# Zlámal Seminar

Summer semester 2018/19

ZLÁMAL SEMINAR takes place usually on Wednesday at 13:00 in the Seminar room 1842 in the 18th floor of A1 building Faculty of Mechanical Engineering BUT, Technická 2.

**Program:**

__Wednesday 13th March 2019 at 1 p.m.__

*Doc. RNDr. Antonín Slavík, Ph.D.*

(Department of Mathematics Education, Faculty of Mathematics and Physics, Chales University in Prague)

**Lotka-Volterra model of competition on graphs**

The lecture will be directed to generalization of the classic Lotka-Volterra model of competition between two biological species. The space-part of the model is represented by a finite graph, where both species can move between vertices along the edges. The model is described by a system of 2N ordinary differential equations, where N is number of vertices. The aim is to study existence of space homogeneous and heterogeneous stationary states and their stability and describe their asymptotic behavior depending on choice of model parameters.__Wednesday April 3rd, 2019 at 1 p.m.__

*Prof. RNDr. Svatoslav Staněk, CSc.*

(Department of Mathematics and Computer Sciences, Faculty of Sciences, Palacký University, Olomouc)

**Non-local Boundary Value Problem for Fractional Differential Equations in Resonance**

The following fractional boundary value problem is studied $$^cD^\alpha u +p(t,u,u')^c D^\beta u= f(t,u,u'), \phi(u)=0, x'(0)=x'(T),$$ where \(^cD\) is Caputa fractional derivaive, \(1<\beta < \alpha \le 2\) and \(\phi: C[0,T] \to \mathbb{R}\) is a functional. This boundary value problem is in resonance. There are introduced conditions for functions \(p,f \in C([0,T] \times \mathbb{R}^2)\), which guarantee existence of the solution. The results are proved by means of combination of Leray-Schauder degree with maximum principle for Caputa fractional derivative and "reduced" initial value problem.__Wednesday April 10th, at 1 p.m.__

*Prof. RNDr. Zdeněk Pospíšil, Dr.*

(Department of Mathematics and Statistics, Faculty of Sciences, Masaryk Univerzity, Brno)

**Discrete Reaction-Dispersion Equations**

The classical reaction-diffusion equation is parabolic partial differential equation with generally non-linear non-homogeneity. It is usually used for spatial population model. The contribution is devoted to a discrete analogy, where the space and time are considered to be discrete; the space is a taken as a graph with dispersion along the edges, the considered reaction at vortices is autocatalytic. The equation can be interpreted as a model of meta-population with separated generations. The contribution aims to find analogy to known results of population ecology – minimal space area necessary for survival, existence and stability of space homogeneous equilibria.

All interested are cordially welcomed.