Reaction-diffusion Research Articles

An innovative Vieta–Fibonacci wavelet collocation method for the numerical solution of three-component Brusselator reaction diffusion system of fractional order

An innovative Vieta–Fibonacci wavelet collocation method for the numerical solution of three-component Brusselator reaction diffusion system of fractional order

Modelling phytoplankton-virus interactions: phytoplankton blooms and lytic virus transmission

A dynamic reaction–diffusion model of four variables is proposed to describe the spread of lytic viruses among phytoplankton in a poorly mixed aquatic environment. The basic ecological reproductive index for phytoplankton invasion and the basic reproduction number for virus transmission are derived to characterize the phytoplankton growth and virus transmission dynamics. The theoretical and numerical results from the model show that the spread of lytic viruses effectively controls phytoplankton blooms. This validates the observations and experimental results of Emiliana huxleyi-lytic virus interactions. The studies also indicate that the lytic virus transmission cannot occur in a low-light or oligotrophic aquatic environment.

Turing instability analysis and parameter identification based on optimal control and statistics method for a rumor propagation system.

This study establishes a reaction-diffusion system to capture the dynamics of rumor propagation, considering two possibilities of contact transmission. The sufficient and necessary conditions for a positive equilibrium point are provided, and the Turing instability conditions for this equilibrium point are derived. Furthermore, utilizing variational inequalities, a first-order necessary condition for parameter identification based on optimal control is established. During the numerical simulation process, the correctness of the Turing instability conditions is verified, and optimal control-based parameter identification is applied to the target pattern. Additionally, statistical methods are employed for pattern parameter identification. The identification results demonstrate that optimal control-based parameter identification exhibits higher efficiency and accuracy. Finally, both theories' parameter identification principles are extended to a small-world network, yielding consistent conclusions with continuous space.

Synchronization of delayed stochastic reaction–diffusion Hopfield neural networks via sliding mode control

Synchronization of stochastic reaction–diffusion Hopfield neural networks with s-delays via sliding mode control is investigated in this article. To begin with, we choose suitable functional space for state variables, then the system is transformed into a functional differential equation in an infinite-dimensional Hilbert space by using appropriate functional analysis technique. Based on above preliminary preparation, sliding mode control (SMC) is constructed to drive the error trajectory into the designed switching surface. Specifically, the switching surface is constructed as linear combination of state variables, which is related to control gains. Then novel SMC law is designed which involving delay, reaction diffusion term, and reaching law. Furthermore, the criterion of mean-square exponential synchronization for stochastic delayed reaction–diffusion Hopfield neural networks with s-delays is given in the form of matrix form. This criterion is less restrictive and easy to check in computer. Meanwhile, a different novel Lyapunov–Krasovskii functional (LKF) mixed with Itô’s formula, Young inequality, Hanalay inequality is employed in this proof procedure. At last, a numerical example is presented to validate the availability of theoretical result. The simulation is based on the finite difference method, and numerical result coincides with the theoretical result proposed.

Strong stability preserving multiderivative time marching methods for stiff reaction–diffusion systems

The present study introduces a new class of unconditionally strong stability preserving (SSP) multi-derivative Runge–Kutta methods for the numerical simulation of reaction–diffusion systems in the stiff regime. The unconditional SSP property of the methods makes them highly efficient for the simulation of reaction–diffusion systems without any restrictive time-step requirements. These methods have been tested for accuracy using L∞ error analysis, and comparisons with existing literature have shown that they perform better even for larger time-steps. In addition, the robustness and efficiency of the derived method are also validated by numerical simulations of the Brusselator, Gray–Scott, and Schnakenberg models.

Spatio-temporal complexity in a prey-predator system with Holling type-IV response and Leslie-type numerical response: Turing and steady-state bifurcations

In this study, we unravel the intricate dynamics of a spatio-temporal prey-predator model featuring a Holling type-IV functional response and Leslie-type predator numerical response under Neumann boundary conditions. Our analysis encompasses the uniform persistence and global asymptotic stability of a positive equilibrium, validated by precise numerical simulations. Additionally, we explore Turing instability and spatial pattern emergence through linear stability analysis. Our primary emphasis lies in the realm of spatio-temporal bifurcation analysis, through which we establish criteria for the presence or absence of non-constant steady states within n-dimensional diffusion models. Moreover, we discern precise conditions governing Hopf bifurcation and steady-state bifurcation in 1-dimensional diffusion models. These findings offer theoretical insights that align with the intricate dynamic patterns observed in our numerical simulations.

