Meiji University Center for Mathematical Modeling and Applications

▼ Abstract:

The Neolithic transition is one of the most significant single developments in human history. Archaeological evidence of Neolithic transition suggests that expanding velocity of farmers is roughly constant. To understand such phenomenon, many theoretical attempts have been progressed through mathematical modeling. The earlier models on expanding farmers in the Neolithic transition assume that the farmers disperse randomly in space and discuss the expanding velocity of farmers.
However, it is known in Archaeology that the farmers have basically a sedentary life style but if the density of the farmers becomes high, some of them disperse randomly. Thus, we could expect that the dispersal of farmers is described by "nonlinear diffusion". Here the question is that what kind of nonlinear diffusion is plausible. In this talk, we apply the singular limit procedure to derive a suitably nonlinear diffusion model and discuss how such sedentary property influence on the expanding farmers’ life. This research has been supported by Meiji Institute for Advanced Studies of Mathematical Sciences, Meiji University.

▼ Abstract:

In 1996, anthropologists K. Aoki, M. Shida and mathematical ecologist N. Shigesada proposed a mathematical model describing the spread of the early farming during the New Stone Age. The model is a three-species reaction-diffusion system involving "Initial Farmers", "Converted Farmers" and "Hunter-Gatherers". This system falls into the category of predator-prey systems, with predators corresponding to "Farmers", and preys corresponding to "Hunter-Gatherers" if the intrinsic growth rates of two types of farmers coincide. By numerical simulations and some formal linearization arguments, they concluded that there are four different types of spreading behaviors depending on the parameter values.
In this talk, we not only give theoretical justification to all of the four types of spreading behaviors observed by Aoki et al, but also present a result on the detailed asymptotic behavior that can hardly be observed by numerical simulation. We also investigate the case where the motility of the hunter-gatherers is different from that of the farmers, which is not discussed in the paper of Aoki et al. We also consider a spatially periodic extension of the model if we have time.

▼ Abstract:

In 2013, H. Berestycki, N. Rodriguez and L. Ryzhik studies a mathematical model describing the spread of criminal activities in society in one space dimension. The model was first proposed by Berestycki and J.-P. Nadal in 2010 and takes the form of a reaction-diffusion system with bistable nonlinearity. The work of Berestycki et al in 2013 establishes results on the existence of traveling waves and on the prevention of propagation of criminal waves.
In this talk, I will review the above work of Berestycki et al and discuss related problems in higher dimensions on the propagation of bistable fronts in the presence of obstacles. This is partly joint work with Henri Berestycki.

▼ Abstract:

In this lecture, I will report on a model aiming at studying the dynamics and spreading of riots, and especially violent social protests. It involves an epidemiological approach for the dynamics with a diffusion interaction term. There is indeed growing evidence that violent social behavior often spreads like epidemics.
I will discuss this model in the setting of the French riots of 2005 that were unique in their extension in space and time. We can compare the outcome of the model with a rather detailed set of data for these riots.
With the aim to provide a general framework to study the diffusion of social norms I will also describe a more general class of reaction-diffusion systems that are relevant in this context.

▼ Abstract:

Since Alan Turing stated the notion of diffusion-induced instability in 1952, it is well known that reaction-diffusion "open" systems generate various types of spatio-temporal patterns and in fact, some systems have been used as models of biological pattern formation. In response to this flow, reaction-diffusion "open" systems have appeared in mathematical communities and became one of the main studies of nonlinear PDEs. As compared to these systems, "closed" systems have been gradually less interesting and have gone from the study of reaction-diffusion systems. However, recently, new pattern formation in propagation phenomena can be observed even in closed systems as the consequence of transient self-organization. In this talk, I emphasize that theoretical understanding of such pattern formation is a very important subject in nonlinear mathematics with application to biology.

▼ Abstract:

In 1996, ecologists K. Aoki, M. Shida and N. Shigesada proposed a mathematical model describing the spread of the early farming during the New Stone Age. This system falls into the category of predator-prey systems, with two types of predators corresponding to "Initial Farmers" and "Converted Farmers", and preys corresponding to "Hunter-Gatherers". By numerical simulations and some formal linearization arguments, they concluded that there are four different types of spreading behaviors depending on the parameter values.

In this talk, we give theoretical justification to all of the four types of spreading behaviors observed by Aoki et al. We also investigate the case where the motility of the hunter-gatherers is larger than that of the farmers, which is not discussed in the paper of Aoki et al. Furthermore, we show that a logarithmic phase drift of the front position occurs as in the scalar KPP equation.

▼ Abstract:

This talk will be devoted to the existence of pulsating travelling front solutions for spatially periodic heterogeneous reaction-diffusion equations in arbitrary dimension, in both bistable and more general multistable frameworks. In the multistable case, the notion of a single front is not sufficient to understand the dynamics of solutions, and we instead observe the appearance of a so-called propagating terrace. This roughly refers to a finite family of stacked fronts connecting intermediate stable steady states and whose speeds are ordered. Surprisingly, for a given equation, the shape of this terrace (i.e., the involved intermediate steady states or even their number) may depend on the direction of the propagation.
The presented results come from a series of works with A. Ducrot, H. Matano and L. Rossi.

▼ Abstract:

In this talk, I will present recent results obtained in collaboration with Adrian Lam (Ohio State University, USA). We are interested in the classical monostable Lotka--Volterra competition--diffusion system of two species and more precisely in the associated Cauchy problem when the initial conditions are null or exponentially decaying in a half-space. Thanks to a sophisticated construction of super- and sub-solutions, we manage to characterize completely the possible pairs of asymptotic spreading speeds for the two species. The main result is unexpected and contradicts a conjecture by Shigesada and Kawasaki: the second invasion is sometimes "nonlocally pulled" with a larger-than-expected speed.

▼ Abstract:

The FitzHugh--Nagumo system has been studied extensively for several decades. It is has been shown numerically that pulses are generated to propagate and then some of the pulses are annihilated after collision. For the mathematical understanding of these complicated dynamics, we investigate the global dynamics of a one-dimensional free boundary problem in the singular limit of a FitzHugh--Nagumo type reaction--diffusion system. We introduce the weak solutions to study the continuation of the classical solutions beyond the annihilation time and apply the notion of symbolic dynamics to classify the type of propagation of interfaces. Then the complete dynamics are obtained. More precisely, we show that the solutions are classified into three categories:

(i) the solution converges uniformly to the resting state;
(ii) the solution converges to a series of traveling pulses propagating in either the same direction or both directions; %with the same wave speed (resp., the two opposite speeds); and
(iii) the solution converges to a propagating wave consisting of multiple traveling pulses and two traveling fronts propagating in both directions.