Search

In this paper, we will build mathematical pelagic fish migration models and inspect simulations to discover the behavior for motion amongst a school. I focus on the Icelandic Capelin, and explore all possible migration patterns and their limitations as governed by mathematical equations and not biological observations. We develop a system of ordinary differential equations from a discrete system for the most general motion. First, we will perturb the system to develop a system of stochastic differential equations to study the unique behavior under naturally occurring external forces such as currents or reefs. Then, we will categorize the found transitory and long term behavior of these systems and compare them to the unperturbed solutions. Finally, we will prove that the system exhibits a subcritical pitchfork bifurcation.

Rotating machinery supported on journal bearings is affected by forces due to rotating unbalance and pressure gradients in the oil film. The interaction of these forces can evoke nonlinear behaviour, including asynchronous motion and even chaos. This work attempts to characterise the sub-synchronous motion of the rigid rotor supported on cylindrical journal bearings due to the abovementioned interaction. The analysis focuses on the rotor behaviour at the rotor speeds lower than the threshold speed for oil whirl, associated with sub-synchronous vibration of magnitude equaling the bearing clearance. It is shown that the sub-synchronous vibration can occur well before reaching the threshold speed and that the underlying period-doubling bifurcation depends on the amount of the rotating unbalance. The rotor response and stability are analysed using a numerical continuation method employing the infinitely short journal bearing model. Continuation results are further validated by time simulations which utilise the finite difference method to compute the hydrodynamic forces. The validation process employs bifurcation diagrams, Poincaré sections and numerical estimates of the largest Lyapunov exponents.

This paper introduces an algorithmic approach to the analysis of bifurcation of limit cycles from the centers of nonlinear continuous differential systems via the averaging method. We develop three algorithms to implement the averaging method. The first algorithm allows one to transform the considered differential systems to the normal form of averaging. Here, we restricted the unperturbed term of the normal form of averaging to be identically zero. The second algorithm is used to derive the computational formulae of the averaged functions at any order. The third algorithm is based on the first two algorithms and determines the exact expressions of the averaged functions for the considered differential systems. The proposed approach is implemented in Maple and its effectiveness is shown by several examples. Moreover, we report some incorrect results in published papers on the averaging method.

In this paper, the periodic solutions of nonlinear mechanical systems are studied starting from the nonlinear state-space model estimated using the nonlinear subspace identification (NSI) technique. In its standard form, NSI needs the input–output data from a nonlinear structure undergoing broadband excitation and requires the prior knowledge of the locations and kind of nonlinearities to be estimated. The method allows the estimation of the nonlinear features of the system and the indirect study of its periodic solutions using a single broadband excitation, without the need of feedback control loops. To this end, the nonlinear frequency response curves of the system are estimated merging the harmonic balance method with the NSI technique and using a continuation approach. Then, a monodromy-based stability analysis is developed in the nonlinear state-space framework to study the stability of the periodic solutions of the system and to track its bifurcations. The method is validated considering conservative nonlinearities on two numerical examples and one experimental application, the latter comprising a double-well oscillator with period-doubling phenomena. The effects of noise and nonlinear modeling errors are also evaluated.

Algebraic structures such as the lattices of attractors, repellers, and Morse representations provide a computable description of global dynamics. In this paper, a sheaf-theoretic approach to their continuation is developed. The algebraic structures are cast into a categorical framework to study their continuation systematically and simultaneously. Sheaves are built from this abstract formulation, which track the algebraic data as systems vary. Sheaf cohomology is computed for several classical bifurcations, demonstrating its ability to detect and classify bifurcations.

This work examines the impact of multiple vaccination strategies on the dynamics of COVID-19 in a population using the Atangana-Baleanu derivative. The existence and uniqueness of solution of the model is proven using Banach’s fixed point theorem. Local and global asymptotic stability of the equilibria of the model is also proven (under some conditions). Conditions for the existence of a unique or multiple equilibria are also derived and the model is shown to undergo backward bifurcation under certain scenarios. Using available data for the Pfizer, Moderna and Janssen vaccination programme for the city of Texas, United States of America from March 13, 2021 to June 29, 2021, the model is fitted using the three data sets. The three vaccination rates ν 1 , ν 2 and ν 3 corresponding to each vaccine as well as the effective contact rate for COVID-19 transmission, β , are estimated. Simulations of the model under different vaccination strategies are carried out. The results show that the three vaccination strategies not only cause significant reduction in the new asymptomatic and vaccinated symptomatic cases but also cause great decrease in the total number of vaccinated symptomatic individuals with severe COVID-19 illness.

