# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

June 2007 , Volume 7 , Issue 4

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2007, 7(4): 683-698 doi: 10.3934/dcdsb.2007.7.683 +[Abstract](1351) +[PDF](212.0KB)
Abstract:
A selection model with $n$ traits is considered. It is assumed that the mortality function is density dependent and that individuals with "weak" traits are able to disperse to a safe refuge patch and avoid competition with individuals carrying the strongest trait. It is shown that if any subpopulation with a "weak" trait does not have a safe refuge then it will become extinct. Therefore, for survival of $n$ traits $n-1$ safe refuge patches are needed. When $n-1$ refuge patches are available global stability of the interior equilibrium is proved provided that the fittest trait is sufficiently better than the other traits. Finally, two special cases with linear and Beverton-Holt density dependent mortality functions are studied in detail.
2007, 7(4): 699-715 doi: 10.3934/dcdsb.2007.7.699 +[Abstract](1602) +[PDF](255.9KB)
Abstract:
We study the long-time asymptotics of a certain class of nonlinear diffusion equations with time-dependent diffusion coefficients which arise, for instance, in the study of transport by randomly fluctuating velocity fields. Our primary goal is to understand the interplay between anomalous diffusion and nonlinearity in determining the long-time behavior of solutions. The analysis employs the renormalization group method to establish the self-similarity and to uncover universality in the way solutions decay to zero.
2007, 7(4): 717-734 doi: 10.3934/dcdsb.2007.7.717 +[Abstract](1510) +[PDF](1511.9KB)
Abstract:
Multiresolution methods such as the wavelet packets or the cosine packets are more and more used in physical applications and in particular in two-dimensional turbulence. The numerical results of the first part of this paper have shown that the wavelet packets decomposition is well suited for studying this kind of problem: the visualization of the vorticity field is better, without any artefacts, than the visualization with the cosine packets filtering. The current second part of the paper is devoted to the physical interpretation of the filtered fields obtained in the first part. Energy and enstrophy spectra as well as energy and enstrophy fluxes are computed to determine the role of each filtered field with respect to the cascades.
2007, 7(4): 735-754 doi: 10.3934/dcdsb.2007.7.735 +[Abstract](1530) +[PDF](253.1KB)
Abstract:
This paper is devoted to study the well-posedness and the asymptotic behavior of a population equation with diffusion in $L^1$. The death and birth rates depend on the age and the spatial variable. Here we allow the birth process to depend also on some modified delay. This paper is a continuation of the studies done by Nickel, Rhandi and Schnaubelt in [28][32][33] and Fragnelli, Maniar, Piazzera and Tonetto in [15][21][29][30].
2007, 7(4): 755-778 doi: 10.3934/dcdsb.2007.7.755 +[Abstract](1351) +[PDF](468.1KB)
Abstract:
We introduce a regularization for the minimal distance maps, giving the locally minimal values of the distance between two points on two confocal Keplerian orbits. This allows to define a meaningful uncertainty for the minimal distance also when orbit crossings are possible, and it is useful to detect the possibility of collisions or close approaches between two celestial bodies moving approximatively on these orbits, with important consequences in the study of their dynamics. An application to the orbit of a recently discovered near-Earth asteroid is also given.
2007, 7(4): 779-792 doi: 10.3934/dcdsb.2007.7.779 +[Abstract](1258) +[PDF](246.0KB)
Abstract:
A chemostat with two organisms competing for a single growth-limiting nutrient controlled by feedback-mediated dilution rate is analyzed. A specific feedback function is constructed which yields circular and elliptical periodic orbits for the limiting system. A theorem on the stabilization of periodic orbits in conservative systems is developed and for a given elliptical orbit, the result is used to modify the chemostat so that the chosen orbit is asymptotically stable. Finally, the feedback function is modified so that finitely many nested periodic orbits of alternating stability exist.
2007, 7(4): 793-806 doi: 10.3934/dcdsb.2007.7.793 +[Abstract](1608) +[PDF](188.7KB)
Abstract:
In this paper we study several aspects of solitary wave solutions of the Ostrovsky equation. Using variational methods, we show that as the rotation parameter goes to zero, ground state solitary waves of the Ostrovsky equation converge to solitary waves of the Korteweg-deVries equation. We also investigate the properties of the function $d(c)$ which determines the stability of the ground states. Using an important scaling identity, together with numerical approximations of the solitary waves, we are able to numerically approximate $d(c)$. These calculations suggest that $d$ is convex everywhere, and therefore all ground state solitary waves of the Ostrovsky equation are stable.
2007, 7(4): 807-823 doi: 10.3934/dcdsb.2007.7.807 +[Abstract](1968) +[PDF](212.1KB)
Abstract:
We carry out error estimation of a class of immersed finite element (IFE) methods for elliptic interface problems with both perfect and imperfect interface jump conditions. A key feature of these methods is that their partitions can be independent of the location of the interface. These quadratic IFE spaces reduce to the standard quadratic finite element space when the interface is not in the interior of any element. More importantly, we demonstrate that these IFE spaces have the optimal (slightly lower order in one case) approximation capability expected from a finite element space using quadratic polynomials.
2007, 7(4): 825-837 doi: 10.3934/dcdsb.2007.7.825 +[Abstract](1542) +[PDF](184.5KB)
Abstract:
This paper focuses on the phase transitions of a 2$\times$2 system of mixed type for viscosity-capillarity with periodic initial-boundary condition in a viscoelastic material. By the Liapunov functional method, we prove the existence, uniqueness, regularity and uniform boundedness of the solution. The results are correct even for large initial data.
2007, 7(4): 839-857 doi: 10.3934/dcdsb.2007.7.839 +[Abstract](1390) +[PDF](1676.0KB)
Abstract:
We present some new results on the asymptotic behavior of the periodic solution to a 2$\times$2 mixed-type system of viscosity-capillarity in a viscoelastic material. We prove that the solution converges to a certain stationary solution as time approaches to infinity, in particular, when the viscosity is large enough or the mean of the initial datum is in the hyperbolic regions, the solution converges exponentially to the trivial stationary solution with it any large initial datum. The location of the initial datum and the amplitude of viscosity play a key role for the phase transitions. Furthermore, we obtain the convergence rate to the stationary solutions. Finally, we carry out numerical simulations to confirm the theoretical predictions.
2007, 7(4): 859-883 doi: 10.3934/dcdsb.2007.7.859 +[Abstract](1304) +[PDF](358.4KB)
Abstract:
We study the asymptotic behavior of the eigenvalues $\beta^\varepsilon$ and the associated eigenfunctions of an $\varepsilon$-dependent Steklov type eigenvalue problem posed in a bounded domain $\Omega$ of $\R^2$, when $\varepsilon \to 0$. The eigenfunctions $u^\varepsilon$ being harmonic functions inside $\Omega$, the Steklov condition is imposed on segments $T^\varepsilon$ of length $O(\varepsilon)$ periodically distributed on a fixed part $\Sigma$ of the boundary $\partial \Omega$; a homogeneous Dirichlet condition is imposed outside. The homogenization of this problem as $\varepsilon \to 0$ involves the study of the spectral local problem posed in the unit reference domain, namely the half-band $G=(-P/2,P/2)\times (0,+\infty)$ with $P$ a fixed number, with periodic conditions on the lateral boundaries and mixed boundary conditions of Dirichlet and Steklov type respectively on the segment lying on $\{y_2=0\}$. We characterize the asymptotic behavior of the low frequencies of the homogenization problem, namely of $\beta^\varepsilon\varepsilon$, and the associated eigenfunctions by means of those of the local problem.
2007, 7(4): 885-905 doi: 10.3934/dcdsb.2007.7.885 +[Abstract](1438) +[PDF](256.7KB)
Abstract:
We investigate the problem of existence of a probabilistic weak solution for the initial boundary value problem for the model doubly degenerate stochastic quasilinear parabolic equation

