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Author: search.filters.author.Dubey, BalramSubject: search.filters.subject.Mathematics

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Now showing 1 - 10 of 12
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    Optimal management of a renewable resource utilized by a population with taxation as a control variable
    (VUP, 2013) Dubey, Balram
    A dynamical model is proposed and analyzed to discuss the effect of population on a resource biomass by taking into account the crowding effect. It is assumed that the resource biomass, which has commercial importance, is subjected to harvesting. The harvesting effort is assumed to be a dynamical variable and taxation is being used as a control variable. The optimal harvesting policy is discussed using the Pontryagin’s maximum principle.
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    A Mathematical Model for Optimal Management and Utilization of a Renewable Resource by Population
    (Hindawi Publishing Corporation, 2013) Dubey, Balram
    A dynamical model is proposed and analyzed to study the effect of the population on the resource biomass by taking into account the crowding effect. Biological and bionomical equilibria of the system are discussed. The global stability behavior of the positive equilibrium is studied via the output feedback control. An appropriate Hamiltonian function is formed for the discussion of optimal harvesting of resource which is utilized by the population using Pontryagin's Maximum Principal. A numerical simulation is performed on the model to analyze the theoretical results.
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    The role of top predator interference on the dynamics of a food chain model
    (Elsevier, 2013-03) Dubey, Balram
    In this paper, the effects of top predator interference on the dynamics of a food chain model involving an intermediate and a top predator are considered. It is assumed that the interaction between the prey and intermediate predator follows the Volterra scheme, while that between the top predator and its favorite food depends on Beddington–DeAngelis type of functional response. The boundedness of the system, existence of an attracting set, local and global stability of non-negative equilibrium points are established. Number of the bifurcation and Lyapunov exponent bifurcation diagrams is established. It is observed that, the model has different types of attracting sets including chaos. Moreover, increasing the top predator interference stabilizes the system, while increasing the normalization of the residual reduction in the top predator population destabilizes the system.
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    Modeling and analysis of an seir model with different types of nonlinear treatment rates
    (World Scientific, 2013) Dubey, Balram; Dubey, Uma S.
    In this study, an SEIR epidemic model is proposed for treatment of infectives considering the development of acquired immunity in recovered individuals. We employed two different types of treatment functions. Stability analysis for disease-free as well as endemic equilibria is performed. It is observed that the existence of unique endemic equilibrium depends on the basic reproductive number R0 as well as on treatment rate. Numerical simulations are performed on the proposed models to support and analyze theoretical findings.
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    Dynamics of Phytoplankton, Zooplankton and Fishery Resource Model
    (AAM, 2014-06) Dubey, Balram
    In this paper, a new mathematical model has been proposed and analyzed to study the interaction of phytoplankton- zooplankton-fish population in an aquatic environment with Holloing’s types II, III and IV functional responses. It is assumed that the growth rate of phytoplankton depends upon the constant level of nutrient and the fish population is harvested according to CPUE (catch per unit effort) hypothesis. Biological and bionomical equilibrium of the system has been investigated. Using Pontryagin’s Maximum Principal, the optimal harvesting policy is discussed. Chaotic nature and bifurcation analysis of the model system for a control parameter have been observed through a numerical simulation
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    A predator–prey interaction model with self and cross-diffusion
    (Elsevier, 2001-07) Dubey, Balram
    In this paper, a mathematical model for a predator–prey interaction with self and cross-diffusion is proposed and analysed. Criteria for local stability, instability and global stability are obtained. The effect of the critical wave length which can drive a system to instability is investigated. The effect of time-varying cross-diffusivity on the stability of the system is also examined.
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    Modelling the Dynamics of a Renewable Resource under Harvesting with Taxation as a Control Variable
    (AAM, 2014-10) Dubey, Balram
    The present paper describes a model of resource biomass and population with a non-linear catch rate function on resource biomass. The harvesting effort is assumed to be a dynamical variable. Tax on per unit harvested resource biomass is used as a tool to control exploitation of the resource. Pontryagin’s Maximum Principle is used to find the optimal control to maintain the resource biomass and population at an optimal level. A numerical simulation is also carried out to support the analytical results.
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    Modeling the effect of pollution on biological species: a socio-ecological problem
    (International Academy of Ecology and Environmental Sciences, 2015) Dubey, Balram
    In this paper, a nonlinear spatial model is proposed and analyzed to study the effect of pollution on biological population. It is assumed that the pollutants enter into the environment not directly by the population but by a precursor produced by the population itself. It is further assumed that larger the population, faster the precursor is produced, and larger the precursor, faster the pollutant is produced. Criteria for nonlinear stability and instability for both spatial and non-spatial models are obtained. The various parameter ranges for stable homogeneous solutions are identified. By the simulation experiments, it is observed that by applying an appropriate effort F, the population density P can be maintained at a higher equilibrium level. It is also shown that the equilibrium level of the concentration of precursor pollutant, concentration of pollutant in the environment and in the population decrease due to the effort F.
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    Role of media and treatment on an SIR model
    (VUP, 2016) Dubey, Balram; Dubey, Uma S.
    In this paper, the impact of awareness programs as well as treatment on an SIR model has been investigated. We assume that the whole population is divided into four compartments, named as susceptible (S), infected (I), aware susceptible (Sa) and recovered (R). Analytical findings and numerical simulations of the model show that if the exposure to the awareness program is high and adequate treatment is available, then the infection can be eliminated. Analysis of the model also depicts that if treatment is not available, then infection is high even if enough awareness is present. But in absence of awareness an infection can not be eliminated inspite of adequate treatment. Effective treatment can led to a diminished level of infection. Stability analysis of the model is investigated by using stability theory of differential equations. Further, numerical simulations are carried out to validate the analytical results.
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    Stability and Hopf-bifurcation in a general Gauss type two-prey and one-predator system
    (Elsevier, 2016-06) Dubey, Balram
    A Gauss type general prey–predator mathematical model is proposed and analysed to study the effect of predation on two competing prey species. The growth rate and functional responses are taken to be general non-linear functions. By analysing the model, local stability of all possible equilibrium points is discussed. By choosing a suitable Lyapunov function the global stability of the system at positive equilibrium point is also found. For the purpose of numerical simulation, growth rates of both prey species are taken to be logistic and the predator's functional response on the prey species are taken as Holling type-II. Taking death rate of the predator as a bifurcation parameter, we observe Hopf-bifurcation of the system. Then we have discussed the stability and direction of the Hopf-bifurcation. We also observed that intra-specific interference factor is an important parameter in governing the dynamics of the system.