The bifurcation structure of the oscillations of ultrasound contrast agents (UCAs) was studied as a function of the driving pressure for excitation frequencies that were determined using the UCAs pressure-dependent resonances $$(f_\mathrm{s})$$ . It was shown that when excited by the $$(f_\mathrm{s})$$ , the UCA can undergo a saddle-node bifurcation (SNB) to higher amplitude oscillations. The driving pressure at which the SNB occurs is controllable and depends on the $$(f_\mathrm{s})$$ magnitude. Utilizing the appropriate $$(f_\mathrm{s})$$ , the scattering cross section of the UCAs can significantly be enhanced (e.g., $$\sim $$ ninefold) while at the same time avoiding potential UCA destruction (by limiting the radial expansion ratio $$

In this paper, we have considered a tumour-immune model with the dosage effect of the targeted chemotherapeutic drug and its efficacy. The chemotherapeutic drug efficacy is characterized via way of means of its impact on the abnormal size of beginning tumours. In pharmacology, the effectiveness of a drug is a very important factor in developing a dosage for any disease. With these points of view regarding drug efficacy, we present a four compartmental mathematical model, viz, the population of tumour cells, T(t), CD8 T-killer cells, E-k(t), CD4 T-helper cells, E-h(t), and the amount of targeted chemotherapeutic drug, C(t), in the bloodstream, at any time t. A qualitative analysis of the proposed model, including the existence and boundedness of the model has been described here. The dynamics of the system are presented by examining the admissibility and stability of the system at trivial, dead, tumour-free, and co-existing equilibrium points with their respective biological significance. The conditions for the local stability of all equilibrium points are derived by using the Jacobian matrix and Routh-Hurwitz's criterion. Numerical calculations have been performed using the Multi-step Differential Transformation Method (Ms-DTM), and are presented and discussed briefly in terms of graphs. We have expressed that the model is suitable for the tumour removal with the passage of time, regardless of the initial population or tumour size. Also, the amount the dosage of the drug with an assigned numerical value for its efficacy is one of the important factors in the treatment. The efficacy term is included to see the effect of drug dosages in CD8 T-killer cells and CD4 T-helper cells compartment.

Comprehensive cancer therapy protocols are generally administered periodically (at discrete time points) and often determined by tumor size. In the literature, a class of mathematical models are proposed, including therapeutic protocols either with pulse therapy at fixed time points or with state-dependent therapy. However, it is clear in practice (in both clinical and experimental contexts) that the treatment process is much more complex combining pulse surveillance (pulse therapy) with a threshold policy (state-dependent implementation of therapy). Here, we propose a novel mathematical model describing tumor size-guided pulse therapies at fixed time points, which is more in line with the design of clinical protocols for cancer therapies. In studying the dynamics of the proposed model, we find a new kind of periodic solution for dynamical systems with pulse intervention, which we denote as a ( l + m ) T periodic solution. In more detail, the study of the existence and stability of semi-trivial periodic solutions uncovers very rich dynamics, including the existence of order-k effector-free periodic solutions. We obtain analytical expressions for these, and establish the existence of order-1 tumor-free periodic solutions. Furthermore, we discuss the existence and stability of positive periodic solutions. We initially establish the existence of stable order-1 periodic solutions through carefully considering the bifurcation near the effector cell-free periodic solution. Our numerical simulations confirm three possibilities: 1) that two different positive periodic solutions can be bi-stable, 2) that the positive equilibrium can be bi-stable with a positive periodic solution, or 3) that the solution is tri-stable with two different positive periodic solutions. Our main results show that no tumor-free periodic solution exists when the threshold policy is feasible, and this may be a reason why frequently a tumor can be only reduced to a certain low level and not eliminated completely. One of the key considerations in the threshold schedule is to determine an optimal threshold of tumor size to balance the therapy frequency (or the times of therapy) and the amplitude of the tumor size, furthermore the optimal therapy should also be individual-based (or personalized). We believe that our modelling approach can be widely applied into many other fields of life science, and provide critical guidance for the implementation of control strategies.

In this paper, we investigate the influence of the nonlocal intraspecific competition on the stability and spatialtemporal dynamics for a diffusive predator–prey system with generalist predator. The stability region and the possible bifurcation curves are determined in the plane consisting of diffusion coefficients of the prey and the predator. Spatially inhomogeneous steady states with different spatial profiles and spatially inhomogeneous periodic solutions with mode-1 as the parameters are varied acrossing the boundaries of the stability region.

The Thomas model is a system of two reaction–diffusion equations which was originally proposed in the context of enzyme kinetics. It was subsequently realised that it offers a plausible chemical mechanism for the generation of coat markings on mammals. To that end previous investigations have focused on establishing the conditions for the Turing instability and on following the associated patterns as the bifurcation parameter increases through the instability. In this paper we use modern ideas from the theory of dynamical systems to systematically investigate the formation of localised structures in this model. The Turing instability is found to exhibit a degeneracy at which it changes from being subcritical to supercritical (or vice versa). Associated with such degenerate points is a heteroclinic connection which is a crucial requirement for the generation of localised patterns. Localised solutions containing a single ‘spike’ are associated with the so-called Belyakov-Devaney transition. Localised solutions containing either multiple ‘peaks’ or multiple ‘holes’ are associated with homoclinic snaking bifurcations. These structures are investigated using continuation software. The snaking bifurcations are found to collapse when the system crosses the Belyakov-Devaney transition. The isolated spike solutions collapse at a codimension two heteroclinic connection to create the so-called collapsed snaking. We show that the temporal stability of the localised solutions depends upon the presence of Hopf bifurcations which in turn are controlled by the diffusion ratio. For small values of the ratio only solutions in the spike region are destabilised by a Hopf bifurcation. For larger values of the diffusion ratio Hopf bifurcations destabilise most of the localised patterns.

