A metapopulation model with Markovian landscape dynamics

We study a variant of Hanski's incidence function model that allows habitat patch characteristics to vary over time following a Markov process. The widely studied case where patches are classified as either suitable or unsuitable is included as a special case. For large metapopulations, we determine a recursion for the probability that a given habitat patch is occupied. This recursion enables us to clarify the role of landscape dynamics in the survival of a metapopulation. In particular, we show that landscape dynamics affects the persistence and equilibrium level of the metapopulation primarily through its effect on the distribution of a local population's life span.Comment: This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0

A comparison of solution methods for Mandelbrot-like polynomials

We compare two different root-finding methods, eigenvalue methods and homotopy methods, using three test problems: Mandelbrot polynomials, Fibonacci-Mandelbrot polynomials, and Narayana-Mandelbrot polynomials. For the eigenvalue methods, using both MATLAB and Maple, we computed the eigenvalues of a specialized recursively-constructed, supersparse, upper Hessenberg matrix, inspired by Piers Lawrence\u27s original construction for the Mandelbrot polynomials, for all three families of polynomials. This led us to prove that this construction works in general. Therefore, this construction is genuinely a new kind of companion matrix. For the homotopy methods, we used a special-purpose homotopy, in which we used an equivalent differential equation to solve for the roots of all three families of polynomials. To solve these differential equations, we used our own ode solver, based on MATLAB\u27s ode45 routine, which has pole-vaulting capabilities. We had two versions of this ode solver: one in MATLAB, and the other in C++ that implements Bailey\u27s ARPREC package