# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

March 2008 , Volume 9 , Issue 2

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2008, 9(2): 199-220 doi: 10.3934/dcdsb.2008.9.199 +[Abstract](2143) +[PDF](273.9KB)
Abstract:
We study the homogenization of an unsteady fluid-structure interaction problem with a scaling corresponding to a long time asymptotic regime. We consider oscillating initial data which are Bloch wave packets corresponding to tubes vibrating in opposition of phase. We prove that the initial displacements follow the rays of geometric optics and that the envelope function evolves according to a Schr ̈odinger equation which can be interpreted as an effect of dispersion.
2008, 9(2): 221-233 doi: 10.3934/dcdsb.2008.9.221 +[Abstract](1909) +[PDF](264.1KB)
Abstract:
Using Lie group theory and canonical transformations, we construct explicit solutions of nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities. We present the general theory, use it to study different examples and use the qualitative theory of dynamical systems to obtain some properties of these solutions.
2008, 9(2): 235-248 doi: 10.3934/dcdsb.2008.9.235 +[Abstract](2056) +[PDF](327.2KB)
Abstract:
The aim of this paper is to study the adaptive synchronization of a class of uncertain chaotic systems in the drive-response framework. A robust adaptive observer-based response system is designed to synchronize a given chaotic system with uncertainties. An improved adaptation law on the upper bound of uncertainties is proposed to guarantee the boundedness of both the synchronization error and the estimated feedback coupling gains when a boundary layer technique is employed. A numerical example of the modified Chua’s circuit is considered to show the efficiency and effectiveness of this scheme.
2008, 9(2): 249-266 doi: 10.3934/dcdsb.2008.9.249 +[Abstract](2248) +[PDF](232.7KB)
Abstract:
In this work a size-structured juvenile-adult population model is considered. The linearized dynamical behavior of stationary solutions is analyzed using semigroup and spectral methods. The regularity of the governing linear semigroup allows us to derive biologically meaningful conditions for the linear stability of stationary solutions. The main emphasis in this work is on juvenile-adult interaction and resulting consequences for the dynamics of the system. In addition, we investigate numerically the effect of a non-zero population inflow, due to an external source of newborns, on the linear dynamical behavior of the system in a special case of model ingredients.
2008, 9(2): 267-279 doi: 10.3934/dcdsb.2008.9.267 +[Abstract](2374) +[PDF](193.8KB)
Abstract:
In this paper we explore the existence of periodic solutions of a nonautonomous semi-ratio-dependent predator-prey dynamical system with functional responses on time scales. To illustrate the utility of this work, we should mention that, in our results this system with a large class of monotone functional responses, always has at least one periodic solution. For instance, this system with some celebrated functional responses such as Holling type-II (or Michaelis-Menten), Holling type-III, Ivlev, $mx$ (Holling type I), sigmoidal [e.g., Real and ${mx^2}/{((A+x)(B+x))}$] and some other monotone functions, has always at least one $\omega$-periodic solution. Besides, for some well-known functional responses which are not monotone such as Monod-Haldane or Holling type-IV, the existence of periodic solutions is proved. Our results extend and improve previous results presented in [4], [10], [22], and [38].
2008, 9(2): 281-308 doi: 10.3934/dcdsb.2008.9.281 +[Abstract](2305) +[PDF](317.4KB)
Abstract:
The aim of this paper is to study a system modelling the flow of an incompressible phase (water) and a compressible phase (gas) in porous media. Two kinds of degeneracy appear for this problem: a dissipative term and an evolution term degenerate with respect to the saturation. Global weak solutions are established for the system by introducing several approximate models. The first one consists in obtaining a non-degenerate dissipative system. The second one is a time discretization method in order to overcome the degeneracy in the evolution term. At this step, the subproblem is a non- degenerate elliptic system which is strongly coupled and highly nonlinear. Then the Leray-Schauder fixed point theorem instead of a classical Schauder fixed point theorem is the key point to solve such a problem.
