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This paper is concerned with a two‐variable Oregonator model for Belousov–Zhabotinsky Reaction. Llibre and Oliveira (2022) proved that Oregonator model with an unstable node or focus has at least one limit cycle in the positive quadrant, and the system has a unique stable limit cycle for some values of the parameters. It is shown in the present paper that the positive equilibrium is not a center, and that if the system has a positive equilibrium which is a weak focus, then its order is at most 2. There exist some parameter values such that system has i(i=1,2)$$ i\left(i=1,2\right) $$ small limit cycle around the positive equilibrium. [ABSTRACT FROM AUTHOR]

Stability of a complex DC bus system (DBS) is a significant design consideration because of its constant power load (CPL) characteristics of the tightly‐regulated load converter. In this paper, the dynamic properties of a stand‐alone buck‐type DBS are investigated based upon the closed‐form large‐signal‐averaged model. The stability‐analysis‐in‐large around the operation point shows that CPLs create a destabilizing effect on the system that can lead to severe voltage oscillation caused by discontinuity‐induced Hopf bifurcations, which are qualitatively different from the bifurcations we know for smooth dynamical systems. Instead, this paper also explores the application of "amplitude death (AD) technique" to stabilize the coupled oscillatory DBSs and show that how this technique with its necessary modifications, for example, the heterogeneity‐induced AD technique characterized by two or more coupled DBSs with mismatched frequency of oscillations can provide an efficient open‐loop control solution for stabilizing the coupled DBSs. Finally, simulations and experimental results are provided to validate all these concepts. [ABSTRACT FROM AUTHOR]

In this paper, we study a predator–prey system with the simplified Holling IV functional response and antipredator behavior such that the adult prey can attack vulnerable predators. The model has been investigated by Tang and Xiao, and the existence and stability of all possible equilibria are determined. In addition, they performed a bifurcation analysis and showed that the system undergoes a Codimension 2 Bogdanov–Takens bifurcation. In this paper, for the same model, we further show that the cusp‐type Bogdanov–Takens bifurcation can be of Codimension 3, which acts as an organizing center for the whole bifurcation set. In addition, we propose the existence of Hopf bifurcation of Codimension 2 and the coexistence of stable limit cycle and unstable limit cycle. In particular, we show that the antipredator behavior has great effect on the dynamics of the model, it may cause the predator population to extinct while the prey population will increase up to the carrying capacity. Numerical simulations including bifurcation diagrams and phase portraits are performed to illustrate and confirm the theoretical results. These results may enrich the dynamics of predator–prey systems. [ABSTRACT FROM AUTHOR]

In this paper, we propose and study a discretized one‐predator two‐prey system along with prey refuge and Michaelis–Menten‐type prey harvesting. The interaction among the species is considered as Holling type III functional response. Firstly, existence and local stability of all the fixed points are derived under certain parametric conditions. Furthermore, a special consideration is made to global asymptotic stability of the interior fixed point. Then, we have shown that the system undergoes different types of bifurcations including transcritical bifurcation, flip bifurcation, and Neimark–Sacker bifurcation by using center manifold theorem, bifurcation theory, and normal form method. Also, Feigenbaum's constant of the system is calculated. It is observed that both harvesting and refuge have a stabilizing effect on the system, and the stabilizing effect of harvesting dominates the stabilizing effect of refuge. Of most interest is the finding of coexisting attractors and multistability. In particular, optimal harvesting policy has been obtained by extension of Pontryagin's maximum principle to discrete system. Finally, some intriguing numerical simulations are provided to verify our analytic findings and rich dynamics of the three species system. [ABSTRACT FROM AUTHOR]

This paper explores the dynamics analysis of a human immunodeficiency virus (HIV) model with saturated cytotoxic T lymphocyte (CTL) immune response and Beddington–DeAngelis infection rate. There are two time delays in the model to describe the time needed for infection of cell and CTL immune response generation, respectively. We obtain two thresholds and three possible equilibria from the model. By analyzing the corresponding characteristic equations, we study the stabilities of equilibrium and the effect of delays on CTL immune response. The results indicate that when immune delay is present, the steady state of equilibrium is disrupted and leads to a Hopf bifurcation. Finally, we use sensitivity analyses to show the effect of parameters on thresholds and numerical simulations to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a predator–prey model with hunting cooperation in predator and anti‐predator behaviors in prey. The conditions for the existence and the stability of the unique positive constant equilibrium are given. It is found that with the increasing of the birth rate r0$$ {r}_0 $$ of the prey, the trivial solution loses its stability, and the semi‐trivial solution emerges and also loses its stability. For the positive constant solution, we find that as the hunting cooperation b$$ b $$ in predator increases or the fear k0$$ {k}_0 $$ decreases, the positive constant equilibrium loses its stability, and Hopf bifurcation occurs. We also derive the existence of limit cycles by Poincaré‐Bendixson theorem. We also study a diffusive model and derive that self‐diffusion can induce Turing instability. Finally, we conduct numerical simulations to present our conclusions. [ABSTRACT FROM AUTHOR]

In this paper, an susceptible–infected–susceptible (SIS) epidemic patch model with media delay is proposed at first. Then the basic reproduction number R0$\mathcal {R}_0$ is defined, and the threshold dynamics are studied. It is shown that the disease‐free equilibrium is globally asymptotically stable if R0<1$\mathcal {R}_0<1$ and the disease is uniformly persistent if R0>1$\mathcal {R}_0>1$. When the dispersal rates of susceptible and infected populations are identical and less than a critical value, it is proved that the limiting model has a unique positive equilibrium. Furthermore, the stability of the positive equilibrium and the existence of local and global Hopf bifurcations are obtained. Finally, some numerical simulations are performed. [ABSTRACT FROM AUTHOR]

