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Regarding delay-induced predator-prey systems, extensive research has focused on the phenomenon of delayed destabilization. However, the question of whether delays contribute to stabilizing or destabilizing the system remains a subtle one. In this paper, the predator-prey interaction with discrete delay involving Ivlev-type functional response is studied by theoretical analysis and numerical simulations. The positivity and boundedness of the solution for the delayed model have been discussed. When time delay is accounted as a bifurcation parameter, stability analysis for the coexistence equilibrium is given in theoretical aspect. Supercritical Hopf bifurcation is detected by numerical simulation. Interestingly, by choosing suitable groups of parameter values, the chaotic solutions appear via a cascade of period-doubling bifurcations, which is also detected. The theoretical analysis and numerical conclusions demonstrate that the delay mechanism plays a crucial role in the exploration of chaotic solutions [ABSTRACT FROM AUTHOR]

In this paper, to investigate the dynamic interplay between supply and demand, with a focus on collectability, a novel mathematical model was introduced via conformable operator. This model considers the possibility that operating expenses or a lack of raw materials causes a manufacturing delay than the supply of goods instantly matching demand. This maturation (delay) is represented by the delay factor (τ) . Stability analysis revolves around the equilibrium point other than zero. Chaotic behavior emerges through Hopf bifurcation at the critical delay parameter value. If this delay parameter is even slightly perturbed, oscillatory limit cycles can be induced in the market dynamics, leading to equilibrium with brisk market expansion, frequent recessions, and sudden collapses. We conducted sensitivity and directional analysis on a number of factors while also examining the stability and duration of the Hopf bifurcation. Numerical findings were validated using MATLAB. Additionally, the Caputo operator was used to examine the fractional of demand and supply dynamics. Importantly, we assumed a pivotal role in advancing fair labor practices and fostering economic growth on a national scale. In this paper, to investigate the dynamic interplay between supply and demand, with a focus on collectability, a novel mathematical model was introduced via conformable operator. This model considers the possibility that operating expenses or a lack of raw materials causes a manufacturing delay than the supply of goods instantly matching demand. This maturation (delay) is represented by the delay factor . Stability analysis revolves around the equilibrium point other than zero. Chaotic behavior emerges through Hopf bifurcation at the critical delay parameter value. If this delay parameter is even slightly perturbed, oscillatory limit cycles can be induced in the market dynamics, leading to equilibrium with brisk market expansion, frequent recessions, and sudden collapses. We conducted sensitivity and directional analysis on a number of factors while also examining the stability and duration of the Hopf bifurcation. Numerical findings were validated using MATLAB. Additionally, the Caputo operator was used to examine the fractional of demand and supply dynamics. Importantly, we assumed a pivotal role in advancing fair labor practices and fostering economic growth on a national scale. [ABSTRACT FROM AUTHOR]

In ecology, the most significant aspect is that the interactions between predators and prey are extremely complicated. Numerous experiments have shown that both direct predation and the fear induced in prey by the presence of predators lead to a reduction in prey density in predator-prey interactions. In addition, a suitable shelter can effectively stop predators from attacking as well as support the persistence of prey population. There has been less exploration of the effects of not only fear but also refuge factors on the dynamics of predator prey interactions. In this paper, we unveil several conclusions about a predator-prey system with fuzzy parameters, considering the cost of fear in two prey species and the effect of shelter on two prey species and one predator. As the first step of the investigation, the boundedness and non-negativity of the solutions to the system are put forward. Using the Jocabian matrix and Lyapunov function methods, we further analyze the existence and stability of the available equilibria and also the existence of Hopf bifurcation, considering the fear parameter as the bifurcation parameter that has been observed by applying the normal form theory. Finally, numerical simulations help us better understand the dynamics of the model, in which some interesting chaotic phenomena are also exhibited. [ABSTRACT FROM AUTHOR]