In the context of prey-predator interaction, Allee effect can have a significant impact and capture the complex dynamics in ecology. In this work, we modify the predator–prey model with component Allee effect for predator reproduction by incorporating the strong Allee effect in prey growth function. We explore the system dynamics in two aspects. Firstly, we study the system dynamics of the model without Allee effect through a comprehensive bifurcation structure and perform the sensitivity analysis of model parameters for fixed coexistence extensively; ii) we analyse the impact of Allee effect on the system dynamics. We determine the number of fixed coexistence points through graphical representation of non-trivial prey and predator nullclines. We study the stability analysis of the fixed coexistence point with the help of the graphical Jacobian method. Interestingly, we observe that initially, a low concentration of prey drives the system toward total extinction and the system will be settled to predator extinction for initially high prey concentration. This system behaviour supplements the existence of bi-stability involving trivial and predator extinction equilibria independent of parametric conditions. The inclusion of the Allee effect enhances the stability behaviour of the proposed model i.e. tetra stable equilibrium points are deduced. We demonstrate the system dynamics through co-dimension one and two bifurcations structure and also show possible phase portraits. Model with Allee effect generates all possible local and global bifurcations namely Hopf bifurcation, saddle–node bifurcation, B-T bifurcation, Bautin bifurcation and homoclinic bifurcation respectively. We observe that low predator reproduction growth rate provides oscillations with low prey densities and high predator reproduction growth rate results in oscillations with high prey densities. We investigate that the low impact of Allee always promotes the persistence of the coexistence. For a model with the Allee effect, we perform sensitivity analysis of model parameters for fixed coexistence points. We demonstrate results analytically and make them more comprehensive, we perform numerical simulation. Moreover, to show the vast applicability of our results, we compare it with the model without Allee effect.

The aim of this paper is to investigate the influence of the coupling of two scales on the dynamics of a piecewise smooth dynamical system. A relatively simple model with two switching boundaries is taken as an example by introducing a nonlinear piecewise resistor and a harmonically changed electric source into a typical Chua’s circuit. Taking suitable values of the parameters, four different types of bursting oscillations are observed corresponding to different values of the exciting amplitude. Regarding the periodic excitation as a slow-varying parameter, equilibrium branches of the fast subsystem as well as the related bifurcations, such as fold bifurcation, Hopf bifurcation, period doubling bifurcation, nonsmooth Hopf bifurcation and nonsmooth fold limit cycle bifurcation, are explored with theoretical and numerical methods. With the help of the overlap of the transformed phase portrait and the equilibrium branches, the mechanism of the bursting oscillations can be analyzed in detail. It is found that for relatively small exciting amplitude, since the trajectory is governed by a smooth subsystem, only conventional bifurcations take place, leading to the transitions between the spiking states and quiescent states. However, with an increase of the exciting amplitude so that the trajectory passes across the switching boundaries, nonsmooth bifurcations occurring at the boundaries may involve the structures of attractors, leading to complicated bursting oscillations. Further increasing the exciting amplitude, the number of the spiking states decreases although more bifurcations take place, which can be explained by the delay effect of bifurcation

Study of a food chain model under uncertainty is quite difficult. Because, in an uncertain food chain model, the biological parameters can’t be determined accurately. The aim of this work is to study the stability and local bifurcations (Saddle–node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation) of an imprecise prey–predator system in an uncertain environment. The proposed imprecise model is formulated by considering two more realistic factors: the effect of a fear factor on the growth rate of prey population and non-linear harvesting of predator population. To study the proposed imprecise system mathematically, the dynamical interactions between the imprecise species are presented by the system of governing interval differential equations. And to study the dynamics of the proposed imprecise system theoretically, it is modelled in a precise way by the linear parametric representation of the interval. Then all the theoretical analyses, including Saddle–node bifurcation, Hopf bifurcation and Bogdanov–Takens (BT) bifurcation of the interior equilibrium point of the proposed imprecise model are discussed in parametric form. To verify all the theoretical analyses of the proposed imprecise model, numerical simulations with interval-valued hypothetical data of the imprecise parameters are performed graphically. Finally, the work is concluded with some biological consequences.

The present paper studies a fractional-order SEIR epidemic model for the transmission dynamics of infectious diseases such as HIV and HBV that spreads in the host population. The total host population is considered bounded, and Holling type-II saturation incidence rate is involved as the infection term. Using the proposed SEIR epidemic model, the threshold quantity, namely basic reproduction number R0, is obtained that determines the status of the disease, whether it dies out or persists in the whole population. The model’s analysis shows that two equilibria exist, namely, disease-free equilibrium (DFE) and endemic equilibrium (EE). The global stability of the equilibria is determined using a Lyapunov functional approach. The disease status can be verified based on obtained threshold quantity R0. If R0 < 1, then DFE is globally stable, leading to eradicating the population’s disease. If R0 > 1, a unique EE exists, and that is globally stable under certain conditions in the feasible region. The Caputo type fractional derivative is taken as the fractional operator. The bifurcation and sensitivity analyses are also performed for the proposed model that determines the relative importance of the parameters into disease transmission. The numerical solution of the model is obtained by the generalized Adams- Bashforth-Moulton method. Finally, numerical simulations are performed to illustrate and verify the analytical results.

