I'm doing some numerical simulations of an exponential growth like system which, for simplicity, has the form:
$$ \frac{dx}{dt}= ax + bxy \quad\quad \frac{dy}{dt}= cy + dxy $$
For some parameter values i get instability in the simulation though I remember reading a paper which used log transformations to prevent this. Any ideas on how I could do this or how to rewrite the equations as:
$$ \frac{d log(x)}{dt}= \ldots \quad\quad \frac{d log(y)}{dt}= \ldots $$
(If you need more information, for example Lyapunov functions, this equation is similar to Lotka-Volterra equation.)
Dividing the first equality by $x$: $$\frac{1}{x} \frac{dx}{dt}=a+by$$ i.e: $$\frac{d \log(x)}{dt}=a+by$$ and similarly: $$\frac{d \log(y)}{dt}=c+bx$$