Solutions of Bracken differential equations In the study of combat models, Bracken proposed a system of coupled differential equations:
$$\frac{dR(t)}{dt}=-gR(t)^qG(t)^p,\frac{dG(t)}{dt}=-rR(t)^pG(t)^q$$ where $p$, $q$, $r$ and $g$ are empirically determined constant. Despite its apparent simplicity, I'm not able to solve the system. Can anyone help me?
Thanks.
 A: Suppose $p,q,r,g\neq0$ :
$\dfrac{dG}{dR}=\dfrac{\dfrac{dG(t)}{dt}}{\dfrac{dR(t)}{dt}}=\dfrac{-rR^pG^q}{-gR^qG^p}=\dfrac{rR^{p-q}}{gG^{p-q}}$
$gG^{p-q}~dG=rR^{p-q}~dR$
$\int gG^{p-q}~dG=\int rR^{p-q}~dR$
$\begin{cases}\dfrac{gG^{p-q+1}}{p-q+1}=\dfrac{rR^{p-q+1}}{p-q+1}+c&\text{when}~p-q\neq-1\\g\ln G=r\ln R+c&\text{when}~p-q=-1\end{cases}$
$\begin{cases}gG^{p-q+1}=rR^{p-q+1}+C_1&\text{when}~p-q\neq-1\\\ln G^g=\ln R^r+c&\text{when}~p-q=-1\end{cases}$
$\begin{cases}G^{p-q+1}=\dfrac{rR^{p-q+1}+C_1}{g}~\text{or}~R^{p-q+1}=\dfrac{gG^{p-q+1}-C_1}{r}&\text{when}~p-q\neq-1\\G^g=C_1R^r~\text{or}~R^r=\dfrac{G^g}{C_1}&\text{when}~p-q=-1\end{cases}$
$\begin{cases}G=\biggl(\dfrac{rR^{p-q+1}+C_1}{g}\biggr)^{\frac{1}{p-q+1}}~\text{or}~R=\biggl(\dfrac{gG^{p-q+1}-C_1}{r}\biggr)^{\frac{1}{p-q+1}}&\text{when}~p-q\neq-1\\G=C_1^{\frac{1}{g}}R^{\frac{r}{g}}~\text{or}~R=\dfrac{G^{\frac{g}{r}}}{C_1^{\frac{1}{r}}}&\text{when}~p-q=-1\end{cases}$
$\therefore\begin{cases}\begin{cases}\dfrac{dG}{dt}=-rG^q\biggl(\dfrac{gG^{p-q+1}-C_1}{r}\biggr)^{\frac{p}{p-q+1}}\\\dfrac{dR}{dt}=-gR^q\biggl(\dfrac{rR^{p-q+1}+C_1}{g}\biggr)^{\frac{p}{p-q+1}}\end{cases}&\text{when}~p-q\neq-1\\\begin{cases}\dfrac{dG}{dt}=-rG^q\dfrac{G^{\frac{pg}{r}}}{C_1^{\frac{p}{r}}}\\\dfrac{dR}{dt}=-gR^qC_1^{\frac{p}{g}}R^{\frac{pr}{g}}\end{cases}&\text{when}~p-q=-1\end{cases}$
$\begin{cases}\begin{cases}\dfrac{dG}{dt}=-\dfrac{G^q(gG^{p-q+1}-C_1)^{\frac{p}{p-q+1}}}{r^{\frac{q-1}{p-q+1}}}\\\dfrac{dR}{dt}=-\dfrac{R^q(rR^{p-q+1}+C_1)^{\frac{p}{p-q+1}}}{g^{\frac{q-1}{p-q+1}}}\end{cases}&\text{when}~p-q\neq-1\\\begin{cases}\dfrac{dG}{dt}=-C_1^{-\frac{p}{r}}rG^{\frac{pg+qr}{r}}\\\dfrac{dR}{dt}=-C_1^{\frac{p}{g}}gR^{\frac{pr+qg}{g}}\end{cases}&\text{when}~p-q=-1\end{cases}$
$\begin{cases}\begin{cases}dt=-r^{\frac{q-1}{p-q+1}}G^{-q}(gG^{p-q+1}-C_1)^{-\frac{p}{p-q+1}}~dG\\dt=-g^{\frac{q-1}{p-q+1}}R^{-q}(rR^{p-q+1}+C_1)^{-\frac{p}{p-q+1}}~dR\end{cases}&\text{when}~p-q\neq-1\\\begin{cases}G^{-\frac{pg+qr}{r}}~dG=-C_1^{-\frac{p}{r}}r~dt\\R^{-\frac{pr+qg}{g}}~dR=-C_1^{\frac{p}{g}}g~dt\end{cases}&\text{when}~p-q=-1\end{cases}$