Recently, many scholars have discovered that fractional calculus possess infinite memory and can better reflect the memory characteristics of neurons. Therefore, this paper studies the Hopf bifurcation of a fractional‐order network with short‐cut connections structure and self‐delay feedback. Firstly, we use the Laplace transform to obtain the characteristic equation of the model, which is the transcendental equation containing four times transcendental item. Secondly, by selecting the communication delay as the bifurcation parameter and the other delay as the constant in its stability interval, the conditions for the occurrence of Hopf bifurcation are established; the bifurcation diagrams are provided to ensure that the derived bifurcation findings are accurate. Thirdly, in the case of identical neurons, the crossing curves method is exploited to the fractional‐order functional function equation to extract the Hopf bifurcation curve. Finally, two numerical examples are employed to confirm the efficiency of the developed theoretical outcomes. [ABSTRACT FROM AUTHOR]

In this paper, we study a delayed HIV infection model with nonmonotonic immune response and perform stability and bifurcation analysis. Our results show that the delayed HIV infection system with nonmonotonic immune response has bistability and stable periodic solution appear. We find that both the uninfected and immune-free equilibria are globally asymptotically stable under certain conditions which are not affected by time delay. However, the time delay makes one immune equilibrium always unstable for τ ≥ 0 and also makes another immune equilibrium appear stability switches; meanwhile, the system will exhibit local Hopf bifurcation, global Hopf bifurcation, and saddle-node-Hopf bifurcation. Numerical simulations are carried out to verify our results. [ABSTRACT FROM AUTHOR]

In the realm of modeling biological systems with memory, particularly those involving intricate interactions like tumor-immune responses, the utilization of multiple time delays and Caputo-type fractional-order derivatives represents a cutting-edge approach. In this research paper, we introduce a novel fractional-order model to investigate the dynamic interplay between tumors and macrophages, a key component of the immune system, while incorporating multiple time delays into our framework. Our proposed model comprises a system of three Caputo-type fractional-order differential equations, each representing distinct cell populations: tumor cells, anti-tumor cells (specifically M1-type macrophages with pro-inflammatory properties), and pro-tumor cells (M2-type macrophages with immune-suppressive characteristics). The stability of equilibria is discussed by analyzing the characteristic equations for each case, and the existence conditions for the Hopf bifurcation are obtained according to the critical values of delay parameters. Furthermore, numerical simulations are presented in order to verify the analytical results obtained for stability and Hopf-bifurcation with respect to the two-time delay parameters τ1 and τ2. The analysis shows the rich dynamics of the model according to the fractional-order parameter and the time delay parameters. [ABSTRACT FROM AUTHOR]

This paper presents a detailed study on the dynamics of a three-species food chain system with memory-based delay under weak Allee effect and middle predator refuge. The local stability analyses of the non-diffusive and memory-based diffusion systems are investigated. Moreover, we give a priori estimates and obtain the existence of the positive constant steady state by applying the comparison theorem. Sufficient conditions for the global stability are established by Barbalat lemma and the Lyapunov function. The theoretical results suggest the joint effect of cross-diffusion and memory-based delay can lead to Hopf bifurcation, which cannot appear in the system with self-diffusion or a small cross-diffusion coefficient. Numerical results verify the validity of the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we study a predator--prey mite model of Leslie type with generalized Holling IV functional response. The model is shown to have very rich bifurcation dynamics, including subcritical and supercritical Hopf bifurcations, degenerate Hopf bifurcation, focustype and cusp-type degenerate Bogdanov--Takens bifurcations of codimension 3, originating from a nilpotent focus or cusp of codimension 3 that acts as the organizing center for the bifurcation set. Coexistence of multiple steady states, multiple limit cycles, and homoclinic cycles is also found. Interestingly, the coexistence of two limit cycles is guaranteed by investigating generalized Hopf bifurcation and degenerate homoclinic bifurcation, and we also find that two generalized Hopf bifurcation points are connected by a saddle-node bifurcation curve of limit cycles, which indicates the existence of global regime for two limit cycles. Our work extends some results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider a predator–prey model with Cosner type functional response and combined harvesting. First, we explore the existence and stability of the equilibria. Then using the center manifold theorem and normal form theory, we investigate codimension one and codimension two bifurcations of the model. The analysis shows that the system has a variety of bifurcation phenomena including transcritical bifurcation, saddle‐node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and homoclinic bifurcation. Our findings indicate that the dynamics with harvesting are significantly richer than the system without harvesting. Finally, numerical simulations are provided to support the analytical results. [ABSTRACT FROM AUTHOR]

Subsynchronous oscillation events of grid‐connected voltage source converter (VSC) pose a potential risk to the security and stability of power systems. It is particularly challenging to reveal the subsynchronous oscillation mechanism precisely because the grid‐connected VSC consists of non‐linear components such as saturation. The describing function and the generalized Nyquist criterion are used in this paper to investigate the bifurcation types and dominant parameters of grid‐connected VSC. First, the condition of a supercritical Hopf bifurcation is investigated, which generates subsynchronous oscillations. Then, the dominant parameters of supercritical Hopf bifurcation are deduced, including the grid strength, the active power reference, or the proportional coefficient of phase‐locked loop. Finally, it is examined how the saturation's limit value impacts the limit cycle's amplitude and stability. The findings demonstrate that a system's stability depends on whether its closed‐loop frequency characteristic curve intersects the describing function's negative reciprocal curve. When the two curves intersect, the supercritical Hopf bifurcation appears, and changing the limit value can affect the limit cycle's amplitude. However, if the two curves do not intersect, altering the limit value cannot cause the supercritical Hopf bifurcation, and the subsynchronous oscillation will not happen. Variations in grid strength, active power reference, or proportional coefficient of phase‐locked loop will significantly affect the Nyquist curve, determining if supercritical Hopf bifurcation appears. [ABSTRACT FROM AUTHOR]

