Verhulst Logistic Model in Modern Biology

Let $y=\phi(t)$ be the the population of a given species at time $t$ . The variation of population is the rate of change of $y$ is proportional to the current value of $y$ that is

$\begin{align*} \frac{dy}{dt} \propto y\\ \Rightarrow \frac{dy}{dt} = ry\\ \end{align}$

Where $r$ is the proportionality constant, called the rate of growth and decline depending on whether it is positive or negative. Let $r>0$ , so the population is growing. This idea was published by the British economist Thomas Malthus (1766-1834) in 1798. From the above equation we get,

$\begin{align*} \frac{dy}{y}=rdt\\ \Rightarrow \int \frac{dy}{y}=r \int dt\\ \Rightarrow \ln y=rt+c \\ \end{align}$

where $c$ is an integrating constant.

$\begin{align*} y=e^{rt+c}\\ \Rightarrow y=e^c e^{rt}\\ \Rightarrow y= Ce^{rt}\\ \end{align}$

If $y(0)=y_0$ then $y_0=Ce^0=C$ which implies

$\begin{align*} y=y(t)=y_0 e^{rt} \end{align}$

Malthus model with $r>0$ predicts that the population will grow exponentially for all time. Equation $y=y(t)=y_0e^{rt}$ is accurate for many populations at least for limited periods of time. However such ideal conditions can not continue indefinitely. Eventually limitations on space, food supply or other resources will reduce the growth rate and bring an end to uninhibited exponential growth.

Considering the fact that the growth rate actually depends on the population. We replace the constant $r$ by a function $h(y)$ and thus we obtain the modified equation

(L) $\begin{equation*} \frac{dy}{dt}=h(y)y \end{equation}$

We now choose $h(y)$ so that $h(y) \cong r>0$ when $y$ is small, $h(y)$ decreases as $y$ grows larger and $h(y)<0$ when $y$ is sufficiently large. Having this properties the simplest function is $h(y)=r-ay$ where $a$ is a positive constant. Thus equation (L) becomes

(M) $\begin{equation*} \frac{dy}{dt}=(r-ay)y \end{equation}$

which is known as Verhulst equation or the logistic equation. P.F.Verhulst (1804-1849) was a Belgian mathematician who introduced equation (M) as a model for human population growth in 1838. He referred to it as a logistic growth hence equation (M) is often called the logistic equation. Due to inadequate census data he was unable to test the accuracy of his model. The equivalent form of the logistic equation is

(N) $\begin{equation*} \frac{dy}{dt}=r(1-\frac{y}{k})y \end{equation}$

where $k=\frac{r}{a}$ . The constant $r$ is called the intrinsic growth rate in the absence of any limiting factors and $k$ is called the saturation level or the environmental carrying capacity for the given species. Now

$\begin{align*} \frac{dy}{dt}=r(1-\frac{y}{k})y \\ \Rightarrow \frac{dy}{(1-\frac{y}{k})y}=rdt \\ \Rightarrow \int \frac{dy}{y} - \int \frac{-\frac{1}{k}}{1-\frac{y}{k}}dy=r \int dt\\ \Rightarrow \ln|y|- \ln|1-\frac{y}{k}|=rt+c\\ \end{align}$