# Is following system of nonlinear ODEs recognized?

The following system of ODEs – is it recognized as distinct system, with meaningful background and uses?

$$\frac{dx}{dt} = – [x(t)]^2 – x(t)y(t)$$
$$\frac{dy}{dt} = – [y(t)]^2 – x(t)y(t)$$

This is probably not needed, but initial conditions: $x(t=0) = x_0, \space y(t=0) = y_0$

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It is a two-dimensional Lotka-Volterra equation. The most general LV-equation has applications in population dynamics, networks and chemical reactions.

I don’t recognise the ODE system, but you can obtain a solution analytically.

Hint: Let $u=1/x$ and $v=1/y$, as though each is a Bernoulli differential equation. You can then show that, if $x_0\neq0$ and $y_0\neq0$, a solution is given by
$$x(t)=\frac{x_0}{1+\left(x_0+y_0\right)t},\\ y(t)=\frac{y_0}{1+\left(x_0+y_0\right)t}.$$

I doubt that it has a name, but you might notice that trajectories are rays from the origin (apart from the fixed points along $x+y=0$).