In this paper, an SIR model with a nonlinear incidence rate is studied. A disease‐free equilibrium E0$$ {E}_0 $$, an endemic equilibrium E1$$ {E}_1 $$, and the basic reproduction number of the model R0$$ {R}_0 $$ are obtained. If R0<1,E0$$ {R}_0&lt;1,{E}_0 $$ is locally asymptotically stable and if R0>1,E1$$ {R}_0&gt;1,{E}_1 $$ is locally asymptotically stable. By Barbalat's lemma, we study the global stability of the model. Transcritical bifurcation analysis is investigated by using the Sotomayor theorem. As the infection rate increases, the asymptotic behavior of the system near E0$$ {E}_0 $$ approaches E1$$ {E}_1 $$ and the system has a transcritical bifurcation. Also, we check the existence of Hopf bifurcation for the given system. In addition, a sensitivity analysis is provided for the basic reproduction number. Our results are supported with numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we study the codimension‐two Turing–Hopf bifurcation of a diffusive Holling–Tanner model with nonlocal effect and digestion time delay. The stability, Turing bifurcation, Hopf bifurcation, and Turing–Hopf bifurcation of this model are first investigated. Then, we derive the algorithm for calculating the normal form of Turing–Hopf bifurcation for this model. At last, we carry out some numerical simulations to verify our theoretical analysis results. The stable positive constant steady state and the stable spatially inhomogeneous periodic solutions are found. Furthermore, the evolution process from unstable spatially inhomogeneous steady states to stable positive constant steady state, the evolution process from unstable spatially inhomogeneous steady states to stable spatially inhomogeneous periodic solutions, and the evolution process from one unstable spatially inhomogeneous periodic solution to another stable spatially inhomogeneous periodic solution are also found. [ABSTRACT FROM AUTHOR]

In this paper, we investigate pattern dynamics in a reaction‐diffusion‐chemotaxis food chain model with predator‐taxis, which extends previous studies of reaction‐diffusion food chain model. By virtue of diffusion semigroup theory, we first prove global classical solvability and boundedness for the considered model over a bounded domain Ω⊂ℝn(n≥1)$$ \Omega \subset {\mathbb{R}}&amp;#x0005E;n\kern0.1em \left(n\ge 1\right) $$ with smooth boundary for arbitrary predator‐taxis sensitivity coefficient. Then the linear stability analysis for the considered model shows that chemotaxis can induce the losing of stability of the unique positive spatially homogeneous steady state via Turing bifurcation and Turing‐spatiotemporal Hopf bifurcation. These bifurcations results in the formation of two kinds of important spatiotemporal patterns: stationary Turing pattern and oscillatory pattern. Simultaneously, the threshold values for Turing bifurcation and Turing‐spatiotemporal Hopf bifurcation are given explicitly. Finally, numerical simulations are performed to illustrate and support our theoretical findings, and some interesting non‐Turing patterns are found in temporal Hopf parameter space by numerical simulation. [ABSTRACT FROM AUTHOR]

In most HIV models, the emergence of backward bifurcation means that the control for basic reproduction number less than one is no longer effective for HIV treatment. In this paper, we study an HIV model with CTL response and cell‐to‐cell transmission by using the dynamical approach. The local and global stability of equilibria is investigated, the relations of subcritical Hopf bifurcation and supercritical bifurcation points are revealed, especially, the so‐called new type bifurcation is also found with two Hopf bifurcation curves meeting at the same Bogdanov–Takens bifurcation point. Forward and backward bifurcation, Hopf bifurcation, saddle‐node bifurcation, Bogdanov–Takens bifurcation are investigated analytically and numerically. Two limit cycles are also found numerically, which indicates that the complex behavior of HIV dynamics. Interestingly, the role of cell‐to‐cell interaction is fully uncovered, it may cause the oscillations to disappear and keep the so‐called new type bifurcation persist. Finally, some conclusions and discussions are also given. [ABSTRACT FROM AUTHOR]

In this paper, a class of respiratory disease model is established by taking into account simultaneously two situations that susceptible individuals directly got sick by inhaling air pollutants and indirectly fallen ill from infection by patients with respiratory diseases. The sufficient conditions for equilibria existence of the system and locally asymptotically stability of endemic equilibria are obtained. The existence of saddle‐node bifurcation is derived by using the Sotomayor theorem. The stability of periodic solution of Hopf bifurcation is determined by calculating the first Lyapunov number. The direct conversion rate and indirect infection rate in the system are selected as bifurcation parameters; the existence of Bogdanov–Takens bifurcation of codimension 2 of the system is proved by calculating the universal unfolding near the cusp. This research shows that direct and indirect ways of getting sick have a significant influence on dynamical behaviors of the respiratory disease system. [ABSTRACT FROM AUTHOR]

