Part I: One-dimensional systems: flows on the line - Chapter 1 s

Prelude: graphical study of one-dimensional differential equations

The graphical study of the logistic equation we did in the previous chapter paves the way for the general case. For if $\dot{x}=f(x)$, plot $\dot{x}$ versus $x$, that is, the graph of $f$. Intersection points with the $x$-axis are fixed points. Between two fixed points, the graph is either over or under the $x$-axis: in the former case $\dot{x}>0$, which means that $x(t)$ increases and can be represented by drawing arrows on the $x$-axis that point to the right; in the latter case $\dot{x}<0$, which means that $x(t)$ decreases and can be represented by drawing arrows on the $x$-axis that point to the left.

Consider for instance the equation

$$ \dot{x}=rx\left(\frac{x}{K_0}-1\right)\left(1-\frac{x}{K}\right) $$

where $K_0$ and $K$ are positive parameters such that $K_{0}\in (0,K)$. This is a modification of the logistic equation in which there is additionally a threshold ($K_0$) to growth. This is used to model the ‘Allee effect’ in population dynamics, that is, the difficulty of finding mates at low densities. Even if one can find explicit solutions for this equation, the graphical study provides effortlessly the main quantitative features of their behavior.

So far, we have left hanging the question of existence and uniqueness of solutions to the system $\dot{x}=f(x)$. We settle this question in the next section and we shall see that this allows to answer questions like: how much time does one need to reach a fixed point?