We present a simple model of population dynamics in the presence of an infection. The model is based on discrete-time equations for sane and infected populations in interaction and correctly describes the dynamics of the epidemic. We find that for some choices of the parameters, the model can...

We present a three-component model which could represent the reaction of the organism to pathogen invasion. A continuous-time (differential) model is constructed first. Its discrete analogue is then derived and is used for numerical simulations which show a great variety of behaviours. We also...

We present the discrete systems which result from the discrete Painlevé equations q-PVI and d-PV associated to the affine Weyl group E7(1). Two different procedures (“limits” and “degeneracies”) are used, giving rise to a host of new discrete Painlevé equations but also to some...

We examine the transition between discrete and ultradiscrete (cellular-automaton-like) systems, the dynamics of which exhibit limit cycles. Motivated by results obtained previously for three-dimensional systems, we consider here a more manageable two-dimensional model. We show that one can...

We consider the autonomous limit of the known families of discrete Painlevé equations. We show how these autonomous systems can be written as special cases of the QRT mapping. For the latter, we present in detail its integration in both the symmetric and asymmetric cases.

We examine a family of three-point mappings that include mappings solvable through linearization. The different origins of mappings of this type are examined: projective equations and Gambier systems. The integrable cases are obtained through the application of the singularity confinement...

A method already introduced by the last two authors for finding the integrable cases of three-dimensional autonomous ordinary differential equations based on the Frobenius integrability theorem is described in detail. Using this method and computer algebra, the so-called three-dimensional...

We study the projective systems in both continuous and discrete settings. These systems are linearizable by construction and thus, obviously, integrable. We show that in the continuous case it is possible to eliminate all variables but one and reduce the system to a single differential equation....

We consider a one-parameter family of two-degrees-of-freedom generalized Toda Hamiltonians. Using Ziglin's theorem with some extensions, we prove the non-existence of an additional single-valued analytic integral except for special values of the parameter.

We use the dressing transformation in order to reconstruct one-dimensional Hamiltonians starting from their spectra. Whenever the given spectrum departs from oscillator-like local behaviour the resulting potential is fractal. An estimate of this fractal dimension is presented.