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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:

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

\begin{equation} \frac{dP}{dt} = ebNP - mP \end{equation}

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:

\begin{equation} [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} \end{equation}

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

\begin{equation} N = \tilde{N}\hat{N}\\P = \tilde{P}\hat{P}\\ t = \tilde{t}\hat{t} \end{equation}

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:

\begin{equation} \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} \end{equation}

\begin{equation} \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} \end{equation}

and after simplifying, we have:

\begin{equation} \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} \end{equation}

\begin{equation} \frac{d\hat{P}}{d\hat{t}} = eb\tilde{N}\hat{N}\tilde{t}\hat{P} - m\tilde{t}\hat{P} \end{equation}

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:

\begin{equation} t = \tilde{t} \hat{t} = \frac{1}{m} \cdot \hat{t} \longrightarrow \hat{t} = t \cdot m \end{equation}

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

Inserting these into the model results in the simplified equations:

\begin{equation} \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} \end{equation}

\begin{equation} \frac{d\hat{P}}{d\hat{t}} = \frac{ebK}{m} \hat{N}\hat{P} - \hat{P} \end{equation}

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.

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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.).

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  • $\begingroup$ 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)? $\endgroup$
    – user_15
    May 2, 2020 at 17:15
  • $\begingroup$ I guess what I am asking is can you clarify what you mean by "as you can see from the first ODE" in your comment? Thanks $\endgroup$
    – user_15
    May 2, 2020 at 17:21
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    $\begingroup$ @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. $\endgroup$ May 3, 2020 at 8:31
  • $\begingroup$ Ah, of course. Thanks a lot for your help. $\endgroup$
    – user_15
    May 3, 2020 at 15:50

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