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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).

    © BrainMass Inc. brainmass.com December 24, 2021, 11:34 pm ad1c9bdddf
    https://brainmass.com/math/algebra/non-dimensionalising-differential-equations-582385

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    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

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 11:34 pm ad1c9bdddf>
    https://brainmass.com/math/algebra/non-dimensionalising-differential-equations-582385

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