Department of Mathematics

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End date: 2009Subject: search.filters.subject.Stability

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Now showing 1 - 10 of 11
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    Simultaneous effect of two toxicants on biological species: a mathematical model
    (World Scientific, 1996) Dubey, Balram
    In this paper, a mathematical model to study the simultaneous effect of two toxicants (one is more toxic than the other) on the growth and survival of a biological species is proposed. The cases of instantaneous spill, constant and periodic emissions of each of the toxicant into the environment are considered. It is shown that in the case of an instantaneous spill of each of the toxicant into the environment, the species after its initial decrease in density may recover to its original level after a period of time, the magnitude of which depends on the toxicity and washout rate of each of the toxicant. However, if both the toxicants are emitted with constant rates, the species in the habitat is doomed to extinction sooner than the case of a single toxicant having the same influx and washout rates as one of them, the extinction rate becoming faster with the increase in toxicity and emission rate of the other toxicant. It is also shown that for a small amplitude periodic emission of the toxicant with a constant mean, the stability behavior of the system is same as that of the case of the constant emission. It is found further through the model study that if suitable efforts are made to reduce the emission rate of each of the toxicant at the source and its concentration in the environment by some removal mechanism, an appropriate level of species density can be maintained.
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    A model for the allelopathic effect on two competing species
    (Elsevier, 2000-05) Dubey, Balram
    In this paper, a mathematical model is proposed and analysed to study the coexistence of two competing plant species in a finite habitat by assuming that each species produces a toxic substance affecting the other species. The diffusion of toxic substances is also considered in the model. It is shown that the usual existence criteria between two competing species in the absence of toxicant may be changed if each species produces toxicant in large amount affecting the other. In case of no diffusion criteria for local stability, instability and global stability of the system are obtained. In case of allelopathy, where one species produces toxicant and affects the other, it is found that the affected species may be driven to extinction. It is also found that diffusion has a stabilizing effect on the system.
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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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    Effects of industrialization and pollution on resource biomass: a mathematical model
    (Elsevier, 2003-09) Dubey, Balram
    In this paper, a mathematical model is proposed and analyzed to study the depletion of resource biomass (plant/tree) due to industrialization and pollution. Industrialization dependent, constant, instantaneous, and periodic emissions of pollutant into the environment are taken into consideration. Criteria for local stability, instability, and global stability of non-negative equilibria are obtained. Numerical simulations are carried out to investigate the dynamics of the system. It is found that in the case of small periodic influx of pollutant into the environment, the resource biomass has a periodic behavior if the depletion rate coefficient of environmental pollutant is small. However, if this coefficient increases beyond a threshold value, then resource biomass converges towards its equilibrium.
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    A model for fishery resource with reserve area
    (Elsevier, 2003-10) Dubey, Balram
    In this paper, we propose and analyse a mathematical model to study the dynamics of a fishery resource system in an aquatic environment that consists of two zones: a free fishing zone and a reserve zone where fishing is strictly prohibited. Biological and bionomic equilibria of the system are obtained, and criteria for local stability, instability and global stability of the system are derived. It is shown that even if fishery is exploited continuously in the unreserved zone, fish populations can be maintained at an appropriate equilibrium level in the habitat. An optimal harvesting policy is also discussed using the Pantryagin's Maximum Principle.
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    Nonlinear models for the survival of two competing species dependent on resource in industrial environments
    (Elsevier, 2003-03) Dubey, Balram
    In this paper, a nonlinear mathematical model is proposed and analysed to study the survival of two competing species dependent on resource in industrial environments with and without diffusion. The competing species are assumed to be either partially dependent, wholly dependent or predating on the resource. A corresponding conservation model is also proposed and analysed to study the importance of regeneration of the exploited resource as well as the control of industrialization. In the case of without diffusion, criteria for survival and extinction of competing species are derived and equilibrium levels of the resource biomass, competing species and industrialization density are compared. In the case of diffusion, it is shown that solutions of the system approach to the steady state more rapidly with the increase in diffusion coefficients. By analysing the conservation model it is shown that if suitable efforts are made to conserve the resource biomass and to control the undesired level of industrialization pressure, an appropriate level of resource biomass density can be maintained and thus the survival of species may be ensured.
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    A model for the effect of time delay on the dynamics of a population living in a polluted environment
    (World Scientific, 2004) Dubey, Balram
    In this paper, a mathematical model is proposed and analyzed to study the effect of time delay on the dynamics of a single-species population living in a polluted environment. It is shown that time delay in the model has destabilizing effect on the system.
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    Modelling the survival of species dependent on a resource in a polluted environment
    (Elsevier, 2006-04) Dubey, Balram
    In this paper, a mathematical model is proposed and analysed to study the survival of species dependent on a resource in a polluted environment. Criteria for instability, local stability and global stability are obtained. The effect of diffusion on the system is also investigated. It is shown that diffusion plays a general role in stabilizing the system.
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    A Prey-Predator Model with a Reserved Area
    (VUP, 2007) Dubey, Balram
    In this paper, a mathematical model is proposed and analysed to study the dynamics of a prey-predator model. It is assumed that the habitat is divided into two zones, namely free zone and reserved zone. Predators are not allowed to enter into the reserved zone. Criteria for the coexistence of predator-prey are obtained. The role of reserved zone is investigated and it is shown that the reserve zone has a stabilizing effect on predator-prey interactions.
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    A mathematical model for the effect of toxicant on the immune system
    (World Scientific, 2007) Dubey, Balram; Dubey, Uma S.
    In this paper, a nonlinear mathematical model is proposed and analyzed to study the effect of environmental toxicant on the immune response of the body. Criteria for local stability, instability and global stability are obtained. It is shown that the immune response of the body decreases as the concentration of environmental toxicant increases, and certain criteria are obtained under which it settles down at its equilibrium level. In the absence of toxicant, an oscillatory behavior of immune system and pathogenic growth is observed. However, in the presence of toxicant, oscillatory behavior is not observed. These studies show that the toxicant may have a grave effect on our body's defense mechanism.