From the perspective of ecological control, harvesting behavior plays a crucial role in the ecosystem natural cycle. This paper proposes a diffusive predator-prey system with predator harvesting to explore the impact of harvesting on predatory ecological relationships. First, the existence and boundedness of system solutions were investigated and the non-existence and existence of nonconstant steady states were obtained. Second, the conditions for Turing instability were given to further investigate the Turing patterns. Based on these conditions, the amplitude equations at the threshold of instability were established using weakly nonlinear analysis. Finally, the existence, direction, and stability of Hopf bifurcation were proven. Furthermore, numerical simulations were used to confirm the correctness of the theoretical analysis and show that harvesting has a strong influence on the dynamical behaviors of the predator-prey systems. In summary, the results of this study contribute to promoting the research and development of predatory ecosystems. [ABSTRACT FROM AUTHOR]

Due to factors such as climate change, natural disasters, and deforestation, most measurement processes and initial data may have errors. Therefore, models with imprecise parameters are more realistic. This paper constructed a new predator-prey model with an interval biological coefficient by using the interval number as the model parameter. First, the stability of the solution of the fractional order model without a diffusion term and the Hopf bifurcation of the fractional order α were analyzed theoretically. Then, taking the diffusion coefficient of prey as the key parameter, the Turing stability at the equilibrium point was discussed. The amplitude equation near the threshold of the Turing instability was given by using the weak nonlinear analysis method, and different mode selections were classified by using the amplitude equation. Finally, we numerically proved that the dispersal rate of the prey population suppressed the spatiotemporal chaos of the model. [ABSTRACT FROM AUTHOR]

In this paper, a delay differential equation model is investigated, which describes the biodegradation of microcystins (MCs) by Sphingomonas sp. and its degrading enzymes. First, the local stability of the positive equilibrium and the existence of the Hopf bifurcation are obtained. Second, the global attractivity of the positive equilibrium is obtained by constructing suitable Lyapunov functionals, which implies that the biodegradation of microcystins is sustainable under appropriate conditions. In addition, some numerical simulations of the model are carried out to illustrate the theoretical results. Finally, the parameters of the model are determined from the experimental data and fitted to the data. The results show that the trajectories of the model fit well with the trend of the experimental data. [ABSTRACT FROM AUTHOR]

The study of harvesting mechanisms in predator-prey systems with an Allee effect on prey has always garnered significant attention. In this paper, the dynamics and harvesting strategies of a predator-prey system are investigated, where the prey is subject to the Allee effect. The positivity and boundedness of solutions, the existence and stability of equilibria are further studied. The existence of a Hopf bifurcation at the interior equilibrium point of the system is investigated and verified by numerical simulations. Furthermore, we investigate the maximum sustainable yield (MSY), maximum sustainable total yield (MSTY) and the optimal economic profit of the proposed system. We find that MSY can be attained through predator harvesting, while MSTY is observed when harvesting efforts are uniform across species. In these situations, the biological system maintains stability. Using the method of control parametrization, the optimal economic profit and harvesting strategy are obtained. The results show that the harvesting efforts can affect the stability of the system, resulting in several interesting biological phenomena. This research provides a theoretical basis for biological resource management. [ABSTRACT FROM AUTHOR]

In this paper, we study a food chain chemostat model with Michaelis-Menten function response and double delays. Applying the stability theory of functional differential equations, we discuss the conditions for the stability of three equilibria, respectively. Furthermore, we analyze the sufficient conditions for the Hopf bifurcation of the system at the positive equilibrium. Finally, we present some numerical examples to verify the correctness of the theoretical analysis and give some valuable conclusions and further discussions at the end of the paper. [ABSTRACT FROM AUTHOR]

