Biological population model and its solution by reduced differential transform method

Transcription

1 Asia Pacific Journal of Engineering Science and Technology () (05) -0 Asia Pacific Journal of Engineering Science and Technology journal homepage: Full length article Biological population model and its solution by reduced differential transform method H. S. Shula, Gufran Mali* Department of Mathematics & Statistics, DDU Gorahpur University, Gorahpur-73009, India (Received: 5 December 05; accepted 8 December 05; published online 30 December 05) ABSTRACT In this paper, an approximate solution of the generalized biological population model (BPM) is proposed by means of a recent semi-analytical method: reduced differential transformation method. The method is very reliable, effective and efficient powerful technique for solving wide range of problems arising in natural sciences. Using the method, it is possible to find an exact solution or a closed approximate solution of any differential equation. The scheme is very easy to implement since it taes very small size of computation contrary to the other methods. Four numerical examples are carried out in order to validate and illustrate the efficiency of the method considering the scenario of both linear and nonlinear biological population model. Keywords: Biological population model; Reduced differential transform method; Exact solution. Introduction The dispersal or emigration plays an important role in the regulation of population of some species. The diffusion of a biological species in a region C is described by the three in region C and time t []: p x, t, the population density; v x, t, the diffusion velocity, and f x, t, the population supply. The population density p x, t expresses the number of individuals per unit volume at the functions of position x x, y position x and time t ; it s integral over any sub region D of region C gives the total population * address: gufranmali@gmail.com 05 Author(s)

2 H.S. Shula, G. Mali Asia Pacific J. Eng. Sci. Tech. () (05) -0 of D at timet, whereas f, x t expresses the rate at which individuals are supplied per unit volume at position x by births and deaths. The diffusion velocity v x, t represents the average velocity of those individuals who occupy the position x at time t, and it describes the flow of population from one point to another point. The entities p, v and f are consistent with the following law of population balance: for every regular sub region D of C and for all time t, d pdv pv. nda fdv, dt () D D D where n is the outward unit normal to the boundary D of D. From Eq. (), it can be seen that the rate of change of population of D plus the rate at which the individuals leave D across its boundary must be equal to the rate at which the individuals are supplied directly to D. By maing the assumptions [] f f p, v p p, () where p 0 for p 0, and is the Laplace operator, the following nonlinear degenerate parabolic partial differential equation for the population density p is found p t where p f p, t 0, x, y R, x y and p is a function of p. Gurney and Nisbet [] applied p, as a special case for the modeling of the population of animals. The movements were made generally either by mature animals driven by mature invaders or by young animals just reaching maturity moving out of their parental territory to establish breeding territory of their own. In both cases, it is much more plausible to assume that they will be directed towards nearby vacant territory. Therefore, in this model, movement will tae place almost exclusively down the population density gradient and will be more rapid at high population densities than at low ones. To model this situation, Gurney and Nisbet [] further considered a wal through a rectangular grid, in which at each step an animal may either stay at its present location or may move in the direction of the lowest population density. The probability distribution between these two possibilities being determined by the magnitude of the population density gradient at the grid side is concerned. This model leads to Eq. (3) with p p, to the following equation p p f p, t 0, x, y R, t with the given initial condition p( x, y,0). Some properties of Eq. (4) such as Holder estimates and its solutions were studied in [3]. Three examples of constitutive equations for f are depicted as (3) (4)

