Global analysis of a predator-prey model with variable predator search rate.

We consider a modified Holling-type II predator-prey model, based on the premise that the search rate of predators is dependent on the prey density, rather than constant. A complete analysis of the global behavior of the model is presented, and shows that the model exhibits a dichotomy similar to the classical Holling-type II model: either the coexistence steady state is globally stable; or it is unstable, and then a unique, globally stable limit cycle exists. We discuss the similarities, but also important differences between our model and the Holling-type II model. The main differences are that: 1. The paradox of enrichment which always occurs in the Holling-type II model, does not always occur here, and 2. Even when the paradox of enrichment occurs, predators can adapt by lowering their search rate, and effectively stabilize the system.

Symmetry breaking and uniqueness for the incompressible Navier-Stokes equations.

The present article establishes connections between the structure of the deterministic Navier-Stokes equations and the structure of (similarity) equations that govern self-similar solutions as expected values of certain naturally associated stochastic cascades. A principle result is that explosion criteria for the stochastic cascades involved in the probabilistic representations of solutions to the respective equations coincide. While the uniqueness problem itself remains unresolved, these connections provide interesting problems and possible methods for investigating symmetry breaking and the uniqueness problem for Navier-Stokes equations. In particular, new branching Markov chains, including a dilogarithmic branching random walk on the multiplicative group (0, ∞), naturally arise as a result of this investigation.