A first tour through examples - Chapter 7

Chaotic attractor in an ‘ecosystem’ with two competing species eaten by a third one

Let us return to ecology to put forward an amazing phenomenon, namely deterministic chaos. We will see other examples later on. Our goal is not to fully analyse this phenomenon here, but rather to show how complex the behavior of an innocent-looking model can be.

We model the interaction between a predator and two competing prey populations in the simplest way. Consider first the two prey populations in the absence of the predator. If $x$ and $y$ denote their densities, then the rates of growth $\dot{x}/x$ and $\dot{y}/y$ have to be deacreasing functions of both $x$ and $y$. This leads to

$$ \begin{cases} \dot{x}=x(1- a_{11} x-a_{12} y)\\ \dot{y}=y(1-a_{21}x-a_{22}y) \end{cases} $$

where $a_{11},a_{12},a_{21}, a_{22}$ are positive parameters. We shall study this competition model later on in this ebook. We see that if one population of preys is absent, then the other one follows the logistic equation. We now add the equation for the predator and modify the previous equations accordingly:

$$ \begin{cases} \dot{x}=x(1- a_{11} x-a_{12} y-a_{13}z)\\ \dot{y}=y(1-a_{21}x-a_{22}y-a_{23}z)\\ \dot{z}=z(-1+a_{31}x+a_{32} y-a_{33}z) \end{cases} $$

where all parameters are positive. We get a system with a three-dimensional phase space. In the following digital experiment, we take

$$ a_{11}=a_{12}=1, a_{13}=10, a_{21}=1.5, a_{22}=a_{23}=1, a_{31}=5, a_{32}=0.5, a_{33}=0.01. $$

We can observe the following features. After an initial transient, the solution, which is the triplet $(x(t),y(t),z(t))$ of the three population densities, settles into an irregular oscillatory regime that seems to persist for all times. These oscillations seem to never repeat exactly: the motion is aperiodic. Geometrically, we see that, after a transient, the trajectory of the system fills a sort of ‘surface’. This surface is rather strange because it attracts all solutions starting from initial conditions in its neighborhood which then wander on it endlessly. As we shall see, the corresponding trajectories cannot cross — as a consequence of a general theorem on differential equations.

Another key phenomenon that we can observe is the ‘sensitive dependence on initial conditions’. This means that two trajectories starting very close together will rapidly diverge from each other, and thereafter have totally different futures. This phenomenon is combined with a confinement of the trajectories. This is the combination of these two ingredients that generates deterministic chaos.
Indeed, sensitive dependence on initial conditions alone is rather uninteresting. Think for instance as the one-dimensional system $\dot{x}=x$. Pick two distinct positive initial conditions $x_0,x_0’$. Then it is obvious that $|x(t)-x’(t)|=|x_0-x_0’| \, e^t$ for all $t\geq 0$. Thus, the solutions separate exponentially fast from each other, but each solution has a trivial behavior (they tend monotonically tend to $+\infty$). This example can easily be extended to any dimension.

This is our first example of an attractor that is neither a fixed point (constant solution) nor a closed trajectory (periodic solution). It is much more complicated, in fact it is a fractal set whose Hausdorff dimension seems to be between $2$ and $3$, but nobody proved it so far. It has zero volume but infinite
surface area!