Single-species Reaction Research Articles

Dynamics of a Reaction–Diffusion–Advection System with Nonlinear Boundary Conditions

In this paper, we consider a single-species reaction–diffusion–advection population model with nonlinear boundary condition in heterogenous space. We not only investigate the existence, nonexistence and stability of positive steady-state solutions through a linear elliptic eigenvalue problem by means of variational approach, but also verify the existence of steady-state bifurcations at zero solution through Crandall and Robinowitz bifurcation theory and discuss the stability of bifurcations, which can lead to Allee effect when the bifurcation is subcritical.

Nonlocal delay driven spatiotemporal patterns in a single-species reaction–diffusion model

This paper presents Turing–Hopf bifurcation analysis and the resulting spatiotemporal dynamics in a single-species reaction–diffusion model with nonlocal delay. A linear stability analysis is performed to find the conditions for the occurrence of Turing–Hopf bifurcation. The weakly nonlinear analysis is then employed to derive the amplitude equations which describe the slow-time evolution of the critical Turing and Hopf modes. By the utilization of amplitude equations, stability conditions for different pattern formations can be obtained, helping to reveal the affluent spatiotemporal patterns near the Turing–Hopf bifurcation and predict when and where these patterns can emerge. Also, our analysis shows the role of the choice of initial conditions in pattern formations. Numerical simulations are given to verify the theoretical results.

Multi-scale random walk models for reactive transport processes in fracture-matrix systems

Random walk methods are particularly well suited for modeling (anomalous) transport processes in complex systems, from heterogeneous porous domains to fractured rocks. Taking full advantage of their multi-scale attributes, these methods can be integrated into multi-scale modeling strategies where they are used for (i) performing small-scale simulations with a high level of description of the heterogeneities, (ii) defining statistical functions that describe these heterogeneities at a larger scale, and (iii) providing upscaled descriptions of the processes while taking into account the impact of the small-scale heterogeneities. In this work, we define such a strategy considering (i) structural heterogeneities with heterogeneous porosity fields incorporated into fracture-matrix systems and (ii) the reactivity of the system with chemical reactions that are not restricted to linear first-order reactions as it is the case in existing upscaled formulations. To this end, we develop two new modeling methods: the Reactive-Time Domain Random Walk (R-TDRW) and Upscaled Reactive TDRW (UR-TDRW) approaches at small and large scale, respectively. The numerical methods and multi-scale strategy are presented with a general formulation and applied to single-species transport and reaction in fracture-matrix systems. We analyze the impact of different levels of structural heterogeneities, and the impact of the reactivity, on the breakthrough curves computed at both scales and on the statistical functions that are used in the multi-scale strategy. This strategy is validated by demonstrating the good agreement between the results obtained at different scales, showing promising applications for future work in large-scale fracture networks.

Travelling wave analysis of cellular invasion into surrounding tissues

Single-species reaction–diffusion equations, such as the Fisher–KPP and Porous-Fisher equations, support travelling wave solutions that are often interpreted as simple mathematical models of biological invasion. Such travelling wave solutions are thought to play a role in various applications including development, wound healing and malignant invasion. One criticism of these single-species equations is that they do not explicitly describe interactions between the invading population and the surrounding environment. In this work we study a reaction–diffusion equation that describes malignant invasion which has been used to interpret experimental measurements describing the invasion of malignant melanoma cells into surrounding human skin tissues Browning et al. (2019). This model explicitly describes how the population of cancer cells degrade the surrounding tissues, thereby creating free space into which the cancer cells migrate and proliferate to form an invasion wave of malignant tissue that is coupled to a retreating wave of skin tissue. We analyse travelling wave solutions of this model using a combination of numerical simulation, phase plane analysis and perturbation techniques. Our analysis shows that the travelling wave solutions involve a range of very interesting properties that resemble certain well-established features of both the Fisher–KPP and Porous-Fisher equations, as well as a range of novel properties that can be thought of as extensions of these well-studied single-species equations. Matlab software to implement all calculations is available at GitHub.

Completely solvable models with nearest-neighbor interactions on one-dimensional lattices

The most general exclusion single species reaction–diffusion models are investigated, which are defined on a one-dimensional lattice, have nearest-neighbor interactions, and their corresponding two-site Hamiltonians are diagonalizable or converted to a Jordan form by a direct product of matrices. The diagonalizable Hamiltonians represent a family of 3-parameter stochastic processes, while those which can be transformed into a Jordan form have 4 free parameters. For the diagonalizable Hamiltonians, all of the eigenvalues and eigenvectors and some correlation functions are explicitly calculated. Also, both the entropy and the entropy production rate in such systems are studied. For those Hamiltonians which can be converted to a Jordan form, some parts of the spectrum are obtained.

