B. Dubey

A predator–prey interaction model with self and cross-diffusion

Abstract 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.

A model for fishery resource with reserve area

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 Pantryagins Maximum Principle.

A model for the allelopathic effect on two competing species

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.

SIMULTANEOUS EFFECT OF TWO TOXICANTS ON BIOLOGICAL SPECIES: A MATHEMATICAL MODEL

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.

EXISTENCE AND SURVIVAL OF TWO COMPETING SPECIES IN A POLLUTED ENVIRONMENT: A MATHEMATICAL MODEL

In this paper, a nonlinear mathematical model to study the effect of a toxicant emitted into the environment from external sources on two competing biological species is proposed and analyzed. The cases of constant emission and instantaneous spill of a toxicant are considered in the model study. In the case of constant emission, it is shown that four usual outcomes of competition between two species may be altered under appropriate conditions which are mainly dependent on emission rate of toxicant into the environment, uptake concentrations of toxicant by the two species and their growth rate coefficients and carrying capacities. However, in the case of instantaneous spill, it is found that if the washout rate of toxicant is large, then the four outcomes of competition exist under usual conditions. It is also pointed out that the survival of the competitors, coexisting in absence of the toxicant, may be threatened if the constant emission of toxicant into their environment continues unabatedly.

A RESOURCE DEPENDENT FISHERY MODEL WITH OPTIMAL HARVESTING POLICY

A dynamic model for a single-species fishery, which depends partially on a logistically growing resource with functional response, is proposed using taxation as control instrument to protect fish population from overexploitation. The analysis of the model shows that both the equilibrium density of fish population as well as the maximum sustainable yield increase as resource biomass density increases. The optimal harvesting policy is also discussed with the help of Pontryagins Maximum Principle. It is found that for the optimum equilibrium value of resource biomass density, the total users cost of harvest per unit effort must be equal to the discounted value of future price at the steady state.

Effects of industrialization and pollution on resource biomass: a mathematical model

Abstract 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.

Effect of changing habitat on survival of species

Abstract In this paper, a mathematical model is proposed to study the growth and existence (survival) of resource-biomass-dependent species in a forested habitat which is being depleted due to the pressure of industrialization (population). It is shown that as the pressure of industrialization increases, the biomass density decreases, leading to lowering of the density of species and its eventual extinction if this pressure continues unabatedly. However, if suitable efforts are made to conserve the resource biomass and to control the pressure of industralization in the forested habitat, the survival of resource-biomass-dependent species can be ensured.

Modelling the depletion and conservation of resources: Effects of two interacting populations

Abstract In this paper, a general mathematical model to study the effects of two interacting populations on the depletion of resources is proposed and analysed. In modelling the system, it is assumed that the resource is a common food for both the populations while one of the population is a supplementary food for the other. A model to conserve the resource is also presented.

A MODEL FOR AN INSHORE-OFFSHORE FISHERY

In this paper, a nonlinear mathematical model to study the dynamics of an inshore-offshore fishery under variable harvesting is proposed and analyzed. Criteria for local stability, instability and global stability of the system are derived. The optimal harvesting policy is discussed by considering taxation as a control instrument. It is shown that the fishery resources can be protected from overexploitation by increasing the tax and discount rates.