We search for limit cycles in the dynamical model of two-species chemical reactions that contain seven reaction rate coefficients as parameters and at least one third-order reaction step, that is, the induced kinetic differential equation of the reaction is a planar cubic differential system. Symbolic calculations were carried out using the Mathematica computer algebra system, and it was also used for the numerical verifications to show the following facts: the kinetic differential equations of these reactions each have two limit cycles surrounding the stationary point of focus type in the positive quadrant. In the case of Model 1, the outer limit cycle is stable and the inner one is unstable, which appears in a supercritical Hopf bifurcation. Moreover, the oscillations in a neighborhood of the outer limit cycle are slow-fast oscillations. In the case of Model 2, the outer limit cycle is unstable and the inner one is stable. With another set of parameters, the outer limit cycle can be made stable and the inner one unstable.

Obesity and diabetes are a progressively more and more deleterious hallmark of modern, well fed societies. In order to study the potential impact of strategies designed to obviate the pathological consequences of detrimental lifestyles, a model for the development of Type 2 diabetes geared towards large population simulations would be useful. The present work introduces such a model, representing in simplified fashion the interplay between average glycemia, average insulinemia and functional beta -cell mass, and incorporating the effects of excess food intake or, conversely, of physical activity levels. Qualitative properties of the model are formally established and simulations are shown as examples of its use.

The most general change of variables theorem for the Riemann integral of functions of a single variable has been published in 1961 (by Kestelman). In this theorem, the substitution is made by an `indefinite integral', that is, by a function of the form (t\mapsto c+\int_a^tg=:G(t)) where (g) is Riemann integrable on ([a,b]) and (c) is any constant. We prove a multidimensional generalization of this theorem for the case where (G) is injective -- using the fact that the Riemann primitives are the same as those Lipschitz functions which are almost everywhere strongly differentiable in ((a,b)). We prove a generalization of Sard's lemma for Lipschitz functions of several variables that are almost everywhere strongly differentiable, which enables us to keep all our proofs within the framework of the Riemannian theory which was our aim.