Jsmithx
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https://isaacphysics.org/questions/m...1-73c31f67f288
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I’m so lost, I haven’t a clue of where to start. There are so many variables and I haven’t a clue what each means.
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davros
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(Original post by Jsmithx)
https://isaacphysics.org/questions/m...1-73c31f67f288
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I’m so lost, I haven’t a clue of where to start. There are so many variables and I haven’t a clue what each means.
What are the original equations you're starting with? This looks just like a change of variable and (possible) application of the chain rule.
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RDKGames
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(Original post by Jsmithx)
https://isaacphysics.org/questions/maths_ch7_5_q5?board=d5187ecf-0f18-4b25-82d1-73c31f67f288

I’m so lost, I haven’t a clue of where to start. There are so many variables and I haven’t a clue what each means.
For the number of weed f, the rate at which it changes over time t is given by

\dfrac{df}{dt} = af - a_1 cf

where c is the number of grass carp which predate upon this grass. The constant a>0 denotes the rate at which the weed grows. More weed means more growth. The second term -a_1 cf describes the rate of 'death' for this weed. It depends on the constant a_1, and effectively the number of predators c, as well as the number of weed f.

Note that if c increases, it means more predators about hence a decrease in weed growth. Also, if f increases, weeds compete against each other for nutrients. So the entire negative effect cf is bunched up into one term.

For the carps, the ODE is

\dfrac{dc}{dt} =  b cf - b_1 c

where b,b_1 are constants of growth and death respectively.

Currently, these models are dimensional. Is the number of weed / carps in hundreds? Thousands? Millions??? Are they densities per some volume of space?

We can bring these models down via a process called non-dimensionalisation whereby we simply scale the variables (in this case f,c,t) in such a way as to eliminate as many constants as possible.

In your link, time is brought down via u=at. This means that we can simply divide our equations through by the constant a and obtain

\dfrac{df}{du} = f - \dfrac{a_1}{a} cf

\dfrac{dc}{du} =  \dfrac{b}{a} cf - \dfrac{b_1}{a} c


We can also scale f = A z and c = B y where A,B are dimensional constants (for us to rewrite in terms of the known constants), and z,y are dimensionless variables.

Substituting these in yields

\dfrac{dz}{du} = z - \dfrac{a_1 B}{a} yz

\dfrac{dy}{du} =  \dfrac{bA}{a} yz - \dfrac{b_1}{a} y



We can bring the coefficient of yz in the first eqn down to -1 if we impose that B = \dfrac{a}{a_1}.

We can also choose A = \dfrac{a}{b} in order to bring the coefficient of yz in the second eqn down to 1.

This leaves us with

\dfrac{dz}{du} = z - yz

\dfrac{dy}{du} =  yz - \dfrac{b_1}{a} y

whereby we can combine the ratio \dfrac{b_1}{a} into a single constant \beta.


Thus, via nondimensionalisation, we have reduced the number of constants in our model down from 4 to only 1. The leftover variables z,y,u do not have any dimensions and can therefore be treated on level ground without having to worry about dimensions and their scaling factors.
Last edited by RDKGames; 8 months ago
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Jsmithx
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(Original post by RDKGames)
For the number of weed f, the rate at which it changes over time t is given by

\dfrac{df}{dt} = af - a_1 cf

where c is the number of grass carp which predate upon this grass. The constant a>0 denotes the rate at which the weed grows. More weed means more growth. The second term -a_1 cf describes the rate of 'death' for this weed. It depends on the constant a_1, and effectively the number of predators c, as well as the number of weed f.

Note that if c increases, it means more predators about hence a decrease in weed growth. Also, if f increases, weeds compete against each other for nutrients. So the entire negative effect cf is bunched up into one term.

For the carps, the ODE is

\dfrac{dc}{dt} =  b cf - b_1 c

where b,b_1 are constants of growth and death respectively.

Currently, these models are dimensional. Is the number of weed / carps in hundreds? Thousands? Millions??? Are they densities per some volume of space?

We can bring these models down via a process called non-dimensionalisation whereby we simply scale the variables (in this case f,c,t) in such a way as to eliminate as many constants as possible.

In your link, time is brought down via u=at. This means that we can simply divide our equations through by the constant a and obtain

\dfrac{df}{du} = f - \dfrac{a_1}{a} cf

\dfrac{dc}{du} =  \dfrac{b}{a} cf - \dfrac{b_1}{a} c


We can also scale f = A z and c = B y where A,B are dimensional constants (for us to rewrite in terms of the known constants), and z,y are dimensionless variables.

Substituting these in yields

\dfrac{dz}{du} = z - \dfrac{a_1 B}{a} yz

\dfrac{dy}{du} =  \dfrac{bA}{a} yz - \dfrac{b_1}{a} y



We can bring the coefficient of yz in the first eqn down to -1 if we impose that B = \dfrac{a}{a_1}.

We can also choose A = \dfrac{a}{b} in order to bring the coefficient of yz in the second eqn down to 1.

This leaves us with

\dfrac{dz}{du} = z - yz

\dfrac{dy}{du} =  yz - \dfrac{b_1}{a} y

whereby we can combine the ratio \dfrac{b_1}{a} into a single constant \beta.


Thus, via nondimensionalisation, we have reduced the number of constants in our model down from 4 to only 1. The leftover variables z,y,u do not have any dimensions and can therefore be treated on level ground without having to worry about dimensions and their scaling factors.
Brilliant, ur an angel! On part E I think I’ve got the ans but it says the dimensions are wrong. u=at butt u/t isn’t allowed eitherName:  6D00D8D4-84EA-4990-B448-D35764A4F837.png
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