A compact Fourth-Order Implicit-Explicit Runge-Kutta Type Method for Solving Diffusive Lotka–Volterra System

This paper aims to developed a high-order and accurate method for the solution of one-dimensional Lotka-Volterra-diffusion with Numman boundary conditions. A fourth-order compact finite difference scheme for spatial part combined with implicit-explicit Runge Kutta scheme in temporal are proposed. Furthermore, boundary points are discretized by using a compact finite difference scheme in terms of fourth order accuracy. A key idea for proposed scheme is to take full advantage of method of line (MOL), this is consequently enabling us to use implicit-explicit Runge Kutta method, that are of fourth order in time. We constructed fourth order accuracy in both space and time and is unconditionally stable. This is consequently leading to a reduction in the computational cost of the scheme. Numerical experiments show that the combination of the compact finite difference with IMEX- RK methods give an accurate and reliable for solving the Lotka-Volterra-diffusion.


Introduction
The finite element (FEM) and compact finite difference (CFDM) methods consider are the most of flexibility common technique used for dealing with partial differential equations. In context of Dirichlet boundary conditions for linear and nonlinear parabolic problems are gaining increasing interest and there is a significant implementation of the method now are understandable and available in the literature [1][2][3][4][5][6][7][8][9] However, there is less progress has been made comparatively in the deriving high order in terms of Neumann boundary conditions [10][11][12]. Cao al el [10] have developed a fourth-order compact finite difference scheme for solving the convection-diffusion equation with Neumann boundary conditions. They used compact finite difference scheme of fourth-order to discrete interior and boundary points. Yao al el [11] derived a fourth-order compact finite difference scheme for solving the model equation of simulated moving bed. They presented two different methods, direct method and pseudo grid point method to address the difficulty of the boundary condition. Fu al el [12] have used a high-order exponential scheme convection-diffusion equation with Neumann boundary conditions. In their analyses, they derived fourth-order compact exponential difference scheme in spatial discretization at all interior and boundary points. The aim of this work is to propose a numerical method for solving (1) and (2) that is fourth-order accurate in both space and time components. The idea is to discretise both spatial derivative and time integration of fourth order accurate by using a compact finite difference approximation and implicitexplicit Runge Kutta method, respectively. This is leading to a nonlinear system of ordinary differential equations. Some numerical methods have been developed to solve the Diffusive Lotka-Volterra System with Neumann boundary conditions, but only first-order or second-order at the  [13,14]. The main difficulties in our work in constructing fourthorder compact method for boundary point and dealing with the nonlinear reaction term. These challenges are addressed by employing techniques introduced by [12] for boundary point and IMEX-RK, introduced by Ruuth and Spiteri (21) for nonlinear term. It is worth noting the main reason of IMEX-RK technique is to lead us utilise implicit methods for linear part and explicit method for nonlinear part.
The rest of this paper is structured as follows. In Section 2, the model problem is introduced with derived fourth order compact scheme for both interior and boundary points. Section 3, implicit explicite Runge kutta method are presented. Numerical Experiments are shown in Section 4, Finally, conclusions are given in Section 5.
Consider the one-dimensional Lotka-Volterra equation with Neumann boundary conditions: with the following initial conditions: and Neumann boundary conditions Here u represents the number of fish (prey), v represents the number of sharks (predator), and is diffusion coefficient of the prey. We also use the parameter is diffusion coefficient of the predator, to scale the predator,  to represent of the prey, to represent the death rate of the predator, and , they represent the rate of interraction between the prey and the predator which facilitates the killing of prey by predator.

Development of fourth-order compact finite differencing schemes 2.1 The interior spatial points
This section aims to discretise on spatial domain, and this is consequently leading to a set of system of nonlinear ordinary differential equations. To do this, we discretise (1) and (2) by a four-order compact difference approximation for the space part setting ( ) and ( ) gives Recalling (1) and (2), gives To obtain higher order scheme yield ( ), this may accomplish by differentiating (3) and (4) respect to gives (5) and (6), and substituting above equations in (5) and (6), reads ( ) If we put (7) along with in (15) and (16), this gives

