Search Results (18)

In this article, the validity of the comparison principle (CP) for weakly coupled quasi-linear cooperative systems with delays is proven. This is a powerful tool for studying the qualitative properties of the solutions. The CP is crucial in the proofs of the existence and uniqueness of weak solutions to cooperative reaction–diffusion systems presented here. Other direct consequences of the CP are the stability of the solution, the attenuation of long time periods, etc. An example model is given by spatial SEIR models with delays. They are suitable for modeling disease spread in space and time and can be described using a weakly coupled cooperative reaction–diffusion system. In this paper, spatial SEIR models with delays are considered in a continuous space. The emphasis is on the qualitative properties of the solutions, which are important for providing a mathematical basis for the model. Full article

This study investigates a model of competition and cooperation between two enterprises with reaction, diffusion, and delays. The stability and Hopf bifurcation for variants with two, one, and no delays are considered by examining a system of delay ODE equations analytically and numerically, applying the Galerkin method. A condition is obtained that helps characterize the existence of Hopf bifurcation points. Full maps of stability analysis are discussed in detail. With bifurcation diagrams, three different cases of delay are shown to determine the stable and unstable regions. It is found that when ${\tau }_i>0$, there are two different stability regions, and that without a delay (${\tau }_i=0$), there is only one stable region. Furthermore, the effects of delays and diffusion parameters on all other free rates in the system are considered; these can significantly affect the stability areas, with important economic consequences for the development of enterprises. Moreover, the relationship between the diffusion and delay parameters is discussed in more detail: it is found that the value of the time delay at the Hopf point increases exponentially with the diffusion coefficient. An increase in the diffusion coefficient can also lead to an increase in the Hopf-point values of the intrinsic growth rates. Finally, bifurcation diagrams are used to identify specific instances of limit cycles, and 2-D phase portraits for both systems are presented to validate all theoretical results discussed in this work. Full article

This paper delves into a comprehensive analysis of a generalized impulsive discrete reaction–diffusion system under periodic boundary conditions. It investigates the behavior of reactant concentrations through a model governed by partial differential equations (PDEs) incorporating both diffusion mechanisms and nonlinear interactions. By employing finite difference methods for discretization, this study retains the core dynamics of the continuous model, extending into a discrete framework with impulse moments and time delays. This approach facilitates the exploration of finite-time stability (FTS) and dynamic convergence of the error system, offering robust insights into the conditions necessary for achieving equilibrium states. Numerical simulations are presented, focusing on the Lengyel–Epstein (LE) and Degn–Harrison (DH) models, which, respectively, represent the chlorite–iodide–malonic acid (CIMA) reaction and bacterial respiration in Klebsiella. Stability analysis is conducted using Matlab’s LMI toolbox, confirming FTS at equilibrium under specific conditions. The simulations showcase the capacity of the discrete model to emulate continuous dynamics, providing a validated computational approach to studying reaction-diffusion systems in chemical and biological contexts. This research underscores the utility of impulsive discrete reaction-diffusion models for capturing complex diffusion–reaction interactions and advancing applications in reaction kinetics and biological systems. Full article

In this paper, we present a method to achieve exponential stability in a class of impulsive delayed neural networks containing parameter uncertainties, time-varying delays, and impulsive effect and reaction–diffusion terms. By using an integro-differential inequality with impulsive initial conditions and employing the M-matrix theory and the nonlinear measure approach, some new sufficient conditions ensuring the global exponential stability and global robust exponential stability of the considered system are derived. In particular, the results obtained are presented by simple algebraic inequalities, which are certainly more concise than the previous methods. By comparisons and examples, it is shown that the results obtained are effective and useful. Full article