3 3 H.S. Shula, G. Mali Asia Pacific J. Eng. Sci. Tech. () (05) -0 (i) f ( p) cp, c = constant, Malthusian Law []. (ii) f ( p) c p c p (iii) f ( p), c, c = positive constants, Verhulst Law []. cp, c 0,0, Porous media [4, 5]. The more general form of f can be taen as ( ) a b f p hp rp p t a b p hp rp, t 0, x, y R,, which yields where h, a, r, b are real numbers. For h c, a, r 0 and h c, a b, r c c, Eq. (5) leads to Malthusian Law and Verhulst Law. The linear and nonlinear population systems were solved in [6, 7] using variational iteration method (VIM), Adomain decomposition method (ADM), and homotopy perturbation method (HPM). A meshless local radial point interpolation numerical method is used by Shivanian [8] to simulate a nonlinear partial integro-differential equation arising in population dynamics. The major drawbacs of these approximate analytical methods are that they require complex and large size of computations. To overcome from these difficulties, reduced differential transform method is employed for solving this mathematical model. The method, introduced by Kesin et al. [9, 0], is a semi-analytical technique for solving linear and nonlinear differential equation, and is widely applicable (refer [-8]). The solution procedure of the method is much simpler than the other methods and it taes very less computational effort. The main purpose of this study is to solve the biological population model using the reduced differential transformation method. The obtained results are compared with those obtained by VIM and ADM and HPM [9-]. (5). Reduced differential transform method In this section, the basic definitions of the reduced differential transformation are w x, t which can be expressed as a product. By means of the properties of the one- revisited. Let us consider a function of two variables of two single-variable functions, i.e., wx, t F xg( t) dimensional differential transformation, the function w x, t can be expressed as i j i j (6) w( x, t) F i x G( j) t W i, j x t, i0 j0 i0 j0 where W i, j F ig( j) is called the spectrum of, Let RD w x t. denotes the reduced differential transform operator, and R D the inverse reduced differential transform operator. The basic definition and operation of the reduced differential transform method is described below. Definition.: Let w x, t be analytic and continuously differentiable with respect to space variable x and time variable t in the domain of interest, the spectrum function

4 4 H.S. Shula, G. Mali Asia Pacific J. Eng. Sci. Tech. () (05) -0 W x wx, t,! t t t0 is the reduced transformed function of w x, t. For convenience, w x, t (7) (lowercase) represents the original function while W x (uppercase) stands for the reduced transformed function. The differential inverse reduced transform of W x is defined as 0 (8) 0 w x, t W x t t. From Eq. (7) and (8), one can observe w x, t w x, t t t. 0 0! t (9) tt 0 From the above definition, it can be noticed that the concept of the reduced differential transform is derived from the two-dimensional differential transform method., and the convolution denotes the reduced differential transform version of the multiplication, the fundamental operations of the reduced differential transform are given in Table. Definition.: Let ux, t R D U x, vx, t R D V x Table : Fundamental operations of the reduced differential transform method. Original function: f x, t RDT function R ( f ( x, t)) F x, t u x, tvx, t D U x V x U x V x r r r0, vx, t U x V x, U x u x t u x t x u x, t t rs u x, t r x t s x U x s! m n m x t, r! x r U s x n 0, otherwise t e! sin wt c w sin c!! x

5 5 H.S. Shula, G. Mali Asia Pacific J. Eng. Sci. Tech. () (05) Computational Examples In this section, we describe the method explained in Section using the following four examples of the biological population model (BPM) to validate the efficiency and reliability of the described method. Example 3.: Consider the linear BPM p p hp, t subject to the initial condition p( x, y,0)= xy. () Applying the described method to Eq. (0), the following recurrence relation is obtained P x, y Pr x, yp r x, y hp x, y. () r0 Using the method to the initial condition (), we find P0 x, y xy. (3) Using Eq. (3) into Eq. (), we found P, 4 h h h x y values successively 3 h h h h P x, y h xy; P x, y xy xy; P3 x, y xy xy;! 6 3! P4 x, y xy xy;...; P x, y xy. 4 4!! Using the differential inverse reduced transform of P x,y, we have p x, y, t P x, y t P x, y P x, y t P x, y t P x, y t h h 3 h xy ht t t... t....! 3!! (0) (4) (5) Fig.. Physical solution profile obtained by RDTM for the Example 3..

