A Nagumo-type model for competing populations with nonlocal coupling.

We consider a model of two competing species with nonlocal competition for resources. The net birthrate is cubic, so that the model allows simulation of the Allee effect, whereby extinction is stable and intermediate populations promote growth, while saturation occurs via cubic competition terms. The model includes both interspecies and intraspecies nonlocal competition which enters via convolution integrals with a specified asymmetric competition kernel function. We introduce two parameters, δ, describing the extent of the coupling, with δ = 0 corresponding to local coupling, and α, describing the extent of the asymmetry, with α = 0 corresponding to symmetric nonlocal interactions. We consider the case where the local model admits a stable coexistence (populations of both species positive) equilibrium solution. We perform a linear stability analysis and show that this solution can be destabilized by sufficient nonlocality, i.e., when δ increases beyond a critical value. We then consider nonlinear patterns, far from the stability boundary. We show that nonlinear patterns consist of arrays of islands, regions of nonzero population, separated by deadzones, where the populations are essentially extinct, (with the array propagating in the case α ≠ 0). The predominant effect of the cubic model is that the islands for the two species are disjoint, so that each species lives in the deadzone of the other species. In addition, some patterns involve both hospitable and inhospitable deadzones, so that islands form in only some of the deadzones.

Propagation failure for a front between stable states in a system with subdiffusion.

The propagation of subdiffusion-reaction fronts is studied in the framework of a model recently suggested by Fedotov [ Phys. Rev. E 81 011117 (2010)]. An exactly solvable model with a piecewise linear reaction function is considered. A drastic difference between the cases of normal diffusion and subdiffusion has been revealed. While in the case of normal diffusion, a traveling wave solution between two locally stable phases always exists, and is unique, in the case of the subdiffusion such solutions do not exist. The numerical simulation shows that the velocity of the front decreases with time according to a power law. The only kind of fronts moving with a constant velocity are waves which propagate solely due to the reaction, with a vanishing subdiffusive flux.

Stability and pattern formation for competing populations with asymmetric nonlocal coupling.

We consider a model of two competing species with asymmetric nonlocal coupling in a competition for resources. The nonlocal coupling is via convolution integrals and the asymmetry is via convolution kernel functions which are not even functions of their arguments. The nonlocality is due to species mobility, so that at any fixed point in space the competition for resources depends not just on the populations at that point but on a suitably weighted average of the populations. We introduce two parameters, δ, describing the extent of the coupling, with δ=0 corresponding to local coupling, and α, describing the extent of the asymmetry, with α=0 corresponding to symmetric nonlocal interactions. We consider the case where the model admits a stable coexistence equilibrium solution. We perform a linear stability analysis and show that this solution can be destabilized by sufficient nonlocality, i.e., when δ increases beyond a critical value. We consider two specific kernel functions, (i) an asymmetric Gaussian and (ii) an asymmetric stepfunction. We compute the stability boundary as a function of α, and for δ beyond the stability boundary we determine unstable wavenumber bands. We compute nonlinear patterns for δ significantly beyond the stability boundary. Patterns consist of arrays of islands, regions of nonzero population, separated by either near-deadzones where the populations are small, but nonzero, or by deadzones where populations are exponentially small and essentially extinct. We find solutions consisting of propagating traveling waves of islands, solutions exhibiting colony formation, where a colony is formed just ahead of an island and eventually grows as the parent island decays, and modulated traveling waves, where competition between the two species allows propagation and inhibits colony formation. We explain colony formation and the modulated traveling waves as due to a positive feedback mechanism associated with small variations in the amplitude of the parent island.

Homoclinic snaking near a codimension-two Turing-Hopf bifurcation point in the Brusselator model.