In this paper, a delayed prey-predator eco-epidemiological model with the nonlinear media is considered. First, the positivity and boundedness of solutions are given. Then, the basic reproductive number is showed, and the local stability of the trivial equilibrium and the disease-free equilibrium are discussed. Next, by taking the infection delay as a parameter, the conditions of the stability switches are given due to stability switching criteria, which concludes that the delay can generate instability and oscillation of the population through Hopf bifurcation. Further, by using normal form theory and center manifold theory, some explicit expressions determining direction of Hopf bifurcation and stability of periodic solutions are obtained. What's more, the correctness of the theoretical analysis is verified by numerical simulation, and the biological explanations are also given. Last, the main conclusions are included in the end. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical behavior of a predator-prey model with discrete time is discussed in terms of both theoretical analysis and numerical simulation. The existence and stability of four equilibria are analyzed. It is proved that the system undergoes Flip bifurcation and Hopf bifurcation around its unique positive equilibrium point using center manifold theorem and bifurcation theory. Additionally, by applying small perturbations to the bifurcation parameter, chaotic cases occur at some corresponding internal equilibria. Finally, numerical simulations are provided with the help of maximum Lyapunov exponent and phase diagrams, which reveal a complex dynamical behavior. [ABSTRACT FROM AUTHOR]

In our paper, a delayed diffusive phytoplankton-zooplankton-fish model with a refuge and Crowley-Martin and Holling II functional responses is established. First, for the model without delay and diffusion, we not only analyze the existence and stability of equilibria, but also discuss the occurrence of Hopf bifurcation by choosing the refuge proportion of phytoplankton as the bifurcation parameter. Then, for the model with delay, we set some sufficient conditions to demonstrate the existence of Hopf bifurcation caused by delay; we also discuss the direction of Hopf bifurcation and the stability of the bifurcation of the periodic solution by using the center manifold and normal form theories. Next, for a reaction-diffusion model with delay, we show the existence and properties of Hopf bifurcation. Finally, we use Matlab software for numerical simulation to prove the previous theoretical results. [ABSTRACT FROM AUTHOR]

Delayed dynamical system plays a vital role in describing the dynamical phenomenon of neural networks. In this article, we proposed a class of new BAM neural networks involving time delay. The traits of solution and bifurcation behavior of the established BAMneural networks involving time delay were probed into. First, the existence and uniqueness is discussed using a fixed point theorem. Second, the boundedness of solution of the formulated BAM neural networks involving time delay was analyzed by applying an appropriate function and inequality techniques. Third, the stability peculiarity and bifurcation behavior of the addressed delayed BAM neural networks were investigated. Fourth, Hopf bifurcation control theme of the formulated delayed BAM neural networks was explored by virtue of a hybrid controller. By adjusting the parameters of the controller, we could control the stability domain and Hopf bifurcation onset, which was in favor of balancing the states of different neurons in engineering. To verify the correctness of gained major outcomes, computer simulations were performed. The acquired outcomes of this article were new and own enormous theoretical meaning in designing and dominating neural networks. [ABSTRACT FROM AUTHOR]

Chaos theory has been widely utilized in password design, resulting in an encryption algorithm that exhibits strong security and high efficiency. However, rapid advancements in cryptanalysis technology have rendered single system generated sequences susceptible to tracking and simulation, compromising encryption algorithm security. To address this issue, we propose an image encryption algorithm based on hybrid heterogeneous time-delay chaotic systems. Our algorithm utilizes a collection of sequences generated by multiple heterogeneous time-delay chaotic systems, rather than sequences from a single chaotic system. Specifically, three sequences are randomly assigned to image pixel scrambling and diffusion operations. Furthermore, the time-delay chaotic system comprises multiple hyperchaotic systems with positive Lyapunov exponents, exhibiting a more complex dynamic behavior than non-delay chaotic systems. Our encryption algorithm is developed by a plurality of time-delay chaotic systems, thereby increasing the key space, enhancing security, and making the encrypted image more difficult to crack. Simulation experiment results verify that our algorithm exhibits superior encryption efficiency and security compared to other encryption algorithms. [ABSTRACT FROM AUTHOR]

