Department of Mathematics

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Subject: search.filters.subject.Hopf-bifurcation

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    Dynamics of tuberculosis transmission with exogenous reinfections and endogenous reactivation
    (Elsevier, 2018-05) Das, Dhiraj Kumar
    We propose and analyze a mathematical model for tuberculosis (TB) transmission to study the role of exogenous reinfection and endogenous reactivation. The model exhibits two equilibria: a disease free and an endemic equilibria. We observe that the TB model exhibits transcritical bifurcation when basic reproduction number . Our results demonstrate that the disease transmission rate and exogenous reinfection rate plays an important role to change the qualitative dynamics of TB. The disease transmission rate give rises to the possibility of backward bifurcation for , and hence the existence of multiple endemic equilibria one of which is stable and another one is unstable. Our analysis suggests that may not be sufficient to completely eliminate the disease. We also investigate that our TB transmission model undergoes Hopf-bifurcation with respect to the contact rate and the exogenous reinfection rate . We conducted some numerical simulations to support our analytical findings.
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    Dynamical behaviour of infected predator–prey eco-epidemics with harvesting effort
    (Springer, 2021-04) Das, Dhiraj Kumar
    This investigation accounts for a predator–prey system where the predator community is affected by infectious disease and also subjected to harvest. The model considers the behavioural change in susceptible predators due to the crowding effect of infected predators. The dynamical characteristics are studied encompassing asymptotic stability of the existing equilibrium points and bifurcation analysis. A sufficient parametric condition for global stability of the interior equilibrium point is investigated using a geometric approach. The system undergoes a Hopf-bifurcation around interior equilibrium point considering disease transmission rate as a bifurcation parameter. An optimal control problem is formulated by considering a time-dependent fishing effort as a control variable. The objective of this optimal control problem is to maximize the present value of the economic revenue obtained by fishing. Finally, several numerical simulations are conducted to visualize our analytical results.
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    A ratio-dependent predator-prey model with delay and harvesting
    (World Scientific, 2010) Dubey, Balram
    In this paper a predator-prey model with discrete delay and harvesting of predator is proposed and analyzed by considering ratio-dependent functional response. Conditions of existence of various equilibria and their stability have been discussed. By taking delay as a bifurcation parameter, the system is found to undergo a Hopf bifurcation. Numerical simulations are also performed to illustrate the results.
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    Modeling and simulation of a wetland park: An application to Keoladeo National Park, India
    (Elsevier, 2017) Dubey, Balram
    In this paper, three mathematical models are proposed and analyzed to study the degradation and conservation of a wetland park. The first model describes the interaction of bird populations with the good biomass. The interaction is considered to be Crowley–Martin type. The good biomass is the cumulative densities of floating vegetation, fishes, waterfowl and other useful species. The model is further extended to study the effect of bad biomass such as Paspalum distichum on the dynamics of the previous system. The Paspalum distichum is a wild grass which depletes the level of oxygen in the open water bodies affecting severely the good biomass and consequently the bird populations. The carrying capacity of bad biomass is one of the most vital parameter. Keeping the low value of the carrying capacity, one can maintain the ‘good health’ of the park. At the situation of ‘bad health’ of the park, some efforts are needed to control the bad biomass. Hence the second model is again extended to control the growth of bad biomass and to maintain the good biomass and the bird populations at an appropriate level. In each case, numerical simulations are carried out to illustrate the analytical results. These models suggest to control the bad biomass in an efficient manner and to maintain the eco-system of the wetland park.
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    Stability and Bifurcation of a Fishery Model with Crowley–Martin Functional Response
    (World Scientific, 2017) Dubey, Balram
    To understand the dynamics of a fishery system, a nonlinear mathematical model is proposed and analyzed. In an aquatic environment, we considered two populations: one is prey and another is predator. Here both the fish populations grow logistically and interaction between them is of Crowley–Martin type functional response. It is assumed that both the populations are harvested and the harvesting effort is assumed to be dynamical variable and tax is considered as a control variable. The existence of equilibrium points and their local stability are examined. The existence of Hopf-bifurcation, stability and direction of Hopf-bifurcation are also analyzed with the help of Center Manifold theorem and normal form theory. The global stability behavior of the positive equilibrium point is also discussed. In order to find the value of optimal tax, the optimal harvesting policy is used. To verify our analytical findings, an extensive numerical simulation is carried out for this model system.
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    Global stability and Hopf-bifurcation of prey-predator system with two discrete delays including habitat complexity and prey refuge
    (Elsevier, 2019-02) Dubey, Balram
    In this paper, we consider a two-dimensional prey-predator system with two delays. One delay is for negative feedback of the prey population and another is for gestation delay of the predator population. The predator is partially dependent on the prey followed by Holling type-II functional response. Due to habitat complexity and prey refuge, the Holling type-II functional response is modified in this work. We discuss the boundedness, permanence, local and global asymptotic behavior of the non-delayed and delayed models. The existence of periodic solutions via Hopf-bifurcation with respect to both the delays is established. The stability and direction of Hopf-bifurcation is also analyzed by using Normal form theory and Centre manifold theory. Lastly, numerical simulations have been carried out to confirm the analytical findings. The main objective of this work is to balance the prey-predator relationship in the presence of habitat complexity, prey refuge and delays.
