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In this manuscript, we consider a neutral functional differential equation. We make some assumptions and formulate conditions that ensure the existence of a zero-Hopf singularity. Using normal form theory and center manifold reduction, we derive the normal form up to third-order terms. [ABSTRACT FROM AUTHOR]

A rigorous description of a period-doubling bifurcation of limit cycles in retarded functional differential equations based on tools of functional analysis and singularity theory is presented. Particularly, sufficient conditions for its occurrence and its normal form coefficients are expressed in terms of derivatives of the operator defining given equations. We also prove the exchange of stability in the case of a non-degenerate period-doubling bifurcation. The approach concerns Fredholm operators, Lyapunov–Schmidt reduction and recognition problem for pitchfork bifurcation. [ABSTRACT FROM AUTHOR]

Mathematical model plays a significant role in describing the mutual interaction of various chemical compositions in chemistry. In this present work, we formulate a new plankton–oxygen dynamical model owning delay. By virtue of fixed point theorem, inequality techniques and construction of function, we set up the conditions on existence and uniqueness, non-negativeness and boundedness of the solution to the formulated plankton–oxygen model. Taking advantage of bifurcation and stability theory of delayed differential equation, we explore the existence of bifurcation and stability for the plankton–oxygen model and set up a novel delay-independent criterion ensuring the existence of bifurcation and stability of the model. Making use of two different extended hybrid controllers, we can successfully control the time of emergence of bifurcation and stability domain of this model. The impact of delay on bifurcation and stability of the model is explored. Software experiment results are provided to sustain the acquired key outcomes. The gained conclusions of this work are perfectly novel and possess immense theoretical significance in adjusting and balancing the concentrations of disparate chemical compositions. [ABSTRACT FROM AUTHOR]

In this article, a diffusive predator-prey model with memory effect and predation-driven Allee effect is considered. Through eigenvalue analysis, the local asymptotic stability of positive constant steady-state solutions is analyzed, and it is found that memory delay affects the stability of positive constant steady-state solutions and induces Hopf bifurcation. The properties of Hopf bifurcating periodic solutions have also been analyzed through the central manifold theorem and the normal form method. Finally, our theoretical analysis results were validated through numerical simulations. It was found that both memory delay and predation-driven Allee effect would cause the positive constant steady-state solution of the model to become unstable, accompanied by the emergence of spatially inhomogeneous periodic solutions. Increasing the memory period will cause periodic oscillations in the spatial distribution of the population. In addition, there would also be high-dimensional bifurcation such as Hopf–Hopf bifurcation, making the spatiotemporal changes of the population more complex.

The present paper is concerned with a generalized logistic reaction–diffusion–advection population model with nonlocal delay effect and subject to homogeneous Dirichlet boundary condition. Normal form of Hopf bifurcation of model at the positive steady-state solution is computed in virtue of the normal form method and the center manifold theorem for partial functional differential equations. Then the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are determined in terms of the obtained normal form. It is shown that Hopf bifurcations of model at the positive steady-state solution are forward and the associated bifurcating periodic solutions are locally orbitally asymptotically stable on the center manifold. [ABSTRACT FROM AUTHOR]

The pattern dynamics of the Brusselator system with cross-diffusion and gene expression time delay are investigated. Conditions for cross-diffusion-driven instability are established, which reveal that the formation of patterns in the system with cross-diffusion does not follow the “short-range activation” and “long-range inhibition” mechanisms. Moreover, the conditions for the occurrence of Turing bifurcation, Hopf bifurcation, Turing–Hopf bifurcation, and double-Hopf bifurcation are obtained using eigenvalue analysis. The algorithms for calculating the normal forms of double-Hopf bifurcation are given using multiple timescale method, with the coefficients expressed explicitly in terms of the original system parameters. An essential contribution of this paper is the effect of the space element taken into account in this method, which results in that the normal forms of the double-Hopf bifurcation for the Brusselator system are divided into two categories: One is 1:3 strong resonance, and the other consists of other strong resonances, weak resonance, and non-resonance. Different types of spatiotemporal patterns are classified according to the exact partition of the parameter space. It is demonstrated that the Brusselator system can exhibit stable spatially inhomogeneous periodic and quasi-periodic orbits. [ABSTRACT FROM AUTHOR]

