I've been trying to non-dimensionalize the differential equations for the predator and prey model. I've written down the procedure below which I was using for non-dimensionalizing the differential equations but I am not sure that whether I'm proceeding in the right direction or not.
We have: $\dfrac {\Bbb dx} {\Bbb dt} = ax - bxy \\ \dfrac {\Bbb dy} {\Bbb dt} = -cy + dxy$,
with the initial conditions $x(0) = x_0, y(0) = y_0$.
$a$ and $c$ have the dimensions of $\dfrac 1 {time}$ whereas $b$ and $d$ have the dimensions of $\dfrac 1 {number \cdot time}$.
Let's call our non-dimensional variables to be $\bar x$ and $\bar y$. Hence, $\bar x = \dfrac x {x_0}$ and $\bar y = \dfrac y {y_0}$. For time, let's call the non-dimensional variable $\bar t = ta$ for the first differential equation and $\bar t = tc$ for the second differential equation. Putting these non-dimensional variables in the original equations, we get:
$\dfrac {\Bbb d \bar x} {\Bbb d \bar t} = \bar x - e \bar x \bar y$, where $e = \dfrac b a x_0$, and
$\dfrac {\Bbb d \bar y} {\Bbb d \bar t} = - \bar y - f \bar x \bar y$, where $f = \dfrac d c x_0$.
So, is this procedure correct or not? Please help me out.
You have two physical parameters, number and time. Let a characteristic number of things in the population be $N$, and a characteristic time scale be $\tau$. Then $x=N\hat x$, $y=N\hat y$ and $t=\tau\hat t$, where hats denote dimensionless variables. Substituting into your system of equations gives $$ \frac{N}{\tau}\frac{\mathrm d \hat x}{\mathrm dt}=aN\hat x-bN^2\hat x\hat y, $$ and $$ \frac{N}{\tau}\frac{\mathrm d \hat y}{\mathrm dt}=-cN\hat y+dN^2\hat x\hat y. $$ Rearranging, $$ \frac{\mathrm d \hat x}{\mathrm dt}=\frac{\tau a}{N}\hat x-bN\tau \hat x\hat y, $$ and $$ \frac{\mathrm d \hat y}{\mathrm dt}=-\frac{\tau c}{N}\hat y+dN\tau \hat x\hat y. $$ You have some options now depending on what you ant your equations to look like. You could make $a\tau=N$ or $c\tau=N$, and $bN\tau=1$ or $dN\tau=1$.