# Institute of Mathematics and Biomathematics

The Department, founded in 2007, focuses on research in Biomathematics. The modelling effort exploits mathematical methodologies based on evolutionary game theory, differential and difference equations, stochastic processes and rule-based computer simulations of interacting individuals.

**Homepage:** https://www.prf.jcu.cz/en/umb**Head of the Department:** Prof. Vlastimil Křivan**E-mail:** krivan@prf.jcu.cz

### Research and education activities

The Institute of Mathematics and Biomathematics offers Bachelor’s and Master’s Programmes in Teaching of Mathematics and a Bachelor’s programme in Applied Mathematics. The current research of the Institute focuses on applied mathematics in biological and technical sciences, differential equations, numerical methods, and geometry.

### Mathematical biology

The research in mathematical biology, conducted jointly with the Biology Centre of the Czech Academy of Sciences, focuses on the development and analysis of mathematical models of behavioural, population, and evolutionary ecology with the aim of understanding mechanisms regulating species biodiversity. Mathematical methodologies based on game theory, differential and difference equations, stochastic processes, and rule-based computer simulations of interacting individuals are used.

### Technical applications

The institute pursues research in the technical applications of mathematics. In particular, this research focuses on the theory and numerical solution of nonlinear differential equations used in the mechanics of solids including elastoplasticity and damage processes in materials. These methods are based on finite element computations, fast solution of linear systems of equations, and error control in terms of posteriori error estimates.

### Geometry and geometric control theory

The main subject of this research is the representation theory of Lie groups and algebras, parabolic geometries and their applications in control theory. These applications include description of kinematic spaces and optimal movement of robots, e.g. tri-segment snake.

### Differential equations

We study the qualitative behaviour of ordinary and partial differential equations. In particular, we focus on the existence and bifurcation of non-trivial solutions with unilateral boundary conditions. We also use Filippov solutions of ordinary differential equations to solve models arising in biology.

**Example of results**

**Publications:**

BEREC L., JANOUŠKOVÁ E. and THEUER M. (2017): Sexually transmitted infections and mate-finding Allee effects*.* *Theoretical Population Biology* 114: 59-69.

BOZORGNIA F. and VALDMAN J. (2017): A FEM approximation of a two-phase obstacle problem and its a posteriori error estimate. *Computers & Mathematics with Applications* 73, No. 3: 419-432.

KŘIVAN V. and CRESSMAN R. (2017): Interaction times change evolutionary outcomes: Two-player matrix games. *Journal of Theoretical Biology* 416: 199–207.

REVILLA T. A. and KŘIVAN V. (2016): Pollinator foraging flexibility and the coexistence of competing plants. *Plus One* 11: e0160076.

ROUBÍČEK T. and VALDMAN J. (2016): Perfect plasticity with damage and healing at small strains, its modelling, analysis, and computer implementation. *SIAM Journal on Applied mathematics* 76, No. 1: 314-340.

MARIANI P., KŘIVAN V., MACKENZIE B.R. and MULLON C. (2016): The migration game in habitat network: the case of tuna. *Theoretical Ecology* 9:219-232.

GREGOROVIČ J. and ZALABOVÁ L. (2016): Geometric properties of homogeneous parabolic geometries with generalized symmetries. *Differential Geometry and its Applications* 49: 388–422.

EISNER J. and VAETH M. (2016): Degree, instability and bifurcation of reaction diffusion systems with obstacles near certain hyperbolas. *Nonlinear Analysis* 135: 158-193.

EISNER J., KUČERA M. and VAETH M. (2016): A variational approach to bifurcation points of a reaction-diffusion system with obstacles and Neumann boundary conditions. *Applications of Mathematics* 61: 1-25.

BEREC L., MAXIN D. and BERNHAUEROVÁ V. (2016): Male-killing bacteria as agents of insect pest control. *Journal of Applied Ecology* 53: 1270-1279.

MAXIN D., BEREC L., BINGHAM A., MOLITOR D. and PATTYSON J. (2015): Is more better? Higher sterilization of infected hosts need not result in reduced pest population size. *Journal of Mathematical Biology* 70:1381-1409.

BERNHAUEROVÁ V. and BEREC L. (2015): Role of trade-off between sexual and vertical routes for evolution of pathogen transmission. *Theoretical Ecology* 8: 23-36.

BERNHAUEROVÁ V., BEREC L. and MAXIN D. (2015): Evolution of early male-killing in horizontally transmitted parasites. *Proceedings of the Royal Society *B 282: 20152068.

GREGOROVIČ J. and ZALABOVÁ L. (2015): On automorphisms with natural tangent action on homogeneous parabolic geometries. *Journal of Lie Theory* 25: 677-715.

EISNER J., KUČERA M. and RECKE L. (2015): Direction and stability of bifurcating solutions for a Signorini problem. *Nonlinear Analysis* 113: 357-371.

KALOVÁ J. and MAREŠ R. (2015): Size dependences of surface tension. *International Journal of Thermophysics* 36: 2862-2868.

HRUBÝ J., VINŠ V., MAREŠ R., HYKL J. and KALOVÁ J. (2014): Surface tension of supercooled water: No inflection point down to –25°C. *Journal of Physical Chemistry Letters* 5: 425–428.

ČERMÁK M., KOZUBEK T., SYSALA S. and VALDMAN J. (2014): A TFETI Domain Decomposition Solver for Elastoplastic Problems. *Applied Mathematics and Computation* 231: 634–653.