Kalyan Das

Chaotic dynamics of a three species prey–predator competition model with bionomic harvesting due to delayed environmental noise as external driving force

We consider a biological economic model based on prey-predator interactions to study the dynamical behaviour of a fishery resource system consisting of one prey and two predators surviving on the same prey. The mathematical model is a set of first order non-linear differential equations in three variables with the population densities of one prey and the two predators. All the possible equilibrium points of the model are identified, where the local and global stabilities are investigated. Biological and bionomical equilibriums of the system are also derived. We have analysed the population intensities of fluctuations i.e., variances around the positive equilibrium due to noise with incorporation of a constant delay leading to chaos, and lastly have investigated the stability and chaotic phenomena with a computer simulation.

HIV/AIDS Model with Delay and the Effects of Stochasticity

We present a deterministic HIV/AIDS model with delay. We then extend the model by adjoining terms capturing stochastic effects. The intensity of the fluctuations in the stochastic system is analytically evaluated using Fourier transform methods. We carry out simulations to assess differences in the dynamical behavior of the deterministic and stochastic models. Simulation results show that they are no significant differences in the behavior of the two models.

Chaotic dynamics of a three species prey-predator competition model with noise in ecology

In this paper we investigate a three species ecosystem consisting of a prey, a predator and a top predator. The predator survives on the prey and the top predator survives on both the prey and the predator. The mathematical model consists of non-linear simultaneous differential equations. All the eight equilibrium points of the model are identified and by using Routh-Hurwitz criteria as well as Lyapunov function, we derive the criteria for local and global stabilities. Also the population intensities of fluctuations (variances) around the positive equilibrium due to noise are computed; their stability is also analyzed with computer simulation using Matlab. Some Numerical simulations for justifying the theoretical analysis are also provided. The main conclusions are supplied in conclusion.

CONTROL OF PARAMETERS OF A DELAYED DIFFUSIVE AUTOTROPH HERBIVORE SYSTEM

This paper deals with the eco-biological dynamics of a delayed diffusive autotroph-herbivore population ecosystem with nutrient recycling. The real situation is represented by a set of two-dimensional nonlinear ordinary differential equations involving autotroph-herbivore biomass. Plant populations undergo critical changes with different amplitude in plant ecology. We propose a description of plant communities as interesting systems which resembles to the behavior of real media. The delay and diffusion parameter have a great role to shape the dynamical features of the system. We have studied the growth of an autotroph and herbivore population depending on the limiting nutrient which is partially recycled through bacterial decomposition. We have analyzed the asymptotic stability and switching to instability with bifurcation of the model system with delay and diffusion where diffusion controls directly the delay parameters of our system. All the analytical results are interpreted ecologically and compared w...

ENDEMIC FLUOROSIS THROUGH DRINKING WATER AND ITS REMEDIAL MEASURE BY BACTERIAL POPULATION — A MATHEMATICAL MODEL

Fluoride-contaminated drinking water is a serious public health hazard in some parts of India. Field surveys have been conducted in fluoride-affected areas of the Birbhum district, West Bengal, India. From these surveys, it is found that in certain locations the drinking water contains less fluoride (within the permissible limit1) than in adjoining areas. We have isolated a strain of Streptococcus species from these wells that has the ability to remove the fluoride ion from the water. We have grown the bacteria in different concentrations of fluoride and monitored the decline in the free fluoride ion concentration. A simplified model of fluoride and bacterial dynamics is proposed. The model is handled both analytically and numerically. For numerical solution mainly our own laboratory data are used. Calibration and validation are performed with different sets of experimental data. Analytical solutions show that the bacterial consumption and growth rates are the most important parameters in the system. Analysis shows asymptotic stability of the system.

Effect of time delay in an autotroph-herbivore system with nutrient cycling

In the present study we consider a mathematical model of a non-interactive type autotroph-herbivore system in which the amount of autotroph biomass consumed by the herbivore is assumed to follow a Holling type II functional response. We have also incorporated discrete time delays in the numerical response term to represent a delay due to gestation, and in the recycling term which represents the time required for bacterial decomposition. We have derived conditions for global asymptotic stability of the model in the absence of delays. Conditions for delay-induced asymptotic stability of the steady state are also derived. The length of the delay preserving stability has been estimated and interpreted ecologically.