Section 2 - The Simplest Case: Inhibited Growth

What happens when b = d = 0?

In layman's terms, this question asks what happens when the two species have no efect on each other. In other words, this isn't mutualism. But it is the foundation of the model that we're using to examine mutualism in later sections.

Notice that there are no cross-terms (terms with both x and y) in the equations that represent this system:

x'=ax-ex^2
y'=cy-fy^2

The kind of growth that this system models is called inhibited growth. In the first system, the term ax causes the population to grow, whereas the second term, ex^2, causes it to decrease. In effect, the more creatures there are, the slower the population grows, until at some point it stabilizes and stops growing.

To find that point, called an equilibrium, we set the equation equal to zero and solve for x:

0 = ax - ex^2

The first solution that jumps to mind is x=0. To find the other solution, assume that x is not zero.

ex^2 = ax
ex = a
x = a/e

So, x=a/e is the second equilibrium for the x' equation. In the same way, the two equilibria for the y' equation are y=0 and y=c/f.

Though we know where the equilibria are and how many of them we have to worry about (four), we still don't know exactly how the system will behave. We can find out by approximating this system with a linear system that behaves similarly. When we do this , we get the following results.

The populations will move away from the equilibrium at (0,0).
They will be attracted to the equilibrium at (a/e, c/f).
They will sometimes be attracted and sometimes be repulsed by the equilibria at (a/e, 0) and (0, c/f), because both of these equilibria are saddles.

If you want to see the mathematics that give us this info, click here .

Throughout these documents, it is very helpful to use DSTool to examine the phase portraits of systems with particular parameters. If you are familiar with DSTool, you can simply launch it now (if it isn't already running) and, for each of the sets of sample parameters given in this system and the next, examine the respective phase portrait. If you are not entirely comfortable with DSTool, jump to DSTool Help for assistance.

To see an example of a system with two species growing independently, use these parameters:

[phase portrait: {a,c,e,f} = {1,2,1,1}]

Birfucations occur when a = c, and hence for the first two equilibria, (a - c)^2 = 0, and it is actually on the "pitchfork."

[phase portrait: {a,c,e,f} = {1,1,1,1}]

Note that it doesn't matter how much of x and y you start with: x and y will always end up at one of the three "attractive" equilibria.

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