# Enhanced Multistage Differential Transform Method: Application to the Population Models

Authors: Younghae Do; Bongsoo Jang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We present an efficient computational algorithm, namely, the enhanced multistage differential transform method (E-MsDTM) for solving prey-predator systems. Since the differential transform method (DTM) is based on the Taylor series, it is difficult to obtain accurate approximate solutions in large domain. To overcome this difficulty, the multistage differential transform method (MsDTM) has been introduced and succeeded to have reliable approximate solutions for many problems. In MsDTM, it is the key to update an initial condition in each subdomain. The standard MsDTM utilizes the approximate solution directly to assign the new initial value. Because of local convergence of the Taylor series, the error is accumulated in a large domain. In E-MsDTM, we propose the new technique to update an initial condition by using integral operator. To demonstrate efficiency of the proposed method, several numerical tests are performed and compared with ones obtained by other numerical methods such as MsDTM, multistage variational iteration method (MVIM), and fourth-order Runge-Kutta method (RK4).


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## Body

## 1. Introduction

The analytical approaches such as the differential transform method (DTM) and the homotopy perturbation method (HPM) have been succeeded to solve differential equations and integral equations in many fields [1–10]. The basic concept of the DTM is based on the Taylor series. Suppose the solution is represented by the Taylor series. DTM provides a recursive relations for the coefficients of the Taylor series which can be solved easily. However, it is different from the conventional high-order Taylor series method which requires a complex symbolic computation. Even if DTM provides a promising algorithm for solving many linear and nonlinear problems arising in various science and engineering fields, it also has some drawbacks. Since DTM is based on the Taylor series which gives a local convergences and it has only a truncated approximation, it does not exhibits a good approximation in a large domain. Recently, the multistage DTM (MsDTM) [11–16] is proposed to accelerate the convergence of the truncated approximation in a large domain as well as to improve the accuracy of the standard DTM. The MsDTM applies the standard DTM in each subdomain. In MsDTM, it is a key to improve accuracy as well as convergence how to assign a new initial condition at each subdomain. In this paper, we demonstrate that the conventional initial updating in MsDTM accumulates an error for the nonlinear systems. In order to overcome those difficulties, a modified version of MsDTM, which is called a enhanced MsDTM (E-MsDTM), is presented. We apply the E-MsDTM to the two prey-predator models. One is a simple prey-predator model [17] as follows:(1.1)ddtx(t)=x(t)(a-by(t)),ddty(t)=-y(t)(c-dx(t)),
where x(t) and y(t) represent the populations of prey and predator such as rabbits and foxes, respectively, at time t and a,b,c, and d are known constants. Another model is a Gauss-type prey-predator model [18] which describes the population dynamics that incorporate functional responses. Specially, we study the following Holling-type II (p=1) and Holling-type III (p=2):(1.2)ddtx(t)=rx(t)(1-x(t)k)-mxp(t)1+xp(t)y(t),ddty(t)=y(t)(μmxp(t)1+xp(t)-D),
where x(t) and y(t) are the population size for the prey and the predator, respectively; r,k,μ, and D are the positive constants representing prey intrinsic growth rate, carrying rate of prey to predator, and death rate of predator, respectively.


