A first tour through examples - Chapter 2

The logistic equation

In the short run, the equation $\dot{x}=ax$ with $a>0$ makes sense to describe our population of bacteria. Indeed, we can imagine that, during a short interval of time, the bacteria do not crowd each other and the population size is $e^{at}x_0$ if there are $x_0$ bacteria at time $0$. But, as time goes on, such an exponential growth would imply that the number of bacteria is exceedingly large (larger than the number of atoms in the universe if we take it literally!).

To eliminate this major drawback, let us assume that the death rate is no more constant but equal to $dx$, so if there are $x$ bacteria, their number increase at a rate $dx^2$. For births, let us assume a constant rate of reproduction $b$, so that if there are $x$ bacteria, their number increase at a rate $bx$. We thus obtain the so-called logistic equation

$$ \dot{x}=bx-dx^2.$$

This is a differential equation for which an explicit solution is known.

By the method of separation of variables, it is not difficult to prove that

$$ x(t)=\frac{x_0 b e^{bt}}{(b-dx_0)+dx_0 e^{bt}}$$

is a solution of the equation $\dot{x}=bx-dx^2$ such that $x(0)=x_0$ is given. It is in fact the only such solution but, at this stage, we lack a fundamental theorem to justify this claim.

But instead of solving the equation, let us see what we can learn directly from it by a graphical approach.
The idea is to plot $\dot{x}$ versus $x$; this is a parabolic curve. We plot only $x\geq 0$ since it makes no sense to think about a negative population (though we can in principle study the logistic equation on the whole real line). Moreover, we draw arrows on the $x$-axis that point to the right when $\dot{x}>0$ and to the left when

A physical way of thinking about the logistic equation is to imagine a particle moving steadily along the $x$-axis with a velocity that varies from place to place, according to the rule $\dot{x}=bx-dx^2$. A particle starting at a point $x$ where $\dot{x}=0$ will stay there for ever. Such points, called fixed points, play a crucial role.

We have two fixed points for the logistic equation: $\bar{x}=0$ and $\bar{x}=b/d$. These are the points at which the parabola intersects the $x$-axis. For these size values, the population size is neither increasing nor decreasing. Since $\dot{x}=bx-dx^2$, the condition that $\dot{x}=0$ is equivalent to the fact that for these two special values the net reproduction/death rates are exactly in balance. We thus have two self-sustaining population sizes: $\bar{x}=0$ and $\bar{x}=b/d$.

We see from the graph that if we start from any population size $x_0>0$, it will tend to $\bar{x}=b/d$. The quantity $b/d$ is called the carrying capacity of the population. At this stage, we cannot say if the carrying capacity is reached in a finite time. We shall see that this is not the case.

We can push further the (...)

We can push further the analysis of the graph to deduce the qualitative shape of the solutions without actually solving the equation! For example, if $x_0<\frac{b}{2d}$, the particle moves faster and faster until it crosses $x=\frac{b}{2d}$, where the parabola reaches its maximum. The the particle slows down and eventually creeps toward $x=\frac{b}{d}$. In biological terms, this means that the population initially grows in an accelerating fashion, and the graph of $x(t)$ is concave up. But, after $x=\frac{b}{2d}$, the derivative $\dot{x}$ begins to decrease, and so $x(t)$ is concave down as it asymptotes to the horizontal line $x=\frac{b}{d}$. Thus the graph of $x(t)$ is S-shaped or sigmoid for $x_0<\frac{b}{2d}$.
Something qualitatively different occurs if the initial condition $x_0$ lies between $\frac{b}{2d}$ and $\frac{b}{d}$; now the solutions are decreasing from the start. Hence these solutions are concave down for all $t$. If the population size initially exceeds carrying capacity $\frac{b}{d}$ ($x_0>\frac{b}{d}$), then $x(t)$ decreases toward $x=\frac{b}{d}$ and is concave up.
Finally, if $x_0=0$ or $x_0=\frac{b}{d}$, then the population stays constant.