Differential equation model and graph question

Question

Consider the model

$dS/dt=f(S)= kS(1-S/N)((S/M)-1)$

Where S is the population of the squirrel, k and M are constant parameter, N is the carrying capacity. Know that the more people move in, the more N decrease.

Suppose M is less than or equal to N, and k,M are fixed value. Sketch the graph for f(S). What value of N does a bifurcation occur?

How can I do that? there are too many unknown value.

Answer 1

You are correct that there are a lot of parameters, but here are some hints.

You are trying to see the general shape of the solution sets for various parameters.

Approach:

• 1) Let $M = N$, solve the model. Fix some values for k, M, N (with M = N) and plot the resulting f(S)
• 2) Let $M < N$, solve the model. Fix some values for k, M, N (with M = N) and plot the resulting f(S).
• 3) Play around with such things as $k = 0$, $M$ very small and very large, $N$ very small and very large. This will let you do a qualitative analysis.

From all of this, see if you can figure out the general shape to the system. You might even find such things as instabilities and you want to see if you can characterize those. It also helps to know general values for the fixed parameters in the model and use those if you know them. This is a qualitative analysis.

Example 1:

Let $M = N = k = 1$, we have:

$$c_1-t = -1/(s(t)-1)-\ln(s(t)-1)+\ln(s(t))$$

If we plot solution curves for various $c_1$, we have:

Notice the general shape of the solution over $t$?

Here is the same thing, except I plotted the direction field (green) alongside many solution curves (blue) for the model.

Example 2: $k = 0$, we get $s(t) = c_1$, a constant function over t.

Example 3: $k = -1$ (does that make sense?), $N = 1$, $M = 1/5$. Here is the phase portrait. Notice that for all those changes, the general behavior did not change that much (direction field changed, but the rest is very similar. What do these curves tell you about the model would be the next question.

You are basically doing a qualitative analysis of the model and seeing if the behaviors can be generalized for various sets of parameter selections (there may be one behavior or several (stable, unstable, indeterminate, for example).