# Partial differential equations in the social and life science: emergent challenges in modeling, analysis, and computations (13w5106)

Arriving in Banff, Alberta Sunday, March 31 and departing Friday April 5, 2013

## Organizers

Rustum Choksi (McGill University)

Maria del Mar Gonzalez (Universidad Autonoma de Madrid)

Maria Gualdani (George Washington University )

Maria Elena Schonbek (University of California Santa Cruz)

## Objectives

The aim of this workshop is twofolds: to disseminate the latest results on modeling, analysis and simulations for partial differential equations that appear as novel mathematical models in many emerging areas of the social and life sciences and to encourage interactions and target new interesting problems and applications . We focus on two central areas: the social and life sciences. Particular topics include

*opinion formations, population dynamics, traffic patterns, behavioral analysis in financial markets, environmental economics, neuroscience*and*crime spots*. While these area are drastically different, one may approach them with similar modeling and analytical techniques. Applications in the social sciences are based on human interactions: most of the time human interactions can be modeled as a large group of agents acting in an homogeneous environment. The description of agents in an homogeneous environment leads to systems of partial differential equations which take into account the interplay between each player and the surrounding environment. Hence collective behaviors and their structure can be described via their universal equilibria. We aim to gather expert in modeling opinion formations, traffic patterns, population dynamics, financial markets, environmental economics and crime hot-spot behaviors. The second area pertains more to a branch of the life sciences. In many situations the dynamic evolution of a system of agents can be driven by a smaller group of agents or by a fluctuating environment. Such situations can be found for example in environmental ecology, neuron networks and molecular motors. It is interesting to notice that the above mentioned applications occur at all different length scale: macroscopic for environmental ecology, microscopic for neuron networks and nanoscale for molecular motors. While such areas are very different, they can be described by similar mathematical models. Systems evolving under the action of a external fluctuation can be described by stochastic differential equations; in some situations such stochastic systems can be rewritten as system of partial differential equation via a diffusion approximation of the mean field limit. In recents years the study of behavior of interacting agents has gained increasing interest in several research fields. In economics, the evolution of social phenomena like financial markets, crime hot-spots, traffic behaviors and opinion formations have been studied quite extensively. In biological programs, the analysis of collective behavior of fishes, bacteria and birds and the study of self-organizing systems have been fundamental research topics. Natural phenomena driven by interactions of agents are present in various real life applications. Dependent on the application, such interactions occur at all length scales, and they can be understood and successfully described by different mathematical tools. One of the most common tools are differential equations of mean field type. Mean field description originates from statistical physics ideas, where interactions between agents (particles, players or bacteria) are studied using a external force field: each agent contributes to the creation of the mean field, and the mean field, in return, influences the behavior of the agents themselves. Examples of mean fields are market prices, epidemic disease to name a few. A different tool often used in systems where interactions can be described via network or agent-based models. One fundamental question is to understand if such models can be rewritten via certain appropriate upscaling (homogenization) processes as effective systems involving differential equations. Several applications can be also modeled by discrete, ill-posed diffusion equations; such equations are quite common in granular flow, as well as in image processing and chemotaxis. They also arise population dynamics and in opinion dynamics, where they describe pairs interaction and compromised-based opinion formations. Mathematical models of partial differential equations which describe behavior of interacting agents usually present analytical and numerical challenges: the difficulties are caused on one side by the nonlinearity and high (differential) order, and consequent lack of classical theory tools. On the other side we often deal with complex systems involving multi-scales. For discrete, ill-posed differential equations, the analysis of the stabilizing effects coming from the discrete scheme is essential to study long time asymptotics. This workshop will place emphasis on the*mathematical structures*which are*common*to all the mentioned research topics. These include the mathematics related to- averaging phenomena and homogenization,
- coarsening
- optimal control and game theories (differential, mean field, etc…)
- diffusion (local and non-local, nonlinear degenerate)
- energy estimates related to stability and extinction in finite time blow-up
- numerical computations.