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Connectivity is the safety network for biodiversity conservation because connected habitats are more effective for saving the species and ecological functions. The nature of coupling for connectivity also plays an important role in the co-existence of species in cyclic-dominance. The rock-paper-scissors game is one of the paradigmatic mathematical model in evolutionary game theory to understand the mechanism of biodiversity in cyclic-dominance. In this paper, the metapopulation model for rock-paper-scissors with mutations is presented in which the total population is divided into patches and the patches form a network of complete graph. The migration among patches is allowed through simple random walk. The replicator-mutator equations are used with the migration term. When migration is allowed then the population of the patches will synchronized and attain stable state through Hopf bifurcation. Apart form this, two phases are observed when the strategies of one of the species mutate to other two species: co-existence of all the species phase and existence of one kind of species phase. The transition from one phase to another phase is taking place due to transcritical bifurcation. The dynamics of the population of species of rock, paper, scissors is studied in the environment of homogeneous and heterogeneous mutation. Numerical simulations have been performed when mutation is allowed in all the patches (homogeneous mutation) and some of the patches (heterogeneous mutation). It has been observed that when the number of patches is increased in the case of heterogeneous mutation then the population of any of the species will not extinct and all the species will co-exist.

One of the simplest examples of a robust heteroclinic cycle involves three saddle equilibria: each one is unstable to the next in turn, and connections from one to the next occur within invariant subspaces. Such a situation can be described by a third-order ordinary differential equation (ODE), and typical trajectories approach each equilibrium point in turn, spending progressively longer to cycle around the three points but never stopping. This cycle has been invoked as a model of cyclic competition between populations adopting three strategies, characterised as Rock, Paper and Scissors. When spatial distribution and mobility of the populations is taken into account, waves of Rock can invade regions of Scissors, only to be invaded by Paper in turn. The dynamics is described by a set of partial differential equations (PDEs) that has travelling wave (in one dimension) and spiral (in two dimensions) solutions. In this paper, we explore how the robust heteroclinic cycle in the ODE manifests itself in the PDEs. Taking the wavespeed as a parameter, and moving into a travelling frame, the PDEs reduce to a sixth-order set of ODEs, in which travelling waves are created in a Hopf bifurcation and are destroyed in three different heteroclinic bifurcations, depending on parameters, as the travelling wave approaches the heteroclinic cycle. We explore the three different heteroclinic bifurcations, none of which have been observed in the context of robust heteroclinic cycles previously. These results are an important step towards a full understanding of the spiral patterns found in two dimensions, with possible application to travelling waves and spirals in other population dynamics models., Comment: 36 pages, 8 figures

Two limit cycles for the continuous model of the rockscissorspaper (RSP) game is constructed.The Hopf bifurcation method is used for the construction of limit cycles.The results give a partial answer to an open question posed by Neumann and Schuster. In this paper we construct two limit cycles with a heteroclinic polycycle for the three-dimensional continuous model of the rockscissorspaper (RSP) game between bacteriocin producing bacteria. Our construction gives a partial answer to an open question posed by Neumann and Schuster (2007) concerning how many limit cycles can coexist for the RSP game.

Classic frameworks of rock-paper-scissors game have been assumed in a closed community that a density of each group is only affected by internal factors such as competition interplay among groups and reproduction itself. In real systems in ecological and social sciences, however, the survival and a change of a density of a group can be also affected by various external factors. One of common features in real population systems in ecological and social sciences is population flow that is characterized by population inflow and outflow in a group or a society, which has been usually overlooked in previous works on models of rock-paper-scissors game. In this paper, we suggest the rock-paper-scissors system by implementing population flow and investigate its effect on biodiversity. For two scenarios of either balanced or imbalanced population flow, we found that the population flow can strongly affect group diversity by exhibiting rich phenomena. In particular, while the balanced flow can only lead the persistent coexistence of all groups which accompanies a phase transition through supercritical Hopf bifurcation on different carrying simplices, the imbalanced flow strongly facilitates rich dynamics such as alternative stable survival states by exhibiting various group survival states and multistability of sole group survivals by showing not fully covered but spirally entangled basins of initial densities due to local stabilities of associated fixed points. In addition, we found that, the system can exhibit oscillatory dynamics for coexistence by relativistic interplay of population flows which can capture the robustness of the coexistence state. Applying population flow in the rock-paper-scissors system can ultimately change a community paradigm from closed to open one, and our foundation can eventually reveal that population flow can be also a significant factor on a group density which is independent to fundamental interactions among groups.

This paper proposes a novel fractional-order delayed Ross-Macdonald model for malaria transmission. This paper aims to systematically investigate the effect of both the incubation periods of Plasmodium and the order on the dynamic behavior of diseases. Utilizing inequality techniques, contraction mapping theory, fractional linear stability theorem, and bifurcation theory, several sufficient conditions for the existence and uniqueness of solutions, the local stability of the positive equilibrium point, and the existence of fractional-order Hopf bifurcation are obtained under different time delays cases. The results show that time delay can change the stability of system. System becomes unstable and generates a Hopf bifurcation when the delay increases to a certain value. Besides, the value of order influences the stability interval size. Thus, incubation periods and the order have a major effect on the dynamic behavior of the model. The effectiveness of the theoretical results is shown through numerical simulations.

We investigate the dynamics of three-strategy (rock–paper–scissors) replicator equations in which the fitness of each strategy is a function of the population frequencies delayed by a time interval $$T$$ . Taking $$T$$ as a bifurcation parameter, we demonstrate the existence of (non-degenerate) Hopf bifurcations in these systems and present an analysis of the resulting limit cycles using Lindstedt’s method.

