I remember the first time I have been surprised by a model. I was working on the conditions under which a mutualist can protect its host from a pathogen, and in particular whether the mutualist can persist or will be displaced by the pathogen (unless there are multiple populations connected by dispersal, the answer is no). What surprised me was how, in the end, the answer to this question depended on the relative value of three parameters. Of course, nothing in modeling should be surprising, because the model encompasses the entirety of its own rules, and so of course the answer is in here, waiting to be found. But where do the models come from?

People. Models are written by people.

And this is what I find fascinating. Modeling is the ultimate exercise in world building. Mark J.P. Wolf wrote that creating a world “renews our vision and gives us new perspective and insight into ontological questions that might otherwise escape our notice within the default assumptions we make about reality”. But as far as ecological models are concerned, these default assumptions are what constitutes the basis of our model. Populations increase in size until they consume the amount of resources that the system receives. Being eaten makes your population smaller. It’s better for your growth rate to be adapted than to not be. Things move around. And yet, despite these being self evident, I am often surprised by what happens when I mix them together.

Isn’t this amazing? That when we pool together the things we know, we create yet more things we didn’t knew yet? Which brings me to the point; modeling, deep down, is experimental work.

What happens if I modify this variable? What happens if I decide to model this process one way or another? By far my favorite example is the way to write the logistic growth of a population. The one most often taught is

$\frac{1}{N}\frac{\text{d}}{\text{dt}}N = r \left(1 - \frac{N}{K}\right)$

where $N$ is the population size, $r$ is the rate at which it can grow, and $K$ is the maximal number of individuals before all the incoming resources are exhausted. But we can write this model in another way, specifically

$\frac{1}{N}\frac{\text{d}}{\text{dt}}N = r - \alpha N$

where $\alpha$ is the rate at which the individuals will compete for the resources. And whereas they would reach $K$ individuals before, they will now reach $r/\alpha$. These are two worlds with very different emphasizes: the first has an upper limit to growth, which is hard-coded. The second also has an upper limit to growth, but this time it emerges from the choice of being explicit about the fact that individuals compete for the resource.

But of course neither of these models are explicit about the fact that resources flow in and out of the system, and so we may want to add an equation for this. And we would need to add a term to explain how the resource is converted into biomass for the population. Should it be fixed, or depend on the metabolic rates? There is a very deep rabbit hole we can bury ourselves in just when deciding how to represent the simple fact that living organisms need to eat in order to grow.

With this in mind, it is hardly surprising to create models whose behavior we cannot anticipate. Once the protocol/model is setup, comes the exploration phase, and the manipulation phase. In a way, speaking about “numerical experiment” is not some sleight of hand designed to make modeling look more practical than what it is; as a modeler, I am experimenting on my system, because although I may have created it, I do not understand it.