6 6 H.S. Shula, G. Mali Asia Pacific J. Eng. Sci. Tech. () (05) -0 In closed form, the exact solution can be expressed as ht p( x, y, t) xye, (6) which is the same exact solution obtained by P. Roul [5] using HPM, and also by Shaeri et al. [4] using VIM and ADM with parameter h 0.0. Fig. depicts the solution profile of Example 3. with h 0.0 at different time levels 0t. Example 3.: Consider the following BPM p p p, t with the initial condition p( x, y,0)= sin xsinh y. (8) Applying RDTM method to Eq. (7), we obtain the following recursive equation P x, y Pr x, yp r x, y P x, y. (9) r0 Using RDTM to the initial condition (8), we get P0 x, y sin xsinh y. (0) Using Eq. (0) in Eq. (9), we get the following P x,y values successively P3 x, y sin xsinh y sin xsinh y; P4 x, y sin xsinh y sin xsinh y; 6 3! 4 4! ;...; P x, y sin xsinh y.! (7) () Fig.. Physical solution behavior obtained by RDTM for the Example 3.. Using the differential inverse reduced transform of P x,y, we get

7 7 H.S. Shula, G. Mali Asia Pacific J. Eng. Sci. Tech. () (05) -0 p x, y, t P x, y t P x, y P x, y t P x, y t P x, y t ! 3!! The exact solution, in closed form, can be given as sin xsinh y t t t... t.... p(,, ) sin sinh t x y t x y e. (3) The same exact solution was obtained by Roul [5] using HPM. Fig. depicts the physical solution behavior of Example 3. at different time levels 0t. () Example 3.3: Consider the nonlinear BPM: p p p 8 p, (4) t 9 with initial condition x y 3 p( x, y,0)= e. Applying the described technique to Eq. (4), we obtain the following iterative formula 8 P x, y Pr x, yp r x, y P x, y Pr x, y P r x, y. r0 9 r0 Now using RDTM to the initial condition (5), we get x y 3 (5) (6) P0 x, y e. (7) Using Eq. (7) in Eq. (6), we get the following P x,y values successively x y x y x y x y x y P x, y e ; P x, y e e ; P x, y e e ;! 6 3! xy x y 3 3 P4 x, y e ;...; P x, y e ;... 4!! Using the differential inverse reduced transform of P x,y, we get p x, y, t P x, y t P x, y P x, y t P x, y t P x, y t e ( t) ! 3!! Hence, the exact solution, in closed form, can be given as x y t t t xyt 3 (8) (9) p( x, y, t) e, (30) which the same is as obtained by Roul [5] using HPM, and also by Shaeri et al. [4] using

8 8 H.S. Shula, G. Mali Asia Pacific J. Eng. Sci. Tech. () (05) -0 VIM and ADM. The physical solution profile is shown in Fig. 3. Fig.3 depicts the solution profile of Example 3.3 at different time levels 0t. Fig. 3. Solution profile obtained by RDTM for the Example 3.3. Example 3.4: Consider the following nonlinear BPM p p hp p t, (3) with initial condition x/ p(,,0)=. x y e (3) The following iterative formula is obtained on applying RDTM to Eq. (3), P x, y Pr x, yp r x, y hp x, y Pr x, y P r x, y. r0 r0 Now using RDTM to the initial condition (3), we obtain x / (33) P0 x, y e. (34) Using Eq. (34) in Eq. (33), we get the following P x,y x / h x / h x / P x, y he ; P x, y e ; P3 x, y e ;! 3! 4 h x/ h x/, ;...;,,... P4 x y e P x y e 4!! Using the differential inverse reduced transform of P x,y values successively, we get (35)

9 9 H.S. Shula, G. Mali Asia Pacific J. Eng. Sci. Tech. () (05) -0 p x, y, t P x, y t P x, y P x, y t P x, y t P x, y t h h h x/ e ( h) t t t... t....! 3!! Thus, the exact solution, in closed form, can be expressed as x ht / p( x, y, t) e. (36) Fig. 4 depicts the solution profile of Example 3.4 with h 0.5,0.5,0.75,.0 at different time levels 0t. 4. Conclusions Fig. 4. Physical solution profile obtained by RDTM for the Example 3.4. In this study, the reduced differential transform method is implemented for a degenerate class of parabolic partial differential equation arising in the spatial diffusion biological populations. The proposed solution is an infinite power series for the appropriate initial condition, and evaluates the solution without any discretization, transformation, perturbation, or restrictive conditions. Computed solutions by the method agree excellently with HPM, VIM and ADM. However, the performed calculations show that the described method needs very small