In the theoretical research of chaotic dynamical system, the different type of bifurcations is a very interesting powerful tool for analyzing the qualitative behavior of chaotic dynamical system; this short paper is devoted to analysis of a simple 2-D symmetry discrete chaotic map with quadratic and cubic nonlinearities. The dynamical behaviors of the map are investigated by mathematical analysis and simulated numerically using package of Matlab . We compute numerically the bifurcation diagram and largest Lyapunov exponent and phase portraits. The research results indicate that there are interesting nonlinear physical phenomena in this simple 2-D symmetry discrete cubic map, such as symmetry bifurcation, Hopf bifurcation, symmetry breaking bifurcation and identical symmetric attractors. The important nonlinear physical phenomena obtained in this paper would benefit the study of the cubic chaotic map and the development of the theory of chaotic discrete dynamical systems. En la investigación teórica de los sistemas dinámicos caóticos, los diferentes tipos de bifurcaciones son una herramienta poderosa muy interesante para analizar el comportamiento cualitativo de los sistemas dinámicos caóticos; este breve artículo está dedicado al análisis de un mapa caótico discreto de simetría bidimensional simple con no linealidades cuadráticas y cúbicas. Los comportamientos dinámicos del mapa se investigan mediante análisis matemático y se simulan numéricamente utilizando el paquete de Matlab . Calculamos numéricamente el diagrama de bifurcación y el mayor exponente de Lyapunov y los retratos de fase. Los resultados de la investigación indican que existen interesantes fenómenos físicos no lineales en este sencillo mapa cúbico discreto de simetría 2-D, como la bifurcación de simetría, la bifurcación de Hopf, la bifurcación de ruptura de simetría y los atractores simétricos idénticos. Los importantes fenómenos físicos no lineales obtenidos en este trabajo beneficiarían el estudio del mapa cúbico caótico y el desarrollo de la teoría de los sistemas dinámicos discretos caóticos.

The role of mutation, which is an error process in gene evolution, in systems of cyclically competing species has been studied from various perspectives, and it is regarded as one of the key factors for promoting coexistence of all species. In addition to naturally occurring mutations, many experiments in genetic engineering have involved targeted mutation techniques such as recombination between DNA and somatic cell sequences and have studied genetic modifications through loss or augmentation of cell functions. In this paper, we investigate nonlinear dynamics with targeted mutation in cyclically competing species. In different ways to classic approaches of mutation in cyclic games, we assume that mutation may occur in targeted individuals who have been removed from intraspecific competition. By investigating each scenario depending on the number of objects for targeted mutation analytically and numerically, we found that targeted mutation can lead to persistent coexistence of all species. In addition, under the specific condition of targeted mutation, we found that targeted mutation can lead to emergences of bistable states for species survival. Through the linear stability analysis of rate equations, we found that those phenomena are accompanied by Hopf bifurcation which is supercritical. Our findings may provide more global perspectives on understanding underlying mechanisms to control biodiversity in ecological/biological sciences, and evidences with mathematical foundations to resolve social dilemmas such as a turnover of group members by resigning with intragroup conflicts in social sciences.

During the outbreak of emerging infectious diseases, media coverage and medical resource play important roles in affecting the disease transmission. To investigate the effects of the saturation of media coverage and limited medical resources, we proposed a mathematical model with extra compartment of media coverage and two nonlinear functions. We theoretically and numerically investigate the dynamics of the proposed model. Given great difficulties caused by high nonlinearity in theoretical analysis, we separately considered subsystems with only nonlinear recovery or with only saturated media impact. For the model with only nonlinear recovery, we theoretically showed that backward bifurcation can occur and multiple equilibria may coexist under certain conditions in this case. Numerical simulations reveal the rich dynamic behaviors, including forward-backward bifurcation, Hopf bifurcation, saddle-node bifurcation, homoclinic bifurcation and unstable limit cycle. So the limitation of medical resources induces rich dynamics and causes much difficulties in eliminating the infectious diseases. We then investigated the dynamics of the system with only saturated media impact and concluded that saturated media impact hardly induces the complicated dynamics. Further, we parameterized the proposed model on the basis of the COVID-19 case data in mainland China and data related to news items, and estimated the basic reproduction number to be 2.86. Sensitivity analyses were carried out to quantify the relative importance of parameters in determining the cumulative number of infected individuals at the end of the first month of the outbreak. Combining with numerical analyses, we suggested that providing adequate medical resources and improving media response to infection or individuals' response to mass media may reduce the cumulative number of the infected individuals, which mitigates the transmission dynamics during the early stage of the COVID-19 pandemic.

The ecological theory of species interactions rests largely on the competition, interference, and predator–prey models. In this paper, we propose and investigate a three-species predator–prey model to inspect the mutual interference between predators. We analyze boundedness and Kolmogorov conditions for the non-spatial model. The dynamical behavior of the system is analyzed by stability and Hopf bifurcation analysis. The Turing instability criteria for the Spatio-temporal system is estimated. In the numerical simulation, phase portrait with time evolution diagrams shows periodic and chaotic oscillations. Bifurcation diagrams show the very rich and complex dynamical behavior of the non-spatial model. We calculate the Lyapunov exponent to justify the dynamics of the non-spatial model. A variety of patterns like interference, spot, and stripe are observed with special emphasis on Beddington–DeAngelis function response. These complex patterns explore the beauty of the spatio-temporal model and it can be easily related to real-world biological systems.

We investigate a population of individuals who play the Rock–Paper–Scissors (RPS) game. The players choose strategies not only by optimizing their payoffs, but also taking into account the popularity of the strategies. For the standard RPS game, we find an asymptotically stable polymorphism with coexistence of all strategies. For the general RPS game we find the limit cycles. Their stability depends exclusively on two model parameters: the sum of the entries of the RPS payoff matrix, and a sensitivity parameter which characterizes the personality of the players. Apart from the supercritical Hopf bifurcation, we found the subcritical bifurcation numerically for some intervals of the parameters of the model.

