Tuesday, April 7, 2015

First Encounters with the Population Dynamics

It all started with a workshop; in fact long before that but I think this is a good opportunity to write the first post of the new blog and stop to make excuses not to write once and for all! That being said, the first post of the 'Turbulent Dynamics' is on a mini-workshop that has been held in Istanbul Center for Mathematical Sciences (IMBM) this week, which was called 'Population Dynamics'.

Population Dynamics is an area of research mainly focusing on biological systems such as ecology, evolution, infectious diseases and so on but clearly there are exceptions regarding its other interesting applications such as linguistics for example. It is a very multi-cultural area in a sense you encounter really different people who are motivated by problems not particularly in their main area but they are willing to apply their own expertise to shed new light and gain new perspectives on them. In our case, the problems generally arise from biological motivations and mostly physicist and applied mathematicians are more than willing to apply various modeling schemes to further understand the underpinnings and mechanisms of the problems in hand.

The mini-workshop was co-organized by Atilla Yılmaz from Bogazici University Mathematics Department and Muhittin Mungan from Physics  Department of the same university. There were four interesting small talks focusing on two main problems, namely The Stochastic Encounter-Mating Model and Ecological Niche Theory.

With his collaborator Onur Gün from Weierstrass Institute of Applied Analysis and Stochastics, Berlin, Germany, A. Yılmaz introduced their recent work on encounter mating model which is by their own words [1]:
... a new model of permanent monogamous pair formation in zoological populations comprised of k types of females and males, which unifies and generalizes the encounter-mating models of Gimelfarb (1988). In our model, animals randomly encounter members of the opposite sex at their so-called firing times to form temporary pairs which then become permanent if mating happens. Given the distributions of the firing times and the mating preferences upon encounter, which depend on the sex and the type of the animals, we analyze the contingency table Q(t) of permanent pair types at any time t.
Both of the talks were quiet rigorous in a sense they derived their results from the basics like classical Lotka-Volterra (LV) equations and imposing stochasticity on them. One of the key aspect of their work is its relation to panmixia, i.e random mating in a population, which is know to be one of the main assumptions in population genetics.

Other talk was given by M. Mungan on Plant-pollinator webs which investigates the relationship between the plants and their pollinators based on a real ecological field data. Again starting with modeling the relationship between the plants and various pollinators with LV equations, the stable configuration of the species abundances are sought and the model predictions and the real data is being compared. It was quiet inspiring to see that with such a sparse data, the model can predict the actual findings pretty well. It was also interesting for me to think about the field work and data-side of all these problems and the relationship between the models and the modeled ecology.

Final talk was reserved for our guest from Uruguay, Institute of Physics, Universidad de la Rep´ublica, Hugo Fort. He introduced the Niche Theory in general, giving many examples from various species and their interaction with other species and their environment. The main part of his talk was evolutionary game theory which regards the reproductive rates of the individuals (i.e their fitness) frequency dependent so that they can be modeled as a simple game with strategies being the species themselves and the payoffs are the interaction coefficients in LV system. Starting from the classical Prisoner's Dilemma game he presented the different simulation results in the parameter space of the game.

This compact mini-workshop gave me the opportunity to listen to interesting talks on topics which I am trying to learn during the last two months. I am pleased in a sense that I could follow the main ideas thoroughly and the key ideas and methods presented were the ones that I was more or less acquainted thanks to our reading group with one of my friend and M. Mungan beginning this semester. More on that in the future posts...


[1] The stochastic encounter-mating model, Onur Gün, Atila Yılmaz, 2014, http://arxiv.org/abs/1408.5036

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