Recently, delayed dynamical model has witnessed a great interest from many scholars in biological and mathematical areas due to its potential application in describing the interaction of different biological populations. In this article, relying the previous studies, we set up two new predator-prey systems incorporating delay. By virtue of fixed point theory, inequality tactics and an appropriate function, we explore well-posedness (includes existence and uniqueness, boundedness and non-negativeness) of the solution of the two formulated delayed predator-prey systems. With the aid of bifurcation theorem and stability theory of delayed differential equations, we gain the parameter conditions on the emergence of stability and bifurcation phenomenon of the two formulated delayed predator-prey systems. By applying two controllers (hybrid controller and extended delayed feedback controller) we can effcaciously regulate the region of stability and the time of occurrence of bifurcation phenomenon for the two delayed predator-prey systems. The effect of delay on stabilizing the system and adjusting bifurcation is investigated. Computer simulation plots are provided to sustain the acquired prime outcomes. The conclusions of this article are completely new and can provide some momentous instructions in dominating and balancing the densities of predator and prey. [ABSTRACT FROM AUTHOR]

We present a stochastic time-delay susceptible-exposed-asymptomatic-symptomvaccinated- recovered (SEAQVR) model with media publicity effect in this study. The model takes into account the impacts of noise, time delay and public sensitivity on infectious illness propagation. The stochastic dynamics of the system are analyzed at the Hopf bifurcation, using time delay and noise intensity as bifurcation parameters, and the theoretical conclusions are validated using numerical simulation. Increasing the time delay and sensitivity coefficient can effectively delay the occurrence of the peak number of infected individuals and mitigate the extent of infection. Additionally, time delay and noise intensity are shown to have specific thresholds, beyond which periodic infections occur. Notably, heightened public sensitivity reduces the threshold for time delay, and media publicity directly affects public sensitivity. The numerical simulation reveals that increasing media publicity intensity does not always yield better results, and that the sensitivity of the public at present is an important reference index for setting an appropriate publicity intensity. [ABSTRACT FROM AUTHOR]

In this work, a Leslie-Gower model with a weak Allee effect on the prey and a fear effect on the predator is proposed. By using qualitative analyses, the local stability of the coexisting equilibrium and the existence of Turing instable are discussed. By analyzing the distribution of eigenvalues, the existence of a Hopf bifurcation is studied by using the gestation time delay as a bifurcation parameter. By utilizing the normal form method and the center manifold theorem, we calculate the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. We indicate that both the weak Allee effect on the prey and fear effect on the predator have an important impact on the dynamical behaviour of the new Leslie-Gower model. We also verify the obtained results by some numerical examples. [ABSTRACT FROM AUTHOR]

This research concerned with a new formulation of a spatial predator-prey model with Leslie-Gower and Holling type II schemes in the presence of prey social behavior. The aim interest here is to distinguish the influence of Leslie-Gower term on the spatiotemporal behavior of the model. Interesting results are obtained as Hopf bifurcation, Turing bifurcation and Turing-Hopf bifurcation. A rigorous mathematical analysis shows that the presence of Leslie-Gower can induce Turing pattern, which shows that this kind of interaction is very important in modeling different natural phenomena. The direction of Turing-Hopf bifurcation is studied with the help of the normal form. The obtained results are tested numerically. [ABSTRACT FROM AUTHOR]

Since the distribution of plankton is always uneven, the nonlocal phytoplankton competition term indicates the spatial weighted mean of phytoplankton density, which is introduced into the plankton model with toxic substances effect to study the corresponding dynamic behavior. The stability of the positive equilibrium point and the existence of Hopf bifurcations are discussed by analysing the distribution of eigenvalues. The direction and stability of bifurcation periodic solution are researched based on an extended central manifold method and normal theory. Finally, spatially inhomogeneous oscillations are observed in the vicinity of the Hopf bifurcations. Through numerical simulation, we can observe that the system without nonlocal competition term only generates homogeneous periodic solution, and inhomogeneous periodic solution will produce only when both diffusion and nonlocal competition exist simultaneously. We can also see that when the toxin-producing rate of each phytoplankton is in an appropriate range, the system with nonlocal competition generates a stability switch with inhomogeneous periodic solution, when the value of time delay is in a certain interval, then Hopf bifurcations will appear, and with the increase of time delay, the quantity of plankton will eventually become stable. [ABSTRACT FROM AUTHOR]

