Nondimensionalization of differential equations – an example using the Lotka-Volterra system of equations

I decided to write about nondimensionalization today since it’s something I only came across recently and found very exciting. It’s apparently a trivial process for a lot of people, but it wasn’t something I was taught how to do during my education so I thought I’d put a short guide out on the interwebs for other people. Nondimensionalization (also referred to as rescaling) refers to the process of transforming an equation to a dimensionless form by rescaling its variables. There are a couple benefits to this:

  • All variables and parameters in the new system are unitless, i.e. scales and units not an issue in the new system and the dynamics of systems operating on different time and/or spatial scales can be compared;
  • The number of model parameters is reduced to a smaller set of fundamental parameters that govern the dynamics of the system; which also means
  • The new model is simpler and easier to analyze, and
  • The computational time becomes shorter.

I will now present an example using a predator-prey system of equations. In my last blogpost, I used the Lotka-Volterra system of equations for describing predator-prey interactions. Towards the end of that post I talked about the logistic Lotka-Volterra system, which is in the following form:image002image004Where x is prey abundance, y is predator abundance, b is the prey growth rate, d is the predator death rate, c is the rate with which consumed prey is converted to predator abundance, a is the rate with which prey is killed by a predator per unit of time, and K is the carrying capacity of the prey given its environmental conditions.

The first step is to define the original model variables as products of new dimensionless variables (e.g. x*) and scaling parameters (e.g. X), carrying the same units as the original variable.image006The rescaled models are then substituted in the original model:
image008image010Carrying out all cancellations and obvious simplifications:image012image014Our task now is to define the rescaling parameters X, Y, and T to simplify our model – remember they have to have the same units as our original parameters.

 

Variable/parameter Unit
x mass 1
y mass 1
t time
b 1/time
d 1/time
a 1/(mass∙time) 2
c mass/mass 3
K mass

There’s no single correct way of going about doing this, but using the units for guidance and trying to be smart we can simplify the structure of our model. For example, setting X=K will remove that term from the prey equation (notice that this way X has the same unit as our original x variable).

image016image018

The choice of Y is not very obvious so let’s look at T first. We could go with both T=1/b or T=1/d. Unit-wise they both work but one would serve to eliminate a parameter from the first equation and the other from the second. The decision here depends on what dynamics we’re most interested in, so for the purposes of demonstration here, let’s go with T=1/b.

image020image022

We’re now left with defining Y, which only appears in the second term of the first equation. Looking at that term, the obvious substitution is Y=b/a, resulting in this set of equations:image024image022

Our system of equations is still not dimensionless, as we still have the model parameters to worry about. We can now define aggregate parameters using the original parameters in such a way that they will not carry any units and they will further simplify our model.

By setting p1=caK/b and p2=d/b we can transform our system to:image024image026a system of equations with no units and just two parameters.

 

1 Prey and predator abundance don’t have to necessarily be measured using mass units, it could be volume, density or something else. The units for parameters a, c, K would change equivalently and the rescaling still holds.

2 This is the death rate per encounter with predator per time t.

3 This is the converted predator (mass) per prey (mass) consumed.

 

4 thoughts on “Nondimensionalization of differential equations – an example using the Lotka-Volterra system of equations

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  3. Thank you so much for this helpful explanation. I have a confusion in one step; you’ve assumed that Y is equal to b/a but the units are not consistent in this way and the last term of the first equation will still have units. May I know how this works?

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