In this paper, to characterize the limited availability of medical resources, we incorporate a saturated treatment rate into a network-based susceptible-infected-susceptible (SIS) epidemic model with time delay and nonlinear incidence rate. Analytical study shows the boundedness of solutions, the basic reproduction number R 0 and equilibrium points of the proposed system. For any infection delay, we perform both local and global stability analyses for the disease-free equilibrium point by analyzing the characteristic equation and using Lyapunov functional. Furthermore, this system exhibits bifurcation behavior at R 0 = 1 due to the introduction of saturated treatment. More precisely, a backward bifurcation takes place from the disease-free equilibrium point when the saturation constant β is sufficiently large. Under the given conditions, the unique disease-spreading equilibrium point is also proved to be locally asymptotically stable. In addition, we analyze an optimal control problem with consideration of two time-dependent control measures. Several numerical simulations are presented to validate the obtained theoretical results.

We study the diffusive generalized logistic problem with constant yield harvesting: { u ″ ( x ) + λ g ( u ) − μ = 0 , − 1 x 1 , u ( − 1 ) = u ( 1 ) = 0 , where λ , μ > 0 . We assume that g satisfies g ( 0 ) = g ( 1 ) = 0 , g ( u ) > 0 on ( 0 , 1 ) , and g is either convex-concave or convex-concave-convex on ( 0 , 1 ) and satisfies certain conditions. We prove that, for any fixed μ > 0 , on the ( λ , ‖ u ‖ ∞ ) -plane, the bifurcation diagram always consists of a continuous, ⊂-shaped curve, and we study the structures and evolution of bifurcation diagrams for varying μ > 0 . We also prove that, for any fixed λ > λ 0 ⁎ for some λ 0 ⁎ > 0 , on the ( μ , ‖ u ‖ ∞ ) -plane, the bifurcation diagram consists of a reversed ⊂-shaped curve with possibly the disjoint union of a strictly increasing curve, and we study the structures and evolution of bifurcation diagrams for varying λ > λ 0 ⁎ . Our results complement those of Hung, Suen, and Wang (2020) [10] in which problem g is either concave or concave-convex on ( 0 , 1 ) .

In the present work, nonlinear dynamics of a simply-supported fluid-conveying composite pipe subjected to axial tension in sub- and super-critical regimes are investigated. The tension acting on the pipe is assumed to comprise an average component and a harmonically changing disturbance component. More specifically, the nonlinear equation of motion is derived based on the Euler-Bernoulli beam theory considering the von Karman nonlinearity for large deformation assumption, which is then truncated via the Galerkin's approach. In addition, the Kelvin-Voigt model is adopted on the pipe materials with visco-elastic structural damping. Afterwards, parametric studies are performed to evaluate the effects of the axial tension and fiber orientation angle on pre- and post-buckling frequencies, critical fluid velocities and static buckling displacements. The global dynamic behaviors of the pipe are studied through the forms of bifurcation diagrams, phase portraits, Poincare maps, power spectral density (PSD) diagrams and linear instability maps, in both sub- and super-critical regimes. The results show that, based on symmetric layers, pipes with smaller fiber orientation angles have larger critical divergence fluid velocities but smaller static buckling displacements. Some interesting dynamical phenomena, including periodic, multi-periodic, quasi-periodic and chaotic motions, can be observed with different fiber orientation angles. Moreover, the system shows ‘soft spring’ characteristic in super-critical regime, compared with the ‘hard spring’ characteristic in sub-critical case.

In this paper, a discrete-time predator-prey model involving prey refuge proportional to predator density is studied. It is assumed that the rate at which prey moves to the refuge is proportional to the predator density. The fixed points, their local stability, and the existence of Neimark-Sacker bifurcation are investigated. At last, the numerical simulations consisting of bifurcation diagrams, phase portraits, and time-series are given to support analytical findings. The occurrence of chaotic solutions are also presented by showing the Lyapunov exponent while some parameters are varied.

This article considers a generalized Logistic reaction–diffusion population model with mixed instantaneous and delayed feedback control and subject to the homogeneous Dirichlet boundary condition on the one-dimensional bounded spatial domain ( 0 , π ) . Existence and asymptotic expression of the small enough positive steady state bifurcating from the zero solution are given in virtue of the implicit function theorem. By choosing the delay τ as the bifurcation parameter and analyzing in detail the associated eigenvalue problem, the local asymptotic stability and the existence of Hopf bifurcation of the small enough positive steady state are obtained. Particularly, the normal form for Hopf bifurcations is derived according to the normal form method and the center manifold theory for partial functional differential equations (PFDEs). It is demonstrated that when the delayed feedback is dominant, the model can undergo a forward Hopf bifurcation at the positive steady state as the delay crosses through monotonically increasing a sequence of critical values and all the bifurcating periodic solutions are locally orbitally asymptotically stable on the center manifold. Specifically, the bifurcating periodic solutions from the first bifurcation value are orbitally asymptotically stable in the entire phase space. The reasonability of the main analytical conclusions is also verified by carrying out some numerical simulations.