In this paper, we analyze the complexity of an eco-epidemiological model for phytoplankton–zooplankton system in presence of toxicity and time delay. Holling type II function response is incorporated to address the predation rate as well as toxic substance distribution in zooplankton. It is also presumed that infected phytoplankton does recover from the viral infection. In the absence of time delay, stability and Hopf-bifurcation conditions are investigated to explore the system dynamics around all the possible equilibrium points. Further, in the presence of time delay, conditions for local stability are derived around the interior equilibria and the properties of the periodic solution are obtained by applying normal form theory and central manifold arguments. Computational simulation is performed to illustrate our theoretical findings. It is explored that system dynamics is very sensitive corresponding to carrying capacity and toxin liberation rate and able to generate chaos. Further, it is observed that time delay in the viral infection process can destabilize the phytoplankton density whereas zooplankton density remains in its old state. Incorporation of time delay also gives the scenario of double Hopf-bifurcation. Some control parameters are discussed to stabilize system dynamics. The effect of time delay on (i) growth rate of susceptible phytoplankton shows the extinction and double Hopf-bifurcation in the zooplankton population, (ii) a sufficiently large value of carrying capacity stabilizes the chaotic dynamics or makes the whole system chaotic with further increment.

In this work, important aspects of bacteriocin producing bacteria and their interplay are elucidated. Various attempts to model the resistant, producer and sensitive Escherichia coli strains in the so-called rock-scissors-paper (RSP) game had been made in the literature. The question arose whether there is a continuous model with a cyclic structure and admitting an oscillatory dynamics as observed in various experiments. The May-Leonard system admits a Hopf bifurcation, which is, however, degenerate and hence inadequate. The traditional differential equation model of the RSP-game cannot be applied either to the bacteriocin system because it involves positive interaction terms. In this paper, a plausible competitive Lotka-Volterra system model of the RSP game is presented and the dynamics generated by that model is analyzed. For the first time, a continuous, spatially homogeneous model that describes the competitive interaction between bacteriocin-producing, resistant and sensitive bacteria is established. The interaction terms have negative coefficients. In some experiments, for example, in mice cultures, migration seemed to be essential for the reinfection in the RSP cycle. Often statistical and spatial effects such as migration and mutation are regarded to be essential for periodicity. Our model gives rise to oscillatory dynamics in the RSP game without such effects. Here, a normal form description of the limit cycle and conditions for its stability are derived. The toxicity of the bacteriocin is used as a bifurcation parameter. Exact parameter ranges are obtained for which a stable (robust) limit cycle and a stable heteroclinic cycle exist in the three-species game. These parameters are in good accordance with the observed relations for the E. coli strains. The roles of growth rate and growth yield of the three strains are discussed. Numerical calculations show that the sensitive, which might be regarded as the weakest, can have the longest sojourn times.

In this paper we analyze the population dynamics of bacteria competing by anti-bacterial toxins (bacteriocins). Three types of bacteria involved in these dynamics can be distinguished: toxin producers, resistant bacteria and sensitive bacteria. Their interplay can be regarded as a R ock- S cissors- P aper - game (RSP). Here, this is modeled by a reasonable three-dimensional Lotka- Volterra ($L$V) type differential equation system. In contrast to earlier approaches to modeling the RSP game such as replicator equations, all interaction terms have negative signs because the interaction between the three different types of bacteria is purely competitive, either by toxification or by competition for nutrients. The model allows one to choose asymmetric parameter values. Depending on parameter values, our model gives rise to a stable steady state, a stable limit cycle or a heteroclinic orbit with three fixed points, each fixed point corresponding to the existence of only one bacteria type. An alternative model, the May - Leonard model, leads to coexistence only under very restricted conditions. We carry out a comprehensive analysis of the generic stability conditions of our model, using, among other tools, the Volterra-Lyapunov method. We also give biological interpretations of our theoretical results, in particular, of the predicted dynamics and of the ranges for parameter values where different dynamic behavior occurs. For example, one result is that the intrinsic growth rate of the producer is lower than that of the resistant while its growth yield is higher. This is in agreement with experimental results for the bacterium Listeria monocytogenes.

When playing games in groups, it is an advantage for individuals to have accurate statistical information on the strategies of their opponents. Such information may be obtained by remembering previous interactions. We consider a rock-paper-scissors game in which agents are able to recall their last m interactions, used to estimate the behavior of their opponents. At critical memory length, a Hopf bifurcation leads to the formation of stable limit cycles. In a mixed population, agents with longer memories have an advantage, provided the system has a stable fixed point, and there is some asymmetry in the payoffs of the pure strategies. However, at a critical concentration of long memory agents, the appearance of limit cycles destroys their advantage. By introducing population dynamics that favors successful agents, we show that the system evolves toward the bifurcation point.

We analyze the replicator-mutator equations for the Rock-Paper-Scissors game. Various graph-theoretic patterns of mutation are considered, ranging from a single unidirectional mutation pathway between two of the species, to global bidirectional mutation among all the species. Our main result is that the coexistence state, in which all three species exist in equilibrium, can be destabilized by arbitrarily small mutation rates. After it loses stability, the coexistence state gives birth to a stable limit cycle solution created in a supercritical Hopf bifurcation. This attracting periodic solution exists for all the mutation patterns considered, and persists arbitrarily close to the limit of zero mutation rate and a zero-sum game., 6 pages, 5 figures

The spatio-temporal arrangement of interacting populations often influences the maintenance of species diversity and is a subject of intense research. Here, we study the spatio-temporal patterns arising from the cyclic competition between three species in two dimensions. Inspired by recent experiments, we consider a generic metapopulation model comprising "rock-paper-scissors" interactions via dominance removal and replacement, reproduction, mutations, pair-exchange and hopping of individuals. By combining analytical and numerical methods, we obtain the model's phase diagram near its Hopf bifurcation and quantitatively characterize the properties of the spiraling patterns arising in each phase. The phases characterizing the cyclic competition away far from the Hopf bifurcation (at low mutation rate) are also investigated. Our analytical approach relies on the careful analysis of the properties of the complex Ginzburg-Landau equation derived through a controlled (perturbative) multiscale expansion around the model's Hopf bifurcation. Our results allows us to clarify when spatial "rock-paper-scissors" competition leads to stable spiral waves and under which circumstances they are influenced by nonlinear mobility., 16 two-column pages, 16 figures

