# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

May 2009 , Volume 11 , Issue 3

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2009, 11(3): 541-561 doi: 10.3934/dcdsb.2009.11.541 +[Abstract](1603) +[PDF](380.1KB)
Abstract:
The paper is devoted to integro-differential equations arising in population dynamics. The integral term describes the nonlocal consumption of resources. We study the Fredholm property of the corresponding linear operators and use it to prove the existence of travelling waves when the support of the integral is sufficiently small. In this case, the integro-differential operator is close to the differential operator and we can use the implicit function theorem. We carry out numerical simulations in order to study the case where the support of the integral is not small. We observe various regimes of wave propagation. Some of them, in particular periodic waves do not exist for the usual reaction-diffusion equation.
2009, 11(3): 563-585 doi: 10.3934/dcdsb.2009.11.563 +[Abstract](1419) +[PDF](259.0KB)
Abstract:
An equation describing the dynamical behaviour of phytoplankton cells is considered in which the effects of cell division and aggregration are incorporated by coupling the coagulation-fragmentation equation with the McKendrick-von Foerster renewal model of an age-structured population. Under appropriate conditions on the model parameters, the associated initial-boundary value problem is shown to be well posed in a physically relevant Banach space.
2009, 11(3): 587-611 doi: 10.3934/dcdsb.2009.11.587 +[Abstract](2512) +[PDF](596.7KB)
Abstract:
In this paper we study the dynamics of a vector-transmitted disease using two deterministic models. First, we look at time dependent prevention and treatment efforts, where optimal control theory is applied. Using analytical and numerical techniques, it is shown that there are cost effective control efforts for treatment of hosts and prevention of host-vector contacts. Then, we considered the autonomous counter part of the mode and we established global stability results based on the reproductive number. The model is applied to study the effects of prevention and treatment controls on a malaria disease while keeping the implementation cost at a minimum. Numerical results indicate the effects of the two controls (prevention and treatment) in lowering exposed and infected members of each of the populations. The study also highlights the effects of some model parameters on the results.
2009, 11(3): 613-627 doi: 10.3934/dcdsb.2009.11.613 +[Abstract](1180) +[PDF](196.7KB)
Abstract:
A nonlinear cyclin content structured cell population model is considered. The population is divided into two types of cells: proliferative and quiescent. Under suitable hypotheses, existence and uniqueness of a steady state of this model is proved by using positive linear semigroup theory.
2009, 11(3): 629-653 doi: 10.3934/dcdsb.2009.11.629 +[Abstract](1259) +[PDF](310.1KB)
Abstract:
We provide sufficient conditions for the existence and Lipschitz continuity of solutions to the constrained Bolza optimal control problem

$\text{minimize}\quad \int_0^T L(x(t),u(t))\dt + l(x(T))$

over all trajectory / control pairs $(x,u)$, subject to the state equation

x'(t)=$f(x(t),u(t))$ for a.e. $t\in [0,T]$
$u(t)\in U$ for a.e. $t\in [0,T]$
$x(t)\in K$ for every $t\in [0,T]$
$x(0)\in Q_0\.$

The main feature of our problem is the unboundedness of $f(x,U)$ and the absence of superlinear growth conditions for $L$. Such classical assumptions are here replaced by conditions on the Hamiltonian that can be satisfied, for instance, by some Lagrangians with no growth. This paper extends our previous results in Existence and Lipschitz regularity of solutions to Bolza problems in optimal control to the state constrained case.

