252 American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 © Global Society of Scientific Research and Researchers http://asrjetsjournal.org/ A Positivity-Preserving Nonstandard Finite Difference Scheme for Parabolic System with Cross-Diffusion Equations and Nonlocal Initial Conditions Marc E. Songolo* Department of Mathematics and Computer Sciences, Université de Lubumbashi, Lubumbashi, R. D. Congo Email: marc.songolo@gmail.com Abstract An important number of ecological phenomena can be modeled using nonlinear diffusion partial differential equations. This paper considers a system of cross-diffusion equations with nonlocal initial conditions. Such equations arise as steady-state equations in an age-structured predator-prey model with diffusion. We use the nonstandard finite difference method developed by Mickens. These types of schemes are made by the following two rules: first, renormalization of step size for the denominator function of representations of derivatives, and secondly, nonlocal representations of nonlinear terms. We obtained a scheme that preserves the positivity of solutions. Furthermore, this scheme is explicit and functional relationship is obtained between time, space, and age step sizes. Keywords: Cross-diffusion equations; finite difference methods; nonlocal initial conditions; nonstandard finite difference schemes; positivity of solutions; predator-prey model. 1. Introduction The application of partial differential equations is very common in the natural and engineering sciences. In particular, the nonlinear modeling of reaction and/or diffusion phenomena [1]. In this paper we consider the equations for studying an age-structure of the predator-prey model, defined by the following parabolic equations with nonlocal initial conditions. The predator population is putting pressure on the prey and all populations suffer fluctuations related to space [2]. ------------------------------------------------------------------ * Corresponding author. http://asrjetsjournal.org/ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 18, No 1, pp 252-258 253 If and are respectively the densities of populations of predator and prey that live in the same spatial region and are structured by age for a maximal age , and spatial position , then the model will be [2] : These equations are subject to the constraints and the conditions to space limitations: and also the additional initial conditions. The Laplacian indicates the spatial movement of the prey population, is the intrinsic dispersion coefficient, which reflects the growing strength of dispersion prey because of interference with the growth of the predator population and is the pressure coefficient of the population of predators. For notational simplicity, we will take . The right hand sides of the system (1) taking into account the interactions within and outside of two specific populations with coefficient and . Equations (3) describe the rates of creation of new individuals to and . The objective of this paper is to construct a finite difference scheme that preserves the positivity of solutions. The method used in this construction is based on nonstandard discretisation technique created by Ronald Mickens. In particular, the scheme must be explicit and requires a functional relationship between time, space and age step sizes. In general, the absence of this restriction leads to the existence of numerical instabilities, i.e. the solutions to the difference equations do not correspond to the solutions of the continuous equations [2]. The remainder of this paper is organized as follows: Section 2 gives a brief overview of the philosophy of construction of nonstandard finite difference scheme (NSFDS), while Section 3 presents the construction of a new NSFDS. Finally, Section 4 provides a summary discussion results. 2. Nonstandard modeling The procedures for constructing NSFDS are stated in [3]. These types of schemes are made by the following two rules [4,5]: first, how to determine the discrete representations of derivatives, and secondly, what is the form of American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 18, No 1, pp 252-258 254 nonlinear terms. The first concept involves the generalization of the representations of derivatives form [2,3]: where is the approximation of , and the denominator function satisfies : In equations (4) and (5), represents the parameters of the differential equation. This construction technique can easily be extended to partial differential equations. The second concept is related to the modeling of nonlinear terms, for example, . In general, the nonlinear terms are replaced by nonlocal representation [4]: Both concepts are discussed in detail in [3]. A good example is the advection-reaction equation [4],[6] where the exact numerical scheme is The denominator function is and is the representation of . The major difficulty in the numerical simulation of partial differential equations of which we do not know the exact solution often leads to consider 3. Nonstandard finite difference scheme In one dimension of space, the system (1) can be written as American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 18, No 1, pp 252-258 255 We now suggest the following scheme for system (11): where we have the following discrete representations : Solving and in (12) we get where and Positivity of solutions provides that American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 18, No 1, pp 252-258 256 which leads to the following conditions and gives Taking , we find that In this paper, we only consider the equality in equation (19) and system (14) becomes : Note that the time step size in (19) is not constant. It is also clear that (20) preserves the positivity of solutions. For simulation, select and then calculate and at every step. 4. Discussion If in (11), then we find a system of reaction-diffusion equations with nonlocal initial conditions [7] The following nonstandard finite difference scheme was constructed for the system (21) in [8] This is in fact the situation when in (14). American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 18, No 1, pp 252-258 257 If more , then we get And considering equality in equation (23), the scheme (22) becomes: where , and 5. Conclusion We have constructed a nonstandard finite difference scheme for a system of nonlinear parabolic cross-diffusion equations with nonlocal initial conditions. This scheme is made by using non local representations of nonlinear terms, and preserves the positivity of solutions. Furthermore, this scheme is explicit and we obtained a variable time step size which depends on space and age step sizes. References [1] Mickens R.E., “Nonstandard finite difference schemes for reaction-diffusion Equations,” Numerical Methods for Partial Differential Equation, vol 15, pp. 201-214, 1999. [2] Walker C., Positive solutions of some system of cross-diffusion equations and nonlocal initial conditions. 2010. [Online] Available: http://www.arxiv.org [November 15, 2010]. [3] Mickens R.E., Nonstandard finite difference models of differential equations, world scientific, 1994. [4] Mickens R.E., “A Nonstandard finite difference schemes for a Fischer PDE Having Nonlinear Diffusion,” Computers and Mathematics with Applications, vol 45, pp. 429-436, 2003. [5] Anguelov R. and Lubuma JMS., “Nonstandard finite difference models by nonlocal approximation,” mathematical and computer in simulation, vol 61 pp. 465-475, 2003. [6] Mickens R.E., “Calculation of Denominator functions in Nonstandard finite Difference schemes for Differential Equations Satisfying a Positivity condition,” Num. Meth. Diff. Eqs, vol 23, pp. 672-691, 2007. http://www.arxiv.org/ American Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2016) Volume 18, No 1, pp 252-258 258 [7] Walker C., On positive solutions of some system of reaction-diffusion equations with nonlocal initials conditions. 2010. [Online] Available: http://www.arxiv.org (March 24, 2010) [8] Songolo M. E., “A positivity-preserving nonstandard finite difference scheme for a system of reaction- diffusion equations with nonlocal initial conditions,” (To be appear in International Journal of Innovation and Applied Sciences in May 2016). http://www.arxiv.org/