This study is devoted to considering dynamic properties of an environmental protection expenditures model for protected areas (PAs) with two delays. It presents an improved understanding of relationship among tourism, environment, and capital stock in PAs and provides a sustainable strategy to keep the balance between them. The model considers the interactions among environmental resource E$$ E $$, visitors V$$ V $$, and capital stock K$$ K $$ in PAs. The tourism income is used partly to increase the capital stock and partly to protect the environment. In this paper, time delays τ1$$ {\tau}_1 $$ and τ2$$ {\tau}_2 $$ are introduced into the system. The technique of stability crossing curves for two delays is proposed to analyze the occurrence of Hopf bifurcation. Numerical simulations are performed for understanding the analytical results. The results show that under some parameter conditions, the share η$$ \eta $$ of total revenue generated by tourism user fees for environmental protection is related to delays τ1$$ {\tau}_1 $$ and τ2$$ {\tau}_2 $$. Moreover, the environment and the capital stock can be maintained persistently if time delay τ1$$ {\tau}_1 $$ is not too large. [ABSTRACT FROM AUTHOR]

In this paper, we study the effect of spatial average and time delay on the dynamics of a diffusive predator–prey model under the Neumann boundary condition. Compared to the model without spatial average, the delay‐induced Hopf bifurcation at the first critical value of delay is nonhomogeneous due to the joint effects of spatial average and delay, and spatiotemporal patterns arise. Also, we show that the spatially homogeneous and nonhomogeneous periodic patterns may switch for different bifurcation parameter values. Moreover, a double Hopf bifurcation occurs at the intersection point of the homogeneous and nonhomogeneous Hopf bifurcation curves, and spatially nonhomogeneous quasi‐periodic patterns can be observed via numerical simulations near the double Hopf bifurcation point. The normal form algorithm for the spatially nonhomogeneous/homogeneous Hopf bifurcation is derived for a general reaction‐diffusion system with spatial average and delay. [ABSTRACT FROM AUTHOR]

In this paper, we investigate bifurcations of stationary solutions of a cross‐diffusion prey–predator system with Ivlev functional response and Neumann boundary conditions. First, we analyze the local stability and existence of a Hopf bifurcation at a coexistence stationary solution. We show that the bifurcating periodic solutions are asymptotically orbitally stable and the bifurcation direction is supercritical when the ratio of the conversion of prey captured by predator to the death rate of predator is in the interval (1,2]$$ \left(1,2\right] $$. Next, we derive sufficient conditions for the existence of a steady‐state bifurcation from a simple eigenvalue. We establish the existence of two intersecting C1$$ {C}^1 $$ curves of steady‐state solutions near the coexistence stationary solution. To illustrate our theoretical results, we give some numerical examples. From numerical simulations we observe that the coexistence stationary solution loses its stability via the Hopf bifurcation and a periodic solution emerges after a short time. Also by taking the bifurcation parameter value in the stable region, the effect of the initial condition disappears over time and the solution returns to the coexistence stationary solution. [ABSTRACT FROM AUTHOR]

In this paper, we explore local dynamics with topological classifications, bifurcation analysis, and chaos control in a discrete‐time COVID‐19 epidemic model in the interior of ℝ+4$$ {\mathbb{R}}_{+}^4 $$. It is explored that for all involved parametric values, discrete‐time COVID‐19 epidemic model has boundary equilibrium solution and also it has an interior equilibrium solution under definite parametric condition. We have explored the local dynamics with topological classifications about boundary and interior equilibrium solutions of the discrete‐time COVID‐19 epidemic model by linear stability theory. Further, for the discrete‐time COVID‐19 epidemic model, existence of periodic points and convergence rate are also investigated. It is also studied the existence of possible bifurcations about boundary and interior equilibrium solutions and proved that there exists no flip bifurcation about boundary equilibrium solution. Moreover, it is proved that about interior equilibrium solution, there exist Hopf and flip bifurcations, and we have studied these bifurcations by utilizing explicit criterion. Moreover, by feedback control strategy, chaos in the discrete COVID‐19 epidemic model is also explored. Finally, theoretical results are verified numerically. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a cannibalistic predator‐prey system with Beddington‐DeAngelis type response function when the infection only exists in the predator species. The predator species are subject to cannibalistic interaction and the infection spreads among the predators via cannibalism. The model is studied in presence of multiple delays for the maturation of prey and predator. Local stability analysis of the predator‐prey model around the biologically realistic two essential steady states is investigated. We also investigate the bifurcation analysis of the proposed model around the co‐existing steady states. Using the normal form method and center manifold theory, the direction of Hopf bifurcation and the Hopf bifurcating periodic solutions are explored. In presence of time delays, we derive the criteria for the permanence of the model. Our findings demonstrate that maturation delays destabilize the system and can produce high periodic oscillations. Cannibalism can act as a self‐regulatory scenario controlling the transmission of disease among the predator species and stabilizing the oscillations in the predator‐prey system. Our theoretical analysis is well supported with numerical simulation. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical behaviors are investigated for a complex network with two independent delays. Instead of taking time delays as bifurcation parameters, we choose probability p$$ p $$ and parameter μ$$ \mu $$ as the control parameters to study their effects on local stability and Hopf bifurcation, respectively. Moreover, the conditions for generating Hopf bifurcation are given. Furthermore, we further discuss the effects of two time delays on the critical values of parameters p$$ p $$ and μ$$ \mu $$. Finally, numerical simulations are used to illustrate the validity of the obtained results. [ABSTRACT FROM AUTHOR]

