Abstract
The consequences of regulatory T cell (Treg) inhibition of interleukine 2 secretion are examined by mathematical modelling. We find a transcritical bifurcation that might explain two alternative scenarios: in one case the appearance of autoimmune responses and in another the suppression of the immune response.

Abstract
We present a new R&D investment function in a Cournot competition model inspired in the logistic equation. We do a full characterization of the associated game and study the short- and long-term economical effects derived from using this new R&D investment function. In particular, we find the existence of regions with multiple Nash investment equilibria. For low production costs, that can correspond to the production of old technologies, the long-term economical effects are not very sensitive to small changes in the efficiency of the R&D programmes neither to small changes in the market structure. However, for high production costs, that can correspond to the production of new technologies, the long-term economical effects are very sensitive to small changes in the efficiency of the R&D programmes and also to small changes in the market structure.

Abstract
We use d'Aspremont and Jacquemin's strategic optimal R&D investment in a duopoly Cournot competition model to construct myopic optimal discrete and continuous R&D dynamics. We show that for some high initial production costs, the success or failure of a firm is very sensitive to small variations in its initial R&D investment strategies.

Abstract
The consequences of regulatory T cell (Treg) inhibition of interleukine 2 secretion are examined by mathematical modelling. We determine the analytic formula that describes the fine balance between Regulatory T cells and T cells at controlled and immune response equilibrium states. We demonstrate that cytokine dependent growth exhibits a quorum T cell population threshold that determines if immune responses develop on activation. We determine the analytic formulas of T cell proliferation thresholds that allow us to study the sensibility of the quorum growth thresholds controlling immune responses.

Abstract
We study a mathematical model of immune response by T cells where the regulatory T cells (Treg) inhibit interleukin 2 secretion. The bystander proliferation to an immune response is modelled. We consider an asymmetry reflecting that the difference between the growth and death rates can be higher for the active T cells and Tregs than for the inactive. This asymmetry leads to a better understanding of the bystander proliferation. An exposure to a pathogen results in an increased proliferation rate of the bystander T cells. If the population of the bystander T cells becomes large enough, autoimmunity can arise, eventually after a long transient period.

Abstract
The consequences of regulatory T cell (Treg) inhibition of interleukine 2 secretion are examined by mathematical modelling. We demonstrate that cytokine dependent growth exhibits quorum T cell population thresholds that determine whether immune responses develop on activation and whether the immune system returns to a control state. We study the effects in the quorum T cell population thresholds, by the T cell maximum growth rate, by the growth rate ratio between Tregs and T cells, by the value of the secretion rate of cytokines, and by the effectiveness of T cell secretion inhibition by Tregs.