Cannibalism is often an extreme interaction in the animal species to quell competition for limited resources. To model this critical factor, we improve the predator–prey model with nonlocal competition effect by incorporating the cannibalism term, and different kernels for competition are considered in this model numerically. We give the critical conditions leading to the double Hopf bifurcation, in which the gestation time delay and the diffusion coefficient were selected as the bifurcation parameters. The innovation of the work lies near the double Hopf bifurcation point, and the stable homogeneous and inhomogeneous periodic solutions can coexist. The theoretical results of the extended centre manifold reduction and normal form method are in good agreement with the numerical simulation. [ABSTRACT FROM AUTHOR]

The Volterra integral equation associated with autonomous Caputo fractional differential equation (FDE) of order α ∈ (0 , 1) in R d was shown by the authors [4] to generate a semi-group on the space C of continuous functions f : R + → R d with the topology uniform convergence on compact subsets. It serves as a semi-dynamical system for the Caputo FDE when restricted to initial functions f(t) ≡ i d x 0 for x 0 ∈ R d . Here it is shown that this semi-dynamical system has a global Caputo attractor in C , which is closed, bounded, invariant and attracts constant initial functions, when the vector field function in the Caputo FDE satisfies a dissipativity condition as well as a local Lipschitz condition. [ABSTRACT FROM AUTHOR]

Characterizing existence or not of periodic orbit is a classical problem, and it has both theoretical importance and many real applications. Here, several new criterions on nonexistence of periodic orbits of the planar dynamical system x ˙ = y , y ˙ = - g (x) - f (x , y) y are obtained and by examples shows that these criterions are applicable, but the known ones are invalid to them. Based on these criterions, we further characterize the local topological structures of its equilibrium, which also show that one of the classical results by Andreev (Am Math Soc Transl 8:183–207, 1958) on local topological classification of the degenerate equilibrium is incomplete. Finally, as another application of these results, we classify the global phase portraits of a planar differential system, which comes from the third question in the list of the 33 questions posed by A. Gasull and also from a mechanical oscillator under suitable restriction to its parameters. [ABSTRACT FROM AUTHOR]

This paper aims to study the sufficient and necessary conditions for a generalized polynomial Liénard system with a unique equilibrium and one periodic annulus. In such a case, there are two types of equilibrium: a centre or a neighbourhood of the equilibrium consisting of a hyperbolic sector and an elliptical sector. The sufficient and necessary conditions for both cases are established and global phase portraits in the Poincaré disc are presented under certain conditions. [ABSTRACT FROM AUTHOR]

The hematological stem cells model is introduced with neutrophil dynamics of two department model setting forth. During the cells differentiation and proliferation process, the neutrophils are functioned with negative feedback with delay history, which contains delayed amplification coefficient. In more general view, the new introduction rate is given to replace the familiar Hill function which is helpful to understand the complex dynamics of neutrophils. The double Hopf bifurcation is calculated with the artificial handtools named DDE-Biftool, which is observed as the self-intersection of Hopf lines. The continuation of periodical solutions arising from Hopf points are done and the longer period solutions are manifested with multi-rhythm and bursting oscillation. The near dynamics of double Hopf points is simulated by DDE-Biftool with different route design, the multi-period attractors, quasi-periodical solutions and chaos are observed. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear boundary-value problem proposed as the simplest model for describing oscillations in a gas flow. We analyze the stability of the trivial (zero) equilibrium state and find a critical value of the speed of the incoming gas flow. Exact solutions of the problem are found in the form of time-periodic functions and their stability is examined. All the results are obtained analytically based on the qualitative theory of infinite-dimensional dynamical systems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of practically susceptible population, which is a fraction of the biologically susceptible population. Assuming that the fraction depends on the severity of the epidemic and the public's level of precaution (as a response of the public to the epidemic), we propose a general framework model with the response level evolving with the epidemic. We firstly verify the well-posedness and confirm the disease's eventual vanishing for the framework model under the assumption that the basic reproduction number R 0 < 1 . For R 0 > 1 , we study how the behavioural response evolves with epidemics and how such an evolution impacts the disease dynamics. More specifically, when the precaution level is taken to be the instantaneous best response function in literature, we show that the endemic dynamic is convergence to the endemic equilibrium; while when the precaution level is the delayed best response, the endemic dynamic can be either convergence to the endemic equilibrium, or convergence to a positive periodic solution. Our derivation offers a justification/explanation for the best response used in some literature. By replacing "adopting the best response" with "adapting toward the best response", we also explore the adaptive long-term dynamics. [ABSTRACT FROM AUTHOR]