$d(|y|^{\alpha -2}y) - [ \sum_{i=1}^{n} \frac{\partial }{\partial x_{i}}( |\frac{\partial y}{\partial x}|^{p-2}\frac{\partial y}{\partial x_{i}}) -c_{1}\|y| ^{2\mu -2}y] dt=fdW$

where $W$ is a $d$-dimensional Wiener process defined on a complete probability space, $f$ is a vector-function, $p$, $\alpha$, $\mu$ are some non negative numbers satisfying appropriate restrictions. The equation arises from a suitable stochastic perturbation of the Darcy Law in the motion of an ideal barotropic gas.

2007, 7(4): 907-929 doi: 10.3934/dcdsb.2007.7.907 +[Abstract](1349) +[PDF](265.9KB)
Abstract:
We continue the study on equilibria of the Smoluchowski equation for dilute solutions of rigid extended (dipolar) nematics and dispersions under an imposed electric or magnetic field [25]. We first provide an alternative proof for the theorem that all equilibria are dipolar with the polarity vector parallel to the external field direction if the strength of the permanent dipole ($\mu$) is larger than or equal to the product of the external field ($E$) and the anisotropy parameter ($\alpha_0$) (i.e. $\mu \ge |\alpha_0| E$). Then, we show that when $\mu < |\alpha_0| E$, there is a critical value $\alpha^$*$\geq 1$ for the intermolecular dipole-dipole interaction strength ($\alpha$) such that all equilibria are either isotropic or parallel to the external field if $\alpha \le \alpha^$*; but nonparallel dipolar equilibria emerge when $\alpha > \alpha^$*. The nonparallel equilibria are analyzed and the asymptotic behavior of $\alpha^$* is studied. Finally, the asymptotic results are validated by direct numerical simulations.

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