Symplectic integrators in the shearing sheet

The shearing sheet is a model dynamical system that is used to study the small-scale dynamics of astrophysical disks. Numerical simulations of particle trajectories in the shearing sheet usually employ the leapfrog integrator, but this integrator performs poorly because of velocity-dependent (Coriolis) forces. We describe two new integrators for this purpose; both are symplectic, time-reversible and second-order accurate, and can easily be generalized to higher orders. Moreover, both integrators are exact when there are no small-scale forces such as mutual gravitational forces between disk particles. In numerical experiments these integrators have errors that are often several orders of magnitude smaller than competing methods. The first of our new integrators (SEI) is well-suited for disks in which the typical inter-particle separation is large compared to the particles' Hill radii (e.g., planetary rings), and the second (SEKI) is designed for disks in which the particles are on bound orbits or the separation is smaller than the Hill radius (e.g., irregular satellites of the giant planets).Comment: 9 pages, 6 figures, accepted for publication in MNRAS, v2: discussion/tests for symmetrized and modified leapfrog integrators adde

Life Stage Simulation Analysis: Estimating Vital-Rate Effects on Population Growth for Conservation

We developed a simulation method, known as life-stage simulation analysis (LSA) to measure potential effects of uncertainty and variation in vital rates on population growth (lambda) for purposes of species conservation planning. Under LSA, we specify plausible or hypothesized levels of uncertainty, variation, and covariation in vital rates fur a given population. We use these data under resampling simulations to establish random combinations of vital rates for a large number of matrix replicates and finally summarize results from the matrix replicates to estimate potential effects of each vital rate on lambda in a probability-based context. Estimates of potential effects are based on a variety of summary statistics, such as frequency of replicates having the same vital rate of highest elasticity, difference in elasticity values calculated under simulated conditions vs, elasticities calculated using mean invariant vital rates, percentage of replicates having positive population growth, and variation in lambda explained by variation in each vital rate. To illustrate, we applied LSA to viral rates for two vertebrates: desert tortoise (Gopherus agassizii) and Greater prairie Chicken (Tympanuchus cupido). Results fur the prairie chicken indicated that a single vital rate consistently had greatest effect on population growth. Results for desert tortoise, however, suggested that a variety of life stages could have strong effects on population growth. Additional simulations for the Greater Prairie Chicken under a hypothetical conservation plan also demonstrated that a variety of vital rates could be manipulated to achieve desired population growth. To improve the reliability of inference, we recommend that potential effects of vital rates on lambda be evaluated using a probability-based approach like LSA. LSA is an important complement to other methods that evaluate vital-rate effects on lambda, including classical elasticity analysis, retrospective methods of variance decomposition, and simulation of the effects of environmental stochasticity