General Reaction-diffusion Equation Research Articles

Fractional Pennes’ Bioheat Equation: Theoretical and Numerical Studies

Abstract In this work we provide a new mathematical model for the Pennes’ bioheat equation, assuming a fractional time derivative of single order. Alternative versions of the bioheat equation are studied and discussed, to take into account the temperature-dependent variability in the tissue perfusion, and both finite and infinite speed of heat propagation. The proposed bioheat model is solved numerically using an implicit finite difference scheme that we prove to be convergent and stable. The numerical method proposed can be applied to general reaction diffusion equations, with a variable diffusion coefficient. The results obtained with the single order fractional model, are compared with the original models that use classical derivatives.

Stability and bifurcation analysis of a reaction–diffusion equation with spatio-temporal delay

The spatio-temporal patterns of a general reaction–diffusion equation with non-local delay are considered. By analyzing the characteristic equations and applying the complex plane theory, the stability of the constant steady state and the possible Hopf bifurcations are obtained with the mean delay as a bifurcation parameter. The absolute stable, conditional stable and unstable region can be explicitly divided by the coefficients of the term without delay and the term with non-local delay. By investigating the effect of any weight function on the dynamics of the system, we establish the existence of spatially homogeneous and inhomogeneous bifurcating periodic solutions. And the algorithm for determining the direction and stability of Hopf bifurcation is derived by computing the normal form on the center manifold. The result shows that, for the case of strong kernel, the average delay may induce the stability switches. Finally, the theoretical results are applied to a diffusive food-limited population model. Numerical simulations show the existence of the orbit connecting the unstable spatially inhomogeneous periodic solutions to stable spatially homogeneous periodic solutions.

We need more empirical investigations and model validation for a better understanding of crime: Comment on “Statistical physics of crime: A review” by M.R. D'Orsogna and M. Perc

Since the seminal works of Wilson and Kelling [1] in 1982, the “broken windows theory” seems to have been widely accepted among the criminologists and, in fact, empirical findings actually point out that criminals tend to return to previously visited locations. Crime has always been part of the urban society’s agenda and has also attracted the attention of scholars from social sciences ever since. Furthermore, over the past six decades the world has experienced a quick and notorious urbanization process: by the eighties the urban population was about 40% of total population, and today more than half (54%) of the world population is urban [2]. The urbanization has brought us many benefits such as better working opportunities and health care, but has also created several problems such as pollution and a considerable rise in the criminal activities. In this context of urban problems, crime deserves a special attention because there is a huge necessity of empirical and mathematical (modeling) investigations which, apart from the natural academic interest, may find direct implications for the organization of our society by improving political decisions and resource allocation. Despite being a naturally interdisciplinary topic, the idea of a physicist studying crime may still cause some surprise (despite the fact that physicists have investigated, more than ever, several systems very far from the traditional domain of physics), but the review by D’Orsogna and Perc [3] shows us that several collective patterns related to crimes are analogous to those exhibited by classical physical systems such as the reaction–diffusion equations, which model the evolution of chemicals under chemical reactions and diffusion, but also describe the evolution of crime hotspots. D’Orsogna and Perc bring us a concise and general view of the recent applications of mathematical methods for modeling crime related problems. The review covers the modeling of crime hotspots by generalized reaction–diffusion equations and by self-exciting point process; presents an overview of the evolutionary game theory for addressing crime as a social dilemma; illustrates the use of network tools for understanding criminal organizations; and, by combining these tools with random walks methods, demonstrates how it is possible to infer the network topology of

Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: an Abel equation based approach.