This paper deals with Kalecki's 1935 business cycle model, where a finite time lag in the investment dynamics is assumed. The time lag is the gestation period elapsing between orders for capital goods and deliveries of finished industrial equipment. Including the actual mainstream theory, the economic literature agrees on the consequences that time lag has on the economic activity. It is a cause of persistent economic fluctuations. Following some recent research lines on this model, here we restate the Kalecki approach, assuming sigmoidal functions in addition to Kalecki's linear treatment and further considering a non-constant capital depreciation. Never made until now, this last assumption is such that to yield, in place of a delayed differential equation, a Volterra delayed integro-differential equation. Taken the time delay and the rate of capital depreciation as critical parameters, a qualitative study of that equation is carried out. We proved that with a small-time lag stable equilibria arise. But, when the delay increases, equilibria are destabilized through Hopf bifurcations and stability switches occur. Consequently, a variety of cyclical behaviors appear. [ABSTRACT FROM AUTHOR]

In this current study, novel fractional neural networks owning delay are formulated. Using Lipschitz condition, we demonstrate that the solution of the formulated fractional delayed neural networks exists and is unique. Applying a reasonable function, we handle the boundedness issue of solution to the formulated fractional delayed neural networks. Exploiting the stability criterion and bifurcation viewpoint of the fractional order delayed dynamical system, we explore the stability and bifurcation phenomenon of the established fractional delayed neural networks. Taking advantage of an adequate hybrid controller, we have efficaciously dominated the stability domain and the time of generation of bifurcation of the formulated fractional delayed neural networks. Ultimately, computer simulation graphs are provided to sustain our acquired outcomes. The acquired theoretical outcomes of this study possess considerable realistic meaning in regulating and controlling neural networks. [ABSTRACT FROM AUTHOR]

The Hopf bifurcation, Turing instability and steady state bifurcation to a fresh-water tussock sedge model with nonlocal interaction under Neumman boundary condition are investigated in this paper. First, we analyze the existence of constant steady states and the effect of the nonlocal term on the its stability and the existence of Hopf bifurcation. Furthermore, the occurrence conditions of Turing instability to such system are studied. Second, we focus on steady state bifurcation to the reaction–diffusion system with nonlocal interaction via Lyapunov–Schmidt reduced method. Finally, numerical simulations have been illustrated to verify our theoretical analysis. [ABSTRACT FROM AUTHOR]

A periodic boundary-value problem for an equation with a deviating spatial argument is considered. Using the Poincaré–Dulac method of normal forms, the method of integral manifolds, and asymptotic formulas, we examine a number of bifurcation problems of codimension 1 and 2. For homogeneous equilibrium states, we analyze possibilities of implementing critical cases of various types. The problem on the stability of homogeneous equilibrium states is studied and asymptotic formulas for spatially inhomogeneous solutions and conditions for their stability are obtained. [ABSTRACT FROM AUTHOR]

In this paper, an age-structured predator–prey system with Beddington–DeAngelis (B–D) type functional response, prey refuge and harvesting is investigated, where the predator fertility function f(a) and the maturation function β (a) are assumed to be piecewise functions related to their maturation period τ . Firstly, we rewrite the original system as a non-densely defined abstract Cauchy problem and show the existence of solutions. In particular, we discuss the existence and uniqueness of a positive equilibrium of the system. Secondly, we consider the maturation period τ as a bifurcation parameter and show the existence of Hopf bifurcation at the positive equilibrium by applying the integrated semigroup theory and Hopf bifurcation theorem. Moreover, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are studied by applying the center manifold theorem and normal form theory. Finally, some numerical simulations are given to illustrate of the theoretical results and a brief discussion is presented. [ABSTRACT FROM AUTHOR]