Effect of bias in a reaction-diffusion system in two dimensions.

We consider a single species reaction diffusion system on a two-dimensional lattice where the particles A are biased to move towards their nearest neighbors and annihilate as they meet. Allowing the bias to take both negative and positive values parametrically, any nonzero bias is seen to drastically affect the behavior of the system compared to the unbiased (simple diffusive) case. For positive bias, a finite number of dimers, which are isolated pairs of particles occurring as nearest neighbors, exist while for negative bias, a finite density of particles survives. Both the quantities vanish in a power-law manner close to the diffusive limit with different exponents. The appearance of dimers is exclusively due to the parallel updating scheme used in the simulation. The results indicate the presence of a continuous phase transition at the diffusive point. In addition, a discontinuity is observed at the fully positive bias limit. The persistence behavior is also analysed for the system.

β-Lactoglobulin heat denaturation: A critical assessment of kinetic modelling

Bovine whey protein β-lactoglobulin (β-lg) will denature above 70 °C. Denaturation may contribute positively or negatively to food functionality, so it is important to control the degree of denaturation during milk processing. This work critically examined the kinetic modelling of β-lg denaturation in water, buffer or milk at 70–150 °C using the general rate law for a single-species reaction and the Arrhenius equation. Several published datasets, originally analysed in linearised form, were re-analysed with nonlinear regression, which preserves the error structure of data and estimates the precision of parameters. This work discusses statistical best practice for modelling the heat denaturation of β-lg, explores the criteria for using 2- or 3-parameter regression equations, and proposes new approaches for modelling the concentration- and temperature-dependence of denaturation rates. These new approaches improved the accuracy and precision of kinetic parameter estimates, and revealed new details of how solution conditions influence β-lg denaturation rates.

Analysis of spatiotemporal patterns in a single species reaction–diffusion model with spatiotemporal delay

Employing the theories of Turing bifurcation in the partial differential equations, we investigate the dynamical behavior of a single species reaction–diffusion model with spatiotemporal delay. The linear stability and the conditions for the occurrence of Turing bifurcation in this model are obtained. Moreover, the amplitude equations which represent different spatiotemporal patterns are also obtained near the Turing bifurcation point by using multiple scale method. In Turing space, it is found that the spatiotemporal distributions of the density of this researched species have spots pattern and stripes pattern. Finally, some numerical simulations corresponding to the different spatiotemporal patterns are given to verify our theoretical analysis.

Two-loop calculation of anomalous kinetics of the reaction A + A → Ø in randomly stirred fluid

The single-species annihilation reaction A + A → Ø is studied in the presence of a random velocity field generated by the stochastic Navier-Stokes equation. The renormalization group is used to analyze the combined influence of the density and velocity fluctuations on the long-time behavior of the system. The direct effect of velocity fluctuations on the reaction constant appears only from the two-loop order, therefore, all stable fixed points of the renormalization group and their regions of stability are calculated in the two-loop approximation in the two-parameter (ε, Δ) expansion. A renormalized integro-differential equation for the number density is put forward which takes into account the effect of density and velocity fluctuations at next-to-leading order. Solution of this equation in perturbation theory is calculated in a homogeneous system.

Field-theoretic technique for irreversible reaction processes

The single-species annihilation reaction A+A->0 is studied in the presence of random advecting field. In order to determine possible infrared behavior of the system all stable fixed points are presented to two-loop approximation in double $(\epsilon,\Delta)$ expansion with the corresponding regions of stability. The main result of this paper is the calculation of all the renormalization constants and the decay exponent to the second-order precision as well as calculation of scaling function the mean particle number to the first order. Effects of random sources and sinks on reaction kinetics in the master-equation description have been investigated in the framework of a field-theoretic model, obtained by the "second quantization" a la Doi of the corresponding master equation. It has been demonstrated that random sources and sinks have a significant effect on the asymptotic behaviour of the model and two universality classes for their description have been identified by the scaling analysis. Results are compared with the Langevin-equation description of the same process.

The spectrum and the phase transition of models solvable through the full interval method

The most general exclusion single species reaction–diffusion models with nearest-neighbor interactions on a one-dimensional lattice are investigated, for which the evolution of full intervals are closed. Using a generating function method, the probability that n consecutive sites be full is investigated. The stationary values of these probabilities, as well as the spectrum of the time translation generator (Hamiltonian) of these are obtained. It is shown that, depending on the reaction rates, the model could exhibit a dynamical phase transition.

