ISSN:

1531-3492

eISSN:

1553-524X

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### Volume 10, 2008

## Discrete & Continuous Dynamical Systems - B

2007 , Volume 7 , Issue 2

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2007, 7(2): 201-218
doi: 10.3934/dcdsb.2007.7.201

*+*[Abstract](402)*+*[PDF](299.2KB)**Abstract:**

In this paper, we analyze theoretically an age structured population model with cannibalism. The model is nonlinear in that cannibalism decreases the birth rate based on total population density. We use degree theory to prove the existence of unique solution. We also investigate the asymptotic stability of the solutions, and prove under special hypotheses, local and global attractivity of a unique nontrivial steady state. We convert the problem to a delay differential equation and prove that quasiconvergence leads to global attraction. Some numerical simulations are presented exhibiting sustained oscillations which may occur when the hypotheses of theoretical analysis are not satisfied.

2007, 7(2): 219-250
doi: 10.3934/dcdsb.2007.7.219

*+*[Abstract](342)*+*[PDF](363.2KB)**Abstract:**

In this paper we propose the analysis of the incompressible non-homogeneous Navier-Stokes equations with nonlinear outflow boundary condition. This kind of boundary condition appears to be, in some situations, a useful way to perform numerical computations of the solution to the unsteady Navier-Stokes equations when the Dirichlet data are not given explicitly by the physical context on a part of the boundary of the computational domain. The boundary condition we propose, following previous works in the homogeneous case, is a relationship between the normal component of the stress and the outflow momentum flux taking into account inertial effects. We prove the global existence of a weak solution to this model both in 2D and 3D. In particular, we show that the nonlinear boundary condition under study holds for such a solution in a weak sense, even though the normal component of the stress and the density may not have traces in the usual sense.

2007, 7(2): 251-273
doi: 10.3934/dcdsb.2007.7.251

*+*[Abstract](710)*+*[PDF](185.5KB)**Abstract:**

This paper is devoted to the study of travelling wave solutions for a simple epidemic model. This model consists in a single scalar equation with age-dependence and spatial structure. We prove the existence of travelling waves for a continuum of admissible wave speeds as well as some qualitative properties, like exponential decay and monotonicity with respect to the direction of front's propagation. Our proofs extensively use the comparison principle that allows us to construct suitable sub and super-solutions or to use the classical sliding method to obtain qualitative properties of the wave front.

2007, 7(2): 275-284
doi: 10.3934/dcdsb.2007.7.275

*+*[Abstract](359)*+*[PDF](337.3KB)**Abstract:**

The standard Melnikov method for analyzing the onset of chaos in the vicinity of a separatrix is used to explore the possibility of suppression of chaos of a certain class of dynamical systems. For a given dynamical system we apply an external perturbation, which we call the stabilizing perturbation, with the goal that after its action the chaos present in the system is suppressed. We apply this method to the nonlinear pendulum as a paradigm, and obtain some analytical expressions for the corresponding external perturbations that eliminate chaotic behavior. Numerical simulations in the pendulum show a complete agreement with the analytical results.

2007, 7(2): 285-291
doi: 10.3934/dcdsb.2007.7.285

*+*[Abstract](394)*+*[PDF](184.3KB)**Abstract:**

In this paper, we study a two-dimensional Burgers--Korteweg-de Vries-type equation with higher-order nonlinearities. A class of solitary wave solution is obtained by means of the Divisor Theorem which is based on the ring theory of commutative algebra. Our result indicates that the presentation of traveling wave solution in [J. Phys. A (Math. Gen.) 35 (2002) 8253--8265] is incorrect; an explanation as to why this is so is given.

2007, 7(2): 293-314
doi: 10.3934/dcdsb.2007.7.293

*+*[Abstract](362)*+*[PDF](319.1KB)**Abstract:**

In systems governing two-dimensional turbulence, surface quasi-geostrophic turbulence, (more generally $\alpha$-turbulence), two-layer quasi-geostrophic turbulence, etc., there often exist two conservative quadratic quantities, one "energy''-like and one "enstrophy''-like. In a finite inertial range there are in general two spectral fluxes, one associated with each conserved quantity. We derive here an inequality comparing the relative magnitudes of the "energy'' and "enstrophy'' fluxes for finite or infinitesimal dissipations, and for hyper or hypo viscosities. When this inequality is satisfied, as is the case of 2D turbulence,where the energy flux contribution to the energy spectrum is small, the subdominant part will be effectively hidden. In sQG turbulence, it is shown that the opposite is true: the downscale energy flux becomes the dominant contribution to the energy spectrum. A combination of these two behaviors appears to be the case in 2-layer QG turbulence, depending on the baroclinicity of the system.

2007, 7(2): 315-350
doi: 10.3934/dcdsb.2007.7.315

*+*[Abstract](360)*+*[PDF](304.4KB)**Abstract:**

In this paper we give a rigorous mathematical proof of the instability of stationary radial flame ball solutions of a three dimensional free boundary model which models the combustion of a gaseous mixture with dust in a microgravity environment.