COVID-19 is a respiratory disease triggered by an RNA virus inclined to mutations. Since December 2020, variants of COVID-19 (especially Delta and Omicron) continuously appeared with different characteristics that influenced death and transmissibility emerged around the world. To address the novel dynamics of the disease, we propose and analyze a dynamical model of two strains, namely native and mutant, transmission dynamics with mutation and imperfect vaccination. It is also assumed that the recuperated individuals from the native strain can be infected with mutant strain through the direct contact with individual or contaminated surfaces or aerosols. We compute the basic reproduction number, R 0 , which is the maximum of the basic reproduction numbers of native and mutant strains. We prove the nonexistence of backward bifurcation using the center manifold theory, and global stability of disease-free equilibrium when R 0 < 1 , that is, vaccine is effective enough to eliminate the native and mutant strains even if it cannot provide full protection. Hopf bifurcation appears when the endemic equilibrium loses its stability. An intermediate mutation rate ν 1 leads to oscillations. When ν 1 increases over a threshold, the system regains its stability and exhibits an interesting dynamics called endemic bubble. An analytical expression for vaccine-induced herd immunity is derived. The epidemiological implication of the herd immunity threshold is that the disease may effectively be eradicated if the minimum herd immunity threshold is attained in the community. Furthermore, the model is parameterized using the Indian data of the cumulative number of confirmed cases and deaths of COVID-19 from March 1 to September 27 in 2021, using MCMC method. The cumulative cases and deaths can be reduced by increasing the vaccine efficacies to both native and mutant strains. We observe that by considering the vaccine efficacy against native strain as 90%, both cumulative cases and deaths would be reduced by 0.40%. It is concluded that increasing immunity against mutant strain is more influential than the vaccine efficacy against it in controlling the total cases. Our study demonstrates that the COVID-19 pandemic may be worse due to the occurrence of oscillations for certain mutation rates (i.e., outbreaks will occur repeatedly) but better due to stability at a lower infection level with a larger mutation rate. We perform sensitivity analysis using the Latin Hypercube Sampling methodology and partial rank correlation coefficients to illustrate the impact of parameters on the basic reproduction number, the number of cumulative cases and deaths, which ultimately sheds light on disease mitigation. [ABSTRACT FROM AUTHOR]

We consider a periodic function p : R → R of minimal period 4 which satisfies a family of delay differential equations 0.1 x ′ (t) = g (x (t - d Δ (x t))) , Δ ∈ R , with a continuously differentiable function g : R → R and delay functionals d Δ : C ([ - 2 , 0 ] , R) → (0 , 2). The solution segment x t in Eq. (0.1) is given by x t (s) = x (t + s) . For every Δ ∈ R the solutions of Eq. (0.1) defines a semiflow of continuously differentiable solution operators S Δ , t : x 0 ↦ x t , t ≥ 0 , on a continuously differentiable submanifold X Δ of the space C 1 ([ - 2 , 0 ] , R) , with codim X Δ = 1 . At Δ = 0 the delay is constant, d 0 (ϕ) = 1 everywhere, and the orbit O = { p t : 0 ≤ t < 4 } ⊂ X 0 of the periodic solution is extremely stable in the sense that the spectrum of the monodromy operator M 0 = D S 0 , 4 (p 0) is σ 0 = { 0 , 1 } , with the eigenvalue 1 being simple. For | Δ | ↗ ∞ there is an increasing contribution of variable, state-dependent delay to the time lag d Δ (x t) = 1 + ⋯ in Eq. (0.1). We study how the spectrum σ Δ of M Δ = D S Δ , 4 (p 0) changes if | Δ | grows from 0 to ∞ . A main result is that at Δ = 0 an eigenvalue Λ (Δ) < 0 of M Δ bifurcates from 0 ∈ σ 0 and decreases to - ∞ as | Δ | ↗ ∞ . Moreover we verify the spectral hypotheses for a period doubling bifurcation from the periodic orbit O at the critical parameter Δ ∗ where Λ (Δ ∗) = - 1 . [ABSTRACT FROM AUTHOR]

