Density-dependent Diffusion Research Articles

Turing bifurcation in a reaction–diffusion system with density-dependent dispersal

Motivated by the recent finding [N. Kumar, G.M. Viswanathan, V.M. Kenkre, Physica A 388 (2009) 3687] that the dynamics of particles undergoing density-dependent nonlinear diffusion shows sub-diffusive behaviour, we study the Turing bifurcation in a two-variable system with this kind of dispersal. We perform a linear stability analysis of the uniform steady state to find the conditions for the Turing bifurcation and compare it with the standard Turing condition in a reaction–diffusion system, where dispersal is described by simple Fickian diffusion. While activator–inhibitor kinetics are a necessary condition for the Turing instability as in standard two-variable systems, the instability can occur even if the diffusion constant of the inhibitor is equal to or smaller than that of the activator. We apply these results to two model systems, the Brusselator and the Gierer–Meinhardt model.

A non-linear degenerate equation for direct aggregation and traveling wave dynamics

The gregarious behavior of individuals of populations is an important factor in avoiding predators or for reproduction. Here, by using a random biased walk approach, we build a model which, after a transformation, takes the general form [u_{t}=[D(u)u_{x}]_{x}+g(u)] . The model involves a density-dependent non-linear diffusion coefficient [D] whose sign changes as the population density [u] increases. For negative values of [D] aggregation occurs, while dispersion occurs for positive values of [D] . We deal with a family of degenerate negative diffusion equations with logistic-like growth rate [g] . We study the one-dimensional traveling wave dynamics for these equations and illustrate our results with a couple of examples. A discussion of the ill-posedness of the partial differential equation problem is included.

Wavespeed in reaction–diffusion systems, with applications to chemotaxis and population pressure

We present a method based on the Melnikov function used in dynamical systems theory to determine the wavespeed of travelling waves in perturbed reaction-diffusion systems. We study reaction-diffusion systems which are subject to weak nontrivial perturbations in the reaction kinetics, in the diffusion coefficient, or with weak active advection. We find explicit formulae for the wavespeed and illustrate our theory with two examples; one in which chemotaxis gives rise to nonlinear advection and a second example in which a positive population pressure results in both a density-dependent diffusion coefficient and a nonlinear advection. Based on our theoretical results we suggest an experiment to distinguish between chemotactic and population pressure in bacterial colonies.

Numerical Simulation of the Current−Voltage Curve in Dye-Sensitized Solar Cells

A theoretical model based on the numerical integration of the continuity equation for electrons with trap-limited density-dependent diffusion and recombination constants is implemented to describe the functioning of dye-sensitized solar cells (DSSC). The application of the model combines recent theory on charge transport in nanocrystalline materials with parameters extracted from five simple measurements: the UV/vis spectrum of the dye in solution, the steady-state current−voltage curve, the open circuit photovoltage versus light intensity curve, photocurrent transient upon switching on an illumination source, and open-circuit voltage decay upon switching off the illumination source. As a novel feature not previously included in this kind of calculations, the model includes an additional term that accounts for the charge transfer from the transparent conducting oxide (TCO) substrate to the electrolyte solution. The general applicability of the model is illustrated by applying it to two different types of cell: a TiO2-based solar cell with an organic solvent electrolyte and a ZnO-based solar cell with a solvent-free electrolyte. It is found that the numerical model is capable of adequately fitting all data for both systems, resulting in quantitative estimates for the main parameters controlling solar cell functioning and efficiency. The results show that it is possible to provide a global description of DSSCs based on fundamental theories for trap-limited transport and recombination using simple experimental techniques available to every solar cell laboratory. The present paper tries to help fill the gap between pure theoreticians and experimentalists working on this kind of system.

Collective diffusion in the presence of substrate inhomogeneities and particle–particle interactions

A novel variational analytic approach to collective diffusion allowing the density dependent collective diffusion coefficient to be calculated in systems of interacting particles adsorbed on a crystalline substrate is presented. The approach, based on a kinetic lattice gas model extracts the diffusion coefficient directly from the master equations which account for the microscopic kinetics of the system in which microscopic processes underlying the diffusion are particle jumps between neighboring adsorption sites. Variational parameters minimizing the evaluated diffusional eigenvalue of the microscopic rate matrix are ‘geometrical’ and ‘correlational’ phases accounting, for the local potential energy landscape experienced by the adsorbed particle and the local correlations, respectively, i.e. an instantaneous occupation pattern of adsorption sites around the particle. Selected results, collective diffusion as a function of particle coverage, for the system of interacting particles adsorbed on a one dimensional non-homogeneous substrate with steps and for a system of non-interacting particles adsorbed on a two dimensional striped–stepped surface are presented and discussed. It is demonstrated in the latter case that the mean field approach which is known in the literature overestimates the diffusion coefficient and corresponds to the variational result in the limit of infinitely fast hopping kinetics in the direction parallel to steps.

