The bistable equation is often used to model the front of a wave that is moving through space (such as, for example, a wave front of electrical activity moving across the surface of the heart, a moving front of rabies infection in a population of foxes, or the front of a forest fire).

Systems like this are called excitable systems; it can take only a small perturbation to give a huge response.

For example, under the right conditions a tiny match can start an enormous forest fire, or a single infected fox can start a population-wide epidemic of rabies.

The equation is dv dt

= v(v − α)(1 − v), for some constant α with 0 < α < 0.5.

a. Set α = 0.2 and solve the bistable equation for v(0) =

0.1, 0.15, 0.25 and 0.3. What do you notice about the longtime behaviour of the solutions? What role is α playing?

Check your answer by trying some more initial conditions above and below α.

b. Set α = 0.01. Before you solve the equations can you predict how the long-time behaviour will depend on the initial condition? Check your prediction by solving the equation numerically.

In ecology, the bistable equation models a phenomenon called the Allee effect after the ecologist Warder Clyde Allee, who noticed that goldfish had a higher death rate when the population was too small.

He concluded that aggregation may improve the survival rate of individuals, and thus that social cooperation could be important for evolution.

c. Do a qualitative analysis (see Section 25.5) of the bistable equation to show the same phenomenon of excitability.

i. What are the steady states? Are they sinks or sources?

ii. Show that a very small change in initial condition can lead to a large change in the solution.

iii. How could you define the "excitability" of the system, and how would this be related to the size of α?