$\begin{cases}\begin{cases}t=-r^{\frac{q-1}{p-q+1}}\int^GG^{-q}(gG^{p-q+1}-C_1)^{-\frac{p}{p-q+1}}~dG+C_2\\t=-g^{\frac{q-1}{p-q+1}}\int^RR^{-q}(rR^{p-q+1}+C_1)^{-\frac{p}{p-q+1}}~dR+C_2\end{cases}&\text{when}~p-q\neq-1\\\begin{cases}\int G^{-\frac{pg+qr}{r}}~dG=-C_1^{-\frac{p}{r}}r\int dt\\\int R^{-\frac{pr+qg}{g}}~dR=-C_1^{\frac{p}{g}}g\int dt\end{cases}&\text{when}~p-q=-1\end{cases}$
$\begin{cases}\begin{cases}t=-r^{\frac{q-1}{p-q+1}}\int^GG^{-q}(gG^{p-q+1}-C_1)^{-\frac{p}{p-q+1}}~dG+C_2\\t=-g^{\frac{q-1}{p-q+1}}\int^RR^{-q}(rR^{p-q+1}+C_1)^{-\frac{p}{p-q+1}}~dR+C_2\end{cases}&\text{when}~p-q\neq-1\\\begin{cases}\begin{cases}\dfrac{G^{-\frac{pg+(q-1)r}{r}}}{-\frac{pg+(q-1)r}{r}}=-C_1^{-\frac{p}{r}}rt+c_2&\text{when}~pg+(q-1)r\neq0\\\ln G=-C_1^{-\frac{p}{r}}rt+c_2&\text{when}~pg+(q-1)r=0\end{cases}\\\begin{cases}\dfrac{R^{-\frac{pr+(q-1)g}{g}}}{-\frac{pr+(q-1)g}{g}}=-C_1^{\frac{p}{g}}gt+c_2&\text{when}~pr+(q-1)g\neq0\\\ln R=-C_1^{\frac{p}{g}}gt+c_2&\text{when}~pr+(q-1)g=0\end{cases}\end{cases}&\text{when}~p-q=-1\end{cases}$
$\begin{cases}\begin{cases}t=-r^{\frac{q-1}{p-q+1}}\int^GG^{-q}(gG^{p-q+1}-C_1)^{-\frac{p}{p-q+1}}~dG+C_2\\t=-g^{\frac{q-1}{p-q+1}}\int^RR^{-q}(rR^{p-q+1}+C_1)^{-\frac{p}{p-q+1}}~dR+C_2\end{cases}&\text{when}~p-q\neq-1\\\begin{cases}\begin{cases}G^{-\frac{pg+(q-1)r}{r}}=C_1^{-\frac{p}{r}}(pg+(q-1)r)t+C_2&\text{when}~pg+(q-1)r\neq0\\G=C_2e^{-C_1^{-\frac{p}{r}}rt}&\text{when}~pg+(q-1)r=0\end{cases}\\\begin{cases}R^{-\frac{pr+(q-1)g}{g}}=C_1^{\frac{p}{g}}(pr+(q-1)g)t+C_2&\text{when}~pr+(q-1)g\neq0\\R=C_2e^{-C_1^{\frac{p}{g}}gt}&\text{when}~pr+(q-1)g=0\end{cases}\end{cases}&\text{when}~p-q=-1\end{cases}$
$\begin{cases}\begin{cases}t=-r^{\frac{q-1}{p-q+1}}\int^GG^{-q}(gG^{p-q+1}-C_1)^{-\frac{p}{p-q+1}}~dG+C_2\\t=-g^{\frac{q-1}{p-q+1}}\int^RR^{-q}(rR^{p-q+1}+C_1)^{-\frac{p}{p-q+1}}~dR+C_2\end{cases}&\text{when}~p-q\neq-1\\\begin{cases}G=\begin{cases}\left(C_1^{-\frac{p}{r}}(pg+(q-1)r)t+C_2\right)^{-\frac{r}{pg+(q-1)r}}&\text{when}~pg+(q-1)r\neq0\\C_2e^{-C_1^{-\frac{p}{r}}rt}&\text{when}~pg+(q-1)r=0\end{cases}\\R=\begin{cases}\left(C_1^{\frac{p}{g}}(pr+(q-1)g)t+C_2\right)^{-\frac{g}{pr+(q-1)g}}&\text{when}~pr+(q-1)g\neq0\\C_2e^{-C_1^{\frac{p}{g}}gt}&\text{when}~pr+(q-1)g=0\end{cases}\end{cases}&\text{when}~p-q=-1\end{cases}$
