G. P. Samanta

Deterministic and stochastic analysis of a prey-dependent predator–prey system

This paper reports on studies of the deterministic and stochastic behaviours of a predator–prey system with prey-dependent response function. The first part of the paper deals with the deterministic analysis of uniform boundedness, permanence, stability and bifurcation. In the second part the reproductive and mortality factors of the prey and predator species are perturbed by coloured noise to simulate a random environment. Using spectral density analysis it has been shown that this system exhibits large fluctuations for high amplitude random forces. When k, the carrying capacity of prey, tends to a particular value k*, this system exhibits abnormally large fluctuations with increasing time with a periodic background noise. A comparative study of the deterministic and stochastic criteria of stability is also established.

Bioeconomic modelling of a three-species fishery with switching effect

This paper aims to study the problem of combined harvesting of a system involving one predator and two prey species fishery in which the predator feeds more intensively on the more abundant species. Mathematical formulation of the optimal harvest policy is given and its solution is derived in the equiblibrium case by using Pontryagins Maximum principle. Dynamic optimization of the harvest policy is also discussed by takingE(t), the combined harvest effort, as a dynamic variable. Biological and bioeconomic interpretations of the results associated with the optimal equilibirum solution are explained. The significance of the constraints required for the existence of an optimal singular control are also given.

Effect of time-delay on a food chain model

This paper aims to study the effect of discrete time-delay on a tritrophic food chain model with Holling type-II functional responses. Dynamical behaviours such as boundedness, stability, persistence and bifurcation of the model are studied. Our analytical findings are illustrated through computer simulation. Biological implications of our analytical findings are addressed critically.

Inventory model with two rates of production for deteriorating items with permissible delay in payments

Goyal (1985) [‘Economic Order Quantity Under Conditions of Permissible Delay in Payments’, Journal of Operational research Society, 36, 35–38] assumed that unit selling price and unit purchasing price are equal. But in real-life the scenario is different. The purpose of this article is to reflect the real life problem by allowing unit selling price and purchasing price to be unequal. Our model is a continuous production control inventory model for deteriorating items in which two different rates of production are available. The results are illustrated with the help of a numerical example. We discuss the sensitivity of the solution together with the changes of the values of the parameters associated with the model. Our model may be applicable in many manufacturing planning situations where management practices for deterioration are stringent; e.g. the two-production rate will be more profitable than the one-production rate in the manufacture of cold, asthma and allergy medicine. Our proposed model might be applicable to develop a prototype advance planning system for those manufacturers to integrate the management science techniques into commercial planning.

Rich dynamics of a food chain model with Hassell-Varley type functional responses

Abstract Complex dynamics of a tritrophic food chain model with Hassell–Varley functional responses is discussed in this paper. Dynamical behaviours such as boundedness, stability and bifurcation of the model are studied. The effect of discrete time-delay on the model is investigated. Various solutions are found using computer simulations. It is discussed how these ideas illuminate some of the observed properties of real populations in the field and explores practical implications.

Dynamical model of a single-species system in a polluted environment

The effect of toxicants on ecological systems is an important issue from mathematical and experimental points of view. Here we have studied dynamical model of a single-species population-toxicant system. Two cases are studied: constant exogeneous input of toxicant and rapidly fluctuating random exogeneous input of toxicant into the environment. The dynamical behaviour of the system is analyzed by using deterministic linearized technique, Lyapunov method and stochastic linearization on the assumption that exogeneous input of toxicant into the environment behaves like ‘Coloured noise’.

Stochastic Gomatam model of interacting species: non-equilibrium fluctuation and stability

The object of the article is to study the stability behaviour of the stochastic version of the Gomatam model of interacting species. The reproductive and mortality factors of the prey and predator species respectively are perturbed by coloured noises due to random environment. Using spectral density analysis we show that this system exhibits large fluctuations for high amplitude random forces. When the intraspecific interaction coefficients tend to zero along a straight line, this system exhibits abnormally large fluctuations with increasing time with a periodic background noise. We also obtain a relation between the real part of the eigen values of the interaction matrix in the deterministic environment and the stability of this system in the stochastic environment.

Sterile insect release method as a control measure of insect pests: A mathematical model

Recently non-conventional approaches of pest control are getting much more importance in different parts of the world. The main reason behind this is the long list of side effects of conventional approaches (use of pesticides etc.). The present paper focuses on one such extremely useful method of insect pest control, namely the Sterile Insect Release Method (SIRM), by using a mathematical model. A blend of dynamical behaviours of the model is studied critically, which, in turn, indicates the relevance of the method. The effect of uncertain environmental fluctuations on both fertile and sterile insects is also investigated. Our analytical findings are verified through computer simulation. Some important restrictions on the parameters of the system are mentioned, which may be implemented for a better performance of SIRM.

Dynamical Behaviour of a Tumor-Immune System with Chemotherapy and Optimal Control

We have considered a tumor growth model with the effect of tumor-immune interaction and chemotherapeutic drug. We have considered two immune components—helper (resting) T-cells which stimulate CTLs and convert them into active (hunting) CTL cells and active (hunting) CTL cells which attack, destroy, or ingest the tumor cells. In our model there are four compartments, namely, tumor cells, active CTL cells, helper T-cells, and chemotherapeutic drug. We have discussed the behaviour of the solutions of our system. The dynamical behaviour of our system by analyzing the existence and stability of the system at various equilibrium points is discussed elaborately. We have set up an optimal control problem relative to the model so as to minimize the number of tumor cells and the chemotherapeutic drug administration. Here we used a quadratic control to quantify this goal and have considered the administration of chemotherapy drug as control to reduce the spread of the disease. The important mathematical findings for the dynamical behaviour of the tumor-immune model with control are also numerically verified using MATLAB. Finally, epidemiological implications of our analytical findings are addressed critically.

Deterministic and stochastic analysis of a prey–predator model with herd behaviour in both

ABSTRACT This paper aims to study the dynamical behaviours of a prey–predator system, where both the prey and predator show herd behaviours. Positivity, boundedness, stability of equilibrium points, et cetera, are discussed in deterministic environment. To incorporate the effect of fluctuating environment, we have perturbed the birth rate of prey species and death rate of predator species by Gaussian white noises. Then the resulting model is cultured by the method of statistical linearization to study the stability and non-equilibrium fluctuation of the populations in stochastic environment. Numerical simulations are carried out to validate our analytical findings. Implications of our analytical and numerical findings are discussed critically.