Whenever a disease emerges, awareness in susceptibles prompts them to take preventive measures, which influence individuals’ behaviors. Therefore, we present and analyze a time-delayed epidemic model in which class of susceptible individuals is divided into three subclasses: unaware susceptibles, fully aware susceptibles, and partially aware susceptibles to the disease, respectively, which emphasizes to consider three explicit incidences. The saturated type of incidence rates and treatment rate of infectives are deliberated herein. The mathematical analysis shows that the model has two equilibria: disease-free and endemic. We derive the basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document}R0 of the model and study the stability behavior of the model at both disease-free and endemic equilibria. Through analysis, it is demonstrated that the disease-free equilibrium is locally 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}$$R_01$$\end{document}R0>1, and linearly neutrally stable when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0=1$$\end{document}R0=1 for the time delay \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varrho >0$$\end{document}ϱ>0. Further, an undelayed epidemic model is studied when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0=1$$\end{document}R0=1, which reveals that the model exhibits forward and backward bifurcations under specific conditions, which also has important implications in the study of disease transmission dynamics. Moreover, we investigate the stability behavior of the endemic equilibrium and show that Hopf bifurcation occurs near endemic equilibrium when we choose time delay as a bifurcation parameter. Lastly, numerical simulations are performed in support of our analytical results.

We study the oscillatory dynamics in the generic three-species rock-paper-scissors games with mutations. In the mean-field limit, different behaviors are found: (a) for high mutation rate, there is a stable interior fixed point with coexistence of all species; (b) for low mutation rates, there is a region of the parameter space characterized by a limit cycle resulting from a Hopf bifurcation; (c) in the absence of mutations, there is a region where heteroclinic cycles yield oscillations of large amplitude (not robust against noise). After a discussion on the main properties of the mean-field dynamics, we investigate the stochastic version of the model within an individual-based formulation. Demographic fluctuations are therefore naturally accounted and their effects are studied using a diffusion theory complemented by numerical simulations. It is thus shown that persistent erratic oscillations (quasi-cycles) of large amplitude emerge from a noise-induced resonance phenomenon. We also analytically and numerically compute the average escape time necessary to reach a (quasi-)cycle on which the system oscillates at a given amplitude., Comment: 25 pages, 9 figures. To appear in the Journal of Theoretical Biology

Considering the influence of environmental capacity and forgetting on rumor spreading, we improve the traditional SIR (susceptible–infected–removed) rumor propagation model and give two dynamic models of rumor propagation in heterogeneous environment and homogeneous environment, respectively. The main purpose of this paper is to make a dynamic analysis of rumor propagation models. In the spatial heterogeneous environment, we have analyzed the uniform persistence of the rumor propagation model and the asymptotic behavior of the positive equilibrium point when the diffusion rate of rumor–susceptible tends to zero. In the spatial homogeneous environment, we discuss the stability of rumor propagation model. Further, optimal control and the necessary optimality conditions are obtained by using the maximum principle. Finally, we study the Hopf bifurcation phenomenon through inducing time delay in the reaction–diffusion model. In addition, the existence of Hopf bifurcation is verified and the influence of diffusion coefficients is studied by numerical simulations.

In this paper, we have considered a deterministic epidemic model with logistic growth rate of the susceptible population, non-monotone incidence rate, nonlinear treatment function with impact of limited hospital beds and performed control strategies. The existence and stability of equilibria as well as persistence and extinction of the infection have been studied here. We have investigated different types of bifurcations, namely Transcritical bifurcation, Backward bifurcation, Saddle-node bifurcation and Hopf bifurcation, at different equilibrium points under some parametric restrictions. Numerical simulation for each of the above-defined bifurcations shows the complex dynamical phenomenon of the infectious disease. Furthermore, optimal control strategies are performed using Pontryagin's maximum principle and strategies of controls are studied for two infectious diseases. Lastly using efficiency analysis we have found the effective control strategies for both cases.

Chronic wasting disease (CWD) is a contagious prion disease among the deer family that has the potential to disrupt the ecosystems where deer occur in abundance. To understand the dynamics of this emerging infectious disease, we consider a simple eco-epidemic model where the host population is infected by CWD. Boundedness of the system is established. The structure of equilibria and their linearized stability are investigated. The persistence condition is discussed. By constructing a suitable Lyapunov function, we discuss the global stability of the endemic equilibrium. Local bifurcation (transcritical) around the boundary equilibria is developed. Sufficient conditions for the existence of Hopf-bifurcation are derived. Further, we have also introduced white type of noise into the system to investigate stochastic stability. This suggests that the deterministic model is robust with respect to stochastic perturbation. Some numerical simulations are performed to validate our results.

A three-dimensional stage-structured predator–prey model is proposed and analyzed to study the effect of predation and cannibalism of the organisms at the highest trophic level with non-constant harvesting. Time lag in maturation of the predator is introduced in the system and conditions for local asymptotic stability of steady states are derived. The length of the delay preserving the stability is also estimated. Moreover, it is shown that the system undergoes a supercritical Hopf bifurcation when the maturation time lag crosses a certain critical value. Computer simulations have been carried out to illustrate various analytical results.

