# Help non-dimensionalizing predator-prey ODE system

I am learning how to non-dimensionalize ODE systems, and I am struggling with a part of the predator-prey non-dimensionalization exercise here: http://wp.auburn.edu/radich/wp-content/uploads/2014/08/Nondimensionalization-Resource-2.pdf

The system of ODE is:

$$$$\frac{dN}{dt} = rN\Big(1 - \frac{N}{K}\Big) - bNP$$$$

$$$$\frac{dP}{dt} = ebNP - mP$$$$

where $$N$$ is the number of prey, $$P$$ is the number of predators, $$r$$ is the intrinsic growth rate of the prey, $$K$$ is the carrying capacity of the prey, $$b$$ is the rate that predators meet and kill prey, $$e$$ is a conversion efficiency from prey to predators, and $$m$$ is the mortality rate of the prey.

First, I'm finding it hard to understand the corresponding dimensions. I think they are:

$$$$[N], [P] = \text{number of animals}\\ [r] = \text{time}^{-1}\\ [K] = \text{number of animals}\\ [b] = \text{time}^{-1}\\ [e] = \frac{\text{number of prey}}{\text{number of predators}}\\ [m] = \text{time}^{-1}$$$$

To non-dimensionalize, I define the old variables as:

$$$$N = \tilde{N}\hat{N}\\P = \tilde{P}\hat{P}\\ t = \tilde{t}\hat{t}$$$$

where the hat terms are the new dimensionless variables, and tilde parameters are the scaling parameters that I understand should have the same dimensions as the original variables.

Inserting these into the equations:

$$$$\frac{d\tilde{N}\hat{N}}{d\tilde{t}\hat{t}} = r\tilde{N}\hat{N}\Big(1 - \frac{\tilde{N}\hat{N}}{K}\Big) - b\tilde{N}\hat{N}\tilde{P}\hat{P}$$$$

$$$$\frac{d\tilde{P}\hat{P}}{d\tilde{t}\hat{t}} = eb\tilde{N}\hat{N}\tilde{P}\hat{P} - m\tilde{P}\hat{P}$$$$

and after simplifying, we have:

$$$$\frac{d\hat{N}}{d\hat{t}} = r\tilde{t}\hat{N}\Big(1 - \frac{\tilde{N}\hat{N}}{K}\Big) - b\tilde{t}\hat{N}\tilde{P}\hat{P}$$$$

$$$$\frac{d\hat{P}}{d\hat{t}} = eb\tilde{N}\hat{N}\tilde{t}\hat{P} - m\tilde{t}\hat{P}$$$$

I understand that if we define $$\tilde{N} = K$$, then it has the same units as the original variable (i.e. number of prey), and simplifies the first equation.

However, the link above then defines $$\tilde{t} = \frac{1}{m}$$, which I see simplifies the equation but doesn't have the same dimensions: $$[t]$$ has dimensions time and $$[m]$$ has dimensions time$$^{-1}$$. So, the relationship between $$t$$ and the new variable becomes:

$$$$t = \tilde{t} \hat{t} = \frac{1}{m} \cdot \hat{t} \longrightarrow \hat{t} = t \cdot m$$$$

which to my mind doesn't define $$\tilde{t}$$ as a dimensionless quality.

Inserting these into the model results in the simplified equations:

$$$$\frac{d\hat{N}}{d\hat{t}} = \frac{r}{m} \hat{N}\Big(1 - \hat{N}) - b \frac{1}{m} \hat{N}\tilde{P}\hat{P}$$$$

$$$$\frac{d\hat{P}}{d\hat{t}} = \frac{ebK}{m} \hat{N}\hat{P} - \hat{P}$$$$

Then, the above link defines $$\tilde{P} = \frac{m}{b}$$, which again simplifies the equation but I don't see how this quantity has the same dimensions as $$P$$... $$\frac{m}{b}$$ has dimensions time$$^{-2}$$ and $$P$$ is the number of predators, so how is $$\hat{P}$$ dimensionless?

I understand how this process simplifies the model, then, but I'm struggling to understand how it is dimensionless.

Any help would be appreciated.

If $$m$$ has dimension 1/time, then surely $$1/m$$ has dimension time, and $$mt$$ is dimensionless, so I don't see what the problem is.
Likewise, the dimension of $$b$$ (as you can see from the first ODE) is “per unit of predator and per unit of time”, 1/(predators$$\times$$time), so $$m/b$$ will be (1/time)/(1/(predators$$\times$$time)) = predators, i.e., it has the same dimensions as whatever unit you use for measuring the amount of predators (number of individuals, biomass in kilograms, etc.).
• Thanks, I see my mistake with $m$. I am struggling with $b$ but I think that's because I really struggle with dimensional analysis in general. I don't intuitively understand why it has the dimension 1/(predators $\times$ time) - is it because b is a composite of the probability of contact between predators and prey (with dimensions 1/time) and the probability the predator consumes the prey (with dimensions 1/predator)? May 2, 2020 at 17:15
• @user_15: The dimension of $dN/dt$ is prey/time, so the dimension of $bNP$ must also be prey/time. Cancelling $N$, you see that $bP$ is 1/time, so $b$ is (1/time)/predators. May 3, 2020 at 8:31