The emergence of vibrations during a roller coaster ride is a recurring phenomenon that can deteriorate the overall experience and accelerate material fatigue. Nevertheless, the causes of such vibrations are still not completely understood and are consequently not fully predictable during the design process. One possible source lies on the irregularities of the rails, acting as a forced external excitation. However, a spectral analysis shows deviations with respect to real measurements [1], as the frequencies are not synchronous with the car velocity divided by the spatial irregularity period. Therefore, there is an indication that the vibrations may be due to self‐excitation and/or that the developed models should consider other phenomena not yet contemplated. This investigation analyses –for the first time to the knowledge of the authors– the plausibility of various causes of self‐excited vibrations during a roller coaster car motion. Given the complex structure of a real roller coaster train and the intricate geometry of the track, the study considers as a first step minimal dynamical models running along a straight track. The cause analyzed in this paper is the contribution of the transversal friction force component that develops in the wheel‐rail contact interface to the destabilization of the longitudinal motion, as well as the effect of the various geometrical and inertial parameters. In this first analysis, no external excitation is considered. For a range of parameters, the minimal system presents a Hopf bifurcation. The results are plausible from the general behavior and thus this cause may be retained as a candidate for the real vibrations occurring in roller coasters. However, the applicability of these preliminary results to general roller coaster motions is not yet confirmed, but a superposition with other causalities may render effective. In further investigations, a complete train motion on curved tracks and external perturbations will be studied to determine the share of the different causes. [ABSTRACT FROM AUTHOR]

In this paper, we propose a host‐parasitoid model with harvest effort. The existence and stability of a positive fixed point are analyzed. The period‐doubling and Neimark–Sacker bifurcations are studied. These analyses are achieved by applying the normal form of the difference‐algebraic system, bifurcation theory, and center manifold theorem. Furthermore, we apply a state‐delayed feedback control strategy to control the complex dynamics of the proposed model. Numerical examples and simulations are given to verify our findings. Owing to the framework of Nicholson–Bailey host‐parasitoid system, the proposed difference‐algebraic model shows rich dynamics compared with the continuous‐time models. [ABSTRACT FROM AUTHOR]

A diffusive predator–prey system with the network connection and harvesting policy is investigated in the present paper. The global existence and boundedness of the positive solutions to the parabolic equations are analyzed. Hereafter, a priori estimates and non‐existence of the non‐constant steady states are investigated for the corresponding elliptic equation. Next, we focus on the network connect model. The stability of the positive equilibrium, the Hopf bifurcation, and the Turing instability of networked system are explored. By using the multiple time scale (MTS), the direction of the Hopf bifurcation is determined. It is found that the networked system may admit a supercritical or subcritical Hopf bifurcation. For the Turing instability, the positive equilibrium will change its stability from an unstable state to a stable one with the change of the connecting probability. That is not the case in the model without network structure. Theoretical results also show that the model can create rich dynamical behaviors and numerical simulations well verify the validity of the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, a fractional Lotka–Volterra mathematical model for a bioreactor is proposed and used to fit the data provided by a bioprocess known as continuous fermentation of Zymomonas mobilis. The model contemplates a time‐delay τ$$ \tau $$ due to the dead‐time (non‐trivial) that the microbe needed to metabolize the substrate. A Hopf bifurcation analysis is performed to characterize the inherent self oscillatory experimental bioprocess response. As consequence, stability conditions for the equilibrium point together with conditions for limit cycles using the delay τ$$ \tau $$ as bifurcation parameter are obtained. Under the assumptions that the use of observers, estimators, or extra laboratory measurements are avoided to prevent the rise of computational or monetary costs, for the purpose of control, we will only consider the measurement of the biomass. A simple controller that can be employed is the proportional action controller u(t)=kpx(t)$$ u(t)={k}_px(t) $$, which is shown to fail to stabilize the obtained model under the proposed analysis. Another suitable choice is the use of a delayed controller u(t)=krx(t−h)$$ u(t)={k}_rx\left(t-h\right) $$ which successfully stabilizes the model even when it is unstable. The delay h$$ h $$ in the feedback control is due to the dead‐time necessary to obtain the measurement of the biomass in the bioreactor by dry weight. Finally, the proposed theoretical results are corroborated through numerical simulations. [ABSTRACT FROM AUTHOR]

The influences of Allee effect and density‐dependent mortality on population growth are of great significance in ecology. In this paper, we first consider a Gause‐type predator–prey model with simplified Holling type IV functional response, strong Allee effect on prey, and density‐dependent mortality of predator. It is shown that the system exhibits rich and complex dynamics like bistability, local and global bifurcations such as transcritical bifurcation, saddle‐node bifurcation, Hopf bifurcation, cusp bifurcation, Bogdanov–Takens bifurcation, Bautin bifurcation, homoclinic bifurcation, saddle‐node bifurcation of limit cycle, and heteroclinic bifurcation. Next, we separately explore the influences of Allee effect and density‐dependent mortality on the dynamics of the same model. The results show that the strong Allee effect induces the occurrence of heteroclinic bifurcation and the reduction of the number of limit cycles, while the density‐dependent mortality stabilizes the system. Since the analytical expressions of the interior equilibria are difficult to derive, the results are verified numerically. [ABSTRACT FROM AUTHOR]

This paper is devoted to the Schnakenberg model with crucial reversible reactions, under Neumann boundary conditions. Existence and uniqueness of the strong solution are obtained on basis of semigroup theory. It is found that the proposed system admits four possible positive constant steady states, and explicit conditions of the stability, Turing instability, steady‐state bifurcation, and Hopf bifurcation are determined. Besides, numerical simulations are given to show the theoretical results and depict the spatiotemporal patterns. [ABSTRACT FROM AUTHOR]