2009, 11(3): 655-668 doi: 10.3934/dcdsb.2009.11.655 +[Abstract](1272) +[PDF](355.0KB)
Abstract:
We consider a control system for a Korteweg-de Vries equation with homogeneous Dirichlet boundary conditions and Neumann boundary control. We address the rapid exponential stabilization problem. More precisely, we build some feedback laws forcing the solutions of the closed-loop system to decay exponentially to zero with arbitrarily prescribed decay rates. We also perform some numerical computations in order to illustrate this theoretical result.
2009, 11(3): 669-690 doi: 10.3934/dcdsb.2009.11.669 +[Abstract](1062) +[PDF](2599.6KB)
Abstract:
An adaptive algorithm for steady convection-diffusion problems that combines a posteriori error estimation with conforming centroidal Voronoi Delaunay triangulations (CfCVDTs) is proposed and tested in two dimensional domains. Different from most current adaptive methods, this algorithm realizes mesh refinement and coarsening implicitly at each level by nodes insertion and redistribution. Especially, the nodes redistribution is implemented through the generation of CfCVDTs with some density function derived from the a posteriori error estimators for the problem. Numerical experiments show that the convergence rates achieved are almost the best obtainable using the linear finite volume discretizations and the resulting meshes always maintain high quality.
2009, 11(3): 691-715 doi: 10.3934/dcdsb.2009.11.691 +[Abstract](995) +[PDF](331.3KB)
Abstract:
Anti-angiogenesis is a novel cancer treatment that targets the vasculature of a growing tumor. In this paper a metasystem is formulated and analyzed that describes the dynamics of the primary tumor volume and its vascular support under anti-angiogenic treatment. The system is based on a biologically validated model by Hahnfeldt et al. and encompasses several versions of this model considered in the literature. The problem how to schedule an a priori given amount of angiogenic inhibitors in order to achieve the maximum tumor reduction possible is formulated as an optimal control problem with the dosage of inhibitors playing the role of the control. It is investigated how properties of the functions defining the growth of the tumor and the vasculature in the general system affect the qualitative structure of the solution of the problem. In particular, the presence and optimality of singular controls is determined for various special cases. If optimal, singular arcs are the central part of a regular synthesis of optimal trajectories providing a full solution to the problem. Two specific examples of a regular synthesis including optimal singular arcs are given.
2009, 11(3): 717-740 doi: 10.3934/dcdsb.2009.11.717 +[Abstract](1281) +[PDF](285.7KB)
Abstract:
The effects of a higher vorticity moment on a variational problem for barotropic vorticity on a rotating sphere is examined rigorously in the framework of the Direct Method. This variational model differs from previous work on the Barotropic Vorticity Equation (BVE) in relaxing the angular momentum constraint, which then allows us to state and prove theorems that give necessary and sufficient conditions for the existence and stability of constrained energy extremals in the form of super and sub-rotating solid-body steady flows. Relaxation of angular momentum is a necessary step in the modeling of the important tilt instability where the rotational axis of the barotropic atmosphere tilts away from the fixed north-south axis of planetary spin. These conditions on a minimal set of parameters consisting of the planetary spin, relative enstrophy and the fourth vorticity moment, extend the results of previous work and clarify the role of the higher vorticity moments in models of geophysical flows.
2009, 11(3): 741-784 doi: 10.3934/dcdsb.2009.11.741 +[Abstract](1136) +[PDF](663.7KB)
Abstract:
The process of phase separation of binary systems is described by the Cahn-Hilliard equation. The main objective of this article is to give a classification on the dynamic phase transitions for binary systems using either the classical Cahn-Hilliard equation or the Cahn-Hilliard equation coupled with entropy, leading to some interesting physical predictions. The analysis is based on dynamic transition theory for nonlinear systems and new classification scheme for dynamic transitions, developed recently by the authors.
2009, 11(3): 785-803 doi: 10.3934/dcdsb.2009.11.785 +[Abstract](1110) +[PDF](399.2KB)
Abstract:
Two identical 2D dynamical systems written in a general form and depending on a parameter were coupled linearly and non-symmetrically. The Hopf bifurcation in the 4D system is studied. New formulae for the computation of the first Lyapunov coefficient are obtained in two cases of Hopf bifurcation. They use only 2D vectors and generalize the expressions of the first Lyapunov coefficient deduced in [13] for symmetrically coupled dynamical systems. In addition, formulae for the computation of the second Lyapunov coefficient are obtained in terms of 2D vectors.
A particular 4D dynamical system obtained by coupling non-symmetrically two identical 2D advertising models is considered. It depends on 4 parameters, two of them being the coupling parameters. A study of the Hopf bifurcation around the symmetric equilibrium point is performed using the general formulae obtained by us for the computation of the Lyapunov coefficients. Even if the Hopf bifurcation in the single system is always supercritical, for the coupled system the Hopf bifurcation can be supercritical, subcritical or degenerate. The generalized Hopf bifurcation (Bautin) is illustrated by our numerical computations using the software winpp [4]. The results obtained when the oscillators are non-symmetrically coupled are compared with those when they are symmetrically coupled.

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