Given $$\frac{dU}{dT}=r_{1}U\left(1-\frac{U}{K_{1}}\right)+\frac{\alpha_{1}UV}{1+\beta_{1}V}$$ $$\frac{dV}{dT}=r_{2}V\left(1-\frac{V}{K_{2}}\right)+\frac{\alpha_{2}UV}{1+\beta_{2}U}$$
Apply the change of variables $u=\frac{U}{K_{1}}$ , $v=\frac{V}{K_{2}}$ , $t=r_{1}T$ to obtain the re-normalized equations
$$\frac{du}{dt}=u\left(1-u+\frac{a_{1}v}{1+b_{1}}\right)$$
$$\frac{dv}{dt}=cv\left(1-v+\frac{a_{2}u}{1+b_{2}u}\right)$$
So far I have gotten to $\frac{du}{dt}=\frac{1}{r_{1}K_{1}}\frac{dU}{dT}$ but I cannot seem to simplify the expression after multiplying this out I get that $\frac{du}{dt}=u(1-u)+\frac{\alpha_{1}UV}{r_{1}k_{1}+r_{1}k_{1}\beta_{1}V}$ if anyone could show me how to do this I would be grateful
Thanks
$$\begin{cases} \frac{dU}{dT}=r_{1}U(1-\frac{U}{K_{1}})+\frac{\alpha_{1}UV}{1+\beta_{1}V} \\ \frac{dV}{dT}=r_{2}V(1-\frac{V}{K_{2}})+\frac{\alpha_{2}UV}{1+\beta_{2}U} \end{cases}$$
$U=K_1u\quad;\quad V=K_2v\quad;\quad T=\frac{t}{r_1}$
$$\begin{cases} r_1K_1\frac{du}{dt}=r_{1}K_1u(1-\frac{K_1u}{K_{1}})+\frac{\alpha_{1}K_1uK_2v}{1+\beta_{1}K_2v} \\ r_1K_2\frac{dv}{dy}=r_{2}K_2v(1-\frac{K_2v}{K_{2}})+\frac{\alpha_{2}K_1uK_2v}{1+\beta_{2}K_1u} \end{cases}$$
$$\begin{cases} \frac{du}{dt}=u(1-u)+\frac{\alpha_{1}uK_2v}{r_1(1+\beta_{1}K_2v)} \\ \frac{dv}{dy}=\frac{r_2}{r_1}v(1-v)+\frac{\alpha_{2}K_1uv}{r_1(1+\beta_{2}K_1u)} \end{cases}$$ Let : $\quad a_1=\frac{\alpha_1K_2}{r_1} \quad\;\quad a_2=\frac{\alpha_2K_1}{r_1}\quad\;\quad c=\frac{r_2}{r_1} \quad\;\quad b_1=\beta_1K_2 \quad\;\quad b_2=\beta_2K_1$
Don't confuse $a$ and $\alpha$.
$$\begin{cases} \frac{du}{dt}=u(1-u)+a_1\frac{uv}{(1+b_1v)} \\ r_1\frac{dv}{dy}=cv(1-v)+a_2\frac{uv}{(1+b_2u)} \end{cases}$$