In this paper, we consider a single species population model over patches with delay and nonlocal interactions, for which no symmetry for the dispersion (connection) matrix is assumed. We show that there exists a positive equilibrium when the dispersal rate is large. We also discuss the stability/instability of this positive equilibrium, establish the threshold dynamics and explore the associated Hopf bifurcation. Moreover, we demonstrate our theoretical results by a nonlocal logistic population model and by the Nicholson's blowflies model. [ABSTRACT FROM AUTHOR]

A large number of articles have been devoted to the study of age-dependent predation interaction. Most of them concentrate on the existence of the solution and the non-trivial periodic solution. The strength of this text is that we mainly investigate the long-time behavior of the solution for a general ratio-dependent predation interaction involving prey-competition and predator-age in the form of an ODE and a PDE. Firstly, we prove that there are some variable values such that this ratio-dependent predation interaction has a unique positive equilibrium age profile with Takens–Bogdanov singularity. Secondly, under fit tiny perturbation, the ratio-dependent predation interaction generates the Takens–Bogdanov bifurcation in a small domain of this positive equilibrium age profile. [ABSTRACT FROM AUTHOR]

Complex metapopulation dynamics research has a profound impact on our understanding of the relationship between species and their habitats. In this paper, the dynamical behaviors of the single-species metapopulation model with reproductive and reaction time delays based on Levins’ model are investigated by analyzing stability charts, rightmost characteristic roots, and bifurcation diagrams of the positive equilibrium. Finally, the theoretical results are compared with the numerical results. [ABSTRACT FROM AUTHOR]

This study considers two time delays applied in a commensalism system with a Beddington–DeAngelis functional response. In contrast with existing literature on commensalism systems, the system considered in the present study has two time delays in one species. The local stability of the positive equilibrium and Hopf bifurcation are investigated. The linearized stability is thoroughly examined. Furthermore, the characteristic equations are investigated, and the time delays are applied as the bifurcation parameter. Eventually, the presence of Hopf bifurcation is demonstrated. The Lyapunov functional is constructed, and the system is shown to have uniform persistence. The consistent persistent domain of the system is obtained by constructing a persistent function. Numerical simulations are conducted, demonstrating the reliability of the derived results. [ABSTRACT FROM AUTHOR]

An elementary inequality is discussed for autonomous Caputo fractional differential equation (FDE) of order α ∈ (0 , 1) in R d for which the vector field satisfies a dissipativity condition. This inequality is fundamental for investigating qualitative and dynamical properties of such equations. Here its use and effectiveness are illustrated to show the global existence and uniqueness of solutions of such equations when the vector field is only locally Lipschitz. In addition, the existence of an absorbing set, which is positively invariant, is established. Finally, it is used to show that an equilibrium solution of a nonlinear Caputo FDE is locally asymptotically stable when the matrix of its linear part is negative definite. [ABSTRACT FROM AUTHOR]

In this paper, we consider the fractional complex Ginzburg–Landau equation with Kerr law and power law nonlinearity. Using the conformable derivative approach and the bifurcation method, we effectively derived new explicit exact parametric representations of solutions (including solitary wave solutions, periodic wave solutions, kink and antikink wave solution, compacton) under different parameter conditions. The quasiperiodic, chaotic behavior and sensitivity analysis of the model is studied for different values of parameters after deploying an external periodic force. Finally, various 2D and 3D simulation figures are plotted to show the physical significance of these exact solutions. [ABSTRACT FROM AUTHOR]

Establishing financial models or economic models to describe economic phenomena in real life has become a heated discussion in society at present. From a mathematical point of view, the exploration on dynamics of financial models or economic models is a valuable work. In this study, we build a new delayed finance model and explore the dynamical behavior containing existence and uniqueness, boundedness of solution, Hopf bifurcation, and Hopf bifurcation control of the considered delayed finance model. By virtue of fixed point theorem, we prove the existence and uniqueness of the solution to the considered delayed finance model. Applying a suitable function, we obtain the boundedness of the solutions for the considered delayed finance model. Taking advantage of the stability criterion and bifurcation argument of delayed differential equation, we establish a delay-independent condition ensuring the stability and generation of Hopf bifurcation of the involved delayed finance model. Exploiting hybrid controller including state feedback and parameter perturbation, we efficaciously adjust the stability region and the time of occurrence of Hopf bifurcation of the involved delayed finance model. The study manifests that time delay is a fundamental parameter in controlling stability region and the time of onset of Hopf bifurcation of the involved delayed finance model. To examine the soundness of established key results, computer simulation figures are concretely displayed. The derived conclusions of this study are perfectly new and has momentous theoretical value in economical operation. [ABSTRACT FROM AUTHOR]

