Chaos!

The Verhulst equation is the name used for the equation we studied in the last section when it is applied to modeling ecological populations. We can study this dynamic system more generally, and then we refer to it as the Logistic Function. We will explore the sensitivity of this equation to different starting values of the population P, and of the fertility factor C.

The Logistic Function

P_n+1 = P_n + CP_n(1-P_n)

The applet below lets us explore the behavior of the Logistic Function. You can click and drag the parabola in the left panel up and down which serves to change the value of C, the fertility. You can also <Shift>-click and drag left and right to adjust the starting value, P₀.

The left panel shows a process called graphical iteration, where the calculation starts with the value of P₀. It then shows a sequence of lines wrapping around and around, connecting the parabola with the diagonal line y=x. This progression represents the iterated behavior of the function.

The right panel shows the value of the equation (say, population) on the Y-axis and the X-axis represents time.

Drag the blue parabola in the left panel up and down and see what happens. By doing this, you can change the value of C from 0 to 4. Make sure to keep P₀ = 0.5 for now. Watch the behavior of the population in the right panel as a function of time. At first, say at C=2.6 the population oscillates briefly but soon converges on a single value. This is seen in the graphical iteration on the left as the red line spiraling in to the single solution where the parabola crosses the diagonal line.

By the time you raise the value of C to 3.2, the population in the right panel is stably oscillating between two values. This is simple periodic behavior, with a period of 2. In the left panel this corresponds to the graphical iteration in the left panel following a simple path between two solutions, the points where the red line touches the parabola.

Drag the parabola further up still, so that C=3.5. Note that the behavior in the right panel is now oscillating between 4 values. This is still simply periodic, but now it has a period of 4. You can see at this point that the graphical iteration in the left panel follows a pathway around two squares, and the path touches the parabola in four points, which correspond to the four different values the system oscillates between in the right panel. Periodicity = 4.

Keeping dragging up the parabola. Between 3.5 and 4.0 the behavior of the Logistic Function is extremely complex. This shows up in the left panel as lines intersecting the parabola in a large number of points, and in the right panel as generally aperiodic (i.e. not periodic) behavior. This means that the population fluctuates erratically, rather than in a simple periodic manner. Although if you look carefully, you can find a number of places where the equation does behave periodically, with various higher-order periods. For instance at C = 3.8375, the equation briefly returns to a stable period 2 behavior.

Question:
Set the values in the Logistic Function to C = 3.6625 and x₀ = 0.5.
What is the periodicity at this point? [ ]

At some ranges, the Logistic Function is exquisitely sensitive to small changes in the value of C. It can also be very sensitive to changes in the starting value of the population. Set the value of C to 3.6625. Hold down <Shift> and drag sideways in the left panel to change the value of P₀. Observe that the behavior of the equation is very sensitive to changes in the starting value of the population. Change the number of iterations in the input field at the top right to 100 to see a longer data set.

Be careful drawing conclusions about the behavior of the equation! Something that might look periodic, may eventually break down into disorderly, irregular behavior. Or, something may start off erratically, but after many iterations it may converge to a stable periodic orbit.

To see this, set the values of the function above to C = 3.775 and P₀ = 0.465625, and the number of iterations to 50. Does the equation look periodic or aperiodic (i.e. not periodic) at these values? [ ____ ]

Now turn up the number of iterations to 140.
Does the equation look it has become periodic or aperiodic? [ _____ ] Now turn up the number of iterations again to 300.
Now does the equation look periodic or aperiodic? [ _____ ]

What lessons can we draw from this behavior about population ecology in the real world? First of all, we must be careful in drawing conclusions, because this equation is a gross over-simplification, dealing with a single species and only two variables. But one lesson is that a system can abruptly change from alternating periodically to behaving unpredictably and erratically. And then switch back abruptly to regular behavior. Chaos theory teaches us to approach dynamic systems cautiously. For while there is often great stability built into networks of interacting systems, it is also true that small changes to a system can spiral out of control, and cause all sorts of unpredictable outcomes.

There are many times when we inadvertently manipulate these parameters in the real world. For instance, when a road is built through a wilderness area, it may only reduce the total area of the wilderness by a miniscule amount. However, roads often functionally subdivide an ecosystem into two disconnected populations of animals. This may have drastic impacts on the population dynamics, because the starting value of each of the two sub-populations is much lower than that of the intact population. Ecologically, two adjacent ecosystems with a barrier between them may behave completely differently from an intact ecosystem.