Analysis of logistic equation pertaining to a new fractional derivative with non-singular kernel

Devendra Kumar,Maysaa Al Qurashi,Jagdev Singh,Dumitru Baleanu

doi:10.1177/1687814017690069

Devendra Kumar, Maysaa Al Qurashi + Show 2 more

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https://doi.org/10.1177/1687814017690069

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Abstract

In this work, we aim to analyze the logistic equation with a new derivative of fractional order termed in Caputo–Fabrizio sense. The logistic equation describes the population growth of species. The existence of the solution is shown with the help of the fixed-point theory. A deep analysis of the existence and uniqueness of the solution is discussed. The numerical simulation is conducted with the help of the iterative technique. Some numerical simulations are also given graphically to observe the effects of the fractional order derivative on the growth of population.

Highlights

The logistic equation describes the population growth
The continuous form of the logistic equation is expressed in the form of nonlinear ordinary differential equation as[1]
The fractional logistic equation and its stability analysis are examined in section ‘‘Fractional model of logistic equation associated with new fractional derivative.’’ In section ‘‘Existence and uniqueness,’’ the existence and uniqueness of the solution are examined

Summary

Introduction

The logistic equation describes the population growth. It was first proposed by Pierre Verhulst that is why it is known as Verhulst model. The mathematical equation is a continuous function of time, but a modified version of the continuous model to a discrete quadratic recurrence model is said to be the logistic map which is extensively used. The continuous form of the logistic equation is expressed in the form of nonlinear ordinary differential equation as[1]. In the above equation (1), N indicates population at time t, l.0 represents Malthusian parameter expressing growth rate of species and K denotes carrying capacity. Equation (2) is said to be logistic equation. Due to its wide applications, many scientists and engineers investigated in this special branch and introduced various

Objectives

Conclusion