Differential equations help!

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.

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.