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Figure 2.13. (a) The geometric solution of the equation x3 no longera steady state, but both xand p change in time. !Slow ax = p. (b) When we append dynamics for p, there is

!Slow

Figure 2.13. (a) The geometric solution of the equation x3 no longera steady state, but both xand p change in time.

ax = p. (b) When we append dynamics for p, there is

Now I want to discuss the solution in a more geometric manner, because learning to think geometrically about these matters is absolutely essential for your understanding of the material. The steady states of the differential equation (2.32) satisfy x3 — ax = 3. In Figure 2.13a, I plotted the curve y = x3 — ax and the line (actually a number of lines) y = 3. Since the steady states correspond to values of x where these are equal, we conclude that the steady states are values of x for which the line and the curve intersect. We also see that there may be just one intersection point (on the left hand branch of the curve or on the right a hand branch), there may be two intersection points (if the line is tangent to the curve) or three intersection points (if the value of p falls between the local maximum and local minimum of the curve). We thus have a geometric interpretation of the cusp. When the horizontal line is tangent to the curve, the system is once again structurally unstable: at the point of tangency there are two steady states, one of which is marginally stable. However, a small change in either of the parameters leads to a situation in which there are either three or one steady states.

But this really is not the situation that I wanted to consider. Rather, I want to consider the situation in which p varies as well. In particular, let us append the equation dp/dt =—ex, in which e is a new parameter, to Eq. (2.32). We will assume that e is small (that is much less than 1), and we know that when e is set equal to 0 we obtain the cusp bifurcation.

The steady state is now x = 0, p = 0, but the dynamics are very interesting. To be explicit, suppose we start on the right hand branch of the cubic, where the line is above the local maximum, as shown in Figure 2.13b. If e were 0, the system would stay there. But since e is not 0, things change. In light of x > 0, p will decline (since its derivative is negative). Thus in the next bit of time, the line will lower a little. Furthermore, now the line is slightly below the cubic and since dx/dt = p — (x3 — ax), x declines slightly too. At this new value of x, dp/dt is still negative, so that both p and x will continue to decline. We will thus slowly move down along the right hand branch of the cubic (Figure 2.13b). For how long will this go on? Until we reach the local minimum of the cubic at x = ^fa. At this point, p is still declining, but once it does so there is no intersection between the line and the curve for positive values of x. We thus predict a rapid transition from the right hand branch of the cubic to the left hand branch. When we get near the left hand branch, x is negative so that dp/dt is positive and p begins to rise. Once again, this happens slowly, along the left hand branch, until the local maximum is crossed, at which point there will be a rapid transition back to the right hand branch of the cubic. In other words, we predict oscillations, and that the oscillations will have a shape that involves a slowly changing component and a rapidly changing component.

In Figure 2.14, I show the numerical solution of the differential equations for the case in which a = 1, e = 0.005 with initial values x(0) = 2 and P(0) = x(0)3 — ax(0). Starting at x(0) = 2, we see a slow decline along the right hand branch of the cubic, until there is a rapid drop, then a slow rise, and oscillations set in. To help make this point clearer, Figures 2.14b and c show just parts of the trajectory; in Figure 2.14c, we most clearly see the slow and fast parts of the oscillation.

Oscillations such as the ones described here are called ''relaxation oscillations" and they arise in many different ecological contexts,

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