Although marine ecosystem is a highly complex phenomena with many non-linearly interacting species, dissolved oxygen and plankton among these have perhaps the most fundamental relationship not only for the protection of marine environment but also for continuation of life on Earth. This paper deals with a generic diffusive model of dissolved oxygen, phytoplankton and zooplankton species, for which constant time delays are incorporated in growth response of phytoplankton and in the gestation time of zooplankton. We mainly focus on the stability analysis of the coexisting states and the existence of Hopf bifurcation through the characteristic equation, where time delay and oxygen production rate are considered as control parameters for all cases. Studying the effect of both time delays on a stable system, we show destabilisation of the system and irregularity in the spatio-temporal dynamical regimes, leading to chaotic oscillations. Although both delay terms have a destabilising effect, our findings indicate that time delay in zooplankton gestation may induce sharp strongly irregular pattern, whereas time delay in phytoplankton growth gives rise to more regular but higher frequency oscillations for oxygen-plankton interactions. The findings of this paper may provide new insights into main environmental issues including global warming and climate change.

This paper [3] proves the existence of the Hopf bifurcation to a periodic solution from a stationary solution in certain problems of fluid dynamics. The results are similar to those already described. For instance, in the subcritical case, the periodic solution is shown to be unstable in the sense of Lyapunov when the real bifurcation parameter (Reynold’s number) is less than the critical value where the bifurcation takes place; it is shown to be (exponentially) stable if this value is greater than the critical value in the supercritical case.

The purpose of this section is to present the general idea that Kirchgassner and Kielhoffer [1] follow in resolving some problems of stability and bifurcation.

Publisher Summary This chapter discusses the stability problems for Hopf bifurcation. It explores the general problem of asymptotic stability of the periodic orbits arising in the Hopf bifurcation. The bifurcating periodic orbits are found to be attracting under the general assumption that 0 is asymptotically stable for μ = 0, which is the critical value of the parameter, and there exists an odd integer h ∈{3, …,k}, such that the character of 0 is recognizable in a suitable sense by the terms of f(0,·) of degree ≤ h. The occurrence of P(h) can be recognized by using a classical procedure of Poincare, and this procedure is reduced to the analysis of linear algebraic systems. With respect to the problem of asymptotic stability of bifurcating periodic orbits, P(h) is not only sufficient but also necessary for the bifurcating periodic orbits to be attracting and for the attractivity to have an appropriate structural character.

One subject of spatial epidemiology is spatial variation in disease risk or incidence. The spread of epidemics can result in strong spatial patterns of such risk or incidence: for example, pathogen dispersal might be highly localized, vectors or reservoirs for pathogens might be spatially restricted, or susceptible hosts might be clumped. Here, spatial pattern of an epidemic model with nonlinear incidence rates is investigated. The conditions for Hopf bifurcation and Turing bifurcation are gained and, in particular, exact Turing domain is found in the two parameters space. Furthermore, numerical results show that force of infection, namely β, plays an important role in the spatial pattern. More specifically, different patterns emerge as β increases. The mathematical analysis and numerical results well extend the finding of pattern formation in the epidemic models and may well explain the field observed in some areas.

There are many harmful effects of smoking. It not only engulfs the health and life of smokers, but also pollutes the air, endangers the life and health of others, and brings a heavy burden to public health. To this end, a delayed smoking model including potential smokers, occasional smokers, smokers, temporary quitters, permanent quitters and smokers with some illness, is investigated in the present paper. Firstly, local stability and existence of Hopf bifurcation of the model is conducted. Secondly, global exponential stability is explored. Lastly, we numerically simulate the correctness of the obtained theoretical results in the paper.

A Leslie–Gower–Holling type II model is modified to introduce a contagious disease in the predator population, assuming that disease cannot propagate to the prey. All the system’s equilibria are determined and the behaviour of the system near them is investigated. The main mathematical issues are global stability and bifurcations for some of the equilibria, together with sufficient conditions for persistence of the ecosystem. Counterintuitive results on the role played by intraspecific competition are highlighted.

Neuromorphic computing can solve computationally hard problems with energy efficiencies unattainable for von Neumann architectures. A locally-active memristor, which possesses the capability to amplify infinitesimal fluctuations in energy and can be used to generate neuromorphic behaviors, is a natural candidate for constructing an electronic equivalent of biological neurons. This paper identifies some unknown neuromorphic dynamics of the Chua corsage memristor (CCM), and shows that the CCM, when biased at the edge of chaos domain, can exhibit rich dynamics of biological neurons. Using Chua’s theories of local activity and edge of chaos, we demonstrate that under the destabilizing of the input voltage and the circuit parameters (inductance or capacitance), two CCM-based circuits can produce thirteen types of neuromorphic behaviors either on, or near the edge of chaos domain via supercritical or subcritical Hopf bifurcation. In addition, we give the conditions to test the edge of chaos of the CCM and the CCM-based circuit only by using the poles and the zero of their admittance functions.

This paper illuminates the issue of bifurcations for a fractional-order bidirectional associative memory neural network(FOBAMNN) with four different delays. On account of the affirmatory presumption, the developed FOBAMNN is firstly transformed into the one with two nonidentical delays. Then the critical values of Hopf bifurcations with respect to disparate delays are calculated quantitatively by establishing one delay and selecting remaining delay as a bifurcation parameter in the transformed model. It detects that the stability of the developed FOBAMNN with multiple delays can be fairly preserved if selecting lesser control delays, and Hopf bifurcation emerges once the control delays outnumber their critical values. The derived bifurcation results are numerically testified via the bifurcation graphs. The feasibility of theoretical analysis is ultimately corroborated in the light of simulation experiments. The analytic results available in this paper are beneficial to give impetus to resolve the issues of bifurcations of high-order FONNs with multiple delays.

