Comparing Models of Growth
by Mike Martin and Helen Byrne

This page allows you to analyze solutions to three different models. Each of the models start with a given differential equation and initial condition. The differential equations themselves have built-in assumptions and an analysis of their solutions relative to the observed phenomena will help to validate (or not) the model themselves. Here are the three differential equations and initial condition and their respective solutions:

The solution to each of the problems are just simple functions and their graphs are shown below. The first of these models, known as the exponential growth law, has an exponential solution, No ek t , and is the red-dashed solution in the graph below. k is per capita reproduction rate and, for the purposes of comparison, is assumed to be positive, producing exponential growth. N is population and it may be expressed as a population count with different units (say, in millions or something) or as a proportion of current population to some level of particular significance. The second model is known as the logistic growth law and typically has a solution (dashed in gray in the graph below) that is s-shaped, or sigmoidal, in nature. Here, θ represents the carrying capacity of the environment and is measured similar to that of N. The third model adds another parameter, α, that can be thought of as controlling the rate at which a population saturates. The solution for the third model is dashed blue in the graph below. The carrying capacity for the system, N = θ, is plotted in green in the graph below.   The inflection points for the models are identified with an exaggerated point/dot.   tinitial and tfinal give the t-interval for plotting. TIme is along the horizontal axis and the population measure, N (t), is along the vertical axis.   Enter the value of the parameters below, then press Evaluate.

k =
α =
θ =
No =
tinitial =
tfinal =


Byrne, H., Cancer Modeling and Simulation, Chapman & Hall/CRC, Boca Raton, FL, 2003.