**Numerical bifurcation analysis of mathematical models with time delays with the package DDE-BIFTOOL**
**Luzyanina T., Sieber J., Engelborghs K., Samaey G., Roose D.**
*Institute of Mathematical Problems of Biology - the branch of Keldysh Institute of Applied Mathematics, 142290 Pushchino, Russia*

Department of Mathematics, University of Exeter, Exeter EX4 4QF, UK

Materialise NV, Technologielaan 15, 3001 Leuven, Belgium

Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200 A, B-3001 Heverlee-Leuven, Belgium

**Abstract**. Mathematical modeling with delay differential equations (DDEs) is widely used for analysis and making predictions in various areas of the life sciences, e.g., population dynamics, epidemiology, immunology, physiology, neural networks. The time delays in these models take into account a dependence of the present state of the modeled system on its past history. The delay can be related to the duration of certain hidden processes like the stages of the life cycle, the time between infection of a cell and the production of new viruses, the duration of the infectious period, the immune period and so on. Due to an infinite-dimensional nature of DDEs, analytical studies of the corresponding mathematical models can only give limited results. Therefore, a numerical analysis is the major way to achieve both a qualitative and quantitative understanding of the model dynamics. A bifurcation analysis of a dynamical system is used to understand how solutions and their stability change as the parameters in the system vary. The package DDE-BIFTOOL is the first general-purpose package for bifurcation analysis of DDEs. This package can be used to compute and analyze the local stability of steady-state (equilibria) and periodic solutions of a given system as well as to study the dependence of these solutions on system parameters via continuation. Further one can compute and continue several local and global bifurcations: fold and Hopf bifurcations of steady states; folds, period doublings and torus bifurcations of periodic orbits; and connecting orbits between equilibria. In this paper we describe the structure of DDE-BIFTOOL, numerical methods implemented in the package and we illustrate the use of the package using a certain DDE system.

**Key words**: *nonlinear dynamics, delay differential equations, stability analysis, periodic solutions, collocation methods, numerical bifurcation analysis, state-dependent delay.*