Spatiotemporal dynamics of prey-predator model incorporating Holling-type II functional response with fear and its carryover effects.

The recent focus in the fields of biology and ecology has centered on the significant attention given to the mathematical modeling and analyzing the spatiotemporal population distribution among species engaged in interactions. This paper explores the dynamics of the temporal and spatiotemporal delayed Bazykin-type prey-predator model, incorporating fear and its carryover effect. In our model, we incorporated a functional response of the Holling-type II. In the temporal model, a detailed dynamic analysis was carried out, investigating the positivity and boundedness of solutions, establishing the uniqueness and existence of positive interior equilibria, and examining both local and global stability. Additionally, we explored the presence of saddle-node, transcritical, and Hopf bifurcations varying attack rate parameter. The delayed system shows highly periodic behavior. Additionally, for the spatiotemporal model, we provide a complete analysis of local and global stability, and we derive the conditions for the existence of Turing instability for both self-diffusion and cross-diffusion, respectively. The two-dimensional diffusive model is further discussed, highlighting various Turing patterns, including holes, stripes, and hot and cold spots, along with their biological significance. Numerical simulations are executed to validate the analytical findings in both temporal and spatiotemporal models.

Emergence of rogue-like waves in a reaction-diffusion system: Stochastic output from deterministic dissipative dynamics.

Rogue waves are an intriguing nonlinear phenomenon arising across different scales, ranging from ocean waves through optics to Bose-Einstein condensates. We describe the emergence of rogue wave-like dynamics in a reaction-diffusion system that arise as a result of a subcritical Turing instability. This state is present in a regime where all time-independent states are unstable and consists of intermittent excitation of spatially localized spikes, followed by collapse to an unstable state and subsequent regrowth. We characterize the spatiotemporal organization of spikes and show that in sufficiently large domains the dynamics are consistent with a memoryless process.

Synchronization in a Three Level Network of All-to-All Periodically Forced Hodgkin–Huxley Reaction–Diffusion Equations

This article focuses on the analysis of dynamics emerging in a network of Hodgkin–Huxley reaction–diffusion equations. The network has three levels. The three neurons in level 1 receive a periodic input but do not receive inputs from other neurons. The three neurons in level 2 receive inputs from one specific neuron in level 1 and all neurons in level 3. The neurons in level 3 (all other neurons) receive inputs from all other neurons in levels 2 and 3. Furthermore, the right-hand side of pre-synaptic neurons is connected to the left-hand side of the post-synaptic neurons. The synchronization phenomenon is observed for neurons in level 3, even though the system is initiated with different functions. As far as we know, it is the first time that evidence of the synchronization phenomenon is provided for spatially extended Hodgkin–Huxley equations, which are periodically forced at three different sites and embedded in such a hierarchical network with space-dependent coupling interactions.

Equivariant Hopf Bifurcation in a Class of Partial Functional Differential Equations on a Circular Domain

Circular domains frequently appear in mathematical modeling in the fields of ecology, biology and chemistry. In this paper, we investigate the equivariant Hopf bifurcation of partial functional differential equations with Neumann boundary condition on a two-dimensional disk. The properties of these bifurcations at equilibriums are analyzed rigorously by studying the equivariant normal forms. Two reaction–diffusion systems with discrete time delays are selected as numerical examples to verify the theoretical results, in which spatially inhomogeneous periodic solutions including standing waves and rotating waves, and spatially homogeneous periodic solutions are found near the bifurcation points.

