i,n are considered as restricted variations, i.e. δñ i,n =. Its stationary conditions can be obtained as 1 + λ 1 t =, λ 1 s + λ 1s s=t =, λ 2 t =, λ 2 s + λ 2s s=t =, λ 3 t =, λ 3 s + λ 3s s=t =. 41 Thus, the Lagrange multipliers are λ 1 s = λ 2 s = λ 3 s = e s+t. The solutions of the linearized versions of are N 1 t = C 1 e t, N 2 t = C 2 e t and N 3 t = C 3 e t. Now taking these as initial estimates and imposing the conditions N 1 =.2, N 2 =.3 and N 3 =.5, for example, give C 1 =.2, C 2 =.3 and C 3 =.5. Thus, the first iteration solutions are N 1,1 t =.2e t +.4e t +.6αe t +.1βe t.4e 2t.6αe 2t.6βe 2t, 42 N 2,1 t =.3e t +.6βe t +.9e t +.15αe t.6βe 2t.9e 2t.15αe 2t, 43 N 3,1 t =.3e t +.6αe t +.15βe t +.9e t.6αe 2t.15βe 2t.25e 2t. 44 Again, the next iterations can be obtained using the computer algebra package Maple. 4. Numerical results and discussion The numerical solutions obtained by using the VIM are compared with the exact solution for the one-species case, and those obtained by ADM and RK4. Table 1 shows comparison between the 2-iteration of VIM, 3-term ADM and the exact solution for the one species in the case b = 1, a = 3 and N =.1. The results show the good accuracy

6 98 B. Batiha et al. / Computers and Mathematics with Applications Table 2 Numerical comparisons in the case b 1 =.1, a 11 =.14, a 12 =.12, b 2 =.8, a 21 =.9, a 22 =.1, N 1 = 4 and N 2 = 1 t ADM, φ 3 2-iteration VIM RK4, h =.1 N 1 N 2 N 1 N 2 N 1 N Table 3 Numerical comparisons when α =.1, β =.1, N 1 =.2, N 2 =.3, N 3 =.5 t ADM, φ 3 4-iteration VIM RK4, h =.1 N 1 N 2 N 3 N 1 N 2 N 3 N 1 N 2 N of VIM. In Table 2 we show the comparison between the 2-iteration of VIM, 3-term ADM and RK4 solutions for the two species in the case b 1 =.1, a 11 =.14, a 12 =.12, b 2 =.8, a 21 =.9, a 22 =.1, N 1 = 4 and N 2 = 1. Clearly, high accuracy is achieved with only two iterations of VIM. The numerical solutions for the three-species case are tabulated in Table 3 in the case α =.1, β =.1, N 1 =.2, N 2 =.3 and N 3 =.5. Again, the numerical results show that VIM is of high accuracy. 5. Conclusions In this paper, the VIM is applied to the solution of nonlinear multispecies Lotka Volterra equations. Comparisons with the Adomian decomposition method and the fourth-order Runge Kutta method show that the VIM is a powerful method for nonlinear equations. The advantage of the VIM over the ADM is that there is no need for the evaluations of the Adomian polynomials and the advantage over the RK4 method is that VIM or variational iteration method gives continuous solutions. Acknowledgement The financial support received from the Academy of Sciences Malaysia under the SAGA grant no. P24c is gratefully acknowledged. References [1] J. Hofbauer, K. Sigmund, The Theory of Evolution and Dynamical Systems, Cambridge University Press, London, 1988.

7 B. Batiha et al. / Computers and Mathematics with Applications [2] S. Olek, An accurate solution to the multispecies Lotka Volterra equations, SIAM Rev [3] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic, Boston, [4] W. Chen, Z. Lu, An algorithm for Adomian decomposition method, Appl. Math. Comput [5] J.H. He, A new approach to nonlinear partial differential equations, Commun. Nonlinear Sci. Numer. Simul [6] M. Inokuti, H. Sekine, T. Mura, General use of the Lagrange multiplier in nonlinear mathematical physics, in: S. Nemat-Nassed Ed., Variational Method in the Mechanics of Solids, Pergamon Press, 1978, pp [7] J.H. He, Approximate analytical solution of Blasius equation, Commun. Nonlinear Sci. Numer. Simul [8] J.H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Engrg [9] J.H. He, Variational iteration method a kind of non-linear analytical technique: some examples, Int. J. Non-Linear Mech [1] J.H. He, Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comput [11] J.H. He, Y.Q. Wan, Q. Guo, An iteration formulation for normalized diode characteristics, Int. J. Circuit Theory Appl [12] J.H. He, Some asymptotic methods for strongly nonlinear equations, Int. J. Modern Phys. B [13] J.H. He, Non-perturbative methods for strongly nonlinear problems, Dissertation, de-verlag im Internet GmbH, Berlin, 26. [14] J.H. He, Variational iteration method Some recent results and new interpretations. J. Comput. Appl. Math. in press. [15] A.A Soliman, Numerical simulation and explicit solutions of KdV Burgers and Lax s seventh-order KdV equations, Chaos Solitons Fractals [16] M.A. Abdou, A.A. Soliman, Variational iteration method for solving Burger s and coupled Burger s equations, J. Comput. Appl. Math [17] M. Moghimi, F.S.A. Hejazi, Variational iteration method for solving generalized Burgers Fisher and Burgers equations, Chaos Solitons Fractals [18] E.M. Abulwafa, M.A. Abdou, A.A. Mahmoud, The solution of nonlinear coagulation problem with mass loss, Chaos Solitons Fractals [19] S. Momani, S. Abuasad, Application of He s variational iteration method to Helmholtz equation, Chaos Solitons Fractals [2] Z.M. Odibat, S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer. Simul [21] N. Bildik, A. Konuralp, The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations, Int. J. Nonlinear Sci. Numer. Simul [22] R.M. May, W.J. Leonard, Nonlinear aspects of competition between three species, SIAM J. Appl. Math