Spatiotemporal Turing-Hopf pinning solutions near the codimension-two Turing-Hopf point of the one-dimensional Brusselator model are studied. Both the Turing and Hopf bifurcations are supercritical and stable. The pinning solutions exhibit coexistence of stationary stripes of near critical wavelength and time-periodic oscillations near the characteristic Hopf frequency. Such solutions of this nonvariational problem are in contrast to the stationary pinning solutions found in the subcritical Turing regime for the variational Swift-Hohenberg equations, characterized by a spatially periodic pattern embedded in a spatially homogeneous background state. Numerical continuation was used to solve periodic boundary value problems in time for the Fourier amplitudes of the spatiotemporal Turing-Hopf pinning solutions. The solution branches are organized in a series of saddle-node bifurcations similar to the known snaking structures of stationary pinning solutions. We find two intertwined pairs of such branches, one with a defect in the middle of the striped region, and one without. Solutions on one branch of one pair differ from those on the other branch by a π phase shift in the spatially periodic region, i.e., locations of local minima of solutions on one branch correspond to locations of maxima of solutions on the other branch. These branches are connected to branches exhibiting collapsed snaking behavior, where the snaking region collapses to almost a single value in the bifurcation parameter. Solutions along various parts of the branches are described in detail. Time dependent depinning dynamics outside the saddle nodes are illustrated, and a time scale for the depinning transitions is numerically established. Wavelength variation within the snaking region is discussed, and reasons for the variation are given in the context of amplitude equations. Finally, we compare the pinning region to the Maxwell line found numerically by time evolving the amplitude equations.

Fronts in anomalous diffusion-reaction systems.

A review of recent developments in the field of front dynamics in anomalous diffusion-reaction systems is presented. Both fronts between stable phases and those propagating into an unstable phase are considered. A number of models of anomalous diffusion with reaction are discussed, including models with Lévy flights, truncated Lévy flights, subdiffusion-limited reactions and models with intertwined subdiffusion and reaction operators.

Nucleation and growth of droplets at a liquid-gas interface.

The nucleation of liquid droplets at a liquid-gas interface from a saturated vapor in the gas phase, as well as the droplet growth after the nucleation are studied. These two processes determine the formation of a regular hexagonal array of drops on the surface of an evaporating film of polymer solution that is used for the fabrication of polymer membranes with a regular microporous structure. The free-energy barrier for the nucleation of a droplet at a liquid-gas interface is found as a function of the droplet radius and the contact angles, and the critical nucleation radius is computed. It is shown that the heterogeneous nucleation is thermodynamically more preferable than the homogeneous one. The role of the line tension between the phases is also estimated. Further growth of a droplet nucleated at the liquid-gas interface is studied. Two growth mechanisms are considered: by the vapor diffusion flux from the gas phase and by the surface diffusion of the vapor molecules adsorbed at the liquid-gas interface outside the droplet. Two cases, corresponding to unsaturated and saturated condensation, are considered. The droplet growth is described by a free-boundary problem which is solved analytically and numerically. The droplet growth exponents at different stages of growth are found.

Gapped gapless packing structures.

The assumption of a gapless packing structure has previously been used to obtain the density and partial coordination numbers of a random mixture of hard spheres in the maximally dense regime. Here we extend the notion of a gapless packing structure to allow the determination of the characteristics of a packing away from maximal density by adding an appropriate number of void spherical elements. A gapless packing is then considered in which the void and solid spherical elements are assumed to be indistinguishable except for the purposes of calculating packing fraction and coordination number. We utilize the notion of specific volume to generate a one-parameter family of void distributions to obtain a set of coupled integral equations, which are solved numerically. Monodisperse and bi-disperse packings are investigated for packing fractions ranging from rho=0.26 to 0.78. Results are shown to be comparable to experiments and the effect of varying packing fraction on coordination numbers is shown to be invariant with respect to number distribution. A linear relationship between coordination number and packing fraction is elucidated for moderate to low packing fractions. Maximum and minimum random packing fractions are also discussed.

The effect of convection on a propagating front with a liquid product: Comparison of theory and experiments.

This work is devoted to the investigation of propagating polymerization fronts converting a liquid monomer into a liquid polymer. We consider a simplified mathematical model which consists of the heat equation and equation for the depth of conversion for one-step chemical reaction and of the Navier-Stokes equations under the Boussinesq approximation. We fulfill the linear stability analysis of the stationary propagating front and find conditions of convective and thermal instabilities. We show that convection can occur not only for ascending fronts but also for descending fronts. Though in the latter case the exothermic chemical reaction heats the cold monomer from above, the instability appears and can be explained by the interaction of chemical reaction with hydrodynamics. Hydrodynamics changes also conditions of the thermal instability. The front propagating upwards becomes less stable than without convection, the front propagating downwards more stable. The theoretical results are compared with experiments. The experimentally measured stability boundary for polymerization of benzyl acrylate in dimethyl formamide is well approximated by the theoretical stability boundary. (c) 1998 American Institute of Physics.