Nonstandard finite difference schemes with application to biological models

This paper deals with the construction of nonstandard finite difference methods for solving a specific Rosenzweig-MacArthur predator-prey model. The reorganization of the denominator of the discrete derivatives and nonlocal approximations of nonlinear terms are used in the design of new schemes. We establish that the proposed nonstandard finite difference methods are elementary stable and satisfy the positivity requirement. We provide some numerical comparisons to illustrate our results.Publisher's Versio

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

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, focused on the prediction of the aphids-ladybirds population densities, that share a prey-predator 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, 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. 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

The Lotka-Volterra Dynamical System and its Discretization

Dynamical systems are a valuable asset for the study of population dynamics. On this topic, much has been done since Lotka and Volterra presented the very first continuous system to understand how the interaction between two species -- the prey and the predator -- influences the growth of both populations. The definition of time is crucial and, among options, one can have continuous time and discrete time. The choice of a method to proceed with the discretization of a continuous dynamical system is, however, essential, because the qualitative behavior of the system is expected to be identical in both cases, despite being two different temporal spaces. In this work, our main goal is to apply two different discretization methods to the classical Lotka-Volterra dynamical system: the standard progressive Euler's method and the nonstandard Mickens' method. Fixed points and their stability are analyzed in both cases, proving that the first method leads to dynamic inconsistency and numerical instability, while the second is capable of keeping all the properties of the original continuous model.Comment: This is a preprint of a paper whose final form is published at [http://dx.doi.org/10.1201/9781003388678-19

(R1504) Second-order Modified Nonstandard Runge-Kutta and Theta Methods for One-dimensional Autonomous Differential Equations

Nonstandard finite difference methods (NSFD) are used in physical sciences to approximate solutions of ordinary differential equations whose analytical solution cannot be computed. Traditional NSFD methods are elementary stable but usually only have first order accuracy. In this paper, we introduce two new classes of numerical methods that are of second order accuracy and elementary stable. The methods are modified versions of the nonstandard two-stage explicit Runge-Kutta methods and the nonstandard one-stage theta methods with a specific form of the nonstandard denominator function. Theoretical analysis of the stability and accuracy of both modified NSFD methods is presented. Numerical simulations that concur with the theoretical findings are also presented, which demonstrate the computational advantages of the proposed new modified nonstandard finite difference methods

Qualitative analysis of a discrete-time phytoplankton–zooplankton model with Holling type-II response and toxicity

[EN]The interaction among phytoplankton and zooplankton is one of the most important processes in ecology. Discrete-time mathematical models are commonly used for describing the dynamical properties of phytoplankton and zooplankton interaction with nonoverlapping generations. In such type of generations a new age group swaps the older group after regular intervals of time. Keeping in observation the dynamical reliability for continuous-time mathematical models, we convert a continuous-time phytoplankton–zooplankton model into its discrete-time counterpart by applying a dynamically consistent nonstandard difference scheme. Moreover, we discuss boundedness conditions for every solution and prove the existence of a unique positive equilibrium point. We discuss the local stability of obtained system about all its equilibrium points and show the existence of Neimark–Sacker bifurcation about unique positive equilibrium under some mathematical conditions. To control the Neimark–Sacker bifurcation, we apply a generalized hybrid control technique. For explanation of our theoretical results and to compare the dynamics of obtained discrete-time model with its continuous counterpart, we provide some motivating numerical examples. Moreover, from numerical study we can see that the obtained system and its continuous-time counterpart are stable for the same values of parameters, and they are unstable for the same parametric values. Hence the dynamical consistency of our obtained system can be seen from numerical study. Finally, we compare the modified hybrid method with old hybrid method at the end of the paper

Numerical convergence of a Telegraph Predator-Prey System

The numerical convergence of a Telegraph Predator-Prey system is studied. This system of partial differential equations (PDEs) can describe various biological systems with reactive, diffusive and delay effects. Initially, our problem is mathematically modeled. Then, the PDEs system is discretized using the Finite Difference method, obtaining a system of equations in the explicit form in time and implicit form in space. The consistency of the Telegraph Predator-Prey system discretization was verified. Next, the von Neumann stability conditions were calculated for a Predator-Prey system with reactive terms and for a Telegraph system with delay. For our Telegraph Predator-Prey system, through numerical experiments, it was verified tat the mesh refinement and the model parameters (reactive constants, diffusion coefficient and delay term) determine the stability/instability conditions of the model. Keywords: Telegraph-Diffusive-Reactive System. Maxwell-Cattaneo Delay. Discretization Consistency. Von Neumann Stability. Numerical Experimentation.Comment: Submited to journal "Semina: Exact and Technological Sciences