Usually, juvenile predators are small in size, comparatively feeble and in the earlier phase of learning the tricky manoeuvre skills of predation. Due to small prey–predator size-ratio, juvenile predators and preys become two deadliest opponents, therefore, whenever both face off, the outcome may be utterly unpredictable. In response to the juvenile predators’ attack, preys also come up with a countermeasure and try to defend themselves but a sharp role-reversal in the predator–prey ecology occurs when preys kill the juvenile predators during the counter-attacking strategies. In this paper, we propose and analyze a biologically motivated delayed predator–prey system to explore the joint impact of counterattacking strategies and multiple delays on the predator–prey interactions. We have introduced the transformation of immature to mature predators including delay $$\tau _1$$ as an average time to reach maturity. Another time delay $$\tau _2$$ owing to the gestation of the adult predators, is taken into account. After the brief discussion of the basic ecosystem properties for the model in absence of delay, we have performed the stability analysis of the delayed system comprehensively. In this context, existence of Hopf bifurcation at the interior equilibrium is established and its stability as well as direction of the Hopf bifurcation is also investigated. Numerical simulations are also performed to corroborate the theoretical findings. Through this work, it has been illustrated that counterattacking of preys suppresses the unstable oscillations in the population, on the other hand, large delays always destabilize the interior steady state. Hence, in the realistic predator–prey ecology, where counterattack and delays are showing opposite behavior in the same ecosystem, the results of this paper are important to disclose the actual crossover and synergies between both the ecological properties.

This paper shows that the dynamics of the Lucas (J Monet Econ, 22:3–42, 1988) endogenous growth model with flow externalities may give rise to a 2-torus, a compact three-dimensional manifold enclosed by a two-dimensional surface. The implications of this result are relevant for many fields of economic theory. It is first of all clear that if we choose to initialize the dynamics in the basin of attraction of this trapping region, a continuum of perfect foresight solutions may be observed. A simple econometric exercise, linking the physical-to-human capital ratio (state variable) to the 5-years forward variance of the growth rate of an unbalanced sample of 183 countries, seems to provide empirical backing for the phenomenon. Other important consequences, relevant from the point of view of endogenous cycles theory, are also scrutinized in the paper.

In this paper, we examine the stability of an endemic equilibrium in a chronological age-structured SIR (susceptible, infectious, removed) epidemic model with age-dependent infectivity. Under the assumption that the transmission rate is a shifted exponential function, we perform a Hopf bifurcation analysis for the endemic equilibrium, which uniquely exists if the basic reproduction number is greater than 1. We show that if the force of infection in the endemic equilibrium is equal to the removal rate, then there always exists a critical value such that a Hopf bifurcation occurs when the bifurcation parameter reaches the critical value. Moreover, even in the case where the force of infection in the endemic equilibrium is not equal to the removal rate, we show that if the distance between them is sufficiently small, then a similar Hopf bifurcation can occur. By numerical simulation, we confirm a special case where the stability switch of the endemic equilibrium occurs more than once.

This paper proposes definitions of implementation and adoption delays, arising from firm and client behaviors, in the context of market share dynamics. Information delay refers to the existence of lags in the information used for the advertising policy, while adoption delay refers to the existence of a lag in the effect of the advertising policy. These are natural lags in the flow of information and have not been considered in several models proposed in the literature. In this paper, these delays are introduced into recently proposed extensions of the Vidale-Wolfe-Deal and Lanchester models of market share dynamics subjected to affine advertising control policies. Conditions for stability of the equilibrium market share are derived. In addition, it is shown that Hopf bifurcations leading to oscillatory behavior exist for certain parameter values, and corresponding conditions for these are given. The main results are: the equilibrium market shares of the extended Vidale-Wolfe-Deal and Lanchester models are both robust to implementation delays, but, in the case of adoption delays, for both models, numerical results show that there is a critical value such that if the sum of the adoption delays exceeds this value, there is an onset of oscillations of market shares, through a Hopf bifurcation.

S-type locally-active memristor (LAM) has a great potential for brain- inspired neuromorphic computing, where the S-type LAM-based oscillator is a fundamental building block. Concerning the S-type LAM, this paper constructs a material-independent model in simple mathematical expression, which can be relatively easily analyzed. By biasing the memristor into the locally- active region, and connecting it with a capacitor, a second-order oscillator can be built. The small-signal equivalent circuit of the memristor and its frequency response are applied to determine the period oscillation frequency range and compensation capacitance. Hopf bifurcation theory is used to analyze oscillation mechanism of the second-order circuit and appropriate capacitance. By adding an extra inductor into the second-order oscillator, a novel third-order chaotic circuit is developed, where a saddle-focus is derived to create chaos. Its dynamic characteristics are investigated via Lyapunov exponents, bifurcation diagrams, dynamic route map, and so on. The local activities of the single memristor, second-order oscillator, and third-order chaotic circuit are verified through the mathematical analysis. Finally, physical circuit realizations of the S- type LAM-based oscillators, including the memristor emulator, are presented. Both simulation and experimental results demonstrate the practicability of the proposed mathematical model and the validity of the theoretical analysis.

This paper expatiates the stability and bifurcation for a fractional-order neural network (FONN) with double leakage delays. Firstly, the characteristic equation of the developed FONN is circumspectly researched by employing inequable delays as bifurcation parameters. Simultaneously the bifurcation criteria are correspondingly extrapolated. Then, unequal delays-spurred-bifurcation diagrams are primarily delineated to confirm the precision and correctness for the values of bifurcation points. Furthermore, it lavishly illustrates from the evidence that the stability performance of the proposed FONN can be demolished with the presence of leakage delays in accordance with comparative studies. Eventually, two numerical examples are exploited to underpin the feasibility of the developed theory. The results derived in this paper have perfected the retrievable outcomes on bifurcations of FONNs embodying unique leakage delay, which can nicely serve a benchmark deliberation and provide a comparatively credible guidance for the influence of multiple leakage delays on bifurcations of FONNs.

