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

Periodic and Chaotic Orbits of a Discrete Rational System

We study a rational planar system consisting of one linear-affine and one linear-fractional difference equation. If all of the system’s parameters are positive (so that the positive quadrant is invariant and the system is continuous), then we show that the unique fixed point of the system in the positive quadrant cannot be repelling and the system does not have a snap-back repeller. By folding the system into a second-order equation, we find special cases of the system with some negative parameter values that do exhibit chaos in the sense of Li and Yorke within the positive quadrant of the plane

A Survey of Extragalactic Faraday Rotation at High Galactic Latitude: The Vertical Magnetic Field of the Milky Way towards the Galactic Poles

We present a study of the vertical magnetic field of the Milky Way towards the Galactic poles, determined from observations of Faraday rotation toward more than 1000 polarized extragalactic radio sources at Galactic latitudes |b| > 77 degs, using the Westerbork Radio Synthesis Telescope and the Australia Telescope Compact Array. We find median rotation measures (RMs) of 0.0 +/- 0.5 rad/m^2 and +6.3 +/- 0.7 rad/m^2 toward the north and south Galactic poles, respectively, demonstrating that there is no coherent vertical magnetic field in the Milky Way at the Sun's position. If this is a global property of the Milky Way's magnetism, then the lack of symmetry across the disk rules out pure dipole or quadrupole geometries for the Galactic magnetic field. The angular fluctuations in RM seen in our data show no preferred scale within the range ~ 0.1 to 25 degs. The observed standard deviation in RM of ~ 9 rad/m^2 then implies an upper limit of ~1microGauss on the strength of the random magnetic field in the warm ionized medium at high Galactic latitudes.Comment: 38 pages, 7 figures, 3 tables Accepted for publication in ApJ, Electronic versions of Tables 1 and 2 are available via email from the first autho