Dynamical behaviors of a constant prey refuge ratio-dependent prey-predator model with Allee and fear effects

In this paper,we consider a nonlinear ratio-dependent prey-predator model with constant prey refuge in the prey population.Both Allee and fear phenomena are incorporated explicitly in the growth rate of the prey population.The qualitative behaviors of the proposed model are investigated around the equilibrium points in detail.Hopf bifurcation including its direction and stability for the model is also studied.We observe that fear of predation risk can have both stabilizing and destabilizing effects and induces bubbling phenomenon in the system.It is also observed that for a fixed strength of fear,an increase in the Allee parameter makes the system unstable,whereas an increase in prey refuge drives the system toward stability.However,higher values of both the Allee and prey refuge parameters have negative impacts and the populations go to extinction.Further,we explore the variation of densities of the populations in different bi-parameter spaces,where the coexistence equilibrium point remains stable.Numerical simulations are carried out to explore the dynamical behaviors of the system with the help of MATLAB software.

Bifurcation and optimal control for an infectious disease model with the impact of information

A nonlinear infectious disease model with information-influenced vaccination behavior and contact patterns is proposed in this paper,and the impact of information related to disease prevalence on increasing vaccination coverage and reducing disease incidence during the outbreak is considered.First,we perform the analysis for the existence of equilibria and the stability properties of the proposed model.In particular,the geometric approach is used to obtain the sufficient condition which guarantees the global asymptotic stability of the unique endemic equilibrium Ee when the basic reproduction number Ro>1.Second,mathematical derivation combined with numerical simulation shows the existence of the double Hopf bifurcation around Ee.Third,based on the numerical results,it is shown that the information coverage and the average information delay may lead to more complex dynamical behaviors.Finally,the optimal control problem is established with information-infuenced vaccination and treatment as control variables.The corresponding optimal paths are obtained analytically by using Pontryagin's maximum principle,and the applicability and validity of virous intervention strategies for the proposed controls are presented by numerical experiments.

Effect of random movement and cooperative hunting in the prey-predator system:A dynamical approach

Self-diffusion prerequisite is obtained as the spreading approach of biological populations.Cooperative hunting is a common behavior in predator populations that promotes predation and the coexistence of the prey-predator system.On the other side,the Allee effect among prey may cause the system to become unstable.In this paper,a difusive prey predator system with cooperative hunting and the weak Allee effect in prey populations is discussed.The linear stability and Hopf-bifurcation analysis had been used to examine the system's stability.From the spatial stability of the system,the conditions for Turing instability have been derived.The multiple-scale analysis has been used to derive the amplitude equations of the system.The stability analysis of these amplitude equations leads to the formation of Turing patterns.Finally,numerical simulations are used to analyze spatial patterns forming in 1-D and 2-D.The studies indicate that the model can generate a complex pattern structure and that self-diffusion has a drastic impacton species distribution.

Vaccination effect on a stochastic epidemic model with healing and relapse

In this work,we consider a stochastic epidemic model with vaccination,healing and relapse.We prove the existence and the uniqueness of the positive solution.We establish sufficient conditions for the extinction and the persistence in mean of the stochastic system.Moreover,we also establish sufficient conditions for the existence of ergodic stationary distribution to the model,which reveals that the infectious disease will persist.The graphical illustrations of the approximate solutions of the stochastic epidemic model have been performed.

Dynamic analysis and bifurcation control of a delayed fractional-order eco-epidemiological migratory bird model with fear effect

In this paper,a new delayed fractional-order model including susceptible migratory birds,infected migratory birds and predators is proposed to discuss the spread of diseases among migratory birds.Fear of predators is considered in the model,as fear can reduce the reproduction rate and disease transmission rate among prey.First,some basic mathematical results of the proposed model are discussed.Then,time delay is regarded as a bifurcation parameter,and the delay-induced bifurcation conditions for such an uncontrolled system are established.A novel periodic pulse feedback controller is proposed to suppress the bifurcation phenomenon.It is found that the control scheme can successfully suppress the bifurcation behavior of the system,and the pulse width can be arbitrarily selected on the premise of ensuring the control effect.Compared with the traditional time-delay feedback controller,the control scheme proposed in this paper has more advantages in practical application,which not only embodies the advantages of low control cost and easy operation but also caters to the periodic changes of the environment.The proposed control scheme,in particular,remains effective even after the system has been disrupted by a constant.Numerical simulation verifies the correctness of the theoretical results.

