Positive and elementary stable nonstandard numerical methods with applications to predator-prey models. (English) Zbl 1087.65068

Dimitrov, Dobromir T.; Kojouharov, Hristo V.

J. Comput. Appl. Math. 189, No. 1-2, 98-108 (2006).

The authors introduce and analyze a class of methods for the numerical solution of predator-prey equations. The methods are constructed in a way that retains some important analytical properties of the systems. Numerical computations indicate the superiority over standard methods that do not follow these construction criteria at least for certain important situations.

Reviewer: Kai Diethelm (Braunschweig)

MSC:

65L05	Numerical methods for initial value problems involving ordinary differential equations
65L12	Finite difference and finite volume methods for ordinary differential equations
34A34	Nonlinear ordinary differential equations and systems
92D25	Population dynamics (general)
65L20	Stability and convergence of numerical methods for ordinary differential equations

Keywords:

finite difference; predator-prey model; positive method; elementary stable method; stability; numerical examples

References:

[1]	Anguelov, R.; Kama, P.; Lubuma, J. M.-S., On non-standard finite difference models of reaction-diffusion equations, J. Comput. Appl. Math., 175, 1, 11-29 (2005) · Zbl 1070.65071
[2]	Anguelov, R.; Lubuma, J. M.-S., Contributions to the mathematics of the nonstandard finite difference method and applications, Numer. Methods Partial Differential Equations, 17, 5, 518-543 (2001) · Zbl 0988.65055
[3]	Anguelov, R.; Lubuma, J. M.-S., Nonstandard finite difference method by nonlocal approximation, Math. Comput. Simul., 61, 3-6, 465-475 (2003) · Zbl 1015.65034
[4]	Brauer, F.; Castillo-Chavez, C., Mathematical Models in Population Biology and Epidemiology (2001), Springer: Springer New York · Zbl 0967.92015
[5]	Burchard, H.; Deleersnijder, E.; Meister, A., A high-order conservative Patankar-type discretization for stiff systems of production-destruction equations, Appl. Numer. Math., 47, 1, 1-30 (2003) · Zbl 1028.80008
[6]	Coddington, E. A.; Levinson, N., Theory of Ordinary Differential Equations (1984), Krieger: Krieger Florida · Zbl 0042.32602
[8]	Dimitrov, D. T.; Kojouharov, H. V., Nonstandard finite-difference schemes for general two-dimensional autonomous dynamical systems, Appl. Math. Lett., 18, 7, 769-774 (2005) · Zbl 1122.65067
[9]	Gumel, A. B.; Mickens, R. E.; Corbett, B. D., A non-standard finite-difference scheme for a model of HIV transmission and control, J. Comput. Methods Sci. Eng., 3, 1, 91-98 (2003) · Zbl 1036.65057
[10]	Holling, C. S., The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Canada, 45, 1-60 (1965)
[11]	Jansen, H.; Twizell, E. H., An unconditionally convergent discretization of the SEIR model, Math. Comput. Simul., 58, 147-158 (2002) · Zbl 0983.92025
[12]	Lubuma, J. M.-S.; Roux, A., An improved theta-method for systems of ordinary differential equations, J. Differential Equations Appl., 9, 11, 1023-1035 (2003) · Zbl 1042.65059
[13]	de Markus, A. S.; Mickens, R. E., Suppression of numerically induced chaos with nonstandard finite difference schemes, J. Comput. Appl. Math., 106, 2, 317-324 (1999) · Zbl 0931.65079
[14]	Mickens, R. E., Nonstandard Finite Difference Model of Differential Equations (1994), World Scientific: World Scientific Singapore · Zbl 0925.70016
[15]	Mickens, R. E., Relation between the time and space step-sizes in nonstandard finite-difference schemes for the Fisher equation, Numer. Methods Partial Differential Equations, 13, 1, 51-55 (1997) · Zbl 0872.65080
[16]	Mickens, R. E., Nonstandard finite difference schemes for differential equations, J. Differential Equations Appl., 8, 9, 823-847 (2002) · Zbl 1010.65032
[17]	Piyawong, W.; Twizell, E. H.; Gumel, A. B., An unconditionally convergent finite-difference scheme for the SIR model, Appl. Math. Comput., 146, 611-625 (2003) · Zbl 1026.92041