S Harris’s research while affiliated with SUNY Ulster and other places

We study diffusive logistic growth with immigration for a habitat surrounded by a hostile environment. The focus of our interest is the effect of immigration on the critical habitat length required for survival of the population. As expected, we find that this is reduced when immigration occurs. We also briefly consider the much simpler case where both immigration and emigration take place in an isolated habitat.

A special family of solutions explicit in the space and time variables is found for the Fisher equation with density-dependent diffusion. The connection with the known travelling wave solution and the initial conditions from which that evolves is also shown.

We apply the method of contraction, familiar in statistical mechanics applications, to reduce the Fisher equation describing population growth and dispersal in the space-time domain to an equation in the time domain. The resulting equation is identical to the well-known logistic equation with an additional correction term that depends on the global solution to the Fisher equation. This equation provides a possible basis for explaining why logistic dynamics has not always described experimental data and also for then formulating models that generalize the logistic equation.

We first determine an approximate traveling wave profile for the Cook model [J. Murray, Mathematical Biology I: An Introduction (Springer, New York, 2002), pp. 471-478] for the case in which the number of dispersers is small relative to the number of nondispersers. The results are consistent with the previous linearized wavefront analysis that predicts, counterintuitively, that relatively few dispersers can drive the population expansion wave with a wavespeed not too different from that for the case of a single dispersing population as described by the Fisher equation. The method of solution differs from that used in the latter case since here the dimensionless wavespeed is close to unity. We next generalize the Cook model to include time-delay effects. While the Cook model, like the Fisher equation, does not adequately describe the wave of advance during the Neolithic transition in Europe, we show that the generalized Cook model provides a close agreement with the historical record.

We extend the previous results, describing the population dispersal that occurs in some insects and small animal populations when this process is not strictly random, by including both the downgradient diffusion and the full Pearl-Verhulst logistic growth term in the equation of evolution. Motivated by the increasing fragmentation of natural habitats that is the result of human activities, we consider a finite habitat surrounded by a hostile environment. Previous work [Phys Rev. E 62, 4032 (2000)] considered only the case of an unbounded habitat, obviating issues concerned with the critical habitat size and the adoption of strategies best suited to achieve lower densities by dispersal through downgradient diffusion.

We consider the Fisher equation and its generalization for an asocial population in a linear, hostile environment. The method of center manifold analysis is used to obtain the time-dependent solution of the former, nonlinear equation. The correct critical habitat size is obtained; in addition, the result for the steady state central density compares favorably with the exact result for relatively large population sizes (up to one half of the carrying capacity). For a model of asocial growth we obtain the expanded criteria for survival. This includes the habitat size, the population size at which positive growth begins, and also the minimum initial central density.

We make use of a one-dimensional mean-field model formulated in previous work to determine the effects of added submonolayer amounts of surfactant during growth on a vicinal surface. We assume that the surfactant acts to increase the activation barrier for surface diffusion and examine the consequences of this relative to average island size and island density for varying surfactant coverage. The results obtained agree qualitatively with experimental results for Sb/Ag/Ag(1 1 1) on flat surfaces.