In this paper, we study a wide spectrum of dynamical features of a prey–predator model with infection in prey population only. In the model formulation, we consider nonlinear incidence for infection transmission and Holling type II as response function of predator. We analyze the model for local stability and Hopf bifurcation, and the theoretical results are validated and extrapolated by numerical simulations. Also, we perform sensitivity analysis to explore the significance of the crucial parameters influencing the densities of prey and predator populations. Our numerical results show that disease persists in the system even if the basic reproduction number is below unity. The rates of predation on susceptible and infected prey populations are shown to dramatically change the behavior of eco-epidemiological system. We extend our model by considering the fact that the newly infected prey becomes productively infectious after the effective contact between susceptible and infectious preys, and some time lag is involved in this biological process; also the conversion process conceals some time. We analyze our eco-epidemiological model with two time delays (incubation delay and predator’s gestation delay) for stability and bifurcation. From numerical simulations, we observe that both the delay parameters substantially affect the stable configuration of the system.

In this paper, a Gause type predator–prey system with constant-yield prey harvesting and monotone ascending functional response is proposed and investigated. We focus on the influence of the harvesting rate on the predator–prey system. First, equilibria corresponding to different situations are investigated, as well as the stability analysis. Then bifurcations are explored at nonhyperbolic equilibria, and we give the conditions for the occurrence of two saddle–node bifurcations by analyzing the emergence, coincidence and annihilation of equilibria. We calculate the Lyapunov number and focal values to determine the stability and the quantity of limit cycles generated by supercritical, subcritical and degenerate Hopf bifurcations. Furthermore, the system is unfolded to explore the repelling and attracting Bogdanov–Takens bifurcations by perturbing two bifurcation parameters near the cusp. It is shown that there exists one limit cycle, or one homoclinic loop, or two limit cycles for different parameter values. Therefore, the system is susceptible to both the constant-yield prey harvesting and initial values of the species. Finally, we run numerical simulations to verify the theoretical analysis.

In this work, a new perturbation approach is developed based on Bogdanov-Takens bifurcation theory, which enables the Poincare-Lyapunov method for switching systems with linear type centers to be applied for studying the center conditions of planar switching polynomial systems associated with a nilpotent equilibrium point. The new method is then applied to consider a class of quadratic switching nilpotent systems, and a complete classification is given on the conditions of the nilpotent equilibrium point to be a center. Moreover, based on one of the center conditions, an example is constructed to show the existence of seven small-amplitude limit cycles around the nilpotent equilibrium point, which is a new lower bound on the number of limit cycles in such systems.

The normal forms of the vector fields with local bifurcations at the equilibrium points can be employed to describe the topological structure in the neighborhood of the critical points. Furthermore, the relationship between the coefficients of the normal form and the original system is very important to understand the behaviors of the practical dynamics. Though many results related to low co-dimensional local bifurcations are presented, the normal forms as well as the computation with high co-dimensional local bifurcations still remain an open problem to be investigated. The main purpose of this paper is to derive the normal form of a vector field with codimension-3 zero-zero-Hopf bifurcation at the origin and develop an uniform program to compute the coefficients of the normal form from a general system. By employing the central manifold theory and the normal form theory, all the expressions of the coefficients of the nonlinear transformations and the normal form up to any desired order related to the local bifurcation are presented, which can be computed via a software program based on the symbolic language Maple, attached in the appendix. Perturbation of the vector field at the bifurcation point can also be derived accordingly, which can be used to explore the topological property of the bifurcation.

In this paper, we study a near-Hamiltonian system on the cylinder. First, we establish some general methods on the existence of limit cycles bifurcating from closed orbits of type II by the Melnikov function method, then we derive the expansions of the first order Melnikov function and consider the bifurcation problem of limit cycles near a double homoclinic loop. As an application, we discuss the number of limit cycles of a class of cylinder pendulum-like systems.

We consider codimension-two focus-fold bifurcation for general piecewise smooth (PWS) planar systems. Some changes of variables and parameters are employed to derive a normal form, which is topologically equivalent to the original system. Making use of the normal form, we verify that observations of focus-fold bifurcation in some special piecewise linear systems also hold for general PWS systems, including the existence of two types of sliding limit cycles and coexistence of crossing and sliding limit cycles. It is shown that two nested crossing limit cycles can be bifurcated from the focus-fold singularity. This result is the improvement of the work [Han and Zhang (2010) [29] ], in which only one bifurcated limit cycle is obtained. In addition, the sliding limit cycles are proved to be convergent in finite time.