Phase transitions and crossovers in reaction-diffusion models with catalyst deactivation

The activity of catalytic materials is reduced during operation by several mechanisms, one of them being poisoning of catalytic sites by chemisorbed impurities or products. Here we study the effects of poisoning in two reaction-diffusion models in one-dimensional lattices with randomly distributed catalytic sites. Unimolecular and bimolecular single-species reactions are considered, without reactant input during the operation. The models show transitions between a phase with continuous decay of reactant concentration and a phase with asymptotic nonzero reactant concentration and complete poisoning of the catalyst. The transition boundary depends on the initial reactant and catalyst concentrations and on the poisoning probability. The critical system behaves as in the two-species annihilation reaction, with reactant concentration decaying as t(-1/4) and the catalytic sites playing the role of the second species. In the unimolecular reaction, a significant crossover to the asymptotic scaling is observed even when one of those parameters is 10% far from criticality. Consequently, an effective power-law decay of concentration may persist up to long times and lead to an apparent change in the reaction kinetics. In the bimolecular single-species reaction, the critical scaling is followed by a two-dimensional rapid decay, thus two crossovers are found.

Slowly oscillating wave solutions of a single species reaction–diffusion equation with delay

We study positive bounded wave solutions u ( t , x ) = ϕ ( ν ⋅ x + c t ) , ϕ ( − ∞ ) = 0 , of equation u t ( t , x ) = Δ u ( t , x ) − u ( t , x ) + g ( u ( t − h , x ) ) , x ∈ R m ( ∗ ) . This equation is assumed to have two non-negative equilibria: u 1 ≡ 0 and u 2 ≡ κ > 0 . The birth function g ∈ C ( R + , R + ) is unimodal and differentiable at 0 and κ. Some results also require the feedback condition ( g ( s ) − κ ) ( s − κ ) < 0 , with s ∈ [ g ( max g ) , max g ] ∖ { κ } . If additionally ϕ ( + ∞ ) = κ , the above wave solution u ( t , x ) is called a travelling front. We prove that every wave ϕ ( ν ⋅ x + c t ) is eventually monotone or slowly oscillating about κ. Furthermore, we indicate c ∗ ∈ R + ∪ { + ∞ } such that Eq. ( ∗ ) does not have any travelling front (neither monotone nor non-monotone) propagating at velocity c > c ∗ . Our results are based on a detailed geometric description of the wave profile ϕ. In particular, the monotonicity of its leading edge is established. We also discuss the uniqueness problem indicating a subclass G of ‘ asymmetric’ tent maps such that given g ∈ G , there exists exactly one positive travelling front for each fixed admissible speed.

Study of transients in the propagation of nonlinear waves in some reaction diffusion systems

We study the transient dynamics of single species reaction diffusion systems whose reaction terms f(u) vary nonlinearly near u ≈ 0, specifically as f(u) ≈ u2 and f(u) ≈ u3. We consider three cases, calculate their traveling wave fronts and speeds analytically and solve the equations numerically with different initial conditions to study the approach to the asymptotic front shape and speed. Observed time evolution is found to be quite sensitive to initial conditions and to display in some cases nonmonotonic behavior, ascribable to the disparity in time scales between the evolution of the front interior and the front tail.

Nonmonotone travelling waves in a single species reaction–diffusion equation with delay

We prove the existence of a continuous family of positive and generally nonmonotone travelling fronts for delayed reaction–diffusion equations u t ( t , x ) = Δ u ( t , x ) − u ( t , x ) + g ( u ( t − h , x ) ) ( ∗ ) , when g ∈ C 2 ( R + , R + ) has exactly two fixed points: x 1 = 0 and x 2 = K > 0 . Recently, nonmonotonic waves were observed in numerical simulations by various authors. Here, for a wide range of parameters, we explain why such waves appear naturally as the delay h increases. For the case of g with negative Schwarzian, our conditions are rather optimal; we observe that the well known Mackey–Glass-type equations with diffusion fall within this subclass of ( ∗ ) . As an example, we consider the diffusive Nicholson's blowflies equation.

Single-species three-particle reactions in one dimension

Renormalization group calculations for fluctuation-dominated reaction-diffusion systems are generally in agreement with simulations and exact solutions. However, simulations of the single-species reactions 3A -->(Ø,A,2A) at their upper critical dimension d(c)=1 have found asymptotic densities argued to be inconsistent with renormalization group predictions. We show that this discrepancy is resolved by inclusion of the leading corrections to scaling, which we derive explicitly and show to be universal, a property not shared by the A+A-->(Ø,A) reactions. Finally, we demonstrate that two previous Smoluchowski approaches to this problem reduce, with various corrections, to a single theory which surprisingly yields the same asymptotic density as the renormalization group.