2007, 7(2): 351-363
doi: 10.3934/dcdsb.2007.7.351

*+*[Abstract](413)*+*[PDF](210.4KB)**Abstract:**

In this paper, we introduce a class of one-dimensional non-autonomous dynamical systems that allow an explicit study of their orbits, of the associated variational equations as well as of certain types of bifurcations. In a special case, the model class can be transformed into the non-autonomous Beverton-Holt equation. We use these model functions for analyzing various notions of non-autonomous transcritical and pitchfork bifurcations that have been recently proposed in the literature.

2007, 7(2): 365-375
doi: 10.3934/dcdsb.2007.7.365

*+*[Abstract](491)*+*[PDF](145.2KB)**Abstract:**

An age structured $s$-$i$-$s$ epidemic model with random diffusion is studied. The model is described by the system of nonlinear and nonlocal integro-differential equations. Finite differences along the characteristics in age-time domain combined with Galerkin finite elements in spatial domain are used in the approximation. It is shown that a positive periodic solution to the discrete system resulting from the approximation can be generated, if the initial condition is fertile. It is proved that the endemic periodic solution is globally stable once it exists.

2007, 7(2): 377-398
doi: 10.3934/dcdsb.2007.7.377

*+*[Abstract](370)*+*[PDF](338.9KB)**Abstract:**

We discuss the applicability of Kolmogorov's theorem on existence of invariant tori to the real Sun-Jupiter-Saturn system. Using computer algebra, we construct a Kolmogorov's normal form defined in a neighborhood of the actual orbit in the phase space, giving a sharp evidence of the convergence of the algorithm. If not a rigorous proof, we consider our calculation as a strong indication that Kolmogorov's theorem applies to the motion of the two biggest planets of our solar system.

2007, 7(2): 399-424
doi: 10.3934/dcdsb.2007.7.399

*+*[Abstract](308)*+*[PDF](404.9KB)**Abstract:**

A family of delay-differential models of the glucose-insulin system is introduced, whose members represent adequately the Intra-Venous Glucose Tolerance Test and allied experimental procedures of diabetological interest. All the models in the family admit positive bounded unique solutions for any positive initial condition and are persistent. The models agree with the physics underlying the experiments, and they all present a unique positive equilibrium point.

Local stability is investigated in a pair of interesting member models: one, a discrete-delays differential system; the other, a distributed-delay system reducing to an ordinary differential system evolving on a suitably defined extended state space. In both cases conditions are given on the physical parameters in order to ensure the local asymptotic stability of the equilibrium point. These conditions are always satisfied, given the actual parameter estimates obtained experimentally. A study of the global stability properties is performed, but while from simulations it could be conjectured that the models considered are globally asymptotically stable, sufficient stability criteria, formally derived, are not actually satisfied for physiological parameters values. Given the practical importance of the models studied, further analytical work may be of interest to conclusively characterize their behavior.

2007, 7(2): 425-440
doi: 10.3934/dcdsb.2007.7.425

*+*[Abstract](401)*+*[PDF](299.6KB)**Abstract:**

We show that in the limit of small Rossby number $\varepsilon$, the primitive equations of the ocean (OPEs) can be approximated by "higher-order quasi-geostrophic equations'' up to an exponential accuracy in $\varepsilon$. This approximation assumes well-prepared initial data and is valid for a timescale of order one (independent of $\varepsilon$). Our construction uses Gevrey regularity of the OPEs and a classical method to bound errors in higher-order perturbation theory.

2007, 7(2): 441-456
doi: 10.3934/dcdsb.2007.7.441

*+*[Abstract](378)*+*[PDF](291.3KB)**Abstract:**

In this paper I will investigate the bifurcation and asymptotic behavior of solutions of the Swift-Hohenberg equation and the generalized Swift-Hohenberg equation with the Dirichlet boundary condition on a one-dimensional domain $(0,L)$. I will also study the bifurcation and stability of patterns in the $n$-dimensional Swift-Hohenberg equation with the odd-periodic and periodic boundary conditions. It is shown that each equation bifurcates from the trivial solution to an attractor $\mathcal A_\lambda$ when the control parameter $\lambda$ crosses $\lambda _{c} $, the principal eigenvalue of $(I+\Delta)^2$. The local behavior of solutions and their bifurcation to an invariant set near higher eigenvalues are analyzed as well.

2007, 7(2): 457-463
doi: 10.3934/dcdsb.2007.7.457

*+*[Abstract](358)*+*[PDF](198.4KB)**Abstract:**

Let $f:\M\to\M$ be a continuous map of a locally compact metric space. Models of interacting populations often have a closed invariant set $\partial \M$ that corresponds to the loss or extinction of one or more populations. The dynamics of $f$ subject to bounded random perturbations for which $\partial \M$ is absorbing are studied. When these random perturbations are sufficiently small, almost sure absorbtion (i.e. extinction) for all initial conditions is shown to occur if and only if $M\setminus \partial M$ contains no attractors for $f$. Applications to evolutionary bimatrix games and uniform persistence are given. In particular, it shown that random perturbations of evolutionary bimatrix game dynamics result in almost sure extinction of one or more strategies.

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