Solve Linear ODE: Find A(B) Equation

[Zeth]

Homework Statement

$$\frac {dA} {dx} = \alpha A - \beta AB$$

$$\frac {dB} {dx} = \gamma B + \delta AB$$

Let A(x) and B(x) be two populations that influence each other and treat A(x) as a function of B, find an ODE for A(B).

This is an extention revision problem so its not in my notes. What is this kind of problem called and a link to an explanation of how this type of equation should be solved would be very appriciated.

 

[siddharth]

These look like nonlinear coupled differential equations. If $$\gamma$$ is -ve, I think this would be the Lotka-Volterra model.

The way I'd do it, is to "linearize" the system of eqns around the critical point by ignoring the non-linear terms.

 

[Zeth]

Yes I noticed that gamma should be negative as well since it doesn't make sense for predators to grow exponentially even if they catch no prey. I'll ask the lecturer about it. And thanks I'm having a look at the wikipage for it now.

"The way I'd do it, is to "linearize" the system of eqns around the critical point by ignoring the non-linear terms."

Say what? I've never heard about that before.

 

[siddharth]

Yes I noticed that gamma should be negative as well since it doesn't make sense for predators to grow exponentially even if they catch no prey. I'll ask the lecturer about it. And thanks I'm having a look at the wikipage for it now.

"The way I'd do it, is to "linearize" the system of eqns around the critical point by ignoring the non-linear terms."

Say what? I've never heard about that before.

For the sake of simplicity, I'm setting $$\alpha=\beta=\gamma=\delta=1$$, and so

$$\frac{dA}{dx} = A - AB = A(1-B)$$

$$\frac{dB}{dx} = -B + AB = B(A-1)$$

The critical points for this set of eqns are (A,B) being (0,0) and (1,1).
In the region very near (0,0), if you drop non-linear terms (as it's negligible), you'll have

$$\frac{dA}{dx} = A$$

$$\frac{dB}{dx} = - B$$

which you can solve for. However, this would obviously hold only very near the critical point.

To solve near (1,1), set u=A-1, v=B-1 and again drop the non-linear terms. The thing is, this would give you an idea of what the trajectories of the solution look like in the phase space of A & B.