We consider quasi-stationary (travelling wave type) solutions to a nonlinear reaction-diffusion equation with arbitrary, autonomous coefficients, describing the evolution of glioblastomas, aggressive primary brain tumors that are characterized by extensive infiltration into the brain and are highly resistant to treatment. The second order nonlinear equation describing the glioblastoma growth through travelling waves can be reduced to a first order Abel type equation. By using the integrability conditions for the Abel equation several classes of exact travelling wave solutions of the general reaction-diffusion equation that describes glioblastoma growth are obtained, corresponding to different forms of the product of the diffusion and reaction functions. The solutions are obtained by using the Chiellini lemma and the Lemke transformation, respectively, and the corresponding equations represent generalizations of the classical Fisher-Kolmogorov equation. The biological implications of two classes of solutions are also investigated by using both numerical and semi-analytical methods for realistic values of the biological parameters.

Stability and bifurcation analysis of a reaction–diffusion equation with distributed delay

Dynamics of a general reaction–diffusion equation with distributed delay are considered. The effects of the weak kernel and the strong kernel on the dynamics of the system are both investigated. By analyzing the characteristic equations in detail and taking the average delay as a bifurcation parameter, the stability of the constant equilibrium and the existence of Hopf bifurcations are obtained. The absolute stability and the conditional stability can be explicitly determined by the coefficients of the linearized system. For the case of the strong kernel, the average delay may induce the stability switches, but it is not able to occur for the case of the weak kernel. The algorithm for determining the direction and stability of the bifurcating periodic solutions is derived. Finally, the obtained theoretical results are applied to several single-species models, and the numerical simulations are illustrated to verify the theoretical results.

Spreading and vanishing behaviors for radially symmetric solutions of free boundary problems for reaction–diffusion equations

We discuss free boundary problems modeling the diffusion of invasive or new species in a multi-dimensional ball or annulus, where unknown functions are population density and the outer boundary representing the spreading front of the species. We define spreading and vanishing to describe the asymptotic behaviors of radially symmetric solutions. The main purpose is to study the underlying principle to determine spreading and vanishing for the free boundary problems of general reaction–diffusion equations. We also focus on the problems with a logistic or bistable reaction term to show dichotomy results, vanishing speed and sufficient conditions for spreading or vanishing.

Simple computation of reaction–diffusion processes on point clouds

The study of reaction-diffusion processes is much more complicated on general curved surfaces than on standard Cartesian coordinate spaces. Here we show how to formulate and solve systems of reaction-diffusion equations on surfaces in an extremely simple way, using only the standard Cartesian form of differential operators, and a discrete unorganized point set to represent the surface. Our method decouples surface geometry from the underlying differential operators. As a consequence, it becomes possible to formulate and solve rather general reaction-diffusion equations on general surfaces without having to consider the complexities of differential geometry or sophisticated numerical analysis. To illustrate the generality of the method, computations for surface diffusion, pattern formation, excitable media, and bulk-surface coupling are provided for a variety of complex point cloud surfaces.

Complex Population Dynamics in Heterogeneous Environments: Effects of Random and Directed Animal Movements

Abstract In this paper, we have investigated the complex dynamics of a one-dimensional spatial nonlinear coupled reaction-diffusion system with a Holling type IV functional response, akin to standard Michaelis-Menten inhibitory kinetics. Prey-taxis is included in a general reaction-diffusion equation to incorporate the active movement of predator species towards regions with high prey concentrations or if the predator is following some sort of cue (such as odor) to find the prey. We have carried out stability analysis of both the non-spatial model without diffusive spreading and of the spatial model. We performed extensive computer simulations to identify various parameter ranges for stable homogeneous solution. Our findings specifically elucidate the role of predator diffusion and prey-taxis in controlling emergent structures, and transitions towards spatio-temporal chaos. We observe that the increasing predator random movement and moderate value of prey-taxis stabilize the system.

Strong solutions for stochastic partial differential equations of gradient type

Unique existence of analytically strong solutions to stochastic partial differential equations (SPDE) with drift given by the subdifferential of a quasi-convex function and with general multiplicative noise is proven. The proof applies a genuinely new method of weighted Galerkin approximations based on the “distance” defined by the quasi-convex function. Spatial regularization of the initial condition analogous to the deterministic case is obtained. The results yield a unified framework which is applied to stochastic generalized porous media equations, stochastic generalized reaction–diffusion equations and stochastic generalized degenerated p-Laplace equations. In particular, higher regularity for solutions of such SPDE is obtained.

