UBC Math Department Colloquium: Fanze Kong

Reaction-diffusion systems serve as paradigms to describe ubiquitous pattern formation phenomena arising from natural science and social science such as biology, ecology, finance,etc., and have been extensively studied over the past several decades. Whereas, while modeling some phenomena such as cellular mobility, population dynamics in the heterogeneous environment and the game among a huge number of players, it seems more reasonable to incorporate the advection term. In this talk, we focus on three classes of R-A-D systems: Keller-Segel (K-S) models, population equations and Mean-field Games (MFG) Systems. K-S systems, one of the bedrocks in math biology, were proposed by E. Keller and L. Segel in the 1970s to model the chemotactic process. In the first part of this talk, we discuss K-S models with logistic growth and consider two distinct regimes: the large chemotactic movement and the small chemical diffusion. We have constructed spiky steady states and shown that K-S models with logistic growth admit rich local dynamics including slow dynamics, nucleation, competition instabilities and oscillation dynamics, around spiky patterns. The second part of the talk will focus on population models arising from ecology. Concerning the effect of large biased movement on the strong Allee effect, we constructed infinitely many localized patterns and analyzed their linearized stability properties. In the last part of this talk, we are concentrated at the stationary MFG systems in the whole space, which are well-used to describe Nash equilibria of differential games among numerous homogeneous agents. In math, with the aid of variational method, we study the existence and blow-up profiles of least energy solutions under the mass critical exponent case, in which the coercive assumptions are imposed on the potential functions.