Positive and elementary stable nonstandard numerical methods with applications to predator–prey models

Dimitrov, Dobromir; Kojouharov, Hristo V.

doi:10.1016/j.cam.2005.04.003

Cited by 82 publications

(60 citation statements)

References 13 publications

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“…On adding the different equations in (28), condition (31) leads to a discrete conservation law of the form (20), which in turn yields the dissipativity of the scheme (28), because…”

Section: Nonstandard Finite Difference Schemesmentioning

confidence: 99%

Dynamically consistent nonstandard finite difference schemes for epidemiological models

Anguelov

Dumont

Lubuma

et al. 2014

Journal of Computational and Applied Mathematics

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This work is the numerical analysis and computational companion of the paper by Kamgang and Sallet (Math. Biosc. 213 (2008), pp. 1-12) where threshold conditions for epidemiological models and the global stability of the disease-free equilibrium (DFE) are studied. We establish a discrete counterpart of the main continuous result that guarantees the global asymptotic stability (GAS) of the DFE for general epidemiological models. Then, we design nonstandard finite difference (NSFD) schemes in which the Metzler matrix structure of the continuous model is carefully incorporated and both Mickens' rules (World Scientific, Singapore, 1994) on the denominator of the discrete derivative and the nonlocal approximation of nonlinear terms are used in an innovative way. As a result of these strategies, our NSFD schemes are proved to be dynamically consistent with the continuous model, i.e., they replicate their basic features, including the GAS of the DFE, the linear stability of the endemic equilibrium (EE), the positivity of the solutions, the dissipativity of the system, and its inherent conservation law. The general analysis is made detailed for the MSEIR model for which the NSFD theta method is implemented, with emphasis on the computational aspects such as its convergence, or local truncation error. Numerical simulations that illustrate the theory are provided.

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“…On adding the different equations in (28), condition (31) leads to a discrete conservation law of the form (20), which in turn yields the dissipativity of the scheme (28), because…”

Section: Nonstandard Finite Difference Schemesmentioning

confidence: 99%

Dynamically consistent nonstandard finite difference schemes for epidemiological models

Anguelov

Dumont

Lubuma

et al. 2014

Journal of Computational and Applied Mathematics

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“…The methods proposed in that paper can be applicable for solving arbitrary two-dimensional autonomous dynamical systems. In another work, Dimitrov and Kojouharov (2006), formulated positive and elementary stable nonstandard finite-difference methods to solve a general class of Rosenzweig-MacArthur predator-prey systems which involve a logistic intrinsic growth of the prey population. Their methods preserve the positivity of solutions and the stability of the equilibria for arbitrary step-sizes, while the approximations obtained by the other numerical methods experience difficulties in preserving either the stability or the positivity of the solutions, or both.…”

Section: Ha Obaid Et Almentioning

confidence: 99%

An unconditionally stable nonstandard finite difference method applied to a mathematical model of HIV infection

Obaid¹,

Ouifki²,

Patidar³

2013

International Journal of Applied Mathematics and Computer Science

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We formulate and analyze an unconditionally stable nonstandard finite difference method for a mathematical model of HIV transmission dynamics. The dynamics of this model are studied using the qualitative theory of dynamical systems. These qualitative features of the continuous model are preserved by the numerical method that we propose in this paper. This method also preserves the positivity of the solution, which is one of the essential requirements when modeling epidemic diseases. Robust numerical results confirming theoretical investigations are provided. Comparisons are also made with the other conventional approaches that are routinely used for such problems.

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“…and F h (F; z k ) approximates the right-hand side of system (2.2) , z k ≈ z (t k ) and t k = t 0 + kh [9]. Definition 2.1.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

“…Because the discrete system obtained by the continuous system has truncation errors. Anguelov and Lubuma [2], Dimitrov and Kojouharov [8,9] and Lubuma and Roux [18], among others, have used Nonstandard Finite Difference Schemes (NSFD), developed by Mickens [20] for designing methods that conserve the local stability of equilibria of the approximated system. However, the NSFD methods, guarantee positive discrete solution for positive initial conditions [11].…”

Section: Introductionmentioning

confidence: 99%

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A nonstandard numerical scheme for a predator-prey model with Allee effect

Ongun¹,

ÖZDOĞAN²

2017

J. Nonlinear Sci. Appl.

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In this paper, we present a Lotka-Volterra predator-prey model with Allee effect. This system with general functional response has an Allee effect on prey population. A nonstandard finite difference scheme is constructed to transform the continuous time predator-prey model with Allee effect into the discrete time model. We use the Schur-Cohn criteria which deal with coefficients of the characteristic polynomial for determining the stability of discrete time system. The proposed numerical schemes preserve the positivity of the solutions with positive initial conditions. The new discrete-time model shows dynamic consistency with continuous-time model.

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Positive and elementary stable nonstandard numerical methods with applications to predator–prey models

Cited by 82 publications

References 13 publications

Dynamically consistent nonstandard finite difference schemes for epidemiological models

Dynamically consistent nonstandard finite difference schemes for epidemiological models

An unconditionally stable nonstandard finite difference method applied to a mathematical model of HIV infection

A nonstandard numerical scheme for a predator-prey model with Allee effect

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