How travelling waves attract the solutions of KPP-type equations

We consider in this paper a general reaction-diffusion equation of the KPP (Kolmogorov, Petrovskii, Piskunov) type, posed on an infinite cylinder. Such a model will have a family of pulsating waves of constant speed, larger than a critical speed c∗. The family of all supercritical waves attract a large class of initial data, and we try to understand how. We describe in this paper the fate of an initial datum trapped between two supercritical waves of the same velocity: the solution will converge to a whole set of translates of the same wave, and we identify the convergence dynamics as that of an effective drift, around which an effective diffusion process occurs. In several nontrivial particular cases, we are able to describe the dynamics by an effective equation.

Microscopic approach to nonlinear reaction-diffusion: The case of morphogen gradient formation

We develop a microscopic theory for reaction-diffusion (RD) processes based on a generalization of Einstein's master equation [Ann. Phys. 17, 549 (1905)] with a reactive term and show how the mean-field formulation leads to a generalized RD equation with nonclassical solutions. For the nth-order annihilation reaction A+A+A+···+A→0, we obtain a nonlinear reaction-diffusion equation for which we discuss scaling and nonscaling formulations. We find steady states with solutions either exhibiting long-range power-law behavior showing the relative dominance of subdiffusion over reaction effects in constrained systems or, conversely, solutions that go to zero a finite distance from the source, i.e., having finite support of the concentration distribution, describing situations in which diffusion is slow and extinction is fast. Theoretical results are compared with experimental data for morphogen gradient formation.

An analytical solution for diffusion and nonlinear uptake of oxygen in a spherical cell

We develop a new analytical solution for a reactive transport model that describes the steady-state distribution of oxygen subject to diffusive transport and nonlinear uptake in a sphere. This model was originally reported by Lin [S.H. Lin, Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics, J. Theor. Biol. 60 (1976) 449–457] to represent the distribution of oxygen inside a cell and has since been studied extensively by both the numerical analysis and formal analysis communities. Here we extend these previous studies by deriving an analytical solution to a generalized reaction–diffusion equation that encompasses Lin’s model as a particular case. We evaluate the solution for the parameter combinations presented by Lin and show that the new solutions are identical to a grid-independent numerical approximation.

Linearizing transformations to a generalized reaction–diffusion system

Linearizing nonlinear differential equations and finding exact solutions has been an interesting subject in the theory of differential equations. For many nonlinear differential equations, so far no general treatment on linearizing transformations and obtaining exact solutions has been formulated. In this article we use the generalized linearizing transformation to derive the first integral of a generalized reaction–diffusion equation with higher order nonlinear rate of growth, which enables us to linearize the associated second-order nonlinear ordinary differential equation to a linear case. Under certain parametric conditions we can also find travelling wave solutions explicitly for the generalized reaction–diffusion equation through using the derived first integral.

Interaction Manifolds for Reaction Diffusion Equations in Two Dimensions

We consider a general planar reaction diffusion equation which we hypothesize has a localized traveling wave solution. Under assumptions which are no stronger than those needed to prove the stability of a single pulse, we prove that the PDE has solutions which are roughly the linear superposition of two pulses, so long as they move along trajectories which are not parallel. In particular, we prove that if the initial data for the equation is close to the sum of two separated pulses, then the solution converges exponentially fast to such a superposition so long as the distance between the two pulses remains sufficiently large.

Non-standard finite-differences schemes for generalized reaction–diffusion equations

Reaction–diffusion equations are commonly used in different science and engineering fields to describe spatial patterns arising from the interaction of chemical or biochemical reactions and diffusive transport mechanisms. In this work we design, in a systematic way, non-standard finite-differences (FD) schemes for a class of reaction–diffusion equations of the form 1 x σ d d x ( x σ d u d x ) = f ( u ( x ) ) , x ∈ [ 0 , 1 ] , where σ is the shape power that accounts for the complexity of the domain geometry. The proposed FD scheme, that is derived from a Green’s function formulation, replicates the underlying geometry and reduces to traditional FD schemes for sufficiently small values of the grid spacing. Numerical results show that the non-standard FD scheme offers smaller approximation errors with respect to traditional schemes, specially for coarse grids.