This paper investigates the effect of a gene expression time delay on the Brusselator model with reaction and diffusion terms in one dimension. We obtain ODE systems analytically by using the Galerkin method. We determine a condition that assists in showing the existence of theoretical results. Full maps of the Hopf bifurcation regions of the stability analysis are studied numerically and theoretically. The influences of two different sources of diffusion coefficients and gene expression time delay parameters on the bifurcation diagram are examined and plotted. In addition, the effect of delay and diffusion values on all other free parameters in this system is shown. They can significantly affect the stability regions for both control parameter concentrations through the reaction process. As a result, as the gene expression time delay increases, both control concentration values increase, while the Hopf points for both diffusion coefficient parameters decrease. These values can impact solutions in the bifurcation regions, causing the region of instability to grow. In addition, the Hopf bifurcation points for the diffusive and non-diffusive cases as well as delay and non-delay cases are studied for both control parameter concentrations. Finally, various examples and bifurcation diagrams, periodic oscillations, and 2D phase planes are provided. There is close agreement between the theoretical and numerical solutions in all cases. Full article

Diffusion is an inevitable important factor in advertising dynamic systems. However, previous literature did not involve this important diffusion factor, and only involved the local stability of the advertising model. This paper develops a global stability criterion for the impulsive advertising dynamic model with a feedback term under the influence of diffusion. Since global stability requires the unique existence of equilibrium points, variational methods are employed to solve it in the infinite dimensional function space, and then a global stability criterion of the system is derived by way of the impulse inequality lemma and orthogonal decomposition of a class of Sobolev spaces. Numerical simulations verify the effectiveness of the proposed method. Full article

This paper proposes a partial differential equation model based on the model introduced by V. A. Kuznetsov and M. A. Taylor, which explains the dynamics of a tumor–immune interaction system, where the immune reactions are described by a Michaelis–Menten function. In this work, time delay and diffusion process are considered in order to make the studied model closer to reality. Firstly, we analyze the local stability of equilibria and the existence of Hopf bifurcation by using the delay as a bifurcation parameter. Secondly, we use the normal form theory and the center manifold reduction to determine the normal form of Hopf bifurcation for the studied model. Finally, some numerical simulations are provided to illustrate the analytic results. We show how diffusion has a significant effect on the dynamics of the delayed interaction tumor–immune system. Full article

In this article, a new competitive neural network (CNN) with reaction-diffusion terms and mixed delays is proposed. Because this network system contains reaction-diffusion terms, it belongs to a partial differential system, which is different from the existing classic CNNs. First, taking into account the spatial diffusion effect, we introduce spatial diffusion for CNNs. Furthermore, since the time delay has an essential influence on the properties of the system, we introduce mixed delays including time-varying discrete delays and distributed delays for CNNs. By constructing suitable Lyapunov–Krasovskii functionals and virtue of the theories of delayed partial differential equations, we study the global asymptotic stability for the considered system. The effectiveness and correctness of the proposed CNN model with reaction-diffusion terms and mixed delays are verified by an example. Finally, some discussion and conclusions for recent developments of CNNs are given. Full article

This paper introduces a novel synchronization scheme for fractional-order neural networks with time delays and reaction-diffusion terms via pinning control. We consider Caputo fractional derivatives, constant delays and distributed delays in our model. Based on the stability behavior, fractional inequalities and Lyapunov-type functions, several criteria are derived, which ensure the achievement of a synchronization for the drive-response systems. The obtained criteria are easy to test and are in the format of inequalities between the system parameters. Finally, numerical examples are presented to illustrate the results. The obtained criteria in this paper consider the effect of time delays as well as the reaction-diffusion terms, which generalize and improve some existing results. Full article

This paper is devoted to studying the dynamics of a delayed reaction-diffusion predator–prey system incorporating the effects of fear and anti-predator behaviour. First, based on its mathematical model, the global attractor is analyzed and the local stability of its positive equilibria is derived. Moreover, the Hopf bifurcation induced by the time delay variable is also investigated. Furthermore, the existence and non-existence of non-constant positive solutions are analyzed. Finally, numerical simulations are presented to validate the theoretical analysis. Full article