The boundary spatial points
The main goal here is to construct a fourth order accurate on the spatial points for the left and right boundaries. This is may have accomplished by using techniques deduced in [12]. For simplicity to present our work, we will derive for and for will follow the same rules, to do this, begin with lift Neumann boundary condition, yields, Substituting (9) and (10) into (19), this becomes Go back to (19), to construct right boundary point, this can be obtained by substituting (9) and (10) in Similarly, for gives Collecting together (17), (18), (21), (22), (23) and (24), this will lead to a system of nonlinear ordinary differential equations as fallows ,

Implicit-Explicit Runge Kutta methods (IMEXRK)
The aim of this section is to combine the high-order compact difference scheme presented in section 2 with IMEXRK schemes to obtain a new and high-order method for solving (1) and (2). The curtail idea of IMEXRK is to split in two methods, the first is implicit Runge Kutta and the second is explicit Runge Kutta method and consequently leading to the linear part is treated by implicit sachem while nonlinear term is treated by explicit method we refer the readers to [15][16][17][18][19][20][21][22][23][24][25] in details. To do this, go back for the system of nonlinear ordinary differential equation in (28) and (29), since A is invertible, and for brevity, we will use the following notions The key idea of IMEX-RK methods is to decompose the right-hand side of (28) and (29) into stiff ( and ) and nonstiff ( and ) terms. Note that the no stiff term is treated by explicit Runge Kutta method while stiff term is treated by single diagonal implicit method. The resulting system of ODE's can be very efficiently integrated using the IMEX approach.
Notes that the matrices = ( ), = 0 for and = ( ̂), where is implicit scheme and is explicit scheme. An IMEX Runge-Kutta scheme is characterized by these two matrices and the coefficient vectors = ( ) , = ( ) . The above methods can be represented by a double tableau in the usual Butcher notation. Using the Kronecker product, this can be written in the Butcher tableau is By using implicit-explicit Runge-Kutta scheme for above, gives

Numerical experiments
We shall now illustrate the performance of a presented method, through an implementation based on the Matlab programming. This section aims to present the relation between sharks and fish populations. Two cases will focus on it in this paper. Firstly, for which conditions sharks avoid killing off the fish population. Secondly, what conditions can sharks and fish coexist so that their populations oscillate, but neither group ever is extinguished. In this article, we will consider different methods of IMEX-RK ( ) the number of p is the order of the scheme, is the number of stage implicit scheme and is the number of stage explicit sachems [19].     Figure 4.1, show that fish population increases, while sharks population remains almost constant in first few time steps with sitting the rate of contact ( ) between the two species is higher compared to the rate of death ( ). However, the Sharks population quickly starts to grow in the areas in the absence of fish, and the result of this can be seen in Figure 4.2. This is consequently leading to Sharks population will dominate the ecosystem and fish population will extinguished with the increased number of sharks as shown in Figure 4.3., 4.4. In the period, fish population start to grown, while the predator population becomes quite small as indicated in Figure 4.5, 4.6. Eventually, both prey and predator repeat themselves, producing a new travelling front which moves again from the top to the bottom. Its formation can be seen in Figure 4.6, 4.7, 4.8. Under normal conditions both prey and predator, always tend to co-exist, no species is extinguished from the ecosystem. This only happens, with the absence of external forces such as human interference in the ecosystem, where fishing is one of the activity in this particular case, which destroy

Conclusion
This paper is devoted to propose a high-order compact scheme for solving the one-dimensional Lotka-Volterra-diffusion equation. The curtail idea is to combine a fourth-order compact finite difference scheme to discretise the spatial derivative and implicit-explicit Runge Kutta method to the time integration, and this leads to nonlinear system of ordinary differential equations. The proposed sachem has fourth order in space and time. The main use for these combinations is in splitting a nonlinear system by a pair of methods ( )where the method is implicit and the method is explicit. An important factor in our proposed method is to reduce the number of iterations and consequently leading to a reduction in the computational cost of the scheme. It is clearly confirmed that the proposed method is a good in term of the computational cost and stability. Numerical experiments through Matlab programming also confirm that the proposed method are reliable and efficient for solving Lotka-Volterra-diffusion equation. This approach may be extended to tackle stochastic equations with growth model [26,27]. Another interesting of this work is to use discontinuous Galerkin methods for estimating this type of problem in terms of ( ) and ( ) [ ]