## 2. Description of Enhanced Multistage Differential Transform Method

Let us describe the basic definition and some properties of the standard DTM. Suppose a functiony(t) is the solution of the following equation:(2.1)ddty(t)=f(t,y(t)),x(0)=c.
Let us define the differential transform Y(k) of y(t) at t=ti as(2.2)Y(k)=1k![dky(t)dtk]t=ti.The differential inverse transform ofY(k) is defined by(2.3)y(t)=∑k=0∞Y(k)(t-ti)k.Suppose thaty(t)=∑k=0∞Y(k)tk. Using some fundamental properties for the DTM, we have the following recursive relation:(2.4)(k+1)Y(k+1)=F(k),Y(0)=c,
where F(k) is the differential transform of f(t,y(t)). One can obtain the Y(k) by solving the recursive relation (2.4). Then y(t) can be approximated by the finite-term Taylor series. In other words,(2.5)y(t)≈∑k=0nY(k)(t-ti)k≡sn(t).It is well known that the approximated error is given by(2.6)|y(t)-sn(t)|≤M(n+1)!|t-ti|n+1,where|x(n+1)(t)|≤M. Thus, it is easy to see that, for the fixed term n, the error is getting bigger if |t-ti| is increasing. To reduce the error, the more terms are added. But it requires cumbersome computational work. Moreover, it is possible that the error blows up even with many terms if the value t is outside of convergence radius of the function y(t).To overcome difficulties arising in the standard DTM, the multistage differential transform method is introduced. The basic idea of the MsDTM is to apply the standard DTM to each subdomain. To describe the MsDTM we consider the equally spaced nodal pointstl; 0=t0<t1<⋯<tN-1<tN=T and tl+1-tl=T/N. On lth subdomain (tl,tl+1)≡Dl, we define y(t)|Dl≡yl(t). The differential transform Yl(k) of yl(t) at t=tl is defined by(2.7)Yl(k)=1k![dkyl(t)dtk]t=tl.The differential inverse transform ofYl(k) is defined by(2.8)yl(t)=∑k=0∞Yl(k)(t-tl)k.Similarly,yl(t) can be approximated by solving the recursive relation (2.4) on Dl. Then we have(2.9)yl(t)≈∑k=0nYl(k)(t-tl)k≡sl,n(t).It is an issue to give the initial conditionyl(tl) in each subdomain Dl. Since y(0)=c in (2.1) is given by the initial condition, we have Y(0)=c. For l>0, we approximate yl(tl) by computing sl-1,n(tl) and assign the approximate value as the initial condition in Dl. Then Yl(0)=sl-1,n(tl). This standard MsDTM has been successful to obtain accurate approximation for many problems in [11–13]. However, it is easy to see that the approximate solution obtained by the standard MsDTM may yield an increasing error due to the nonlinearity because the DTM employs only finite term Taylor series for the given problems. For instance, let us consider(2.10)y′(t)=y(t)m.Applying the DTM gives(2.11)(∑k=0∞Y(k)(t-ti)k)′=(∑k=0∞Y(k)(t-ti)k)m≡∑k=0∞F(k)(t-ti)k.
The basic properties of DTM have the following recursive relation for m=3:(2.12)(k+1)Y(k+1)=∑s=0k[∑r=0sY(r)Y(s-r)]Y(k-s)=F(k).
One can solve the above recursive relation easily to obtain the differential transforms Y(k),k=0,…,n. Then we can approximate(2.13)y(t)≈∑k=0nY(k)(t-ti)k.Since the recursive relation (2.12) is derived by equating the coefficients for (t-ti)k in (2.11), the differential transforms Y(k) are determined by solving(2.14)(k+1)Y(k+1)=F(k),k=0,1,…,n.However, the right-hand side of (2.11) has the coefficients of (t-ti)k, k=0,1,…,3n so that as many as 2n coefficients of (t-ti)k, k=n+1,…,3n are not considered to approximate y(t). Thus, it could have a poor approximation y(t). It is supposed to take care of all terms in obtaining the differential transform Y(k) especially for the nonlinear problem.In what follows we describe some modifications of MsDTM, which is called the Enhanced MsDTM. Suppose thatyl-1(t) is approximated by the standard MsDTM, that is, yl-1(t)≈sl-1,n(t). Then, the solution yl(t) can be determined by solving the following problem:(2.15)ddtyl(t)=f(t,yl(t)),yl(tl)=sl-1,n(tl).
Applying the DTM to (2.15) gives the differential transform Yl(k). Then yl(t) is approximated by(2.16)yl(t)≈∑k=0nYl(k)(t-ti)k≡sl,n(t).
In the standard MsDTM, yl(tl+1)≈sl,n(tl+1) which gives Yl+1(0). As mentioned before, it could give inaccurate approximate solution yl+1(t). Here, taking the integral operator to (2.15), the solution yl(t) can be represented by(2.17)yl(t)=yl(tl)+∫tltf(τ,yl(τ))dτ.Thus, we have(2.18)yl(tl+1)=yl(tl)+∫tltl+1f(τ,yl(τ))dτ.Sinceyl(t)≈sl,n(t) in (2.16), we can approximate the value of yl(tl+1) by(2.19)yl(tl+1)=yl(tl)+∫tltl+1f(τ,sl,n(τ))dτ.
Then, yl(tl+1)=Yl+1(0). For each subdomain Dl, repeat this process (2.19) to approximate yl(t). Thus, the solution y(t) can be approximated by(2.20)y(t)≈∑l=0N-1sl,n(t)χ(t-tl)χ(tl+1-t),where(2.21)χ(t-tl)={1,t≥tl,0,otherwise.


