Biological Differential Equation

[Kreizhn]

Homework Statement

I have a system of coupled differential equations of the form
[tex] \frac{dR_1}{dt} = R_1^0 \cdot g\left( \frac{R_2}{K_R} \right) - R_1 [/tex]
[tex] \frac{dR_2}{dt} = R_2^0 \cdot g\left( \frac{R_1}{K_R} \right) - R_1 [/tex]
where
[tex] 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 } [/tex]
where f << 1 is a constant, [itex] R_1^0 [/itex] is the steady state level of [itex] R_1 [/itex] in the absence of [itex] R_2 [/itex] and vice versa.

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

The Attempt at a Solution

It seems to me that an obvious choice for a single solution would be to set [itex] R_i^0 =0 [/itex] 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.

 

[Kreizhn]

Nevermind, solved it.

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