Biomathematics / Maths - behind the scenes / Maths is fun

See you later, alligator!

Recently I explained what a mathematical model is. I thought you might be interested in learning how mathematical modelling works in practice. My first encounter with this process was a research project that considered… a population of crocodiles.

I find crocodiles really fascinating. They’ve lived on the Earth for millions of years, thus we can call them “living fossils”. Other dinosaur genera died out but crocodiles survived and even haven’t changed much since then. Scientists aren’t sure why that happened, so I decided to try to explain it myself. My work was strongly based on the excellent book by J.D. Murray: “Mathematical Biology: I. An Introduction”. By the way, this volume contains a wide range of problems: from the spread of infectious diseases to predicting the divorce. Applied mathematicians don’t know what boredom means!

The hypothesis is that crocodiles succeeded due to their peculiar way of sex determination. Namely, for most species, the sex of the hatchling is determined by genes during the conception. A boy or a girl, just like that. Crocodiles are different. Their sex depends on the environment in which they’re hatched, especially on the temperature of egg incubation. We call this phenomenon “temperature sex determination” (TSD).

How do we model such thing? A good practice is to start small, from a very simple model. Murray proposed to divide the natural habitat of crocodiles into three regions:

  1. Wet marsh – here only females are born;
  2. Dry marsh – here 50% of hatchling are female and 50% are male;
  3. Levees – here only males are born.

Does the nature really work this way, i.e. can we find such three regions in the real world? Of course not! We can only observe that further from the wet marsh fewer females are born. But remember that the model is just a simplification of the reality.

Now we need to think about some assumptions. How do we know where a female crocodile will lay the eggs? We need to make use of the literature about ecology (or talk to our friends who are experts in this area). Observations show that females tend to go back to the habitat similar to the one where they were born. Thus we can assume that the most desired region to lay eggs is the wet marsh.

However, this way all the females went to the first region, so only females would be born, no males would be left to fertilize these females and the whole population would die out. Thus we need to assume that all regions have some capacity, i.e. the number of eggs that can be laid there. When there are no spaces left in the first region, females go to the second one. When the dry marsh is also full, they need to migrate to the levees.

Later comes the technical part. We need to come up with equations describing how populations of males and females in each region evolve with time. We also have to figure out parameters (constant values) describing different parts of the model, for example region capacities. In order to do that, we either use observations from the real world or statistical methods.

When the model is ready, it’s useful to run some computer simulations to check if they agree with the observations (or, if observations aren’t available, with the common sense). Sometimes a graphical representation provided by a computer can tell us more about the model than arduous calculations.

If we’re not satisfied with the model’s performance, we can modify it. If we’re satisfied… we can modify it as well to include more details or add extra components. For example, in my research I decided to see what happens when we periodically change the capacity of the first region. This can happen in the nature, because in many natural habitats of crocodiles seasonal floodings can occur, moving the regions far from the river.

We could imagine many extensions to such model. For example, I assumed that all events are deterministic, which means that given the state of population in the beginning, in principle we can predict what happens at any given time in the future. Of course the nature doesn’t work this way, so we could add some uncertainty to this model.

Another idea would be to model also the age structure of the population. In the simple model there is an unrealistic assumption that all crocodiles can reproduce, also very young and very old ones. Dividing the population into fertile and infertile individuals could be another step.

I hope this article helped you see how applied mathematicians work. We try to help scientists in different areas draw conclusions from their observations. In this case we were able to suggest that one of the reasons why crocodiles are still on the Earth is TSD. More precisely, it allows their population to stay female-dominated, which appears to be a good thing (of course!). This makes sense: in case of a catastrophe, few males can fertilize many females and the population regenerates very quickly.

Conclusion? Females rule! At least according to this model 🙂

 

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