## 3. Application

To apply the MsDTM, we consider a subdomain with uniform lengthT/N=h stated in the previous section. For each subdomain (tl,tl+1)≡Dl, we apply the DTM to the model problem (1.1), then the following recursive relation is obtained:(3.1)(k+1)Xl(k+1)=aXl(k)-b∑r=0kXl(r)Yl(k-r),(k+1)Yl(k+1)=-cYl(k)+d∑r=0kYl(r)Xl(k-r).
In order to compare the numerical performance for the enhanced MsDTM, we demonstrate comparative approximations with the standard MsDTM, the multistage variational iteration method (MVIM) [17, 19, 20], and the fourth-order Runge-Kutta method (RK4). In MVIM, for each subdomain Dl, the solutions xl(t) and yl(t) can be determined by(3.2)xl(t)=lim⁡n→∞xl,n(t),yl(t)=lim⁡n→∞yl,n(t),where(3.3)xl,n+1(t)=xl,n(t)+∫tltλ1(τ)(ddtxl,n(τ)-axl,n(τ)+bxl,n(τ)yl,n(τ)),yl,n+1(t)=yl,n(t)+∫tltλ2(τ)(ddtyl,n(τ)-cyl,n(τ)+dxl,n(τ)yl,n(τ)).Here,λ1,λ2 are general Lagrange multiplier which are determined by solving the stationary problem. Here, the multipliers are given by [17](3.4)λ1(τ)=-exp⁡[a(t-τ)],λ2(τ)=-exp⁡[c(τ-t)].In order to apply DTM to the Holling-type prey-predator model (1.2) we let ϕ(t) be the functional response of predator to prey as(3.5)ϕ(t)=mxp(t)1+xp(t).For each subdomainDl, using the basic properties of DTM to (1.2) gives the following recursive relation:(3.6)(k+1)Xl(k+1)=rXl(k)-rk∑ikXl(i)Xl(k-i)-∑ikΦl(i)Xl(k-i),(k+1)Yl(k+1)=μ∑ikYl(i)Φ(k-i)-DYl(k),where forp=1,(3.7)Φl(k)={11+Xl(k)(mXl(k)),k=0,11+Xl(0)(mXl(k)-∑i=1kXl(i)Φl(k-i)),k>0,and forp=2(3.8)Φl(k)={11+Xl2(k)(mXl2(k)),k=0,11+Xl2(0)(m∑i=0kXl(i)Xl(k-i)-∑j=1k∑i=0jXl(j)Xl(j-i)Φl(k-j)),k>0.


