Non standard finite difference scheme preserving dynamical properties

Cresson, Jacky; Pierret, Frédéric

doi:10.1016/j.cam.2016.02.007

Cited by 45 publications

(33 citation statements)

References 15 publications

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“…The numerical integration of ordinary differential equations using traditional methods could produce different solutions from those of the original ODE [2,14]. In particular, using a discretization step-size larger than some relevant time scale, is possible to obtain solutions that may not reflect the dynamics of the original system.…”

Section: Discrete-time Modelmentioning

confidence: 99%

Aphids, Ants and Ladybirds: a mathematical model predicting their population dynamics

Gabbriellini¹

2019

Open j. Math. Sci

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The interaction between aphids, ants and ladybirds has been investigated from an ecological point of view since many decades, while there are no attempts to describe it from a mathematical point of view. This paper introduces a new mathematical model to describe the within-season population dynamics in an ecological patch of a system composed by aphids, ants and ladybirds, through a set of four differential equations. The proposed model is based on the Kindlmann and Dixon set of differential equations [5], focused on the prediction of the aphids-ladybirds population densities, that share a preypredator relationship. The population of ants, in mutualistic relationship with aphids and in interspecific competition with ladybirds, is described according to the Holland and De Angelis mathematical model [8], in which the authors faced the problem of mutualistic interactions in general terms. The set of differential equations proposed here is discretized by means the Nonstandard Finite Difference scheme, successfully applied by Gabbriellini to the mutualistic model [4]. The constructed finite-difference scheme is positivity-preserving and characterized by four nonhyperbolic steady-states, as highlighted by the phase-space and time-series analyses. Particular attention is dedicated to the steady-state most interesting from an ecological point of view, whose asymptotic stability is demonstrated via the Centre Manifold Theory. The model allows to numerically confirm that mutualistic relationship effectively influences the population dynamic, by increasing the peaks of the aphids and ants population densities. Nonetheless, it is showed that the asymptotical populations of aphids and ladybirds collapse for any initial condition, unlike that of ants that, after the peak, settle on a constant asymptotic value. arXiv:1905.05244v1 [q-bio.PE]

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Section: Discrete-time Modelmentioning

confidence: 99%

Aphids, Ants and Ladybirds: a mathematical model predicting their population dynamics

Gabbriellini¹

2019

Open j. Math. Sci

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“…We use different numerical schemes to integrate these equations [12][13][14][15]. A natural question on numerical schemes can be the following despite the convergence analysis: Do the numerical schemes preserve the dynamical properties of the initial system [16]?…”

Section: Introductionmentioning

confidence: 99%

A reliable numerical analysis for stochastic dengue epidemic model with incubation period of virus

2019

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This article represents a numerical analysis for a stochastic dengue epidemic model with incubation period of virus. We discuss the comparison of solutions between the stochastic dengue model and a deterministic dengue model. In this paper, we have shown that the stochastic dengue epidemic model is more realistic as compared to the deterministic dengue epidemic model. The effect of threshold number R 1 holds in the stochastic dengue epidemic model. If R 1 < 1, then situation helps us to control the disease while R 1 > 1 shows the persistence of disease in population. Unfortunately, the numerical methods like Euler-Maruyama, stochastic Euler, and stochastic Runge-Kutta do not work for large time step sizes. The proposed framework of stochastic nonstandard finite difference scheme (SNSFD) is independent of step size and preserves all the dynamical properties like positivity, boundedness, and dynamical consistency.

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“…Up to date NSFD schemes became a power and efficient tool for simulating dynamical systems, especially in converting continuous models to dynamically consistent discrete counterparts [2,3,7,9,10,11,12,13,25,28,31,32,35]. The majority of these models are met in physics, mechanics, chemistry, biology, epidemiology, finance, .…”

Section: Introductionmentioning

confidence: 99%

Nonstandard finite difference schemes for a general predator–prey system

Dang

Hoang

2019

Journal of Computational Science

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In this paper we transform a continuous-time predator-prey system with general functional response and recruitment for both species into a discrete-time model by nonstandard finite difference scheme (NSFD). The NSFD model shows complete dynamic consistency with its continuous counterpart for any step size. Especially, the global stability of a non-hyperbolic equilibrium point in a particular case of parameters is proved by the Lyapunov stability theorem. The performed numerical simulations confirmed the validity of the obtained theoretical results.

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Non standard finite difference scheme preserving dynamical properties

Cited by 45 publications

References 15 publications

Aphids, Ants and Ladybirds: a mathematical model predicting their population dynamics

Aphids, Ants and Ladybirds: a mathematical model predicting their population dynamics

A reliable numerical analysis for stochastic dengue epidemic model with incubation period of virus

Nonstandard finite difference schemes for a general predator–prey system

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