Distributed delay is included in a simple predator–prey model in the prey-to-predator biomass conversion term. The delayed term includes a delay-dependent "discount" factor that ensures the predators that do not survive the delay interval, do not contribute to growth of the predator population. A simple model was chosen so that without delay all solutions converge to a globally asymptotically stable equilibrium in order to show the possible effects of delay on the dynamics. If the co-existence equilibrium does not exist, the dynamics of the system is identical to its non-delayed analog. However, with delay, there is a delay-dependent threshold for the existence of the co-existence equilibrium. When the co-existence equilibrium exists, unlike the dynamics of the model without delay, a much wider range of dynamics is possible, including a strange attractor and bi-stability, although the system is uniformly persistent. A bifurcation theory approach is taken, using both the mean delay and the predator death rate as bifurcation parameters. We consider the gamma and the uniform distributions as delay kernels and show that the "discounting" term ensures that the Hopf bifurcations occur in pairs, as was observed in the analogous system with discrete delay (i.e., using the Dirac delta distribution). We show that there are certain features common to all distributions, although the model with different kernels can have a significantly different range of dynamics. In particular, the number of bi-stabilities, the sequence of bifurcations, the criticality of the Hopf bifurcations, and the size of the stability regions can differ. Also, the width of the interval over which the delay history is nonzero seems to have a significant effect on the range of dynamics. Thus, ignoring the delay and/or not choosing the right delay kernel might result in inaccurate modelling predictions. [ABSTRACT FROM AUTHOR]

In this paper, we show the existence of a Hopf bifurcation in a delayed single population model with patch structure. The effect of the dispersal rate on the Hopf bifurcation is considered. Especially, if each patch is favorable for the species, we show that when the dispersal rate tends to zero, the limit of the Hopf bifurcation value is the minimum of the "local" Hopf bifurcation values over all patches. On the other hand, when the dispersal rate tends to infinity, the Hopf bifurcation value tends to that of the "average" model. [ABSTRACT FROM AUTHOR]

This paper considers a chemical reaction-diffusion model for studying pattern formation with the Sel'kov–Schnakenberg model. Firstly, the stability conditions of the positive equilibrium and the existing conditions of the Hopf bifurcation are established for the local system. Then, Turing instability (diffusion-driven), which causes the spatial pattern is investigated and the existing condition of the Turing bifurcation is obtained. In addition, the dynamic behaviors near the Turing bifurcation are also studied by employing the method of weakly nonlinear analysis. The theoretical analysis shows that spatio-temporal patterns change from the spot, mixed (spot-stripe) to stripe with the variation of parameters, which can be verified by a series of numerical simulations. These numerical simulations give a visual representation of the evolution of spatial patterns. Our results not only explain the evolution process of reactant concentration, but also reveal the mechanism of spatio-temporal patterns formation. [ABSTRACT FROM AUTHOR]

Effects of predator-taxis on the dynamics of a predator–prey model with Michaelis–Menten type nonlinear harvesting are considered in this paper. Through theoretical analysis of quasilinear parabolic equations, the local existence, global existence and boundedness of solutions to the system are first established. Then the formation mechanisms of spatiotemporal patterns in such model are explored. It is found that only the attractive predator-taxis will lead to the Turing bifurcation and the corresponding spatiotemporal solutions; meanwhile, no such phenomenon occurs with the repulsive taxis or in the absence of predator-taxis. Moreover, the diffusion ratio can affect the Turing bifurcation thresholds, and nonlinear harvesting can affect the spatial distribution of prey and predator species. Further, the stability of the nonconstant steady state bifurcated from the Turing bifurcation is analyzed through the amplitude equation, so that the direction of the Turing bifurcation is determined. Effectiveness of the analysis is illustrated in the numerical simulations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the global dynamics of a continuous planar piecewise linear differential system with three zones, where the dynamic of the one of the exterior linear zones is saddle and the remaining one is anti-saddle. We give all global phase portraits in the Poincaré disc and the complete bifurcation diagram including boundary equilibrium bifurcation curves, degenerate boundary equilibrium bifurcation curves, homoclinic bifurcation curves and double limit cycle bifurcation curves. Its application in a second-order memristor oscillator is shown. Finally, some numerical phase portraits are demonstrated to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