Cryptococcal superinfection is a rare but potentially fatal complication, especially if its detection and subsequent treatment are delayed. Histopathological findings of pulmonary parenchyma from a deceased patient with these complications were acquired. Quite interestingly, only a minimal inflammatory reaction could be seen in an individual with no previously known immune suppression, indicating a disturbance of the immune system. This finding was well in concordance with the described changes in cellular immunity in COVID-19. We report the case of a 60 year old male with critical coronavirus disease 2019 (COVID-19) complicated by cryptococcal pneumonia and multiorgan failure. Both X-ray and CT scans revealed lung infiltrates corresponding with COVID-19 infection early after the onset of symptoms. Despite receiving standard treatment, the patient progressed into multiple organ failure, requiring mechanical ventilation, circulatory support, and haemodialysis. Cryptococcus neoformans was detected by subsequent BAL, and specific antifungal treatment was instituted. His clinical status deteriorated despite all treatment, and he died of refractory circulatory failure after 21 days from hospital admission. Histopathological findings confirmed severe diffuse alveolar damage (DAD) caused by COVID-19 and cryptococcal pneumonia. Timely diagnosis of cryptococcal superinfection may be challenging; therefore, PCR panels detecting even uncommon pathogens should be implemented while taking care of critical COVID-19 patients. Full article

The study considers a nonlinear multi-parameter reaction–diffusion system of two Lotka–Volterra-type equations with several delays. It treats both cases of different diffusion coefficients and identical diffusion coefficients. The study describes a few different techniques to solve the system of interest, including (i) reduction to a single second-order linear ODE without delay, (ii) reduction to a system of three second-order ODEs without delay, (iii) reduction to a system of three first-order ODEs with delay, (iv) reduction to a system of two second-order ODEs without delay and a linear Schrödinger-type PDE, and (v) reduction to a system of two first-order ODEs with delay and a linear heat-type PDE. The study presents many new exact solutions to a Lotka–Volterra-type reaction–diffusion system with several arbitrary delay times, including over 50 solutions in terms of elementary functions. All of these are generalized or incomplete separable solutions that involve several free parameters (constants of integration). A special case is studied where a solution contains infinitely many free parameters. Along with that, some new exact solutions are obtained for a simpler nonlinear reaction–diffusion system of PDEs without delays that represents a special case of the original multi-parameter delay system. Several generalizations to systems with variable coefficients, systems with more complex nonlinearities, and hyperbolic type systems with delay are discussed. The solutions obtained can be used to model delay processes in biology, ecology, biochemistry and medicine and test approximate analytical and numerical methods for reaction–diffusion and other nonlinear PDEs with delays. Full article

A new model of viral infection spreading in cell cultures is proposed taking into account virus mutation. This model represents a reaction-diffusion system of equations with time delay for the concentrations of uninfected cells, infected cells and viral load. Infection progression is characterized by the virus replication number $R_v$, which determines the total viral load. Analytical formulas for the speed of propagation and for the viral load are obtained and confirmed by numerical simulations. It is shown that virus mutation leads to the emergence of a new virus variant. Conditions of the coexistence of the two variants or competitive exclusion of one of them are found, and different stages of infection progression are identified. Full article

This paper has dealt with a tracking control problem for a class of unstable reaction–diffusion system with time delay. Iterative learning algorithms are introduced to make the infinite-dimensional repetitive motion system track the desired trajectory. A new Lyapunov–Krasovskii functional is constructed to deal with the time-delay system. Picewise distribution functions are applied in this paper to perform piecewise control operations. By using Poincaré–Wirtinger inequality, Cauchy–Schwartz inequality for integrals and Young’s inequality, the convergence of the system with time delay using iterative learning schemes is proved. Numerical simulation results have verified the effectiveness of the proposed method. Full article

This paper describes the stability and Hopf bifurcation analysis of the Brusselator system with delayed feedback control in the single domain of a reaction–diffusion cell. The Galerkin analytical technique is used to present a system equation composed of ordinary differential equations. The condition able to determine the Hopf bifurcation point is found. Full maps of the Hopf bifurcation regions for the interacting chemical species are shown and discussed, indicating that the time delay, feedback control, and diffusion parameters can play a significant and important role in the stability dynamics of the two concentration reactants in the system. As a result, these parameters can be changed to destabilize the model. The results show that the Hopf bifurcation points for chemical control increase as the feedback parameters increase, whereas the Hopf bifurcation points decrease when the diffusion parameters increase. Bifurcation diagrams with examples of periodic oscillation and phase-plane maps are provided to confirm all the outcomes calculated in the model. The benefits and accuracy of this work show that there is excellent agreement between the analytical results and numerical simulation scheme for all the figures and examples that are illustrated. Full article