On solutions to a FitzHugh–Rinzel type model

Ricerche di Matematica - A ternary autonomous dynamical system of FitzHugh–Rinzel type is analyzed. The system, at start, is reduced to a nonlinear integro differential equation. The...     

On a normed version of a Rogers–Shephard type problem

A translation body of a convex body is the convex hull of two of its translates intersecting each other. In the 1950s, Rogers and Shephard found the extremal values, over the family of n-dimensional convex bodies, of the maximal volume of the translation bodies of a given convex body. In our paper, we introduce a normed version of this problem, and for the planar case, determine the corresponding quantities for the four types of volumes regularly used in the literature: Busemann, Holmes–Thompson, and Gromov’s mass and mass*. We examine the problem also for higher dimensions, and for centrally symmetric convex bodies.     

On a phase field model of Cahn–Hilliard type for tumour growth with mechanical effects

Mechanical effects have mostly been neglected so far in phase field tumour models that are based on a Cahn–Hilliard approach. In this paper we study a macroscopic mechanical model for tumour growth in which cell–cell adhesion effects are taken into account with the help of a Ginzburg–Landau type energy. In the overall model an equation of Cahn–Hilliard type is coupled to the system of linear elasticity and a reaction–diffusion equation for a nutrient concentration. The highly non-linear coupling between a fourth-order Cahn–Hilliard equation and the quasi-static elasticity system lead to new challenges which cannot be dealt within a gradient flow setting which was the method of choice for other elastic Cahn–Hilliard systems. We show existence, uniqueness and regularity results. In addition, several continuous dependence results with respect to different topologies are shown. Some of these results give uniqueness for weak solutions and other results will be helpful for optimal control problems.     

Quantum revivals of a non-Rabi type in a Jaynes–Cummings model

We study full revivals (e.g., the reappearance in the unitary evolution) of quantum states in the Jaynes–Cummings model with the rotating wave approximation. We prove that in the case of a zero detuning in subspaces generated by two adjacent pairs of energy levels, full revival does not exist for any values of the parameters. In contrast, the set of parameters that allows full revival is everywhere dense in the set of all parameters in the case of a nonzero detuning. The nature of these revivals differs from Rabi oscillations for a single pair of energy levels. In more complex subspaces, the presence of full revival reduces to particular cases of the tenth Hilbert problem for rational solutions of systems of nonlinear algebraic equations, which has no algorithmic solution in the general case. Non-Rabi revivals become partial revivals in the case where the rotating wave approximation is rejected.     

On well-posedness of a dispersive system of the Whitham–Boussinesq type

The initial-value problem for a particular bidirectional Whitham system modelling surface water waves is under consideration. This system was recently introduced in Dinvay (2018). It is numerically shown to be stable and a good approximation to the incompressible Euler equations. Here we prove local in time well-posedness. Our proof relies on an energy method and a compactness argument. In addition some numerical experiments, supporting the validity of the system as an asymptotic model for water waves, are carried out.     

On the asymptotic behaviour of a pantograph–type difference equation

In this paper we study the asymptotic behaviour of solutions of the pantograph-type differnce equation, and obtain aymptotic estimates, which can imply asymptotic stability or stability of solutions     

On a fourth-order quasilinear elliptic equation of concave–convex type

We consider a fourth-order quasilinear equation depending on a nonnegative parameter λ and with subcritical or critical growth. Such equation is equivalent to a Hamiltonian system and the main goal of this work is to prove the existence of at least two positive and infinitely many solutions for such equation when the parameter λ is positive and small enough. This work was supported by FAPESP grant #07/54872-8.     

On perturbation of the Lotka–Volterra model

In this work, a general perturbed model is investigated and equilibrium is analyzed. The Lotka–Volterra version of the known system is considered. Local and global stability are established.     

The Cauchy problem for a Bardina–Oldroyd model to the incompressible viscoelastic flow

In this paper, we study the Cauchy problem for a regularized viscoelastic fluid model in space dimension two, the Bardina–Oldroyd model, which is inspired by the simplified Bardina model for the turbulent flows of fluids, introduced by Cao et al. (2006). In particular, we obtain the local existence of smooth solutions to this model via the contraction mapping principle. Furthermore, we prove the global existence of smooth solutions to this system.     

On selective harvesting of an inshore–offshore fishery: a bioeconomic model

A bioeconomic model is developed for the selective harvesting of a single species, inshore–offshore fishery, assuming that the growth of the species is governed by the Gompertz law. The dynamical system governing the fishery is studied in depth; the local and global stability of its non-trivial steady state are examined. Existence of a bionomic equilibrium is established under different parametric considerations. The optimal harvest policy is discussed by invoking Pontryagin's Maximum Principle. Lastly, the results are illustrated with the help of a numerical example.     

Chaotic dynamics of a discrete prey–predator model with Holling type II

A discrete-time prey–predator model with Holling type II is investigated. For this model, the existence and stability of three fixed points are analyzed. The bifurcation diagrams, phase portraits and Lyapunov exponents are obtained for different parameters of the model. The fractal dimension of a strange attractor of the model was also calculated. Numerical simulations show that the discrete model exhibits rich dynamics compared with the continuous model, which means that the present model is a chaotic, and complex one.     

On reaction–diffusion equations derived from a PEPA model

This work presents an extension of the fluid approximation to the diffusion approximation for a PEPA model. The diffusion approximation leads to a reaction–diffusion system in which the spatial or location information is considered, while it is ignored in the fluid approximation approach. The fundamental results for the solution of the reaction–diffusion equations, such as the existence, uniqueness, positivity and convergence, have been established. Numerical simulations are also given to illustrate the analytical results.     

On singularity formation of a 3D model for incompressible Navier–Stokes equations

We investigate the singularity formation of a 3D model that was recently proposed by Hou and Lei (2009) in [15] for axisymmetric 3D incompressible Navier–Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier–Stokes equations is that the convection term is neglected in the 3D model. This model shares many properties of the 3D incompressible Navier–Stokes equations. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the 3D inviscid model for a class of initial boundary value problems with smooth initial data of finite energy. We also prove the global regularity of the 3D inviscid model for a class of small smooth initial data.