Dynamics and data fitting of a time-delayed SIRS hepatitis B model with psychological inhibition factor and limited medical resources

Hepatitis B is an infectious disease worthy of attention.Considering the incubation period,psychological inhibition factor,vaccine,limited medical resources and horizontal transmission,an SIRS model is proposed to describe hepatitis B transmission dynamics.In order to describe the behavior changes caused by people's psychological changes,the non-monotonic incidence rate is adopted in the model.We use the saturated treatment rate to describe the limited medical resources.Mathematical analysis shows the existence conditions of the equilibria,forward or backward bifurcation,Hopf bifurcation and the Bogdanov-Takens bifurcation.During the observation of the case data of hepatitis B in China,it is found that there are mainly three features,periodic outbreaks,aperiodic outbreaks,and periodic outbreaks turns to aperiodic outbreaks.According to the above features,we select three different representative regions,Jiangxi,Zhejiang province and Beijing,and then use our model to fit the actual monthly hepatitis B case data.The basic reproduction numbers that we estimated are 1.7712,1.4805 and 1.4132,respectively.The results of data fitting are consistent with those of theoretical analysis.According to the sensitivity analysis of Ro,we conclude that reducing contact,increasing treatment rate,strengthening vaccination and revaccinating can effectively prevent and control the prevalence of hepatitis B.

Bifurcations,chaos analysis and control in a discrete predator-prey model with mixed functional responses

Many discrete systems have more distinctive dynamical behaviors compared to continuous ones,which has led lots of researchers to investigate them.The discrete predatorprey model with two different functional responses(Holling type I and II functional responses)is discussed in this paper,which depicts a complex population relationship.The local dynamical behaviors of the interior fixed point of this system are studied.The detailed analysis reveals this system undergoes flip bifurcation and Neimark-Sacker bifurcation.Especially,we prove the existence of Marotto's chaos by analytical method.In addition,the hybrid control method is applied to control the chaos of this system.Numerical simulations are presented to support our research and demonstrate new dynamical behaviors,such as period-10,19,29,39,48 orbits and chaos in the sense of Li-Yorke.

A class of natural pinus koraiensis population system with time delay and diffusion term

In this paper,we consider the long-term sustainability of the northeast Korean pine.We propose a class of natural Korean pine population system with time delay and diffusion term.First,by analyzing the roots distribution of the characteristic equation,we study the stability of the model system with diffusion terms and prove the occurrence of Hopf bifurcation.Second,we introduce lactation time delay into a population model with a diffusion term,based on stability theory of ordinary differential equation,norm form methods and center manifold theorem,the stability of bifurcating periodic solutions and the relevant formula for the direction of Hopf bifurcation are given.Finally,some numerical simulations are given.

Dynamics of a delay-induced prey-predator system with interaction between immature prey and predators

In biological pest control systems,several pests(including insects,mites,weeds,etc.)are controlled by biocontrol agents that rely primarily on predation.Following this biocontrol management ecology,we have created a three-tier prey-predator model with prey phase structure and predator gestation delay.Several studies have demonstrated that predators with Holling type-II functional responses sometimes consume immature prey.A study of the well-posedness and local bifurcation(such as saddle-node and transcritical)near the trivial and planer equilibrium points is carried out.Without any time lag,the prey development coeficient has a stabilizing impact,while increasing attack rate accelerates instability.Energy transformation rate and handling time are shown to cause multiple stability switches in the system.Numerical results demonstrate time delay is the key destabilizer that destroys stability.Our model can replicate more realistic events by including time-dependent factors and exploring the dynamic behavior of nonautonomous systems.In the presence of time delay,sufficient conditions of permanence and global attractivity of the nonautonomous system are derived.Finally,MATLAB simulations are performed to validate the analytical findings.

Severe acute respiratory syndrome-coronavirus-2(SARS-COV-2)infection of pneumocytes with vaccination and drug therapy:Mathematical analysis and optimal control

We propose a mathematical model studying a within-host infection dynamics of SARSCoV-2 in pneumocytes.This model incorporates immune response,vaccination and antiviral drugs.The crucial properties of the model-the existence,positivity and boundary of solutions are established.Equilibrium points and the basic reproduction number are calculated.The stability of each equilibrium point is analyzed.Optimal control is applied to the model by adding three control variables:vaccination,treatment by Favipiravir and treatment by Molnupiravir.Numerical results show that each individual control could reduce SARS-CoV-2 infection in some aspects;however,with a combination of three controls,we obtain the best results in reducing SARS-CoV-2 infection.This study has emphasized the importance of prevention by vaccine and the antiviral treatments.