Exact Solutions and Invariant Sets to General Reaction-Diffusion Equation

In this paper, we introduce new invariant sets, and the invariant sets and exact solutions to general reaction-diffusion equation are discussed. It is shown that there exist a class of exact solutions to the equations that belong to the invariant sets.

Generalization of the reaction-diffusion, Swift-Hohenberg, and Kuramoto-Sivashinsky equations and effects of finite propagation speeds

The work proposes and studies a model for one-dimensional spatially extended systems, which involve nonlocal interactions and finite propagation speed. It shows that the general reaction-diffusion equation, the Swift-Hohenberg equation, and the general Kuramoto-Sivashinsky equation represent special cases of the proposed model for limited spatial interaction ranges and for infinite propagation speeds. Moreover, the Swift-Hohenberg equation is derived from a general energy functional. After a detailed validity study on the generalization conditions, the three equations are extended to involve finite propagation speeds. Moreover, linear stability studies of the extended equations reveal critical propagation speeds and unusual types of instabilities in all three equations. In addition, an extended diffusion equation is derived and studied in some detail with respect to finite propagation speeds. The extended model allows for the explanation of recent experimental results on non-Fourier heat conduction in nonhomogeneous material.

Analysis of nonlocal neural fields for both general and gamma-distributed connectivities

This work studies the stability of equilibria in spatially extended neuronal ensembles. We first derive the model equation from statistical properties of the neuron population. The obtained integro-differential equation includes synaptic and space-dependent transmission delay for both general and gamma-distributed synaptic connectivities. The latter connectivity type reveals infinite, finite, and vanishing self-connectivities. The work derives conditions for stationary and nonstationary instabilities for both kernel types. In addition, a nonlinear analysis for general kernels yields the order parameter equation of the Turing instability. To compare the results to findings for partial differential equations (PDEs), two typical PDE-types are derived from the examined model equation, namely the general reaction–diffusion equation and the Swift–Hohenberg equation. Hence, the discussed integro-differential equation generalizes these PDEs. In the case of the gamma-distributed kernels, the stability conditions are formulated in terms of the mean excitatory and inhibitory interaction ranges. As a novel finding, we obtain Turing instabilities in fields with local inhibition–lateral excitation, while wave instabilities occur in fields with local excitation and lateral inhibition. Numerical simulations support the analytical results.

Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations

We study entire solutions of a scalar reaction-diffusion equationof 1-space dimension. Here the entire solutions are meant by solutionsdefined for all $(x,t)\in\mathbb R^2$.Assuming that the equation has traveling front solutions and using thecomparison argument, we prove the existence of entire solutions whichbehave as two fronts coming from the both sides of $x$-axis.A key idea for the proof of the main results is to characterize theasymptotic behavior of the solutions as $t\to-\infty$ in terms ofappropriate subsolutions and supersolutions. This argument can applynot only to a general bistable reaction-diffusion equation but also tothe Fisher-KPP equation.We also extend our argument to the Fisher-KPP equation withdiscrete diffusion.

On the stationary solutions of generalized reaction diffusion equations with $p\& q$-Laplacian

In this paper, we analyse a family of stationary nonlinear equations with $p\& q$- Laplacian $-\Delta_p u -\Delta_q u=\lambda c(x,u)$ which have a wide spectrum of applications in many areas of science. We introduce a new type of variational principles corresponding to this family of equations. Using this formalism, we exhibit intervals for the scalar parameter $\lambda$ where there exist positive solutions of the considered problems. Furthermore, we prove, in another interval, the nonexistence of nontrivial solutions. These results are different from those of existence and nonexistence for stationary equations with single Laplacian.