# Simplifying a system of differential equations

Given is the following system of differential equations which describes the behaviour of a biological system:

$\dfrac{dR_1}{dt} = \alpha_1 \dfrac{P_2^4}{a^4+P_2^4} - R_1$

$\dfrac{dP_1}{dt} = \alpha_2 \dfrac{1}{1+R_1^4} - P_1$

$\dfrac{dR_2}{dt} = \alpha_3 \dfrac{P_1^4}{a^4+P_1^4} - R_2$

$\dfrac{dP_2}{dt} = \alpha_4 \dfrac{1}{1+R_2^4} - P_2$

I've read that one can simplify this system preserving the crucial characteristics by reducing it to the following model (there' s not mentioned why):

$\dfrac{dR_1}{dt} = \beta_1 \dfrac{R_2^4}{b^4+R_2^4} - R_1$

$\dfrac{dR_2}{dt} = \beta_2 \dfrac{R_1^4}{b^4+R_1^4} - R_2$

I can't see why this works. Does anyone know?

• integrate the P's and substitute into the respective R's, solve for R
– JMP
Sep 27, 2016 at 19:48
• But how to integrade the $P's$? There are $R's$ in the respective equation, too. Sep 27, 2016 at 20:14
• $\frac{dR}{dt}+R=0$ gives $R=e^{-x}$
– JMP
Sep 27, 2016 at 20:45
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