ISSN:

1531-3492

eISSN:

1553-524X

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### Volume 10, 2008

## Discrete & Continuous Dynamical Systems - B

November 2002 , Volume 2 , Issue 4

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2002, 2(4): 473-482
doi: 10.3934/dcdsb.2002.2.473

*+*[Abstract](1578)*+*[PDF](139.6KB)**Abstract:**

Optimal control theory is applied to a system of ordinary differential equations modeling a two-strain tuberculosis model. Seeking to reduce the latent and infectious groups with the resistant-strain tuberculosis, we use controls representing two types of treatments. The optimal controls are characterized in terms of the optimality system, which is solved numerically for several scenarios.

2002, 2(4): 483-494
doi: 10.3934/dcdsb.2002.2.483

*+*[Abstract](747)*+*[PDF](191.4KB)**Abstract:**

We study the linear and nonlinear stability of stationary solutions of the forced two-dimensional Navier-Stokes equations on the domain $[0,2\pi]\times[0,2\pi/\alpha]$, where $\alpha\in(0,1]$, with doubly periodic boundary conditions. For the linear problem we employ the classical energy--enstrophy argument to derive some fundamental properties of unstable eigenmodes. From this it is shown that forces of pure $x_2$-modes having wavelengths greater than $2\pi$ do not give rise to linear instability of the corresponding primary stationary solutions. For the nonlinear problem, we prove the equivalence of nonlinear stability with respect to the energy and enstrophy norms. This equivalence is then applied to derive optimal conditions for nonlinear stability, including both the high- and low-Reynolds-number limits.

2002, 2(4): 495-520
doi: 10.3934/dcdsb.2002.2.495

*+*[Abstract](640)*+*[PDF](420.6KB)**Abstract:**

The purpose of this paper is to study the mathematical properties of the solutions of a model for bacteria and virulent bacteriophage system in a chemostat. A general model was first proposed by Levin, Stewart and Chao [13] and then, a specific one, by Lenski and Levin [12]. The numerical simulations come from the experimental data referred in [12,13]. In our Knowledge the analysis presented herefollowing is the first mathematical attempt to analyse the model of bacteria and virulent bacteriophage and presents two fresh frontiers: 1) the modeling of delay (latency period) incorporating the realistic through time death rate in linear stability analysis brings to characteristic equations with delay dependent parameters for which only recently Beretta and Kuang [5] provided a geometric stability switch criterion which application is presented along the paper; 2) the modelling of the dynamics through three full delay stages can be reduced to two using the integral representation for the density of infected bacteria. The basic properties of the model which are investigated are the existence of equilibria, positive invariance and boundedness of solutions and permanence results. Second, using the geometric stability switch criterion in the delay differential system with delay dependent parameters, we present the local asymptotic stability of the equilibria by analyzing the corresponding characteristic equation which coefficients depend on the time delay (the latency period). Numerical simulations are presented to illustrate the results of local stability. Then, we study the global asymptotic stability of the boundary equilibria via Liapunov functional method. Finally, we give a discussion about the model.

2002, 2(4): 521-540
doi: 10.3934/dcdsb.2002.2.521

*+*[Abstract](617)*+*[PDF](3741.4KB)**Abstract:**

We numerically investigate the dynamics of a symmetric rigid body with a fixed point in a small analytic external potential (equivalently, a fast rotating body in a given external field) in the light of previous theoretical investigations based on Nekhoroshev theory. Special attention is posed on "resonant" motions, for which the tip of the unit vector $\mu$ in the direction of the angular momentum vector can wander, for no matter how small $\varepsilon$, on an extended, essentially two-dimensional, region of the unit sphere, a phenomenon called "slow chaos". We produce numerical evidence that slow chaos actually takes place in simple cases, in agreement with the theoretical prediction. Chaos however disappears for motions near proper rotations around the symmetry axis, thus indicating that the theory of these phenomena still needs to be improved. An heuristic explanation is proposed.

2002, 2(4): 541-560
doi: 10.3934/dcdsb.2002.2.541

*+*[Abstract](683)*+*[PDF](242.7KB)**Abstract:**

In this paper we investigate global stability for a differential equation containing a positively homogeneous nonlinearity. We first consider perturbations of the infinitesimal generator of a strongly continuous semigroup which has a simple dominant eigenvalue. We prove that for "small" perturbation by a positively homogeneous nonlinearity the qualitative properties of the linear semigroup persist. From this result, we deduce a global stability result when one adds a certain type of saturation term. We conclude the paper by an application to a phenotype structured population dynamic model.

2002, 2(4): 561-574
doi: 10.3934/dcdsb.2002.2.561

*+*[Abstract](830)*+*[PDF](166.8KB)**Abstract:**

In this article we study the stability properties of two different configurations in nematic liquid crystals. One of them is the static configuration in the presence of magnetic fields. The other one is the Poiseuille flow under the model of Ericksen for liquid crystals with variable degree of orientation [E, 91]. In the first case, we show that the planar radial symmetry solution is stable with respect to the small external magnetic field. Such phenomenon illustrates the competition mechanism between the magnetic field and the strong anchoring boundary conditions. In the Poiseuille flow case, we show that the stationary configuration obtained from our previous works [C-L, 99] [C-M, 96] is stable when the velocity gradient is small.

2002, 2(4): 575-590
doi: 10.3934/dcdsb.2002.2.575

*+*[Abstract](545)*+*[PDF](408.7KB)**Abstract:**

We propose a reaction-diffusion extension of a two species ecotoxicological model with time-delays proposed by Chattopadhyay

*et al*(1997). Each species has the capacity to produce a substance toxic to its competitor, and a distributed time-delay is incorporated to model lags in the production of toxin. Additionally, nonlocal spatial effects are present because of the combination of delay and diffusion. The stability of the various uniform equilibria of the model are studied by using linearised analysis, on an infinite spatial domain. It is shown that simple exponentially decaying delay kernels cannot destabilise the coexistence equilibrium state. In the case of a finite spatial domain, with purely temporal delays, a nonlinear convergence result is proved using ideas of Lyapunov functionals together with invariant set theory. The result is also applicable to the purely temporal system studied by other investigators and, in fact, extends their results.

2002, 2(4): 591-607
doi: 10.3934/dcdsb.2002.2.591

*+*[Abstract](721)*+*[PDF](215.7KB)**Abstract:**

We study the existence, uniqueness and long time behaviour of a system consisting of the viscous Burgers' equation coupled to a kinetic equation. This system models the motion of a dispersed phase made of inertial particles immersed in a fluid modelled by the Burgers' equation. The initial conditions are in $L^\infty+W^{1,1}(\mathbb{R}_x)$ for the fluid and in the space $\mathcal {M}(\mathbb{R}_x\times\mathbb{R}_v\times\mathbb{R}_r)$ of bounded measures for the dispersed phase. This means that the limiting case where the particles are regarded as point particles is taken into account. First, we prove the existence and uniqueness of solutions to the system by using the regularizing properties of the viscous Burgers' equation. Then, we prove that the usual stability properties of travelling waves for the viscous Burgers' equation is not affected by the coupling with a small mass of inertial particles.

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