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    Global analysis of a delayed stage structure prey–predator model with Crowley–Martin type functional response
    (Elsevier, 2019-08) Dubey, Balram
    A stage structure prey–predator model that consists of a system of three nonlinear ordinary differential equations in the presence of discrete time delay is proposed and analysed in this paper. The prey population is divided into two categories: immature and mature prey. The predator population depends on mature prey only and that followed by Crowley–Martin type functional response. We analyse positivity, boundedness and existence of equilibrium points. The local and global stability behaviour of the delayed and non-delayed system are also analysed. Considering delay as a bifurcation parameter, the Hopf-bifurcation is also examined for this system. Then we discuss the stability and direction of Hopf-bifurcation using Normal form theory and Centre manifold theory. Numerical simulation is carried out to verify our analytical findings. We observe that, for a set of values of parameters, the bifurcated periodic solution is supercritical, stable with decreasing period and as the time delay increases, interior equilibrium point disappears. Model of this type may be considered to save the immature prey from the predator population and to maintain the prey–predator relation.
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    Modeling the Effect of Fear in a Prey–Predator System with Prey Refuge and Gestation Delay
    (World Scientific, 2019) Dubey, Balram
    Recently, some field experiments and studies show that predators affect prey not only by direct killing, they induce fear in prey which reduces the reproduction rate of prey species. Considering this fact, we propose a mathematical model to study the fear effect and prey refuge in prey–predator system with gestation time delay. It is assumed that prey population grows logistically in the absence of predators and the interaction between prey and predator is followed by Crowley–Martin type functional response. We obtained the equilibrium points and studied the local and global asymptotic behaviors of nondelayed system around them. It is observed from our analysis that the fear effect in the prey induces Hopf-bifurcation in the system. It is concluded that the refuge of prey population under a threshold level is lucrative for both the species. Further, we incorporate gestation delay of the predator population in the model. Local and global asymptotic stabilities for delayed model are carried out. The existence of stable limit cycle via Hopf-bifurcation with respect to delay parameter is established. Chaotic oscillations are also observed and confirmed by drawing the bifurcation diagram and evaluating maximum Lyapunov exponent for large values of delay parameter.
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    Stability and Bifurcation of a Prey-Predator System with Additional Food and Two Discrete Delays
    (Tech Science Press, 2021) Dubey, Balram
    In this paper, the impact of additional food and two discrete delays on the dynamics of a prey-predator model is investigated. The interaction between prey and predator is considered as Holling Type-II functional response. The additional food is provided to the predator to reduce its dependency on the prey. One delay is the gestation delay in predator while the other delay is the delay in supplying the additional food to predators. The positivity, boundedness and persistence of the solutions of the system are studied to show the system as biologically well-behaved. The existence of steady states, their local and global asymptotic behavior for the non-delayed system are investigated. It is shown that (i) predator’s dependency factor on additional food induces a periodic solution in the system, and (ii) the two delays considered in the system are capable to change the status of the stability behavior of the system. The existence of periodic solutions via Hopf-bifurcation is shown with respect to both the delays. Our analysis shows that both delay parameters play an important role in governing the dynamics of the system. The direction and stability of Hopf-bifurcation are also investigated through the normal form theory and the center manifold theorem. Numerical experiments are also conducted to validate the theoretical results.
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    Stability switching and chaos in a multiple delayed prey-predator model with fear effect and anti-predator behaviour, Mathematics and Computers in Simulation
    (Elsevier, 2021-10) Dubey, Balram
    Recent studies demonstrate that the density of prey population is not only affected by direct killing by the predator, but the fear in prey caused by predator also reduces it by cutting down the reproduction rate of prey community, and prey shows anti-predator behavior in response to this fear. In this study, we propose a prey–predator model with fear in prey due to predator and anti-predator behavior by prey against the predator with fear response delay and gestation delay. It is assumed that the predator consumes prey via simplified Holling Type-IV functional response. We evaluate the equilibrium points and study the local and global stability behavior of the system around them. It is observed that our system undergoes Hopf-bifurcation with respect to the fear parameter. Moreover, the system shows the attribute of bi-stability involving two stable equilibriums. Further, we study the dynamics of the delayed system by incorporating fear response delay and gestation delay. We observe that the delayed system suffers Hopf-bifurcation with respect to both delays. Using the normal form method and center manifold theory, the direction and stability of Hopf-bifurcation are studied. Chaotic behavior for delayed system is observed for large values of fear response delay. All these findings are supported by numerical simulation.