Hurst exponents for interacting random walkers obeying nonlinear Fokker–Planck equations

Anomalous diffusion of random walks has been extensively studied for the case of non-interacting particles. Here we study the evolution of nonlinear partial differential equations by interpreting them as Fokker–Planck equations arising from interactions among random walkers. We extend the formalism of generalized Hurst exponents to the study of nonlinear evolution equations and apply it to several illustrative examples. They include an analytically solvable case of a nonlinear diffusion constant and three nonlinear equations which are not analytically solvable: the usual Fisher equation which contains a quadratic nonlinearity, a generalization of the Fisher equation with density-dependent diffusion constant, and the Nagumo equation which incorporates a cubic rather than a quadratic nonlinearity. We estimate the generalized Hurst exponents.

Coexistence problem for a prey–predator model with density-dependent diffusion

We study a prey–predator model with nonlinear diffusions. In a case when the spatial dimension is less than 5, a universal bound for coexistence steady-states is found. By using the bound and the bifurcation theory, we obtain the bounded continuum of coexistence steady-states.

Stationary Patterns for a Lotka-Volterra Cooperative Model with a Density-Dependent Diffusion Term

This paper is concerned with positive stationary solutions for a Lotka-Volterra cooperative model with a density-dependent diffusion term of a fractional type. The existence of stationary patterns is proven by the presence of density-dependent diffusion. Our proof is based on the Leray-Schauder degree theory and some a priori estimates. We also derive a certain limiting system which positive stationary solutions satisfy.

Traveling wave solutions in parametric forms for a diffusion model with a nonlinear rate of growth

In this paper, we study a model of insect and animal dispersal where both density-dependent diffusion and nonlinear rate of growth are present. We analyze the existence of bounded traveling wave solution under certain parametric conditions by using the qualitative theory of dynamical systems. An explicit traveling wave solution is obtained by means of the first integral method. Traveling wave solutions in parametric forms for three particular cases are established by the Lie symmetry method.

Explicit Separation of Growth and Motility in a New Tumor Cord Model

We investigate a new model of tumor growth in which cell motility is considered an explicitly separate process from growth. Bulk tumor expansion is modeled by individual cell motility in a density-dependent diffusion process. This model is implemented in the context of an in vivo system, the tumor cord. We investigate numerically microscale density distributions of different cell classes and macroscale whole tumor growth rates as functions of the strength of transitions between classes. Our results indicate that the total tumor growth follows a classical von Bertalanffy growth profile, as many in vivo tumors are observed to do. This provides a quick validation for the model hypotheses. The microscale and macroscale properties are both sensitive to fluctuations in the transition parameters, and grossly adopt one of two phenotypic profiles based on their parameter regime. We analyze these profiles and use the observations to classify parameter regimes by their phenotypes. This classification yields a novel hypothesis for the early evolutionary selection of the metastatic phenotype by selecting against less motile cells which grow to higher densities and may therefore induce local collapse of the vascular network.

Interacting random walkers and non-equilibrium fluctuations

We introduce a model of interacting Random Walk, whose hopping amplitude depends on the number of walkers/particles on the link. The mesoscopic counterpart of such a microscopic dynamics is a diffusing system whose diffusivity depends on the particle density. A non-equilibrium stationary flux can be induced by suitable boundary conditions, and we show indeed that it is mesoscopically described by a Fourier equation with a density dependent diffusivity. A simple mean-field description predicts a critical diffusivity if the hopping amplitude vanishes for a certain walker density. Actually, we evidence that, even if the density equals this pseudo-critical value, the system does not present any criticality but only a dynamical slowing down. This property is confirmed by the fact that, in spite of interaction, the particle distribution at equilibrium is simply described in terms of a product of Poissonians. For mesoscopic systems with a stationary flux, a very effect of interaction among particles consists in the amplification of fluctuations, which is especially relevant close to the pseudo-critical density. This agrees with analogous results obtained for Ising models, clarifying that larger fluctuations are induced by the dynamical slowing down and not by a genuine criticality. The consistency of this amplification effect with altered coloured noise in time series is also proved.

Analytical and numerical solutions of the density dependent diffusion Nagumo equation

In this Letter, we obtained solutions to a class of density dependent diffusion Nagumo equations. In particular, series solutions are obtained, along with a bound for the range of the convergence. Also, numerical solutions are obtained by the Runge–Kutta–Fehlberg 45 method. Moreover, the dependence of the traveling wave solutions on various parameters is discussed. Furthermore, we compare the series solutions with the numerical solutions to validate the numerical method. The results obtained in this study reveal many interesting behaviors that warrant further study on the Nagumo equation.

Effect of a cutoff on pushed and bistable fronts of the reaction-diffusion equation

We give an explicit formula for the change of speed of pushed and bistable fronts of the reaction-diffusion equation when a small cutoff is applied to the reaction term at the unstable or metastable equilibrium point. The results are valid for arbitrary reaction terms and include the case of density-dependent diffusion.

