A Discrete Equivalent of the Logistic Equation

Eugenia N. Petropoulou

doi:10.1155/2010/457073

Abstract

A discrete equivalent and not analogue of the well-known logistic differential equation is proposed. This discrete equivalent logistic equation is of the Volterra convolution type, is obtained by use of a functional-analytic method, and is explicitly solved using the z-transform method. The connection of the solution of the discrete equivalent logistic equation with the solution of the logistic differential equation is discussed. Also, some differences of the discrete equivalent logistic equation and the well-known discrete analogue of the logistic equation are mentioned. It is hoped that this discrete equivalent of the logistic equation could be a better choice for the modelling of various problems, where different versions of known discrete logistic equations are used until nowadays.

Highlights

The well-known logistic differential equation was originally proposed by the Belgian mathematician Pierre-Francois Verhulst 1804–1849 in 1838, in order to describe the growth of a population P t under the assumptions that the rate of growth of the population was proportional to
Equation 1.2 can be regarded as a Bernoulli differential equation or it can be solved by applying the simplest method of separation of variables
1.2 can be considered as a simple differential equation, in the sense that it is completely solvable by use of elementary techniques of the theory of differential equations, it has tremendous and numerous applications in various fields

Summary

Introduction

The well-known logistic differential equation was originally proposed by the Belgian mathematician Pierre-Francois Verhulst 1804–1849 in 1838, in order to describe the growth of a population P t under the assumptions that the rate of growth of the population was proportional to. When this problem is “translated” into mathematics, results to the differential equation dP t dt rP t − Pt K P0. P0, Advances in Difference Equations where t denotes time, P0 is the initial population, and r, K are constants associated with the growth rate and the carrying capacity of the population. The first application of 1.2 was already mentioned, and it is connected with population problems, and more generally, problems in ecology. Other applications of 1.2 appear in problems of chemistry, medicine especially in modelling the growth of tumors , pharmacology especially in the production of antibiotic medicines 1 , epidemiology 2, 3 , atmospheric pollution, flow in a river 4 , and so forth

Objectives

Discussion

Conclusion