The phenomenon produced by the Hopf bifurcations is of notable importance. In fact, a Hopf bifurcation—guaranteeing the existence of an unsteady periodic solution of the linearized problem at stake—is also an optimum limit cycle candidate of the nonlinear associated problem and, if non linearly globally attractive, is an absorbing set and an effective limit cycle. The present paper deals with the onset of Hopf bifurcations in thermal magnetohydrodynamics (MHD). Precisely, it is devoted to characterization—via a simple formula—of the Hopf bifurcations threshold in horizontal plasma layers between rigid planes, heated from below and embedded in a constant transverse magnetic field. This problem, remarked clearly and notably by the Nobel Laureate Chandrasekhar (Nature 175:417–419, 1955), constitutes a difficulty met by him and—for plasma layers between rigid planes electricity perfectly conducting—is, as far as we know, still not removed. Let $$m_0$$ m 0 be the thermal conduction rest state and let $$P_r, P_m, R, Q$$ P r , P m , R , Q , be the Prandtl, the Prandtl magnetic, the Rayleigh and the Chandrasekhar number, respectively. Recognized (according to Chandrasekhar) that the instability of $$m_0$$ m 0 via Hopf bifurcation can occur only in a plasma with $$P_m>P_r$$ P m > P r , in this paper it is shown that the Hopf bifurcations occur if and only if $$\begin{aligned} Q>Q_c=\displaystyle \frac{4\pi ^2[1+P_r(\mu /2\pi )^4]}{P_m-P_r}, \end{aligned}$$ Q > Q c = 4 π 2 [ 1 + P r ( μ / 2 π ) 4 ] P m - P r , with $$ \mu =7.8532$$ μ = 7.8532 . Moreover, the critical value of R at which the Hopf bifurcation occurs is characterized via the smallest zero of the second invariant of the spectrum equation governing the most destabilizing perturbation. The critical value of Q, in the free-rigid and rigid-free cases is shown to be $$\displaystyle \frac{1}{4}$$ 1 4 of the previous value.

Magnetic-liquid double-suspension bearing (MLDSB) is composed of an electromagnetic supporting system and a hydrostatic supporting system. Due to its greater supporting capacity and stiffness, it is appropriate for middle-speed applications, overloading, and frequent starting. However, because it contains two sets of systems, its structure and rotor support system are more complex. It contains strong nonlinear links. When the parameters of the system change, the bearing rotor may feature Hopf bifurcation, resulting in system flutter and reducing the operational stability of the magnetic fluid double-suspension bearing rotor, which has become one of the key problems restricting its development and application. As key parameters of MLDSB, the coil current and oil film thickness exert a major impact on Hopf bifurcation. Therefore, the mathematical model of MLDSB is established in this paper, and the border and direction of Hopf bifurcation, period, and amplitude of limit cycle are analyzed. The calculation, simulation, and experimental results show that when the coil current and oil film thickness of the bearing system are greater than the boundary value of the Hopf bifurcation, Hopf bifurcation will occur, resulting in the vibration of the bearing rotor and affecting the stability of the system. In addition, when analyzing the combined effects of coil current and oil film thickness on the Hopf bifurcation of the system, it was found that the boundary value of Hopf bifurcation in the system is reduced compared with when it is are affected solely due to the coupling of the two parameters. The period, amplitude and vibration speed of limit cycle increase with increases in the coil current and oil film thickness. Hopf bifurcation experiment was conducted on MLDSB testing system. The results show that Hopf bifurcation does not occur when i0 < 0.5 A, the bearing rotor operates stably in the balanced position, i0 > 1.0 A, Hopf bifurcation occurs in the system, and the bearing rotor vibrates with equal amplitude, which reduces the stability of operation. The research in this paper can provide a theoretical reference for the Hopf bifurcation analysis of MLDSB.

In this paper, a model of active control system with state-dependent actuation time delay is investigated. Galerkin projection scheme is used to obtain its low dimensional approximation system. State-dependency and effect on dynamics and performance of actuation delay in this active control system are considered. The following results are obtained. Firstly, a state-dependent actuation delay is proposed, which has never been reported in the previous works. It displays that actuation delay is state-dependent. Secondly, employing Galerkin projection scheme, low dimensional approximation system of this delayed active control system is obtained. At last, its stability and Hopf bifurcation are investigated by Routh-Hurwitz Criterion and Hopf bifurcation theory. Using software WinPP, numerical simulation is executed and supports the theoretical results in this paper.

Coherence resonance (CR), which is typical nonlinear phenomenon that noise plays construct roles, has been widely investigated in single neurons at resting state instead of bursting behavior when noise free. In the present paper, multiple CRs evoked from bursting instead of resting state and the underlying bifurcation mechanism are studied in a theoretical neuron model composed of fast subsystem and slow variable, which is very important for the bursting neurons receiving strong synaptic noise in the central nervous system. With increasing noise intensity within a range, the bursting period decreases to induce the burst number increases firstly and then becomes irregular, which results in the increase firstly and then decrease in the corresponding peak of the power spectrum of membrane potentials, i.e., CR is evoked from bursting instead of resting state. With further increasing noise intensity, two other peaks exhibit CR phenomenon, one at middle frequency and middle noise intensity and the other at large frequency and strong noise intensity, which shows that multiple CRs are evoked from bursting. Furthermore, the bifurcation mechanism of the latter two resonant peaks is acquired with fast–slow variable dissection method. With increasing the value of slow variable, the fast subsystem exhibits a fold bifurcation of limit cycle and a subcritical Hopf bifurcation to a stable focus, and the imaginary part of the characteristic value of the focus increases, which implies that spikes evoked from the focus exhibit increasing frequency. The trajectory of the deterministic bursting in phase space locates between the two bifurcations and alternates between the coexisting limit cycle and focus of the fast subsystem. With increasing noise intensity, the value of slow variable of the stochastic bursting increases and moves to a range of the focus of the fast subsystem, and the spikes of the stochastic bursting are evoked from the focus by noise. Therefore, the stronger the noise intensity, the larger the value of slow variable, and the higher frequency the spikes, due to the changes of characteristic values, which is the cause that the peaks with middle and high frequency appear at middle and strong noise intensity, respectively. The results of the present paper extend the concept of CR to multiple CRs evoked from bursting, provide the bifurcation mechanism underlying the multiple CRs, and imply multiple chances to utilize noise to enhance information of bursting neurons.

