See You Later, Alligator!

What do mathematics and the fashion have in common? Models. They represent the reality, for example… evolution of crocodile populations.

Why have crocodiles survived millions of years while other dinosaur genera died out? Possibly thanks to the temperature sex determination. Usually the sex of the hatchling is determined by genes during conception. However, for crocodiles it depends on the temperature of egg incubation.

A mathematician J.D. Murray proposed a very simple model of this phenomenon. He divided the natural habitat of crocodiles into three regions: wet marsh (only females are born); dry marsh (half of hatchling are female); levees (only males). Of course we cannot find such three distinctive regions in nature. Models are just a simplification of the reality and indeed we observe that further from the wet marsh fewer females are born.

How do we know where a female crocodile will lay eggs? They tend to go back to the habitat similar to the one where they were born, so we can assume that the most desired region to lay eggs is the wet marsh. If all the females went to the first region, only females would be born, no males would be left to fertilize them, so 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.

Now the maths comes in: 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 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, 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 periodic floods change the capacity of the first region.

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. Adding some uncertainty might improve the model performance.

Another idea would be to model also the age structure of the population. Note that we assumed that all crocodiles can reproduce, also very young and very old ones. Dividing the population into fertile and infertile individuals could make the model more realistic.

Now you have an idea 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 Earth is thanks to temperature sex determination. 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.

About the Author

Paula RowińskaPaula Rowińska is a PhD student of the Mathematics of Planet Earth Centre for Doctoral Training at Imperial College London. Her research interests include statistics and stochastic modelling applied to Earth sciences and finance. Currently she is studying how renewable energy sources influence electricity prices. She also engages in science communication, for example by blogging about science: www.paularowinska.wordpress. com. She participated in FameLab Poland, FameLab UK and Three Minute Thesis competition. Soon she’ll give a TEDx talk in London. Paula is fighting the stereotype that the only world mathematicians know is the world of numbers. Somehow she manages to hold the presidency of the Imperial College student chapter or the Society for Industrial and Applied Mathematics, practice piano, dance, sing, write, teach maths, learn German, see more plays in London theatres than she can actually afford, organise conferences and still keep working on her research.

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