In the paper, to better understand the dynamic relationship between Microcystis aeruginosa and filter-feeding fish, we have constructed a new aquatic ecological model to describe the aggregation effect of Microcystis aeruginosa. Mathematical theory works mainly investigated the critical threshold condition through the discovery of transcritical bifurcation, saddle–node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation. The numerical simulation works mainly revealed that Microcystis aeruginosa aggregation effect has played an important role in the dynamic relationship with the aid of bifurcation analysis, which also in turn proved the validity of theoretical derivation. The results showed that if the Microcystis aeruginosa aggregation area became larger and larger, and exceeded a critical threshold, the filter-feeding fish would eventually go extinct. Furthermore, it should be stressed that Microcystis aeruginosa aggregation can effectively control the feeding dynamic behavior of filter-feeding fish and provide shelter from predators. Finally, all these results were expected to be useful in studying population dynamics of some aquatic ecosystems.

The spread of worm virus has brought great loss to our production and life. In this paper, a new Vulnerable-Exposed-Infectious-Quarantined-Secured (VEIQS) worm propagation model with a saturated incidence and two delays is proposed. The local stability of the worm-existence equilibrium and the occurrence of Hopf bifurcation at the critical values of the two delays are obtained by regarding different combinations of time delays as bifurcation parameters. It shows that the model is ideal stable when the time delay is below the critical value and a Hopf bifurcation occurs when the time delay is above the critical value. In particular, direction and stability of the Hopf bifurcation are determined by using the center manifold theorem. Finally, some numerical simulations are presented in order to verify the analytical results.

In this paper, we study the global bifurcation of infinity of a class of nonlinear eigenvalue problems for fourth-order ordinary differential equations with nondifferentiable nonlinearity. We prove the existence of two families of unbounded continuance of solutions bifurcating at infinity and corresponding to the usual nodal properties near bifurcation intervals.

In this paper, we study the Turing–Hopf bifurcation in the predator–prey model with cross-diffusion considering the individual behaviour and herd behaviour transition of prey population subject to homogeneous Neumann boundary condition. Firstly, we study the non-negativity and boundedness of solutions corresponding to the temporal model, spatiotemporal model and the existence and priori boundedness of solutions corresponding to the spatiotemporal model without cross-diffusion. Then by analysing the eigenvalues of characteristic equation associated with the linearized system at the positive constant equilibrium point, we investigate the stability and instability of the corresponding spatiotemporal model. Moreover, by calculating and analysing the normal form on the centre manifold associated with the Turing–Hopf bifurcation, we investigate the dynamical classification near the Turing–Hopf bifurcation point in detail. At last, some numerical simulations results are given to support our analytic results.

In this paper, we study the global dynamics of a nonsmooth Rayleigh–Duffing equation $$\ddot{x}+a{\dot{x}}+b{\dot{x}}|{\dot{x}}|+cx+{\mathrm {d}}x^3=0$$ for the case $$d>0$$ , i.e., the focus case. The global dynamics of this nonsmooth Rayleigh–Duffing oscillator for the case $$d

A minimal filling of a finite metric space is a weighted graph of a minimal possible weight spanning this space so that the weight of any path in it is not less than the distance between its ends. Bifurcation diagrams of types and the weight of minimal fillings for four points of the Euclidean plane are built in the present work.

This paper presents an alternative approach to stochastic analysis of nonlinear dynamic systems. It exploits this approach to analyze the response of electrostatic MEMS bifurcation sensors to a combination of deterministic excitation, mechanical-thermal noise, and electrical-thermal noise. The analytical approach combines the methods of multiple scales and stochastic averaging of the amplitude, to derive the stochastic Ito differential equations describing the modulations of the sensor amplitude and phase difference in the presence of thermal noise and the Fokker–Planck–Kolmogorov (FPK) equation governing the stationary probability density function (PDF) of the stochastic response. Good agreement is found between the predictions of the derived modulation equations and the original equation of motion. The scope of the FPK equation applicability to the noise excitation levels is examined. The impact of the additive noise, arising from mechanical-thermal and electrical-thermal noise, on the sensor response is found to dominate that of the multiplicative noise, arising from the electrical-thermal noise. PDFs of the response are used to investigate the stochastic switching between the co-existing orbits of the bifurcation sensor under the interaction between the excitation frequency and noise intensity. We found that the stochastic switching is activated when the margins of stability of both orbits become comparable to the size of noise-driven motions. Variations in the mean and variance of the amplitude within the hysteretic region can be exploited as sensitive indicators of the stochastic switching. Finally, our results suggest the possibility of implementing a novel highly sensitivity ‘noise-aware’ bifurcation sensor that exploits the quantitative change in the mean amplitude (or RMS) of the sensor states within the frequency range of stochastic switching to detect mass change or gas concentration.

In order to analyze the effect of vaccination in a population with the presence of viruses, a variation of the SIR (Susceptible-Infected-Removed) model is proposed taking into account social distancing and the effect of the vaccine. The equilibrium points of the proposed model are calculated and the stability analysis of the system is carried out. For the proposed model, disease-free equilibrium point and endemic equilibrium point are found and the conditions of existence are discussed. For the disease-free equilibrium point the bifurcation conditions are derived and simulations show that reducing the vaccination effort can lead the disease-free equilibrium to the endemic equilibrium. From the theoretical analysis, a minimum value of effort is obtained to guarantee a disease-free equilibrium point. Simulations were carried out from the value obtained from Rv to validate the theoretical results.

