Non-dimensionalising Differential Equations
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).
© BrainMass Inc. brainmass.com December 24, 2021, 11:34 pm ad1c9bdddfhttps://brainmass.com/math/algebra/non-dimensionalising-differential-equations-582385
SOLUTION This solution is FREE courtesy of BrainMass!
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
© BrainMass Inc. brainmass.com December 24, 2021, 11:34 pm ad1c9bdddf>https://brainmass.com/math/algebra/non-dimensionalising-differential-equations-582385