Stability and bifurcation analysis of a size-stage-structured cooperation model

In this paper,we propose a size-stage-structured cooperation model which has two distinct life stages in facultative cooperator.The primary feature of this model is to consider size structure,stage structure and obligate and facultative symbiosis at the same time in a cooperation system.We use the method of characteristic to show that this new model can be reduced to a threshold delay equations(TDEs)model,which can be further transformed into a functional differential equations(FDEs)model by a simple change of variables.Such simplification allows us to apply the classical theory of FDEs and establish a set of sufficient conditions to investigate the qualitative analysis of solutions of the FDEs model,including the global existence and uniqueness,positivity and boundedness.What's more,we use the geometric criteria to get the conclusions about stability and Hopf bifurcation of positive equilibrium because the coefficients of the characteristic equation depend on the bifurcation parameter.Finally,numerical simulations are carried out as supporting evidences of our analytical results.Our results show that the presence of size structure and stage structure plays an important role in the dynamic behavior of the model.

Dynamical study of varicella-zoster virus model in sense of Mittag-Leffler kernel

The primary varicella-zoster virus(VzV)infection that causes chickenpox(also known as varicella),spreads quickly among people and,in severe circumstances,can cause to fever and encephalitis.In this paper,the Mittag-Leffler fractional operator is used to examine the mathematical representation of the vzV.Five fractional-order differential equations are created in terms of the disease's dynamical analysis such as S:Susceptible,V:Vaccinated,E:Exposed,I:Infectious and R:Recovered.We derive the existence criterion,positive solution,Hyers-Ulam stability,and boundedness of results in order to examine the suggested fractional-order model's wellposedness.Finally,some numerical examples for the VzV model of various fractional orders are shown with the aid of the generalized Adams-Bashforth-Moulton approach to show the viability of the obtained results.

Influence of toxic substances on dynamical behavior of a delayed diffusive predator-prey model

In this paper,we propose and investigate a delayed diffusive predator-prey model affected by toxic substances.We first study the boundedness and persistence property of the model.By analyzing the associated characteristic equation,we obtain the conditions for the existence of steady state bifurcation,Hopf bifurcation and Turing bifurcation.Furthermore,we also study the Hopf bifurcation induced by the delay.Finally,our theoretical results are verified by numerical simulation.The numerical observation results are in good agreement with the theoretically predicted results.Theoretical and numerical simulations indicate that toxic substances have a great impact on the dynamics of the system.

Global dynamics of a Leslie-Gower predator-prey model in open advective environments

This paper investigates the global dynamics of a reaction-diffusion-advection Leslie-Gower predator-prey model in open advective environments.We find that there exist critical advection rates,intrinsic growth rates,diffusion rates and length of the domain,which classify the global dynamics of the Leslie-Gower predator-prey system into three scenarios:coexistence,persistence of prey only and extinction of both species.The results reveal some significant differences with the classical specialist and generalist predator-prey systems.In particular,it is found that the critical advection rates of prey and predator are independent of each other and the parameters about predation rate have no influence on the dynamics of system.The theoretical results provide some interesting highlights in ecological protection in streams or rivers.

The global stability and optimal control of the COVID-19 epidemic model

This paper considers stability analysis of a Susceptible-Exposed-Infected-Recovered-Virus(SEIRV)model with nonlinear incidence rates and indicates the severity and weakness of control factors for disease transmission.The Lyapunov function using Volterra-Lyapunov matrices makes it possible to study the global stability of the endemic equilibrium point.An optimal control strategy is proposed to prevent the spread of coronavirus,in addition to governmental intervention.The objective is to minimize together with the quantity of infected and exposed individuals while minimizing the total costs of treatment.A numerical study of the model is also carried out to investigate the analytical results.

