Enhanced Multistage Differential Transform Method: Application to the Population Models

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).

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:𝑑𝑥(𝑡)(𝑡)(𝑡),𝑑𝑑𝑡=𝑥𝑎−𝑏𝑦𝑦(𝑡)(𝑡(𝑡,𝑑𝑡=−𝑦)(𝑐−𝑑𝑥)) (1.1) 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):𝑑𝑥(𝑡)(𝑡)𝑥(𝑡)𝑑𝑡=𝑟𝑥1−𝑘−𝑚𝑥^𝑝(𝑡)1+𝑥^𝑝(𝑡)𝑦(𝑡),𝑑𝑦(𝑡)(𝑡)𝜇𝑑𝑡=𝑦𝑚𝑥^𝑝(𝑡)1+𝑥^𝑝(𝑡)−𝐷, (1.2) 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 function 𝑦(𝑡) is the solution of the following equation:𝑑𝑦(𝑡)(𝑡)(0)𝑑𝑡=𝑓𝑡,𝑦,𝑥=𝑐. (2.1) Let us define the differential transform 𝑌(𝑘) of 𝑦(𝑡) at 𝑡=𝑡_𝑖 as𝑌(𝑘)=1𝑑𝑘!^𝑘𝑦(𝑡)𝑑𝑡^𝑘_{𝑡=𝑡𝑖}. (2.2)

The differential inverse transform of 𝑌(𝑘) is defined by𝑦(𝑡)=^∞_𝑘=0𝑌(𝑘)𝑡−𝑡_𝑖^𝑘. (2.3)

Suppose that 𝑦(𝑡)=∑^∞_𝑘=0𝑌(𝑘)𝑡^𝑘. Using some fundamental properties for the DTM, we have the following recursive relation:(𝑘+1)𝑌(𝑘+1)=𝐹(𝑘),𝑌(0)=𝑐, (2.4) 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,𝑦(𝑡)≈^𝑛_𝑘=0𝑌(𝑘)𝑡−𝑡_𝑖^𝑘≡𝑠_𝑛(𝑡). (2.5)

It is well known that the approximated error is given by𝑦(𝑡)−𝑠_𝑛(𝑡≤𝑀)(𝑛+1)!𝑡−𝑡_𝑖^𝑛+1, (2.6)

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−𝑡_𝑖| 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 points 𝑡_𝑙; 0=𝑡₀<𝑡₁<⋯<𝑡_𝑁−1<𝑡_𝑁=𝑇 and 𝑡_𝑙+1−𝑡_𝑙=𝑇/𝑁. On 𝑙th subdomain (𝑡_𝑙,𝑡_𝑙+1)≡𝐷_𝑙, we define 𝑦(𝑡)|_𝐷𝑙≡𝑦_𝑙(𝑡). The differential transform 𝑌_𝑙(𝑘) of 𝑦_𝑙(𝑡) at 𝑡=𝑡_𝑙 is defined by𝑌_𝑙(𝑘)=1𝑑𝑘!^𝑘𝑦_𝑙(𝑡)𝑑𝑡^𝑘_{𝑡=𝑡𝑙}. (2.7)

The differential inverse transform of 𝑌_𝑙(𝑘) is defined by𝑦_𝑙(𝑡)=^∞_𝑘=0𝑌_𝑙(𝑘)𝑡−𝑡_𝑙^𝑘. (2.8)

Similarly, 𝑦_𝑙(𝑡) can be approximated by solving the recursive relation (2.4) on 𝐷_𝑙. Then we have𝑦_𝑙(𝑡)≈^𝑛_𝑘=0𝑌_𝑙(𝑘)𝑡−𝑡_𝑙^𝑘≡𝑠_𝑙,𝑛(𝑡). (2.9)

It is an issue to give the initial condition 𝑦_𝑙(𝑡_𝑙) in each subdomain 𝐷_𝑙. Since 𝑦(0)=𝑐 in (2.1) is given by the initial condition, we have 𝑌(0)=𝑐. For 𝑙>0, we approximate 𝑦_𝑙(𝑡_𝑙) by computing 𝑠_{𝑙−1,𝑛}(𝑡_𝑙) and assign the approximate value as the initial condition in 𝐷_𝑙. Then 𝑌_𝑙(0)=𝑠_{𝑙−1,𝑛}(𝑡_𝑙). 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 consider𝑦′(𝑡)=𝑦(𝑡)^𝑚. (2.10)

