556,887 research outputs found
Discrete Nonlinear Planar Systems and Applications to Biological Population Models
We study planar systems of difference equations and applications to biological models of species populations. Central to the analysis of this study is the idea of folding - the method of transforming systems of difference equations into higher order scalar difference equations. Two classes of second order equations are studied: quadratic fractional and exponential.
We investigate the boundedness and persistence of solutions, the global stability of the positive fixed point and the occurrence of periodic solutions of the quadratic rational equations. These results are applied to a class of linear/rational systems that can be transformed into a quadratic fractional equation via folding. These results apply to systems with negative parameters, instances not commonly considered in previous studies. We also identify ranges of parameter values that provide sufficient conditions on existence of chaotic and multiple stable orbits of different periods for the planar system.
We study a second order exponential difference equation with time varying parameters and obtain sufficient conditions for boundedness of solutions and global convergence to zero. For the autonomous case, we show occurrence of multistable periodic and nonperiodic orbits. For the case where parameters are periodic, we show that the nature of the solutions differs qualitatively depending on whether the period of the parameters is even or odd.
The above results are applied to biological models of populations. We investigate a broad class of planar systems that arise in the study of stage-structured single species populations. In biological contexts, these results include conditions on extinction or survival of the species in some balanced form, and possible occurrence of complex and chaotic behavior. Special rational (Beverton-Holt) and exponential (Ricker) cases are considered to explore the role of inter-stage competition, restocking strategies, as well as seasonal fluctuations in the vital rates
Global attractivity of a higher order nonlinear difference equation with unimodal terms
In the present paper, we study the asymptotic behavior of the following higher order nonlinear difference equation with unimodal terms where , and are constants with , , and , is decreasing, and is a positive integer. We obtain some new sufficient conditions for the global attractivity of positive solutions of the equation. Applications to some population models are also given
Instability, Intermittency and Multiscaling in Discrete Growth Models of Kinetic Roughening
We show by numerical simulations that discretized versions of commonly
studied continuum nonlinear growth equations (such as the Kardar-Parisi-Zhang
equation and the Lai-Das Sarma equation) and related atomistic models of
epitaxial growth have a generic instability in which isolated pillars (or
grooves) on an otherwise flat interface grow in time when their height (or
depth) exceeds a critical value. Depending on the details of the model, the
instability found in the discretized version may or may not be present in the
truly continuum growth equation, indicating that the behavior of discretized
nonlinear growth equations may be very different from that of their continuum
counterparts. This instability can be controlled either by the introduction of
higher-order nonlinear terms with appropriate coefficients or by restricting
the growth of pillars (or grooves) by other means. A number of such
``controlled instability'' models are studied by simulation. For appropriate
choice of the parameters used for controlling the instability, these models
exhibit intermittent behavior, characterized by multiexponent scaling of height
fluctuations, over the time interval during which the instability is active.
The behavior found in this regime is very similar to the ``turbulent'' behavior
observed in recent simulations of several one- and two-dimensional atomistic
models of epitaxial growth. [pacs{61.50.Cj, 68.55.Bd, 05.70.Ln, 64.60.Ht}]Comment: 47 pages + 26 postscript figures, submitted to Phys. Rev.
Backward error analysis for multisymplectic discretizations of Hamiltonian PDEs
Several recently developed multisymplectic schemes for Hamiltonian PDEs have
been shown to preserve associated local conservation laws and constraints very
well in long time numerical simulations. Backward error analysis for PDEs, or
the method of modified equations, is a useful technique for studying the
qualitative behavior of a discretization and provides insight into the
preservation properties of the scheme. In this paper we initiate a backward
error analysis for PDE discretizations, in particular of multisymplectic box
schemes for the nonlinear Schrodinger equation. We show that the associated
modified differential equations are also multisymplectic and derive the
modified conservation laws which are satisfied to higher order by the numerical
solution. Higher order preservation of the modified local conservation laws is
verified numerically.Comment: 12 pages, 6 figures, accepted Math. and Comp. Simul., May 200
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