The global burden of illegal drug-related death and disability continues to be a public health threat in developed and developing countries. Hence, a fractional-order mathematical modeling approach is presented in this study to examine the consequences of illegal drug usage in the community. Based on epidemiological principles, the transmission mechanism is the social interaction between susceptible and illegal drug users. A pandemic threshold value (Λ) is provided for the illegal drugusing profession, which determines the stability of the model. The Lyapunov function is employed to determine the stability conditions of illegal drug addiction equilibrium point of society. Finally, the proposed model has been extended to include time lag in the delayed illegal drug transmission model. The characteristic equation of the endemic equilibrium establishes a set of appropriate conditions for ensuring local stability and the development of a Hopf bifurcation of the model. Finally, numerical simulations are performed to support the analytical results. [ABSTRACT FROM AUTHOR]

In this work, a five-parameter jerk system with one hyperbolic sine nonlinearity is proposed, in which ε is a small parameter, and a, b, c, d are some other parameters. For ε=0, the system is Z2 symmetric. For ε≠0, the system loses the symmetry. For the symmetrical case, the pitchfork bifurcation and Hopf bifurcation of the origin are studied analytically by Sotomayor's theorem and Hassard's formulas, respectively. These bifurcations can be either supercritical or subcritical depending on the governing parameters. In comparison, it is much more restrictive for the origin of the Lorenz system: Only a supercritical pitchfork bifurcation is available. Thus, the symmetrical system can exhibit very rich local bifurcation structures. The continuation of local bifurcations leads to the main contribution of this work, i.e., the discovery of two basic mechanisms of chaotic motions for the jerk systems. For four typical cases, Cases A–D, by varying the parameter a, the mechanisms are identified by means of bifurcation diagrams. Cases A and B are Z2 symmetric, while Cases C and D are asymmetric (caused by constant terms). The forward period-doubling routes to chaos are observed for Cases A and C; meanwhile, the backward period-doubling routes to chaos are observed for Cases B and D. The dynamical behaviors of these cases are studied via phase portraits, two-sided Poincaré sections and Lyapunov exponents. Using Power Simulation (PSIM), a circuit simulation model for a chaotic jerk system is created. The circuit simulations match the results of numerical simulations, which further validate the dynamical behavior of the jerk system. [ABSTRACT FROM AUTHOR]

A delayed diffusive predator-prey system with nonlocal competition and generalist predators is considered. The local stability of the positive equilibrium and Hopf bifurcation at positive equilibrium is studied by using time delay as a parameter. In addition, the property of Hopf bifurcation is analyzed using the center manifold theorem and normal form method. It is determined that time delays can affect the stability of the positive equilibrium and induce spatial inhomogeneous periodic oscillation of prey and predator population densities. [ABSTRACT FROM AUTHOR]

The current emergence of coronavirus (SARS-CoV-2 or COVID-19) has put the world in threat. Social distancing, quarantine and governmental measures such as lockdowns, social isolation, and public hygiene are helpful in fighting the pandemic, while awareness campaigns through social media (radio, TV, etc.) are essential for their implementation. On this basis, we propose and analyse a mathematical model for the dynamics of COVID-19 transmission influenced by awareness campaigns through social media. A time delay factor due to the reporting of the infected cases has been included in the model for making it more realistic. Existence of equilibria and their stability, and occurrence of Hopf bifurcation have been studied using qualitative theory. We have derived the basic reproduction number (R0) which is dependent on the rate of awareness. We have successfully shown that public awareness has a significant role in controlling the pandemic. We have also seen that the time delay destabilizes the system when it crosses a critical value. In sum, this study shows that public awareness in the form of social distancing, lockdowns, testing, etc. can reduce the pandemic with a tolerable time delay. [ABSTRACT FROM AUTHOR]