Applying the DTM gives^∞_𝑘=0𝑌(𝑘)𝑡−𝑡_𝑖^𝑘′=^∞_𝑘=0𝑌(𝑘)𝑡−𝑡_𝑖^𝑘𝑚≡^∞_𝑘=0𝐹(𝑘)𝑡−𝑡_𝑖^𝑘. (2.11) The basic properties of DTM have the following recursive relation for 𝑚=3:(𝑘+1)𝑌(𝑘+1)=^𝑘_𝑠=0^𝑠_𝑟=0𝑌(𝑟)𝑌(𝑠−𝑟)𝑌(𝑘−s)=𝐹(𝑘). (2.12) One can solve the above recursive relation easily to obtain the differential transforms 𝑌(𝑘),𝑘=0,…,𝑛. Then we can approximate𝑦(𝑡)≈^𝑛_𝑘=0𝑌(𝑘)𝑡−𝑡_𝑖^𝑘. (2.13)

Since the recursive relation (2.12) is derived by equating the coefficients for (𝑡−𝑡_𝑖)^𝑘 in (2.11), the differential transforms 𝑌(𝑘) are determined by solving(𝑘+1)𝑌(𝑘+1)=𝐹(𝑘),𝑘=0,1,…,𝑛. (2.14)

However, the right-hand side of (2.11) has the coefficients of (𝑡−𝑡_𝑖)^𝑘, 𝑘=0,1,…,3𝑛 so that as many as 2𝑛 coefficients of (𝑡−𝑡_𝑖)^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 that 𝑦_𝑙−1(𝑡) is approximated by the standard MsDTM, that is, 𝑦_𝑙−1(𝑡)≈𝑠_{𝑙−1,𝑛}(𝑡). Then, the solution 𝑦_𝑙(𝑡) can be determined by solving the following problem:𝑑𝑦𝑑𝑡^𝑙(𝑡)=𝑓𝑡,𝑦_𝑙(𝑡),𝑦_𝑙𝑡_𝑙=𝑠_{𝑙−1,𝑛}𝑡_𝑙. (2.15) Applying the DTM to (2.15) gives the differential transform 𝑌_𝑙(𝑘). Then 𝑦_𝑙(𝑡) is approximated by𝑦_𝑙(𝑡)≈^𝑛_𝑘=0𝑌_𝑙(𝑘)𝑡−𝑡_𝑖^𝑘≡𝑠_𝑙,𝑛(𝑡). (2.16) In the standard MsDTM, 𝑦_𝑙(𝑡_𝑙+1)≈𝑠_𝑙,𝑛(𝑡_𝑙+1) which gives 𝑌_𝑙+1(0). As mentioned before, it could give inaccurate approximate solution 𝑦_𝑙+1(𝑡). Here, taking the integral operator to (2.15), the solution 𝑦_𝑙(𝑡) can be represented by𝑦_𝑙(𝑡)=𝑦_𝑙𝑡_𝑙+^𝑡_𝑡𝑙𝑓𝜏,𝑦_𝑙(𝜏)𝑑𝜏. (2.17)

Thus, we have𝑦_𝑙𝑡_𝑙+1=𝑦_𝑙𝑡_𝑙+^𝑡𝑙+1_𝑡𝑙𝑓𝜏,𝑦_𝑙(𝜏)𝑑𝜏. (2.18)

Since 𝑦_𝑙(𝑡)≈𝑠_𝑙,𝑛(𝑡) in (2.16), we can approximate the value of 𝑦_𝑙(𝑡_𝑙+1) by𝑦_𝑙𝑡_𝑙+1=𝑦_𝑙𝑡_𝑙+^𝑡𝑙+1_𝑡𝑙𝑓𝜏,𝑠_𝑙,𝑛(𝜏)𝑑𝜏. (2.19) Then, 𝑦_𝑙(𝑡_𝑙+1)=𝑌_𝑙+1(0). For each subdomain 𝐷_𝑙, repeat this process (2.19) to approximate 𝑦_𝑙(𝑡). Thus, the solution 𝑦(𝑡) can be approximated by𝑦(𝑡)≈^𝑁−1_𝑙=0𝑠_𝑙,𝑛(𝑡)𝜒𝑡−𝑡_𝑙𝜒𝑡_𝑙+1−𝑡, (2.20)

where𝜒𝑡−𝑡_𝑙=1,𝑡≥𝑡_𝑙,0,otherwise. (2.21)

