Logistics Growth Model

Population Biology Model

A model in population biology is one in which the population of a species increases at a rate proportional to that population until it becomes so large that individual organisms start to compete with one another for space and/or food. In a simple model, the effects of this competition are proportional to the likelihood that one organism will encounter another - a likelihood that is proportional to the square of the population. This yields the following ODE:

$$\frac{d N}{d t} = r N(t) ( Np - N(t) )$$ (1)

where $N(t)$ is the population, $r$ the population growth rate and $Np$ is a stable population according to the environment - at this population the increase rate vanishes. This model is an initial value problem with $N(0)=No$. The solution will depend on $r$ and $Np$.

   

This model proves equivalent to our previous simple Aids Epidemic model.

Scaled Model - Universal Behavior

Dividing the ODE by $Np$ yields

$$\frac{1}{Np}\frac{d N}{d t} = r Np \frac{N(t)}{Np} ( 1 - \frac{N(t)}{Np} )$$ (2)

Changing variables, $\frac{N(t)}{Np} \rightarrow N(t)$ and $t ( r Np ) \rightarrow t$, and renaming, yields

$$\frac{d N}{d t} = N(t) ( 1 - N(t) )$$ (3)

and the initial condition $N(0)=\frac{No}{Np}$.

  

The ODE thus becomes parameter free through these variable and function scalings.

Solution - Universal Sigmoid curve

The Maple worksheet logistic_growth.mws solves this ODE. It also shows how Maple allows checking the validity of an analytical solution.

  

The solution of the scaled equation depends only on the scaled initial value $N(0)$ - all physical parameters in the ODE having been scaled away. This can can be studied via the Maple solution or a C code. You can modify the Aids Epidemics C code to do the latter.

   

Plot the solutions for scaled initial values for $N(0)$ in a range $N(0)=[0.05,2.0]$. Explain the behavior of these solutions both biologically and mathematically.