This study concentrates on the dynamics of a prey–predator model incorporating the fear effect in the birth and death rate of prey, whose physiological changes may give rise to undirect predation. In the presence and absence of time delay, single parameter numerical continuation with respect to two parameters, that are (i) fear level in the death rate of prey and (ii) conversion efficiency by which new predators are introduced through prey consumption in the system, is performed. Basic results on extinction and delay‐driven Hopf bifurcation criteria are investigated. Then, the model is extended to involve the spatial dynamics with and without time delay. Theoretical results for orientation and stability of Hopf bifurcation in spatial system are provided by applying the normal form recipe and also the center manifold theory. Classical reaction–diffusion‐type models, incorporating self‐diffusion, can induce regular (periodic) and irregular (chaotic) oscillations in space. However, space periodic oscillations are not common in prey–predator interactions. Here, it is shown that the dynamics of only diffusion involved model is periodically arranged in space and time. However, introducing a very small value of time delay in predator maturation, spatial dynamics with chaos is initiated as a result of the joint effect of time delay and diffusion. This reassures that time delay has a crucial role in population dynamics incorporated with the role of indirect predation and gives some useful intuition into underlying species interactions. Theoretical results of the model in this paper are supported with numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we consider a predator–prey model with Beddington–DeAngelis functional response and predator competition, which is a five‐parameter family of planar vector field. It is shown that the model can undergo a sequence of bifurcations including focus type degenerate Bogdanov–Takens bifurcation of codimension 3 and Hopf bifurcation of codimension at least 2 as the parameters vary. Our theoretical results indicate that predator competition can cause richer dynamics such as two limit cycles enclosing one or three hyperbolic positive equilibria and three kinds of homoclinic orbits (homoclinic to hyperbolic saddle, saddle‐node, or neutral saddle). Moreover, there exists a threshold value m0$$ {m}_0 $$ for predator capturing rate m$$ m $$, below or equal to which the predators always tend to extinction, above which the predators and preys will coexist in the form of multiple steady states or periodic oscillations for all positive initial populations. Numerical simulations are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Since the pioneering work of Turing, it has been known that diffusion can destablise a homogeneous solution that is stable in the underlying model in the absence of diffusion. The destabilisation of the homogeneous solutions leads to the generation of patterns. In recent years, techniques have been developed to analyse so‐called localised spatial structures. These are solutions in which the spatial structure occurs in a localised region. Unlike Turing patterns, they do not spread out across the whole domain. We investigate the existence of localised structures that occur in two predator‐prey models. The functionalities chosen have been widely used in the literature. The existence of localised spatial structures has not investigated previously. Indeed, it is easy to show that these models cannot exhibit the Turing instability. This has perhaps led earlier researchers to conclude that interesting spatial solutions can therefore not occur for these models. The novelty of our paper is that we show the existence of stationary localised patterns in systems which do not undergo the Turing instability. The mathematical tools used are a combination of Linear and weakly nonlinear analysis with supporting numerical methods. By combining these methods, we are able to identify conditions for a wide range of increasing exotic behaviour. This includes the Belyckov‐Devaney transition, a codimension two spatial instability point and the formation of localised patterns. The combination of spectral computations and numerical simulations reveals the crucial role played by the Hopf bifurcation in mediating the stability of localised spatial solutions. Finally, numerical solutions in two spatial dimensions confirms the onset of intricate spatio‐temporal patterns within the parameter regions identified within one spatial dimension. [ABSTRACT FROM AUTHOR]

This paper examines the viscous flow resistance in branching tubes as applied to simplified models of the lungs and compares the results of computational fluid dynamics simulations for a range of conditions with measurement data. The results are in good agreement with the available measurement data for both inspiration and expiration. A detailed sensitivity analysis of the dissipation and viscous resistance in a branch then examines the ratio of the viscous resistance to that for a fully developed Poiseuille flow, Z. As other researchers have noted, the calculated resistances give lower values than those from the standard correlation of Pedley et al. The results demonstrate that the resistance is sensitive to the velocity profile upstream of the bifurcations and explain from fluid dynamical considerations the apparent sensitivity of the resistance to the generation number of the branch. The paper also suggests a revised value for the calibration constant in the expression for Z. Finally, a limited set of results are presented for junction losses, and for expiration. [ABSTRACT FROM AUTHOR]

In this paper, we improve a new mathematical model associated with glue applying of particleboard. First, we study the existence and stability of the equilibria and the existence of fold, Hopf, and Bogdanov–Takens bifurcations in above system. Second, the normal forms of Hopf bifurcation and Bogdanov–Takens bifurcation are derived, and the classifications of local dynamics near above bifurcation critical values are analyzed. Then, numerical simulation results show that the glue flow system associated with glue applying of particleboard exists stable equilibrium, stable periodic‐1, periodic‐2, and periodic‐4 solutions, and chaotic attractor phenomenon from a sequence of period‐doubling bifurcations. Finally, we compare the dynamical phenomena of glue flow system with frequency water supply system, showing that cubic terms which the latter lacks, could influence the dynamical behaviors of glue flow system. [ABSTRACT FROM AUTHOR]

