## 数理人口学・数理生物学セミナー

### 2016年06月01日(水)

16:30-17:30   数理科学研究科棟(駒場) 128演習室号室

A variational problem associated with the minimal speed of traveling waves for the spatially
periodic KPP equation (ENGLISH)
[ 講演概要 ]
We consider a spatially periodic KPP equation of the form
$$u_t=u_{xx}+b(x)u(1-u).$$
This equation is motivated by a model in mathematical ecology describing the invasion of an alien species into spatially periodic habitat. We deal with the following variational problem:
$$\underset{b\in A_i}{\mbox{Maximize}}\ \ c^*(b),\ i=1,2,$$
where $c^*(b)$ denotes the minimal speed of the traveling wave of the above equation, and sets $A_1$, $A_2$ are defined by
$$A_1:=\{b\ |\ \int_0^Lb=\alpha L,||b||_{\infty}\le h \},$$
$$A_2:=\{b\ |\ \int_0^Lb^2=\beta L\},$$
with $h>\alpha>0$ and $\beta>0$ being arbitrarily given constants. It is known that $c^*(b)$ is given by the principal eigenvalue $k(\lambda,b)$ associated with the one-dimensional elliptic operator under the periodic boundary condition:
$$-L_{\lambda,b}\psi=-\frac{d^2}{dx^2}\psi-2\lambda\frac{d}{dx}\psi-(b(x) +\lambda^2)\psi\ \ (x\in\mathbb{R}/L\mathbb{Z}).$$
It is important to note that, in one-dimensional reaction-diffusion equations, the minimal speed $c^*(b)$ coincides with the so-called spreading speed. The notion of spreading speed was introduced in mathematical ecology to describe how fast the invading species expands its territory. In other words, our goal is to find an optimal coefficient $b(x)$ that gives the fastest spreading speed under certain given constraints and to study the properties of such $b(x)$.