In this paper, an SWEIQR epidemic model with transport-related infection is proposed. The model considers inter-patch travel with entry-departure screening. The reproduction number, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_{ed}^\phi $$\end{document}Redϕ, is computed and analyzed with respect to awareness and screening parameters. The analytic computations show that the disease-free equilibrium in the absence of travel is globally asymptotically stable when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_\omega \le 1$$\end{document}Rω≤1 and unstable otherwise. The trans-critical bifurcation occurs at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_\omega =1$$\end{document}Rω=1 and the locally stable endemic equilibrium point appears if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_\omega >1$$\end{document}Rω>1 near to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_\omega =1$$\end{document}Rω=1. The numerical simulations are performed to verify the analytical computation and explore the dynamic behavior with respect to different model parameters. The result shows that disseminating awareness through the population reduces the spread of disease. Furthermore, the full model results show that the departure screening may reduce the spread of disease in each patch.

Using a novel formulation based on flow force, we prove the existence of a global continuum of periodic traveling wave solutions to the irrotational water wave problem in finite depth where the flow force is held fixed, instead of fixing the mass flux. We further establish the limiting behaviour of solutions in this continuum where the flow force remains fixed throughout the continuum of the solutions. The global solutions, with a fixed flow force, show the possibility that a weak stagnation point can be approached at the surface, which is a characteristic of large amplitude irrotational water waves. The formulation is suitable for applications to numerical studies on water waves.

In this work new global dynamic bifurcation theorems are established for local semiflows on complete metric spaces. These theorems are applied to the nonlinear evolution equation u t + A u = f λ ( u ) in a Banach space X, where A is a sectorial operator with compact resolvent, and f λ ( 0 ) = 0 for all λ ∈ R . In particular, if f λ takes the form f λ ( u ) = λ u + f ( u ) , it is shown that the global dynamic bifurcation branch of a bifurcation point ( 0 , λ 0 ) is necessarily unbounded without assuming the “crossing odd-multiplicity” condition. It has long been recognized that dynamical methods can be helpful in solving static problems. As an example, the bifurcation of elliptic equations on bounded domains associated with homogenous Dirichlet boundary condition is addressed. By considering the corresponding nonclassical parabolic flows and applying the global dynamic bifurcation theorems given here for local semiflows, some new results with global features are derived.

application/pdf, Nonlinear Dynamics. 2021, 106 (3), P.2363-2376

In the present work the spontaneous dynamics of a ring of N Chua's oscillators, mutually coupled through a resistor R in anearest-neighbor configuration, is investigated numerically for different strengths of the coupling. A transition from periodic to chaoticglobal dynamics is observed when the coupling decreases below a critical value and complex patterns in the spatiotemporal dynamics of thering emerge for a small coupling interval after the transition to chaos. The recovered behavior, as well asthe value of the criticalthreshold, appears to be independent of the size of the ring. We also propose aninterpretation of this property, which relates the regularsynchronized dynamics of the ring to the dynamics of the isolated oscillator. Finally,for the ring of the coupled oscillator, a theoretical wavedispersion relation is calculated and successfully compared with the results of thenumerical simulations, analyzed by classicaltechniques adopted for turbulent flows.

Malaria, one of the longest-known vector-borne diseases, poses a major health problem in tropical and subtropical regions of the world. Its complexity is currently being exacerbated by the emerging COVID-19 pandemic and the threats of its second wave and looming third wave. We formulate and analyze a mathematical model incorporating some epidemiological features of the co-dynamics of both malaria and COVID-19. Sufficient conditions for the stability of the malaria only and COVID-19 only sub-models’ equilibria are derived. The COVID-19 only sub-model has globally asymptotically stable equilibria while under certain condition, the malaria-only could undergo the phenomenon of backward bifurcation whenever the sub-model reproduction number is less than unity. The equilibria of the dual malaria-COVID19 model are locally asymptotically stable as global stability is precluded owing to the possible occurrence of backward bifurcation. Optimal control of the full model to mitigate the spread of both diseases and their co-infection are derived. Pontryagin’s Maximum Principle is applied to establish the existence of the optimal control problem and to derive the necessary conditions for optimal control of the diseases. Though this is not a case study, simulation results to support theoretical analysis of the optimal control suggests that concurrently applying malaria and COVID-19 protective measures could help mitigate their spread compared to applying each preventive control measure singly as the world continues to deal with this unprecedented and unparalleled COVID-19 pandemic.

The structure of solution branches of the equation e 2 u x x = u 3 − λ u + δ g ( x ) , for x ∈ R is studied, for real constants, e, λ and δ, where g ( x ) is periodic in x and u satisfies Neumann boundary conditions. For α : = u ( 0 ) bifurcation structure in the α–δ plane occurs that for small e leads to families of singular asymptotic solutions with interior transition layers between outer solutions. In contrast, bifurcation in the α–λ plane produces families of oscillatory solutions that for small e have large numbers, n, of oscillations between envelope outer solutions. The n label of the solution branch is defined by its local bifurcation at e 2 n 2 = λ > 0 . Further the bifurcation structure on the n branch is classified by the integer n being either even or odd. When n is even the pitchfork bifurcation in the α–δ plane leads to connected branches of solutions in the α–λ–δ space. But when n is odd the presence of g ( x ) leads to cusp catastrophes (for small e) in the α–δ plane and these give rise to disconnected (or isolated) branches of solution in the α–λ plane that appear to be torn from the main solution branch. The integer n acts as a count of the number of oscillations on these solution branches. As e 2 λ − 1 → 0 we see layer behaviour occurring about the top and bottom outer solutions and as e 2 λ − 1 → ∞ we find convergence to the middle outer solution.