3. Application

To apply the MsDTM, we consider a subdomain with uniform length 𝑇/𝑁=ℎ stated in the previous section. For each subdomain (𝑡_𝑙,𝑡_𝑙+1)≡𝐷_𝑙, we apply the DTM to the model problem (1.1), then the following recursive relation is obtained:(𝑘+1)𝑋_𝑙(𝑘+1)=𝑎𝑋_𝑙(𝑘)−𝑏^𝑘_𝑟=0𝑋_𝑙(𝑟)𝑌_𝑙(),()𝑌𝑘−𝑟𝑘+1_𝑙(𝑘+1)=−𝑐𝑌_𝑙(𝑘)+𝑑^𝑘_𝑟=0𝑌_𝑙(𝑟)𝑋_𝑙(𝑘−𝑟). (3.1) 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 𝐷_𝑙, the solutions 𝑥_𝑙(𝑡) and 𝑦_𝑙(𝑡) can be determined by𝑥_𝑙(𝑡)=lim_𝑛→∞𝑥_𝑙,𝑛(𝑡),𝑦_𝑙(𝑡)=lim_𝑛→∞𝑦_𝑙,𝑛(𝑡), (3.2)

where𝑥_𝑙,𝑛+1(𝑡)=𝑥_𝑙,𝑛(𝑡)+^𝑡_𝑡𝑙𝜆₁(𝜏)𝑑𝑥𝑑𝑡^𝑙,𝑛(𝜏)−𝑎𝑥_𝑙,𝑛(𝜏)+𝑏𝑥_𝑙,𝑛(𝜏)𝑦_𝑙,𝑛(𝜏),𝑦_𝑙,𝑛+1(𝑡)=𝑦_𝑙,𝑛(𝑡)+^𝑡_𝑡𝑙𝜆₂(𝜏)𝑑𝑦𝑑𝑡^𝑙,𝑛(𝜏)−𝑐𝑦_𝑙,𝑛(𝜏)+𝑑𝑥_𝑙,𝑛(𝜏)𝑦_𝑙,𝑛(𝜏). (3.3)

Here, 𝜆₁,𝜆₂ are general Lagrange multiplier which are determined by solving the stationary problem. Here, the multipliers are given by [17]𝜆₁(𝜏)=−exp[𝑎(𝑡−𝜏)],𝜆₂(𝜏)=−exp[𝑐(𝜏−𝑡)]. (3.4)

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 as𝜙(𝑡)=𝑚𝑥^𝑝(𝑡)1+𝑥^𝑝(𝑡). (3.5)

For each subdomain 𝐷_𝑙, using the basic properties of DTM to (1.2) gives the following recursive relation:(𝑘+1)𝑋_𝑙(𝑘+1)=𝑟𝑋_𝑙(𝑘)−𝑟𝑘^𝑘_𝑖𝑋_𝑙(𝑖)𝑋_𝑙(𝑘−𝑖)−^𝑘_𝑖Φ_𝑙(𝑖)𝑋_𝑙(),()𝑌𝑘−𝑖𝑘+1_𝑙(𝑘+1)=𝜇^𝑘_𝑖𝑌_𝑙(𝑖)Φ(𝑘−𝑖)−𝐷𝑌_𝑙(𝑘), (3.6)

where for 𝑝=1,Φ_𝑙(𝑘)=11+𝑋_𝑙(𝑘)𝑚𝑋_𝑙(𝑘1),𝑘=0,1+𝑋_𝑙(0)𝑚𝑋_𝑙(𝑘)−^𝑘_𝑖=1𝑋_𝑙(𝑖)Φ_𝑙(𝑘−𝑖),𝑘>0, (3.7)

and for 𝑝=2Φ_𝑙(𝑘)=11+𝑋²_𝑙(𝑘)𝑚𝑋²_𝑙(𝑘1),𝑘=0,1+𝑋²_𝑙(0)𝑚^𝑘_𝑖=0𝑋_𝑙(𝑖)𝑋_𝑙(𝑘−𝑖)−^𝑘𝑗_𝑗=1𝑖=0𝑋_𝑙𝑗𝑋_𝑙𝑗−𝑖Φ_𝑙𝑘−𝑗,𝑘>0. (3.8)

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.

Figure 1 

Population of 𝑥(𝑡) and 𝑦(𝑡) by RK4 (ℎ=0.0001) 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 2 

Comparison of absolute errors in case 1.

Figure 3 

Population of 𝑥(𝑡) and 𝑦(𝑡) by RK4 (ℎ=0.0001)   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) right

Figure 4 

Comparison of absolute errors in case 2.

Figure 5 

Population of 𝑥(𝑡) and 𝑦(𝑡) by RK4 (ℎ=0.0001) 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) right

Figure 6 

Comparison of absolute errors in case 3.

Figure 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=32√2/7−0.1.

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=32√2/7+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.

Acknowledgments

B. Jang is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (no. 2010–0013297), and Y. Do is supported by WCU (World Class University) program through the Korea Science and Engineering Foundation funded by the Ministry of Education, Science, and Technology (no. R32-2009-000-20021-0).