The housefly is a dominant species in most places and is widely distributed worldwide. However, it is becoming more troublesome that they are not only closely related to human life but can also seriously endanger human health and spread many diseases. Thus, in this paper, a more exhaustive dynamical explanation involving houseflies' DDE and reduced ODE models is elucidated by employing graphical methods, in which these conclusions are the essence of the predecessors. Hereafter, the houseflies model is considered as a houseflies-age-structured model with a biologically more realistic fertility function by using the theory of integrated semigroups, after which this model is transformed as a non-densely defined Cauchy problem. We also investigate model's equilibria and linearized equation, including the characteristic equation about equilibria and the local stability of the boundary equilibrium. Then, the geometric stability switch criteria are used to examine the Hopf bifurcation of the positive equilibrium, whose characteristic equation coefficients depend on the bifurcation parameter. Additionally, numerical simulations enrich the effectiveness of our model. [ABSTRACT FROM AUTHOR]

We consider a scalar delay differential equation x ˙ (t) = - d x (t) + f ((1 - α) ρ x (t - τ) + α ρ x (t - 2 τ)) with an instant mortality rate d > 0 , the nonlinear Rick reproductive function f, a survival rate during all development stages ρ , and a proportion constant α ∈ [ 0 , 1 ] with which population undergoes a diapause development. We consider global continuation of a branch of periodic solutions locally generated through the Hopf bifurcation mechanism, and we establish the existence of periodic solutions with periods within (3 τ , 6 τ) for a wide range of parameter values. We show this existence of periodic solutions not only for the delay τ near the first critical value τ ∗ when a local Hopf bifurcation takes place near the positive equilibrium, but for all τ > τ ∗ . We obtain this (global) existence of periodic solutions by using the equivalent-degree based global Hopf bifurcation theory, coupled with an application of the Li–Muldowney technique to rule out periodic solutions with period 3 τ . We conduct some numerical simulations to illustrate that this global continuation is completely due to the diapause-delay since solutions of the delay differential equation with only normal development delay in the given biologically realistic range all converge to the positive equilibrium. [ABSTRACT FROM AUTHOR]

This paper is to study the global dynamics of a van der Pol-Duffing oscillator with indefinite degree x ˙ = y , y ˙ = a x + x 2 n + 1 - δ (b + x 2 m) y , where a , b , δ ∈ R , m , n ∈ N + and δ ≠ 0 . By qualitative and bifurcation analysis, the oscillator contains abundant nonlinear phenomena, including the heteroclinic bifurcation, degenerate Hopf bifurcation, bifurcation of equilibria at infinity and pitchfork bifurcation. [ABSTRACT FROM AUTHOR]

This paper performs an in-depth qualitative analysis of the dynamic behavior of a diffusive SIS epidemic system with delay in heterogeneous environment subject to homogeneous Neumann boundary condition. Firstly, we explore the principal eigenvalue to obtain the stability of the disease-free equilibrium (DFE) and the effect of the nonhomogeneous coefficients on the stable region of the DFE. Secondly, we obtain the existence, multiplicity and explicit structure of the endemic equilibrium (EE), i.e., spatially nonhomogeneous steady-state solutions, by using the implicit function theorem and Lyapunov-Schmidt reduction method. Furthermore, by analyzing the distribution of eigenvalues of infinitesimal generators, the stability of EE and the existence of Hopf bifurcations at EE are given. Finally, the direction of Hopf bifurcation and stability of the bifurcating periodic solution are obtained by virtue of normal form theory and center manifold reduction. [ABSTRACT FROM AUTHOR]