Bifurcation theory has been highly popular in the analysis of mathematical models. However, stability and bifurcation analyses are only for asymptotic dynamics while applied scientists care more about transient dynamics. In this paper, we first rigorously analyze Holling–Tanner model with generalist predators who have alternative food sources, and then discuss transient dynamics via a changing environment. For a constant environment, we provide a complete bifurcation analysis with high codimension. It is shown that the highest codimension of a nilpotent cusp is 3, and the model can undergo degenerate Bogdanov–Takens bifurcation of codimension 3. Moreover, by using resultant elimination to solve the semialgebraic varieties of Lyapunov coefficients, we show that a center‐type equilibrium is a weak focus with order at most 2, and the model can exhibit Hopf bifurcation of codimension 2. Our results indicate that generalist predators can cause not only richer dynamics and bifurcations, but also the extinction of prey for some positive initial densities. Numerical simulations, including the coexistence of a limit cycle and a homoclinic cycle, tristability, two limit cycles, are presented to illustrate the theoretical results. In a changing environment, the populations start along one stable state but can track unstable states or oscillations when the system crosses a bifurcation point, and then tend to another stable state or oscillations. This tracking on transient dynamics predicts regime shifts under environmental changes. When environmental conditions vary, the populations can track unstable states in the constant environment. The rate of environmental change determines how long the system tracks an unstable state although finally the solution under environmental change is attracted to a stable steady state or limit cycle. Finally, we focus on a periodic environment and find that the populations converge to a periodic solution or an invariant torus depending on both the initial environmental capacity and the amplitude of periodic fluctuation. [ABSTRACT FROM AUTHOR]

The paper discusses the effects of constant prey harvesting on the dynamics of a modified Leslie–Gower predator–prey model. Prey harvesting will lead to enriched dynamics to help us realize ecological phenomena such as the extinction of some species with a positive initial density when the harvesting rate is larger than a critical level. All these may be very useful for biological management. Based on these reasons, firstly, we analyze the existence conditions and stability of different equilibrium points to predict the eventual state of the given system. In particular, the existence of cusps of Codimensions 2 and 3 is proved. Secondly, we successfully demonstrate the presence of Hopf bifurcation and saddle‐node bifurcation for specified parameter values. To prove the limit cycle stability of Hopf bifurcation, we use the first Lyapunov coefficient K$$ K $$ for illustration. As well as calculating the universal expansion near the cusp, the Bogdanov–Takens bifurcations of Codimensions 2 and 3 are investigated. If the parameters are chosen fittingly, there will be a stable or an unstable limit cycle, a homoclinic loop, two limit cycles, or both a limit cycle and a homoclinic loop simultaneously in the system. Finally, numerical simulations are performed using MATLAB to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with a diffusive predator–prey model with prey‐taxis and prey‐stage structure under the homogeneous Neumann boundary condition. The stability of the unique positive constant equilibrium of the predator–prey model is derived. Hopf bifurcation and steady‐state bifurcation are also concluded. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a fractional differential‐algebraic model with two discrete delays and double linear species harvesting. One of the delays denotes the time taken for digestion of the prey, and the other reflects the negative feedback of the predator's density. Taking into account of the economic profit, linear predator harvesting and prey harvesting have been incorporated into the proposed predator–prey system. From the biological viewpoint, we discuss the sufficient conditions for the existence of nontrivial positive equilibrium point. By jointly using the stability theory of fractional differential‐algebraic equations and the bifurcation theory, we study the delay‐induced instability and Hopf bifurcations. Finally, the correctness of the theoretical analysis is verified by numerical simulations, and the effects of delays, economic profit, and fractional exponents on the stability of the system are explored. [ABSTRACT FROM AUTHOR]

A general reaction‐diffusion Brusselator model subject to homogeneous Neumann boundary conditions is investigated in this paper. First, the stability of the unique positive equilibrium is studied, and we identify the existence of the Hopf bifurcation. Then, occurrence conditions of the Turing instability, the Turing‐Turing, and the Turing‐Hopf bifurcations are established. To explore the spatiotemporal solutions resulting from the bifurcation, the amplitude equations of the Turing‐Turing and the Turing‐Hopf bifurcations are established via the method of multiple time scale. It is found that the model admits the nonconstant steady state, the mixed nonconstant steady state, the spatially homogeneous periodic solution, and the spatially nonhomogeneous periodic solution. As such, the nonhomogeneous spatiotemporal solutions appear in the model. In the end, numerical simulations verify the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

Using sterile insect release to guide the occurrence of the mate‐finding Allee effect in pest populations is considered a pest management strategy. In this paper, a pest management model is established with the mate‐finding Allee effect induced by releasing sterile pests. In addition, the natural enemies of pests and spraying insecticide are also taken into account to achieve integrated pest management. For the model without delay, although the origin is not well‐defined in our model, we pay attention to the dynamics near the origin and find it is always an attractor. This means the extinction of pests is closely related to their initial scale. Based on that, we conduct a detailed qualitative study on other equilibria. Furthermore, the model with a time delay is also observed. The stability of equilibrium points, Hopf bifurcation, and Bogdanov–Takens bifurcation are all carried out. Through numerical simulation, the results indicate that adding the mate‐finding Allee effect can significantly reduce the demand for pesticides and reduce the environmental damage from pesticides, comparing with the traditional approach (i.e., combining natural enemies and pesticides). We also find that a time delay can switch the stability of the equilibrium points and promote the extinction of pests. [ABSTRACT FROM AUTHOR]

In the present paper, we present the solution of four problems drawn from the chemical and biochemical engineering field of study. These problems illustrate various important aspects of nonlinear dynamics such as limit cycles, quasi‐periodic and chaotic behaviors, time series and phase portraits, power spectra, the time‐delay reconstruction diagrams, Hopf bifurcation, bifurcation diagrams, steady state multiplicity... Among a large number of examples, we have selected the following case studies because they illustrate the basic nonlinear dynamics concepts listed previously: (i) The glycolytic oscillator model, first suggested by Sel'kov in 1968 in order to elucidate the mechanism that living cells use to obtain energy from sugar breakdown; (ii) The Oregonator model derived by Field, Körös, and Noyes in the early 1970s, which elucidates the famous oscillatory behavior observed in the Belousov–Zhabotinsky reaction; (iii) The steady state multiplicity in a biochemical reactor for both the Monod and substrate inhibition kinetics; and (iv) The three‐variable autocatalator, first proposed by Peng, Scott, and Showalter in 1990. Finally, we conclude this paper by sharing our experience teaching this subject to graduate students at King Fahd University of Petroleum & Minerals (KFUPM). [ABSTRACT FROM AUTHOR]