Complex bioindication and environmental stress assessment

Abstract The indicative quality of bioindicators, ranging from organelles, organs or single organisms to complex ecosystems, depends on inherent ecophysiological properties, population dynamics, and stress reactions with regard to physical and chemical changes in site conditions as described in Section 2 of the present contribution. Section 3 provides a systematic review of both typology and rational selection of bioindicators on the species, population, biocenotic, and ecosystem levels. It is to show that the primary task of bioindicators is the general determination of physiological effects in the sense of strain reactions rather than the direct measurement of environmental concentrations of stressors. Thus, in early recognition perspective the lack of specificity has the advantage of a broad-based caveat, inducive to subsequent systematic search for quantitative causal inter-relationships. A further advantage of biomonitoring is its comparatively low cost on the one hand and the integrative recording character on the other. Contrary to these positive aspects of bioindicator use there is, however, an essential deficiency resulting from the highly variable susceptibility of the test species exposed to stressors, which leads to difficulties in comparing specific effect data. In view of such problems, the possibilities of fuzzy logic approaches for evaluative data interpretation and inter- and intraspecific comparison purposes are emphasized. Active and passive biomonitoring approaches on the basis of single-species reactions yield spatially valid data only on condition the underlying sampling networks are implemented in compliance with geostatistical requirements or the corresponding test methodologies of variogram analysis and kriging procedures, respectively. Analogously, also the selection of complex bioindicators such as biocenoses or ecosystems must be based on rigid criteria of spatial and temporal representativeness. The last section then is a critical comparative appraisal of the problems encountered in biomonitoring, which leads to a set of suggestions for improving both the technical practicability and the data quality of biomonitoring approaches.

Applications of field-theoretic renormalization group methods to reaction–diffusion problems

We review the application of field-theoretic renormalization group (RG) methods to the study of fluctuations in reaction-diffusion problems. We first investigate the physical origin of universality in these systems, before comparing RG methods to other available analytic techniques, including exact solutions and Smoluchowski-type approximations. Starting from the microscopic reaction-diffusion master equation, we then pedagogically detail the mapping to a field theory for the single-species reaction k A -> l A (l < k). We employ this particularly simple but non-trivial system to introduce the field-theoretic RG tools, including the diagrammatic perturbation expansion, renormalization, and Callan-Symanzik RG flow equation. We demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative eps = d_c - d expansions with respect to the upper critical dimension d_c. With these basics established, we then provide an overview of more sophisticated applications to multiple species reactions, disorder effects, L'evy flights, persistence problems, and the influence of spatial boundaries. We also analyze field-theoretic approaches to nonequilibrium phase transitions separating active from absorbing states. We focus particularly on the generic directed percolation universality class, as well as on the most prominent exception to this class: even-offspring branching and annihilating random walks. Finally, we summarize the state of the field and present our perspective on outstanding problems for the future.

Wavefronts and global stability in a time-delayed population model with stage structure

We formulate and study a one–dimensional single–species diffusive–delay population model. The time delay is the time taken from birth to maturity. Without diffusion, the delay differential model extends the well–known logistic differential equation by allowing delayed constant birth processes and instantaneous quadratically regulated death processes. This delayed model is known to have simple global dynamics similar to that of the logistic equation. Through the use of a sub/supersolution pair method, we show that the diffusive delay model continues to generate simple global dynamics. This has the important biological implication that quadratically regulated death processes dramatically simplify the growth dynamics. We also consider the possibility of travelling wavefront solutions of the scalar equation for the mature population, connecting the zero solution of that equation with the positive steady state. Our main finding here is that our fronts appear to be all monotone, regardless of the size of the delay. This is in sharp contrast to the frequently reported findings that delay causes a loss of monotonicity, with the front developing a prominent hump in some other delay models.

Single-species reactions on a random catalytic chain

We present an exact solution for a catalytically-activated annihilation A + A \to 0 reaction taking place on a one-dimensional chain in which some segments (placed at random, with mean concentration p) possess special, catalytic properties. Annihilation reaction takes place, as soon as any two A particles land from the reservoir onto two vacant sites at the extremities of the catalytic segment, or when any A particle lands onto a vacant site on a catalytic segment while the site at the other extremity of this segment is already occupied by another A particle. We find that the disorder-average pressure $P^{(quen)}$ per site of such a chain is given by $P^{(quen)} = P^{(lan)} + \beta^{-1} F$, where $P^{(lan)} = \beta^{-1} \ln(1+z)$ is the Langmuir adsorption pressure, (z being the activity and \beta^{-1} - the temperature), while $\beta^{-1} F$ is the reaction-induced contribution, which can be expressed, under appropriate change of notations, as the Lyapunov exponent for the product of 2 \times 2 random matrices, obtained exactly by Derrida and Hilhorst (J. Phys. A {\bf 16}, 2641 (1983)). Explicit asymptotic formulae for the particle mean density and the compressibility are also presented.