2008, 9(2): 309-320 doi: 10.3934/dcdsb.2008.9.309 +[Abstract](1838) +[PDF](184.5KB)
Abstract:
We examine quasi-static evolution of crystals in three dimensions. We assume that the Wulff shape is a prism with a hexagonal base. We include the Gibbs-Thomson law on the crystal surface and the so-called Stefan condition. We show local in time existence of solutions assuming that the initial crystal has admissible shape.
2008, 9(2): 321-351 doi: 10.3934/dcdsb.2008.9.321 +[Abstract](1697) +[PDF](393.3KB)
Abstract:
We consider a mathematical model that describes the competition of three species for a single nutrient in a chemostat in which the dilution rate is assumed to be controllable by means of state dependent feedback. We consider feedback schedules that are affine functions of the species concentrations. In case of two species, we show that the system may undergo a Hopf bifurcation and oscillatory behavior may be induced by appropriately choosing the coefficients of the feedback function. When the growth of the species obeys Michaelis-Menten kinetics, we show that the Hopf bifurcation is supercritical in the relevant parameter region, and the bifurcating periodic solutions for two species are always stable. Finally, we show that by adding a third species to the system, the two-species stable periodic solutions may bifurcate into the coexistence region via a transcritical bifurcation. We give conditions under which the bifurcating orbit is locally asymptotically stable.
2008, 9(2): 353-374 doi: 10.3934/dcdsb.2008.9.353 +[Abstract](1828) +[PDF](1562.9KB)
Abstract:
We propose a two-prey, one-predator model involving the effect of the carcass. We consider a commensal interaction that a prey species eats the remains of the other prey species’ carcass given by their predator. Under some biological assumptions, we construct two ODE models. We analyze the linear stability and prove the permanence of the two models. We also show that the effect of the remains of the carcass leads to chaotic dynamics for biologically reasonable choices of parameters by numerical simulations. Finally, we discuss the dynamical results and the coexistent regions of the three species.
2008, 9(2): 375-396 doi: 10.3934/dcdsb.2008.9.375 +[Abstract](1877) +[PDF](437.5KB)
Abstract:
We prove that a polygonal scatterer in $\mathbb{R}^2$, possibly consisting of finitely many sound-soft and sound-hard polygons, is uniquely determined by a single far-field measurement.
2008, 9(2): 397-413 doi: 10.3934/dcdsb.2008.9.397 +[Abstract](1753) +[PDF](337.3KB)
Abstract:
In some cases of delay differential equations (DDEs), a delay-dependant coefficient is incorporated into models which takes the form of a function of delay quantity. This brings forth frequent stability-switch phenomena. A geometrical stability criterion is developed on the two-parameter plane for analyzing Hopf bifurcations of equilibria. It is shown that the increasing direction of parameter $\sigma$ would confirm bifurcation directions (from stable one to unstable one, or whereas) at the critical delay values. These lead to the definite partition of stable and unstable regions on the $(\sigma-\tau)$ plane. Several examples are given to illustrate how to use this method to detect both Hopf and double Hopf bifurcations.
2008, 9(2): 415-429 doi: 10.3934/dcdsb.2008.9.415 +[Abstract](2812) +[PDF](174.8KB)
Abstract:
Discrete-time Lotka-Volterra competition models are obtained by applying nonstandard finite difference (NSFD) schemes to the continuous-time counterparts of the model. The NSFD methods are noncanonical symplectic numerical schemes when applying to the predator-prey model $x'=x-xy$ and $y'=-y+xy$. The local dynamics of the discrete-time model are analyzed and compared with the continuous model. We find the NSFD schemes that preserve the local dynamics of the continuous model. The local stability criteria are exactly the same between the continuous model and the discrete model independent of the step size. Two specific discrete-time Lotka-Volterra competition models by NSFD schemes that preserve positivity of solutions and monotonicity of the system are also given. The two discrete-time models are dynamically consistent with their continuous counterpart.

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