Some quantities in reaction-diffusion models from cellular biology or ecology depend on the spatial average of density functions instead of local density functions. We show that such nonlocal spatial average can induce instability of constant steady state, which is different from classical Turing instability. For a general scalar equation with spatial average, the occurrence of the steady state bifurcation is rigorously proved, and the formula to determine the bifurcation direction and the stability of the bifurcating steady state is given. For the two-species model, spatially non-homogeneous time-periodic orbits could arise due to spatially non-homogeneous Hopf bifurcation from the constant equilibrium. Examples from a nonlocal cooperative Lotka-Volterra model and a nonlocal Rosenzweig-MacArthur predator-prey model are used to demonstrate the bifurcation of spatially non-homogeneous patterns. [ABSTRACT FROM AUTHOR]

We propose and investigate a mathematical model on interaction between tumor and the immune system, where the regulation of PD-1/PD-L1 and the stimulation delay of tumor antigen for the immune system are considered. Though delay will not change the structure of equilibria, the global dynamics in the case without delay is simple compared with that in the case with delay. Theoretic analysis and numerical simulations show that the incorporation of delay leads to complex dynamics, including the appearance of oscillating solutions, periodic solutions from Hopf bifurcation, and homoclinic orbits, etc. The effect of the immunotherapy including anti-PD-1/PD-L1 inhibitor and tumor vaccine is also discussed. [ABSTRACT FROM AUTHOR]

A periodic boundary-value problem for an equation with deviating spatial argument is considered. This equation describes the phase of a light wave in light resonators with distributed feedback. Optical systems of this type are used in computer technologies and in the study of laser beams. The boundary-value problem was considered for two values of spatial deviations. In the work, bifurcation problems of codimensions 1 and 2 were analyzed by various methods of studying dynamical systems, for example, the method of normal Poincaré–Dulac forms, the method of integral manifolds, and asymptotic formulas. The problem on the stability of certain homogeneous equilibrium states is examined. Asymptotic formulas for spatially inhomogeneous solutions and conditions for their stability are obtained. [ABSTRACT FROM AUTHOR]

It is shown that the attractor of an autonomous Caputo fractional differential equation of order α ∈ (0 , 1) in R d whose vector field has a certain triangular structure and satisfies a smooth condition and dissipativity condition is essentially the same as that of the ordinary differential equation with the same vector field. As an application, we establish several one-parameter bifurcations for scalar fractional differential equations including the saddle-node and the pichfork bifurcations. The proof uses a result of Cong & Tuan [2] which shows that no two solutions of such a Caputo FDE can intersect in finite time. [ABSTRACT FROM AUTHOR]

Based on the study of the plankton population system, a diffusive toxic plankton model with Tissiet type functional response function and predation delay is proposed. Firstly, the sufficient conditions for locally asymptotic stability of the diffusion system without delay at the positive equilibrium are given, the existence conditions of Hopf bifurcation caused by diffusion are given, and the conditions under which diffusion makes spatially homogeneous and nonhomogeneous periodic solutions bifurcate from the positive constant equilibrium are given. Secondly, the time delay effect on the plankton reaction–diffusion system is studied, the existence of Hopf bifurcation at the positive equilibrium induced by delay is discussed. By applying the central manifold theory and normal form method of partial functional differential equations, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are studied. Finally, the reliability of theoretical research is verified by numerical simulation. [ABSTRACT FROM AUTHOR]

Inorganic arsenic causes carcinogenesis in a large part of the world. Its potential is elicited by the generation of ROS, which leads to damages to DNA, lipid and protein. Black tea, an antioxidant, can mitigate such deleterious effects by quenching ROS. We study Arsenic-toxicity and its amelioration by black tea in a colony of albino mice: a homology exists between the protein coding regions of mice and human. We observe that black tea has salutary effects on tumor-growth: it arrests damaged cell growth and produces early saturation of the damage. The experimental data obtained by us are modelled with dynamical equations. This is followed by a search for steady states and their stability analysis.

We study a fractional-order delayed predator-prey model with Holling–Tanner-type functional response. Mainly, by choosing the delay time $\tau $ as the bifurcation parameter, we show that Hopf bifurcation can occur as the delay time $\tau $ passes some critical values. The local stability of a positive equilibrium and the existence of the Hopf bifurcations are established, and numerical simulations for justifying the theoretical analysis are also presented. [ABSTRACT FROM AUTHOR]