Modelling dispersal of populations and genetic information by finite element methods

This paper shows how biological population dynamic models in the form of partial differential equations can be applied to heterogeneous landscapes. The systems of coupled partial differential equations presented combine dispersal, growth, competition and genetic interactions. The equations belong to the class of reaction diffusion equations and are strongly non-linear. Realistic biological dispersal behaviour is introduced by density dependent diffusion coefficients and chemotaxis terms, which model the active movement along gradients of environmental variables. The resulting non-linear initial boundary value problems are solved for geometries of heterogeneous landscapes, which determine model parameters such as diffusion coefficients, habitat suitability and land use. Geometry models are imported from a geographical information system into a general purpose finite element solver for systems of coupled PDEs. The importance of spatial heterogeneity is demonstrated for management of biological control by sterile males and for risk management of GMO crops.

Positive solutions of quasilinear parabolic systems with nonlinear boundary conditions

The aim of this paper is to investigate the existence, uniqueness, and asymptotic behavior of solutions for a coupled system of quasilinear parabolic equations under nonlinear boundary conditions, including a system of quasilinear parabolic and ordinary differential equations. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system as well as the uniqueness of a positive steady-state solution. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients D i ( u i ) may have the property D i ( 0 ) = 0 for some or all i. Our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. It is shown that the time-dependent solution converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a porous medium type of problem, a heat-transfer problem, and a two-component competition model in ecology. These applications illustrate some very interesting distinctive behavior of the time-dependent solutions between density-independent and density-dependent diffusions.

Macroscopic description of particle systems with nonlocal density-dependent diffusivity

In this paper we study macroscopic density equations in which the diffusion coefficient depends on a weighted spatial average of the density itself. We show that large differences (not present in the local density-dependence case) appear between the density equations that are derived from different representations of the Langevin equation describing a system of interacting Brownian particles. Linear stability analysis demonstrates that under some circumstances the density equation interpreted with the Ito calculus has pattern solutions that never appear for the kinetic interpretation, which is the other one typically appearing in the context of nonlinear diffusion processes. We also introduce a discrete-time microscopic model of particles that confirms the results obtained at the macroscopic density level.

Persistence in reaction diffusion models with weak allee effect

We study the positive steady state distributions and dynamical behavior of reaction-diffusion equation with weak allele effect type growth, in which the growth rate per capita is not monotonic as in logistic type, and the habitat is assumed to be a heterogeneous bounded region. The existence of multiple steady states is shown, and the global bifurcation diagrams are obtained. Results are applied to a reaction-diffusion model with type II functional response, and also a model with density-dependent diffusion of animal aggregation.

Simulation of structured populations in chemically stressed environments

A heterogenous environment usually impacts, and sometimes determines, the structure and function of organisms in a population. We simulate the effects of a chemical on a population in a spatially heterogeneous environment to determine perceived stressor and spatial effects on dynamic behavior of the population. The population is assumed to be physiologically structured and composed of individuals having both sessile and mobile life history stages, who utilize energetically-controlled, resource-directed, chemical-avoidance advective movements and are subjected to random or density dependent diffusion. From a modeling perspective, the presence of a chemical in the environment requires introduction of both an exposure model and an effects module. The spatial location of the chemical stressor determines the exposure levels and ultimately the effects on the population while the relative location of the resource and organism determines growth. We develop a mathematical model, the numerical analysis for this model, and the simulation techniques necessary to solve the problem of population dynamics in an environment where heterogeneity is generated by resource and chemical stressor. In the simulations, the chemical is assumed to be a nonpolar narcotic and the individuals respond to the chemical via both physiological response and by physical movement. In the absence of a chemical stressor, simulation experiments indicate that despite a propensity to move to regions of higher resource density, organisms need not concentrate in the vicinity of high levels of resource. We focus on the dynamical variations due to advection induced by the toxicant. It is demonstrated that the relationship between resource levels and toxicant concentrations is crucial in determining persistence or extinction of the population.

A Fisher/KPP-type equation with density-dependent diffusion and convection: travelling-wave solutions

This paper concerns processes described by a nonlinear partial differential equation that is an extension of the Fisher and KPP equations including density-dependent diffusion and nonlinear convection. The set of wave speeds for which the equation admits a wavefront connecting its stable and unstable equilibrium states is characterized. There is a minimal wave speed. For this wave speed there is a unique wavefront which can be found explicitly. It displays a sharp propagation front. For all greater wave speeds there is a unique wavefront which does not possess this property. For such waves, the asymptotic behaviour as the equilibrium states are approached is determined.

Wave speeds of density dependent Nagumo diffusion equations – inspired by oscillating gap-junction conductance in the islets of Langerhans

The equation ut = (D(u)ux)x + f(u) arises in several biological examples and is known to have wave solutions for appropriate D and f. We give here a new formula for finding an approximation to the wave speed, relevant for comparing experiments with model simulations. This is done in details for the simple example D(u) = u + k and an N-shaped f, derived from a model of coupled pancreatic beta-cells, where the coupling conductance follows the electrical activity as it is found in experiments. On the way, we claim that the wave speed does not depend on the parameter g(K,ATP), mimicking the glucose concentration in the islet, in sharp contrast to the claim set forth in the article by Aslanidi et al.