Dynamic analysis of a latent HIV infection model with CTL immune and antibody responses

This paper develops a mathematical model to investigate the Human Immunodeficiency Virus(HIV)infection dynamics.The model includes two transmission modes(cell-to-cell and cell-free),two adaptive immune responses(cytotoxic T-lymphocyte(CTL)and antibody),a saturated CTL immune response,and latent HIV infection.The existence and local stability of equilibria are fully characterized by four reproduction numbers.Through sensitivity analyses,we assess the partial rank correlation coefficients of these reproduction numbers and identify that the infection rate via cell-to-cell transmission,the number of new viruses produced by each infected cell during its life cycle,the clearance rate of free virions,and immune parameters have the greatest impact on the reproduction numbers.Additionally,we compare the effects of immune stimulation and cell-to-cell spread on the model's dynamics.The findings highlight the significance of adaptive immune responses in increasing the population of uninfected cells and reducing the numbers of latent cells,infected cells,and viruses.Furthermore,cell-to-cell transmission is identified as a facilitator of HIV transmission.The analytical and numerical results presented in this study contribute to a better understanding of HIV dynamics and can potentially aid in improving HIV management strategies.

Asymptotic behavior of an SIQR epidemic model driven by Lévy jumps on scale-free networks

To investigate the effect of information transmission,Levy jumps and contact heterogeneity of individuals on the asymptotic behavior of epidemic,a stochastic SIQR epidemic model with non-monotone incidence rate and Levy jumps on scale-free networks is constructed.At first,the global dynamics of the deterministic model is studied by constructing appropriate Lyapunov functions.Then the stochastic model is made in accordance with the ecological significance,the existence and uniqueness of the global positive solution of the stochastic SIQR model is manifested.Next,by constructing suitable stochastic Lyapunov functions and applying Ito formula with jump,the asymptotic behavior of solutions of stochastic model around equilibrium of the corresponding deterministic model is checked.At last,the correctness of the analytical results is verified by numerical simulations.

Pattern dynamics and bifurcation in delayed SIR network with diffusion network

The spread of infectious diseases often presents the emergent properties,which leads to more dificulties in prevention and treatment.In this paper,the SIR model with both delay and network is investigated to show the emergent properties of the infectious diseases'spread.The stability of the SIR model with a delay and two delay is analyzed to illustrate the effect of delay on the periodic outbreak of the epidemic.Then the stability conditions of Hopf bifurcation are derived by using central manifold to obtain the direction of bifurcation,which is vital for the generation of emergent behavior.Also,numerical simulation shows that the connection probability can affect the types of the spatio-temporal patterns,further induces the emergent properties.Finally,the emergent properties of COVID-19 are explained by the above results.

Fear and delay effects on a food chain system with two kinds of different functional responses

For food chain system with three populations,direct predation is the basic interaction between species.Different species often have different predation functional responses,so a food chain system with Holling-II response for middle predator and Beddinton-DeAngelis response for top predator is proposed.Apart from direct predation,predator population can significantly impact the survival of prey population by inducing the prey's fear,but the impact often possesses a time delay.This paper is concentrated to explore how the fear and time delay affect the system stability and the species persistence.By use of Lyapunov functional method and bifurcation theory,the positiveness and boundedness of solutions,local and global behavior of species,the system stability around the equilibrium states and various kinds of bifurcation are investigated.Numerically,some simulations are carried out to validate the main findings and the critical values of the bifurcation parameters of fear and conversion rate are obtained.It is observed that fear and delay can not only stabilize,but also destabilize the system,which depends on the magnitude of the fear and delay.The system varies from unstable to stable due to the continuous increase of the prey's fear by middle predator.Small fear induced by top predator or small delay of the prey's fear can stabilize the system,while they are sufficiently large,the system stability is to be destroyed.Simultaneously,the conversion rate can also change the stability and even make the species to be extinct.Some rich dynamics like multiple stabilities and various types of bistability behaviors are also exhibited,which results in the convergence of the species from one stable equilibrium to another.

Global dynamics analysis for a nonlinear HTLV-I model with logistic proliferation and CTL response

It is beyond dispute that cytotoxic T-lymphocytes(CTLs)exert a vital function in the host's antiviral defense mechanism.With the idea of the above factor and the logistic proliferation of CD4+T-cells,we establish a HTLV-I(human T-cell leukemia virus type-I)mathematical model.First,two threshold parameters Ro and Re(the basic reproduction numbers for viral infection and CTL immune response,respectively)are obtained.Second,sufficient criteria for local and global asymptotic stabilities of the feasible equilibria of the model are deduced,respectively.Third,the sensitivity analyses of Ro and Rc are performed to better understand the effective strategies for HTLV-I infection.Finally,not only numerical simulations are given to illustrate the stability conclusions,but also the biological significance is stated.