A numerical investigation of a well-known nonlinear Newell-Whitehead-Segel equation using the rank polynomial of the star graph

Mathematical models of pattern formation are indispensable tools in various fields, from developmental biology to ecology, providing insights into complex phenomena and contributing to our understanding of the natural world. These patterns have been extensively studied using reaction-diffusion and NewellWhiteheadSegel models. This article intended to find an approximate numerical solution to the NewellWhiteheadSegel equation. The appearance of stripe patterns in two-dimensional systems is explained in nonlinear systems using the NewellWhiteheadSegel equation. Based on the function basis of rank polynomials of star graphs and the well-posed operational matrices, the rank polynomial collocation method is constructed. The alleged rank polynomial collocation method created a system of nonlinear algebraic equations from the nonlinear NewellWhiteheadSegel equation. The nonlinear NewellWhiteheadSegel equation solution is approximated by solving the resulting system via Newton’s Raphson method. Numerical instances are provided to illustrate the validity and effectiveness of the technique. Verification of accuracy is accomplished by calculating error norms. The obtained numerical results show a reasonable degree of consistency with the findings reported in the current literature. The scheme’s primary benefit is the algorithm’s ease of implementation.

Asymptotic Behavior of Global Solution of an Anomalous Gray Scott Model

The object of this paper is to prove that existence global and asymptotic behavior of solutions for anomalous coupled reaction diffusion system (Gray Scott model) with homogeneous Neumann boundary conditions. The existence and uniqueness of the local solution are given by the Banach fixed point theorem. However, the asymptotic behavior is investiged by technique semi group estimates and the Sobolev embedding theorem

Spatiotemporal dynamics analysis and parameter optimization of a network epidemic-like propagation model based on neural network method

In this paper, a reaction-diffusion model is established to study the dynamic behavior of rumor propagation. Firstly, we consider the existence of the positive equilibrium points. Then, we perform a stability analysis to study the conditions for the occurrence of Turing instability. Secondly, we use multiscale analysis to derive the expression of the amplitude equation. In the process of numerical simulation, the reality is considered. It shows that controlling the spread rate of rumor and the number of new Internet users have a great effect on curbing the spread of online rumor. Furthermore, it is proved that the analysis of amplitude equation plays a decisive role in the formation of Turing patterns. We also discuss the phenomenon of Turing patterns when the network structure changes and verify the rationality of the model by Monte Carlo method. Finally, we consider two methods based on statistical principle and convolutional neural network severally to identify the parameters of the reaction-diffusion system with Turing instability by using stable patterns. The statistical principle-based method offers superior accuracy, whereas the convolutional neural network-based approach significantly reduces recognition time and cuts down time costs.

A toxicant-taxis reaction–diffusion model for the interaction between two species and a toxicant in an aquatic ecosystem

In this paper, we develop a toxicant-taxis reaction–diffusion model that describes the interaction dynamics between two species and a toxicant in a polluted aquatic ecosystem. We analyze the existence and stability of the equilibria of the corresponding space-absent ordinary differential equation system. We then perform linear stability analysis for our reaction–diffusion system to obtain sufficient conditions on model parameters under which the stable spatially homogeneous steady states become unstable. Numerical simulations are performed to illustrate that spatially inhomogeneous patterns and spatiotemporal patterns between the two species and the toxicant will typically emerge. We also numerically analyze the effects of spatially varying toxicant input and the intensity of toxicant-taxis on the population persistence and species diversity. It turns out that heterogeneous toxicant inputs and substantial toxicant taxis can boost the coexistence of two species, thereby promoting species diversity in ecosystems. By comparing the impact of a toxicant on a single species versus its impact on two species, we find that the collective sharing of the toxic burden among multiple species proves advantageous for biodiversity.

Mean-squared exponential stability of distributed almost periodic sequence for nonlocal space-time discrete shunting inhibitory cellular neural networks with Brownian motions

The discrete-time and discrete-space model for stochastic nonlocal shunting inhibitory cellular neural networks with reaction diffusions are modelled in the first time. Owing to the consideration of spatial variables, the discrete model derived in this article is more complex than the traditional ordinary difference model. In accordance with the constant-variation-formula for discrete-time and discrete-space model, Banach contracting mapping principle, the method of proof by contradiction and stochastic calculus, we also obtain the existence of a unique bounded almost periodic sequence for the discrete model, which is exponentially stable in the mean-square sense. Noting that it is the first time to consider the dynamics of almost periodicity in distribution of time-space discrete neural network models. The practicability of the present results is demonstrated by means of an illustration.

