## Mathematical Biology Seminar

Seminar information archive ～09/22｜Next seminar｜Future seminars 09/23～

### 2016/06/01

16:30-17:30 Room #128演習室 (Graduate School of Math. Sci. Bldg.)

A variational problem associated with the minimal speed of traveling waves for the spatially

periodic KPP equation (ENGLISH)

**Xiao Dongyuan**(Graduate School of Mathematical Sciences, The University of Tokyo)A variational problem associated with the minimal speed of traveling waves for the spatially

periodic KPP equation (ENGLISH)

[ Abstract ]

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 $b(x)$.

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 $b(x)$.