We investigate dynamics and bifurcations in a mathematical model that captures electrochemical experiments on arrays of microelectrodes. In isolation, each individual microelectrode is described by a one-dimensional unit with a bistable current-potential response. When an array of such electrodes is coupled by controlling the total electric current, the common electric potential of all electrodes oscillates in some interval of the current. These coupling-induced collective oscillations of bistable one-dimensional units are captured by the model. Moreover, any equilibrium is contained in a cluster subspace, where the electrodes take at most three distinct states. We systematically analyze the dynamics and bifurcations of the model equations: We consider the dynamics on cluster subspaces of successively increasing dimension and analyze the bifurcations occurring therein. Most importantly, the system exhibits an equivariant transcritical bifurcation of limit cycles. From this bifurcation, several limit cycles branch, one of which is stable for arbitrarily many bistable units.

This research is chiefly concerned with the stability and Hopf bifurcation for newly established fractional-order neural networks involving different types of delays. By means of an appropriate variable substitution, equivalent fractional-order neural network systems involving one delay are built. By discussing the distribution of roots of the characteristic equation of the established fractional-order neural network systems and selecting the delay as bifurcation parameter, a novel delay-independent bifurcation condition is derived. The investigation verifies that the delay is a significant parameter which has an important influence on stability nature and Hopf bifurcation behavior of neural network systems. The computer simulation plots and bifurcation graphs effectively illustrate the reasonableness of the theoretical fruits.

In this work, a sufficiently simple model allows one to perform detailed analytic studies to gain insights for the dynamical mechanisms in the general model of gene expression. Through theoretical analysis and numerical simulation, we find that the transcription and translation delays as a bifurcation parameter can drive the system to undergo a Hopf bifurcation, thereby producing oscillation behavior. Meanwhile, the length of these delays can determine the amplitude and period of the oscillations. Moreover, an optimal rate of model parameters is also essential to induce the limit-cycle oscillations. In particular, the transcription factor concentration is called a signal that not only induces twice Hopf bifurcation, but also affects the role of the delay in the system, that is, the varying time delay leads to subcritical Hopf bifurcation and bistability under the small signal strength. On the contrary, supercritical Hopf bifurcation is generated. Furthermore, two-parameter diagram is drawn to further explore how time delay and signal strength coordinate to regulate the oscillation behavior of the system. In addition, the stability and the direction of periodic solutions bifurcating from Hopf bifurcations are determined through the explicit formulae and the global existence of periodic solutions is discussed to verify the authenticity of the simulation. These results may be helpful for us to fully grasp oscillation dynamics and provide clues for further exploring the molecular mechanism of biological rhythms.

In this article, it is formulated a predator-prey model of two predators consuming a single limited prey resource. On the other hand, two predators have to compete with each other for survival. The predation function for two predators is assumed to be different where one predator uses Holling type I while the other uses Holling type II. It is also assumed that the fear effect is considered in this model as indirect influence evoked by both predators. Non-negativity and boundedness is written to show the biological justification of the model. Here, it is found that the model has five equilibrium points existed under certain condition. We also perform the local stability analysis on the equilibrium points with three equilibrium points are stable under certain conditions and two equilibrium points are unstable. Hopf bifurcation is obtained by choosing the consumption rate of the second predator as the bifurcation parameter. In the last part, several numerical solutions are given to support the analysis results.

Using a nonlocal version of the center manifold theorem and a normal form reduction, we prove the existence of small-amplitude generalized solitary-wave solutions and modulated solitary-wave solutions to the steady gravity-capillary Whitham equation with weak surface tension. Through the application of the center manifold theorem, the nonlocal equation for the solitary wave profiles is reduced to a four-dimensional system of ODEs inheriting reversibility. Along particular parameter curves, relating directly to the classical gravity-capillary water wave problem, the associated linear operator is seen to undergo either a reversible $0^{2+}(i k_0)$ bifurcation or a reversible $(i s)^2$ bifurcation. Through a normal form transformation, the reduced system of ODEs along each relevant parameter curve is seen to be well approximated by a truncated system retaining only second-order or third-order terms. These truncated systems relate directly to systems obtained in the study of the full gravity-capillary water wave equation and, as such, the existence of generalized and modulated solitary waves for the truncated systems is guaranteed by classical works, and they are readily seen to persist as solutions of the gravity-capillary Whitham equation due to reversibility. Consequently, this work illuminates further connections between the gravity-capillary Whitham equation and the full two-dimensional gravity-capillary water wave problem., 35 pages, 4 figures. Some mathematical typos fixed