## 4. Numerical Experiment and Discussion

In this section, several numerical tests are demonstrated to show the effectiveness of the E-MsDTM. For the simple prey-predator model (1.1), the constants in three different cases are shown in Table 1. Figure 1 shows the population x(t) and y(t) for the case 1 by using RK4 with time step h=0.0001 in (0,1). With a dense time step h<0.001 all approximations are fitted well in graph. Thus it is difficult to show the effectiveness for the proposed method, E-MsDTM. In order to show the performance of each method clearly, we fix the time-step h=0.01 at MsDTM, MVIM, and E-MsDTM. Figure 2 shows the absolute error of the approximations obtained by RK4 and other methods with one iteration (a) and two iterations (b). Since the solution is stable for t>0.2 in Figure 1, all methods give reliable approximations for t>0.2. However, MVIM and MsDTM give a large error compared with the result by E-MsDTM in (0,0.2). Moreover, E-MsDTM gives an accurate approximation in all domain (0,1). In case 2, all numerical results present the similar behavior shown in case 1. See Figures 3 and 4. In order to see the effectiveness of the E-MsDTM in a more complex population system, we consider the population model in case 3 in a large domain (0,50). Here we used a RK4 at h=0.0001 and all other methods at h=0.01. As seen in Figure 5 this system exhibits a oscillatory dynamics as well as many peaks. In Figure 6 the absolute errors are demonstrated as the number of iteration is increasing. All approximations have large errors at peaks, but still the E-MsDTM yields a better performance than others. In order to see the performance associated to the number of iteration, the maximum absolute errors between RK4 and other methods are presented in Figure 7. As the number of iteration is increasing, it is shown that MVIM and MsDTM give similar performance. However, the accuracy of the E-MsDTM is dramatically increasing when the number of iteration is bigger than 2.Table 1
Various parameters.
Caseabcdx(0)y(0)1110.11141820.11111610320.1110.10.2Figure 1
Population ofx(t) and y(t) by RK4 (h=0.0001) in case 1.Comparison of absolute errors in case 1.
(a)
Absolute error with one iteration:x(t) (left), y(t) (right)(b)
Absolute error with two iterations:x(t) (left), y(t) (right)Figure 3
Population ofx(t) and y(t) by RK4 (h=0.0001)   in case 2.Comparison of absolute errors in case 2.
(a)
Absolute error with one iteration:x(t) (left), y(t) right(b)
Absolute error with two iterations:x(t) (left), y(t) rightFigure 5
Population ofx(t) and y(t) by RK4 (h=0.0001) in case 3.Comparison of absolute errors in case 3.
(a)
Absolute error with one iteration:x(t) (left), y(t) right(b)
Absolute error with two iterations:x(t) (left), y(t) right(c)
Absolute error with three iterations:x(t) (left), y(t) rightFigure 7
Maximum error with the number of iteration.For the Holling-type prey-predator model (1.2), the parameters are set by r=1,m=1,μ=9  and D=1. For the Holling-type II (p=1), it has been known [18] that the model (1.2) has at least one limit cycle if k>17 and no limit cycle if k≤17 and the critical point (x*,y*), where (x*)p=D/(mμ-D),  y*=rμx*(1-x*/k)/D is globally asymptotically stable if 8<k≤17 and unstable if k>17. The results obtained by the nonstandard numerical scheme [18],  MsDTM and E-MsDTM are demonstrated in Figures 8 and 9. Here, the time step h=0.01 is used in nonstandard scheme, h=0.25 in both MsDTM and E-MsDTM. Only one iteration is employed in both MsDTM, and E-MsDTM. As shown in Figures 8 and 9, all results by nonstandard scheme and E-MsDTM are good agreement. However, the numerical results by MsDTM present a wrong dynamical behavior. For Holling-type III (p=2), it is shown in Figures 10 and 11 that all numerical results by nonstandard scheme and E-MsDTM show the accurate dynamical behavior, but the result by MsDTM does not. Moreover, E-MsDTM gives the approximate solutions with a coarse time step h=0.125, which shows that E-MsDTM is very effective method.Figure 8
x(t)(Top) and y(t) (Bottom); nonstandard scheme (Left, h=0.01), MsDTM (Middle, h=0.25), E-MsDTM (Right, h=0.25); k=16.9.Figure 9
x(t)(Top) and y(t) (Bottom); nonstandard scheme (Left, h=0.01), MsDTM (Middle, h=0.25), E-MsDTM (Right, h=0.25); k=17.1.Figure 10
x(t)(Top) and y(t) (Bottom); nonstandard scheme (Left, h=0.01), MsDTM (Middle, h=0.2), E-MsDTM (Right, h=0.2); k=322/7-0.1.Figure 11
x(t)(Top) and y(t) (Bottom); nonstandard scheme (Left, h=0.01), MsDTM (Middle, h=0.125), E-MsDTM (Right, h=0.125)  k=322/7+0.1.


## 5. Conclusion

In this paper, we propose an efficient computational algorithm, namely the enhanced multistage differential transform method for solving several prey-predator systems. The basic idea of the MsDTM is to apply the standard DTM to each sub-domain by updating an initial condition. However, it is shown that the standard initial updating gives a numerical inefficiency for the complex nonlinear systems. In E-MsDTM, an initial condition at each sub-domain is updated by using the integral operator which is equivalent to the system. For all illustrative examples the E-MsDTM yields accurate and stable numerical solutions in comparison with the ones obtained by other methods such as the standard MsDTM and MVIM. Also, the E-MsDTM gives almost the same results at low computational cost compared with ones obtained by RK4 and nonstandard numerical scheme. Thus, it can be concluded that the E-MsDTM is an effective numerical tool for solving differential equations. Here, all computations are performed by using Mathematica 8.0.



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*Source: 253890-2012-06-04.xml*