We deal with a fractional‐order nonlinear financial model with time delay in this paper. The quantitative analysis of the model has been done in which the asymptotic stability of the equilibrium points of the model have been discussed. Furthermore, the Hopf bifurcation analysis of the model has been done under the impact of time delay. The stability of the model has been studied with the reproduction number less than or greater than 1. The analysis of the model has been done by using the Laplace transformation technique. The analysis shows that the fractional‐order model with a time delay can adequately enhance the elements and fortify the results for either stable or unstable criteria. All unstable cases in the nonlinear financial system are converted to stable cases for fractional order under neighborhood points. The reproduction ranges of all parameters have been discussed. The paper examines the impact of time delay and the importance of determining essential parameters such as saving amount by using Hopf bifurcation. Finally, numerical simulations have been carried out by MATLAB to illustrate our derived results. The financial system is verified under the theoretical outcomes from parameter estimation. [ABSTRACT FROM AUTHOR]

This paper studies the nonlinear modeling and operation stability of variable speed pumped storage power station (PSPS). Firstly, basic equations of variable speed PSPS are established. Nonlinear state equation in the form of relative deviation value is derived by considering supplementary conditions. Then, Hopf bifurcation analysis of variable speed PSPS is carried out. Stable domain of variable speed PSPS is studied. Finally, influence of factors on stability is analyzed. Dynamic response of variable speed PSPS is investigated. The results indicate that variable speed PSPS is a five‐order nonlinear dynamic system. Nonlinear characteristics of variable speed PSPS reflect the nonlinear coupling effect between unit speed and rotor current. The emerged Hopf bifurcation is supercritical. Stable domain locates at the lower side of bifurcation line. Stable domain intuitively reflects and evaluates the stability of variable speed PSPS. When state point locates at stable domain, variable speed PSPS is stable and dynamic response of state variables is damped oscillation. A smaller flow inertia of penstock, a greater head loss of penstock, and a greater slip ratio are favorable for stability of variable speed PSPS. Stable domain under positive disturbance of active power is larger than that under negative disturbance of active power. [ABSTRACT FROM AUTHOR]

In this paper, we develop an age‐structured model with two infectious stages to study the effect of infection age on dynamic properties. By reformulating the model as a non‐densely defined Cauchy problem and applying theorem related with Hille–Yosida operator, the threshold dynamics are investigated. Theoretical analysis shows that destabilization of the age‐dependent endemic equilibrium can occur due to the perturbation of critical infection age that separates the two infectious stages. Furthermore, numerical simulations are conducted to illustrate the dynamical behaviors of stability switching. [ABSTRACT FROM AUTHOR]

With desertification becoming more and more serious all over the world, it is crucial to understand the intrinsic relationship between vegetation and water (soil water and surface water). In this paper, a cross‐diffusion model with mixed delays is established to describe the dynamics of vegetation, soil water, and surface water in the case of water shortage. The local stability conditions of positive equilibrium point is established by using Routh–Hurwitz criterion. Then, the conditions for generating Hopf bifurcation are studied. In the end, numerical simulations are carried out to illustrate the effectiveness of the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fractional‐order delay differential model for tuberculosis (TB) transmission with the effects of endogenous reactivation and exogenous reinfections. We investigate the qualitative behaviors of the model throughout the local stability of the steady states and bifurcation analysis. A discrete time delay is introduced in the model to justify the time taken for diagnosis of the disease. Existence and positivity of the solutions are investigated. Some interesting sufficient conditions that ensure the local asymptotic stability of infection‐free and endemic steady states are studied. The fractional‐order TB model undergoes Hopf bifurcation with respect to time delay and disease transmission rate. The presence of fractional order and time delay in the model improves the model behaviors and develops the stability results. A numerical example is provided to support our theoretical results. [ABSTRACT FROM AUTHOR]

Since population behaviors possess the characteristic of history memory, we, in this paper, introduce time fractional‐order derivatives into a diffusive Gause‐type predator‐prey model, which is time fractional‐order reaction‐diffusion equations and a generalized form of its corresponding first‐derivative model. For this kind of model, we prove the existence and uniqueness of a global positive solution by using the theory of evolution equations and the comparison principle of time fractional‐order partial differential equations. Besides, we obtain the stability and Hopf bifurcation of the Gause‐type predator‐prey model in the forms of the time fractional‐order ordinary equations and of the time fractional‐order reaction‐diffusion equations, respectively. Our results show that the stable region of the parameters in these 2 models can be enlarged by the time fractional‐order derivatives. Some numerical simulations are made to verify our results. [ABSTRACT FROM AUTHOR]

The present paper concerns with the dynamics of a fractional neural network involving three neurons. Firstly, the bifurcation point is identified for which Hopf bifurcations may occur by taking the system parameter as a bifurcation parameter via the stability analysis of fractional systems. It is indicated that the system parameter can significantly affect the dynamical properties of such network. Secondly, the impact of the order on the bifurcation point is carefully examined. It is found that the occurrence of bifurcation is delayed as the order increases as long as the other system parameters are established. Finally, a numerical example is exploited to verify the efficiency of theoretical results. [ABSTRACT FROM AUTHOR]