The perfect mixing paradox and the logistic equation: Verhulst vs. Lotka

A theoretical analysis of density-dependent population dynamics in two patches sheds novel light on our understanding of basic ecological parameters. Firstly, as already highlighted in the literature, the use of the traditional r-K parameterization for the logistic equation (due to Lotka and Gause) can lead to paradoxical situations. We show that these problems do not exist with Verhulst's original formulation, which includes a quadratic “friction” term representing intraspecific competition (parameter α) instead of the so-called carrying capacity K. Secondly, we show that the parameter α depends on the number of patches, or more generally on area. This is also the case of all parameters that quantify the interaction strengths between individuals, either of the same species or of different species. The consequence is that estimates of interaction strength will vary when population size is measured in absolute terms. In order to obtain scale-invariant parameter estimates, it is essential to express population abundances as densities. Also, the interaction parameters must be reported with all explicit units, such as (m2·individual−1·d−1), which is rarely the case

The spin dependence of high energy proton scattering

Motivated by the need for an absolute polarimeter to determine the beam polarization for the forthcoming RHIC spin program, we study the spin dependence of the proton-proton elastic scattering amplitudes at high energy and small momentum transfer.We examine experimental evidence for the existence of an asymptotic part of the helicity-flip amplitude phi_5 which is not negligible relative to the largely imaginary average non-flip amplitude phi_+. We discuss theoretical estimates of r_5, essentially the ratio of phi_5 to phi_+, based upon extrapolation of low and medium energy Regge phenomenological results to high energies, models based on a hybrid of perturbative QCD and non-relativistic quark models, and models based on eikonalization techniques. We also apply the model-independent methods of analyticity and unitarity.The preponderence of evidence at available energy indicates that r_5 is small, probably less than 10%. The best available experimental limit comes from Fermilab E704:those data indicate that |r_5|<15%. These bounds are important because rigorous methods allow much larger values. In contradiction to a widely-held prejudice that r_5 decreases with energy, general principles allow it to grow as fast as ln(s) asymptotically, and some models show an even faster growth in the RHIC range. One needs a more precise measurement of r_5 or to bound it to be smaller than 5% in order to use the classical Coulomb-nuclear interference technique for RHIC polarimetry. As part of this study, we demonstrate the surprising result that proton-proton elastic scattering is self-analysing, in the sense that all the helicity amplitudes can, in principle, be determined experimentally at small momentum transfer without a knowledge of the magnitude of the beam and target polarization