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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.

Alex M.
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reaper
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  • Dedimensionalizing an equation can have several meanings, please explain yours. Related: http://math.stackexchange.com/questions/1497616 – Did Nov 22 '15 at 09:16
  • Since $\bar t$ must be the same in both equations (time is the same for both predator and prey), you cannot have it as $at$ in one equation and as $ct$ in the other (unless $a=c$, which is probably not what you have there). – Alex M. Nov 22 '15 at 09:20
  • But, don't we have to non-dimensionalize every differential equation according to the parameters contained in every differential equation, separately? – reaper Nov 22 '15 at 09:23
  • Aye, but these aren't separate differential equations, they're coupled, and have meaning together. A more accurate way to put it would be that one must 'non-dimensionalise' every system of differential equations according to the parameters contained therein. – stochasticboy321 Nov 22 '15 at 09:28
  • So, what should be the non-dimensionalized differential equations now? – reaper Nov 22 '15 at 09:40

1 Answers1

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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 $dN\tau=1$.

David
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