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Mean field methods for stochastic population dynamics

Dr Elena Issoglio (SoM), Dr Jan Palczewski (SoM)

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Population dynamics are widely studied by using both deterministic and stochastic models that aim to describe interactions between large heterogeneous groups of individuals. Some of the objectives of these studies are: establishing the long term behaviour of the whole population (e.g., which species will thrive, and which ones will disappear), or determining the expected survival time of a particular species. Other interesting questions in community ecology include identifying which models predict a sensible relative species abundance (RSA), which is used to calculate the probability that a certain species has a given number of individuals alive. In studies of natural environments, it is important to answer questions of this kind as accurately as possible, in order to make informed decisions concerning human interventions aimed, e.g., at preserving fauna. 

A substantial chunk of current research around population dynamics hinges on numerical methods and theoretical studies of various mathematical models. This is challenging because of the complexity of modelling interactions between large groups of individuals. In this project we will study emerging area of modelling in which individuals are not only affected by the rest of population but also subject to random perturbations of their state. The use of stochastic models is valuable since it is virtually impossible to include all possible factors that can affect the dynamic of a population in a tractable model. In general, the term “stochastic” is broadly understood as some sort of “randomness” in the system under consideration. Recently developed methods of McKean-Vlasov equations as limits of the population dynamics when their size grows to infinity enables tractable studies of stochastic population models similarly as the master equation is used in classical models. 

The candidate will study interacting particle systems that describe stochastic interaction of individuals and their mean-field limit from a mathematical perspective, hence using tools such as stochastic analysis, function spaces, probability, numerical analysis and so on. One should note that these models can describe a wide range of interacting individuals, from animals (as we see in more detail in the example below) to humans, from more complex human structures (such as banks) to neurons in the brain, or even inanimate particles (like gas). 

A general plan for the PhD project could be 

1. Formulate (in a rigorous mathematical way) a stochastic interacting particle system for a large population, that describes a class biological/social problems in the realm of population and behavioural ecology 

2. Study this large interacting system, in particular find conditions under which the system has a solution which is unique. Study also the regularity properties of this solution 

3. Postulate the mean-field (or McKean-Vlasov) approximating equation and solve it, that is study existence, uniqueness and regularity of the solution 

4. Investigate the limiting behaviour of the large interacting system as the number of individuals tends to infinity with the aim to show that the limiting system is indeed the one postulated above. Study also the rate of convergence of the approximation 

5. Devise numerical schemes to solve the system and the mean-field equation numerically, and show convergence of the numerical schemes Implement the numerical schemes 

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Related undergraduate subjects:

  • Mathematics