Precise rates of propagation in reaction–diffusion equations with logarithmic Allee effect

This paper focuses on propagation phenomena in reaction–diffusion equations with a weakly degenerate monostable nonlinearity. The kind of reaction term we consider can be seen as an intermediate between the classical logistic one (or Fisher–KPP) and more usual power laws that usually model Allee effects. We investigate the effect of the decay rate of the initial data on the propagation rate. When the tail of the initial data is sub-exponential, both finite speed propagation and acceleration may happen. We derive the exact separation between the two situations. When the initial data is sub-exponentially unbounded, acceleration unconditionally occurs. Estimates for the locations of the level sets are expressed in terms of the decay of the initial data. In addition, sharp exponents of acceleration for initial data with sub-exponential and algebraic tails are given. Numerical simulations are presented to illustrate the above findings.

Global attractivity for reaction–diffusion equations with periodic coefficients and time delays

In this paper, we provide sharp criteria of global attraction for a class of non-autonomous reaction–diffusion equations with delay and Neumann conditions. Our methodology is based on a subtle combination of some dynamical system tools and the maximum principle for parabolic equations. It is worth mentioning that our results are achieved under very weak and verifiable conditions. We apply our results to a wide variety of classical models, including the non-autonomous variants of Nicholson’s equation or the Mackey–Glass model. In some cases, our technique gives the optimal conditions for the global attraction.

Attached Flows for Reaction–Diffusion Processes Described by a Generalized Dodd–Bullough–Mikhailov Equation

This paper uses the attached flow method for solving nonlinear second-order differential equations of the reaction–diffusion type. The key steps of the method consist of the following: (i) reducing the differentiability order by defining the first derivative of the variable as a new variable called the flow and (ii) a forced decomposition of the derivative-free term so that the flow appears explicitly in it. The resulting reduced equation is solved using specific balancing rules. Only step (i) would lead to an Abel-type equation with complicated integral solutions. Completed with (ii) and with the graduation procedure, the attached flow method used in the paper, without requiring such a great effort, allows for the obtaining of accurate analytical solutions. The method is applied here to a subclass of reaction–diffusion equations, the generalized Dodd–Bulough–Mikhailov equation, which includes a translation of the variable and nonlinearities up to order five. The equation is solved for each order of nonlinearity, and the solutions are discussed following the values of the parameters involved in the equation.

On wavefront patterns in a fractional reaction–diffusion model for predator–prey system with anti-predator behavior

AbstractIn recent studies, much attention has been paid to reaction–diffusion systems with anomalous diffusion. In this paper, we investigate the formation of wavefront patterns in a predator–prey model with anti-predator behavior under the influence of anomalous subdiffusion. We use methods of traveling wave analysis and numerical integration to establish the existence of traveling wavefront solutions. Further, obtained traveling wavefront solutions are validated through direct computer simulations of time-dependent solutions for fractional partial differential equation system. It is found that wavefronts exist in a range of system parameters, which travel faster in the subdiffusive system than in the normal diffusive one.

Research on Pattern Dynamics Behavior of a Fractional Vegetation-Water Model in Arid Flat Environment

In order to stop and reverse land degradation and curb the loss of biodiversity, the United Nations 2030 Agenda for Sustainable Development proposes to combat desertification. In this paper, a fractional vegetation–water model in an arid flat environment is studied. The pattern behavior of the fractional model is much more complex than that of the integer order. We study the stability and Turing instability of the system, as well as the Hopf bifurcation of fractional order α, and obtain the Turing region in the parameter space. According to the amplitude equation, different types of stationary mode discoveries can be obtained, including point patterns and strip patterns. Finally, the results of the numerical simulation and theoretical analysis are consistent. We find some novel fractal patterns of the fractional vegetation–water model in an arid flat environment. When the diffusion coefficient, d, changes and other parameters remain unchanged, the pattern structure changes from stripes to spots. When the fractional order parameter, β, changes, and other parameters remain unchanged, the pattern structure becomes more stable and is not easy to destroy. The research results can provide new ideas for the prevention and control of desertification vegetation patterns.