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# Non-dimensionalising Differential Equations

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Two species with populations N1(t) and N2(t) interact according to the model:

(dN_1/d_t)=r_1N_1(1-(N_1/K_1) - b_12(N_2/K_1),

(dN_2/d_t)=r_2N_2(1-b_21(N_1/K_2),

Non-dimensionalise the system (7) so that:

(d_u/d_t) = u (1- u - au), (d_u/d_t) = pv (1-Bu),

finding explicit formulae for the new constraints p, a, and B in terms of the constants of equation (7).

https://brainmass.com/math/algebra/non-dimensionalising-differential-equations-582385

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See attached

The equations are:
(1.1)
We set the non dimensional variables
(1.2)
Where are scaling constants with the appropriate dimensions.
Then, using the chain rule:
(1.3)
And by the same token:
(1.4)
Plugging all this into (1.1a) we get:

(1.5)

And the second equation becomes:

(1.6)
So now the two equations are:
(1.7)

If we define:
(1.8)

Then equation (1.7a) becomes:
(1.9)

Now from (1.8) we see that
(1.10)
Plugging this back into (1.7b) we get:

(1.11)
So if we define:
(1.12)
Then (1.11) becomes:
(1.13)

Summary
The equations

Can be non-dimensionalised by choosing

And the equations transform into

Where

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