10 0 H.S. Shula, G. Mali Asia Pacific J. Eng. Sci. Tech. () (05) -0 size of computation compared to HPM, VIM and ADM, and therefore, it is a very effective and efficient powerful mathematical tool for finding the approximate exact solutions for a wide range of problems arising in applied sciences and engineering fields. References [] M.E. Gurtin, R.C. Maccamy, On the diffusion of biological population, Math. BioSci., 54 (33) (977) [] W.S.C. Gurney, R.M. Nisbet, The regulation of inhomogenous populations, J. Theor. Bio., 5 (975) [3] Y.G. Lu, Holder estimate of solutions of biological population equations, Appl. Math. Lett., 3 (000) 3-6. [4] J. Bear, Dynamics of fluids in porous media, American Elsevier, New Yor (97). [5] A. Oubo, Diffusion and Ecological problem, Mathematical Models, Biomathematics 0, Springer, Berlin (980). [6] F. Shaeri, M. Dehghan, Numerical solution of a biological population model using He s variational iteration method, Comput. Math. Appl., 54 (006) [7] P. Roul, Application of homotopy perturbation method to biological population model, Appl. Applied Math., 0 (00) [8] E. Shivanian, Analysis of meshless local radial point interpolation (MLRPI) on a nonlinear partial integrodifferential equation arising in population dynamics, Engineering Analysis with Boundary Elements 37 () (03) [9] Y. Kesin, G. Oturanc, Reduced differential transform method for partial differential equations, Int. J. Nonlinear Sci. Numer. Simul., 0 (6) (009) [0] Y. Kesin, G. Oturanc, Reduced differential transform method: a new approach to factional partial differential equations, Nonlinear Sci. Lett. A, (00) 6-7. [] R. Abazari, M. Ganji, Extended two-dimensional DTM and its application on nonlinear PDEs with proportional delay, Int. J. Comput. Math., 88 (8) (0) [] P.K. Gupta, Approximate analytical solutions of fractional Benney-Lin equation by reduced differential transform method and the homotopy perturbation method, Comp. and Math. appl., 58 (0) [3] R. Abazari, M. Abazari, Numerical simulation of gerneralized Hirota-Satsuma coupled KdV equation by RDTM and comparison with DTM, Commun. Nonlinear Sci. Numer. Simulat., 7 (0) [4] V. K. Srivastava, M.K. Awasthi and M. Tamsir, RDTM solution of Caputo time fractional order hyperbolic telegraph equation, AIP Advances 3, 034 (03). [5] V. K. Srivastava, M.K. Awasthi, R.K. Chauraisa and M. Tamsir, The Telegraph Equation and Its Solution by Reduced Differential Transform Method, Modelling and Simulation in Engineering, Vol. 03, Article ID (03). [6] V. K. Srivastava, N. Mishra, S. Kumar, B. K. Singh, M. K. Awasthi, Reduced differential transform method for solving (+n)-dimensional Burgers Equation, Egyptian Journal of Basic and Applied Sciences, () (04) 5-9. [7] V. K. Srivastava, M. K. Awasthi, R. K. Chaurasia, B. K. Singh, Reduced differential transform method to solve two and three dimensional second order hyperbolic telegraphic equations, Journal of King Saud University - Engineering Sciences, In Press, 04, DOI:0.06/j.jsues [8] V. K. Srivastava, M. K. Awasthi, S. Kumar, Analytical approximations of two and three dimensional time-fractional telegraphic equation by reduced differential transform method, Egyptian Journal of Basic and Applied Sciences, () (04) [9] V. K. Srivastava, M. K. Awasthi, (+n) - dimensional Burgers equation and its analytical solution: A comparative study of HPM, ADM and DTM, Ain Shams Engineering Journal, 5() (04) [0] S. Abbasbandy, E. Shivanian, Application of the Variational Iteration Method to Nonlinear Volterra's Integro-Differential Equations, Zeitschrift für Naturforschung A 63 (9), (008). [] S. Abbasbandy, E. Shivanian, Application of Variation Iteration Method for nth-order Integro-Differential Equations, Zeitschrift für Naturforschung A 64, (009).