A correction is noted to the previously published paper (Zou and Lv, 2017). Regularity, positive recurrence and stationary distribution are considered in this paper. A complete threshold analysis of strong stochastic persistence and extinction is investigated. Stochastic Hopf bifurcation is considered from the viewpoint of numerical simulations.

In this paper, we mainly investigate a delayed predator–prey model with stage structure for prey and Ivlev-type functional response. Four assumptions about this model are made as follows: (1) there are a single predator and a single prey population in the model; (2) the prey is divided by the age into two stage structures, immature and mature; (3) the trophic transfer from prey to predator is incomplete; (4) there are two time delays due to the time of maturation of prey and the time of gestation of predator. Some properties of equilibria for the model without delays are provided. Further, the stability of equilibria and existence of Hopf bifurcation are studied by discussing the different cases of time delays for the model with delays. We observe that delays can cause a stable equilibrium to become unstable one, even occur Hopf bifurcation when delays pass through their corresponding critical values. Meanwhile, we derive explicit formulae to determine the properties of Hopf bifurcation such as the direction of Hopf bifurcation and the stability of periodic solutions. Numerical simulations of all theoretical analyses are given for verifying our theoretical results. In the present work, we demonstrate that the validity and universality of oscillations induced by the time delays both theoretically and numerically. These results of this paper may be helpful for us to further understand the role of the critical values of time delays in stabilizing the predator–prey model.

This paper is concerned with the stability and Hopf bifurcation of fractional-order neural networks with discrete and distributed delays. The novelty of this paper is to take into account the discrete time delay and the distributed time delay for fractional-order systems. By introducing two virtual neurons to the original network, a new four-neuron network only involving discrete delays is formed. The sum of discrete delays is adopted as the bifurcation parameter to demonstrate the existence of Hopf bifurcation. It is found that the critical value of bifurcation can be effectively manipulated by choosing appropriate system parameters and order. Finally, numerical simulations are executed to substantiate the theoretical results and describe the relationships between the parameters and the onset of bifurcation.

This paper examines economic cycles that do not depend on exogenous economic actions. More precisely, the paper develops a positive model of government behavior in order to define the intertemporal fiscal policies that are optimal for a country and which, determine both the optimal budget level and the optimal level of environmental quality. For this purpose, we establish an optimal control model involving intertemporal subsidy strategies characteristically used by an authoritarian government similar to those found in central Europe. It will be shown that by applying the Hopf bifurcation theorem,a cyclical strategy – that is, waves of regulation, environmental subsidies alternating with deregulation and cuts in social programs – may represent an optimal policy. In this paper, we propose an extremely simple optimal control model which is applied to budget surpluses and environmental subsidies. We investigate the cyclical environmental policies as applied through the bifurcation theorem. A number of propositions are stated during the solution process. The first proposition sets the rules in the model parameters in order to establish the cyclical policies. A second proposition sets the relationship between the discount rate and the opportunity cost of capital, which is used to determine a taxing strategy which becomes the future optimal subsidy policy.

In this paper, we address the problem of bifurcation control for a delayed neuron system. By introducing a new fractional-order Proportional-Derivative (PD) feedback controller, this paper aims to control the stability and Hopf bifurcation through adjusting the control gain parameters. The order chosen in PD controller is different with that of the integer-order neuron system. Sufficient conditions for guaranteeing the stability and generating Hopf bifurcation are constructed for the controlled neuron system. Finally, numerical simulation results are illustrated to verify our theoretical derivations and the relationships between the onset of the Hopf bifurcation and the gain parameters are obtained.

Neurons contain a large number of ions inside and outside the cell, and the transmembrane currents formed by the movement of these ions cause membrane potential fluctuations and induce electromagnetism inside and outside the cell. In addition, any change in external electromagnetic fields can cause changes in the membrane potential of the neurons. Therefore, based on the three-dimensional Hindmarsh — Rose (HR) neuron model, a five-dimensional neuron model with time delay is developed in this paper by introducing flux and electric field variables and considering the resulting time delay. First, the Hopf bifurcation theory is used to demonstrate the local stability of the system at the equilibrium point at different time delays. Then, the stability of the Hopf bifurcation and its direction are proved by using the central flow shape theorem. Finally, the existence of the Hopf bifurcation is proved using the phase diagram and the bifurcation diagram, and the effects of several important parameters on the model are investigated by numerical simulations using time series plots, ISI bifurcation plots and two-parameter bifurcation plots. The model is found to be accompanied by chaotic and chaos-free plus-periodic bifurcation structures, mixed-mode discharges and other phenomena. Also, its discharge pattern can be controlled after adding time delay. The results of this paper provide help to the pathogenic mechanism and control of neurological diseases.

In this paper, a single-species population model with distributed delay and Michaelis-Menten type harvesting is established. Through an appropriate transformation, the mathematical model is converted into a two-dimensional system. Applying qualitative theory of ordinary differential equations, we obtain sufficient conditions for the stability of the equilibria of this system under three cases. The equilibrium A1 of system is globally asymptotically stable when br−c&gt, 0 and η&lt, 0. Using Poincare-Bendixson theorem, we determine the existence and stability of limit cycle when br−c&gt, 0 and η&gt, 0. By computing Lyapunov number, we obtain that a supercritical Hopf bifurcation occurs when η passes through 0. High order singularity of the system, such as saddle node, degenerate critical point, unstable node, saddle point, etc, is studied by the theory of ordinary differential equations. Numerical simulations are provided to verify our main results in this paper.

This paper addresses the problem of chaos control in an economic mathematical dynamical model. By regarding the control variables as the bifurcation parameters, the stability of equilibria and the existence of Hopf bifurcations of the relevance feedback system are investigated, and the criterion of controllability for the chaotic system is obtained based on a time-delayed feedback control technique. Furthermore, numerical simulations are provided to demonstrate the feasibility of our methods and results.