I was struggling with a modeling problem in a game.Many soldiers fight with each other, the number of casualties to the enemy is related to the number of quadratic, then we can make the following differential equation: $$ x'(t)=-ay(t)^2 $$ $$ y'(t)=-bx(t)^2 $$ a and b are constants
Because of the actual business requirements, I wanted to find an asymptotic analytic solution,please help
The function $V(x,y)=bx^3-ay^3$ is a conserved quantity/first integral of the system: $$ \dot V= 3bx^2\dot x-3ay^2\dot y=3bx^2(-ay^2)-3ay^2(-bx^2)=0. $$ It means that the point $(x,y)$ on a plane will stay on the level curve $bx^3-ay^3=C$ during the battle:
Let the initial values of $x$ and $y$ be $x_0$ and $y_0$. At any moment of the battle $$ bx^3-ay^3=bx_0^3-ay_0^3, $$ so if the side $y$ wins, the final values are $x_{final}=0$ and $$y_{final}=\sqrt[3]{\frac{ay_0^3-bx_0^3}{a}},$$ if the side $x$ wins, then $y_{final}=0$ and $$x_{final}=\sqrt[3]{\frac{bx_0^3-ay_0^3}{b}}.$$