# Biological Differential Equation

1. Jan 25, 2010

### Kreizhn

1. The problem statement, all variables and given/known data
I have a system of coupled differential equations of the form
$$\frac{dR_1}{dt} = R_1^0 \cdot g\left( \frac{R_2}{K_R} \right) - R_1$$
$$\frac{dR_2}{dt} = R_2^0 \cdot g\left( \frac{R_1}{K_R} \right) - R_1$$
where
$$g\left( \frac{ R_i}{K_r} \right) = \frac{ 1 + f\cdot \left[ \frac{ R_i}{K_R} \right]^2 }{1 + \left[ \frac{R_i}{K_r} \right]^2 }$$
where f << 1 is a constant, $R_1^0$ is the steady state level of $R_1$ in the absence of $R_2$ and vice versa.

I need to show (graphically) that if we are free to manipulate $R_1^0, R_2^0$ then this can lead to one or three solutions that simultaneously satisfy both equations.

3. The attempt at a solution
It seems to me that an obvious choice for a single solution would be to set $R_i^0 =0$ which will decouple the systems and make them decreasing exponentials. However, other than this I am unsure how to determine that there are three solutions, let alone what it means to do this graphically.

2. Jan 25, 2010

### Kreizhn

Nevermind, solved it.

The equilibrium points are the intersections of the steady state solutions.