Mathematics

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The paper deals with integrated pest management (IPM) of a single density-dependent pest affecting a single crop. The model is basically a prey-predator model, predator being the natural enemy of the pest. In addition, it also contains the growth equation of the affected crop. This dynamic model is subjected to biological control in the form of release of additional predators and chemical control in the form of spraying of pesticides. To consider the natural side effect of pesticide on fish living in the soil water, the model is further improved to accommodate the growth equation of the fishes in the soil water. The paper considers the optimal analysis of the model under two control parameters, one is the spraying of pesticide and other one is the release of predators; such optimal analysis under multi-control parameters is completely new of its kind. © 2007 Springer Science+Business Media, Inc.

(SoNS) School of Natural Sciences

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Bulletin of Mathematical Biology

A more realistic approach to pest-management problem

The interrelationship of the host-virus-environment is of critical importance in determining the amount of virus production per insect and its consequences in the overall infection process in insect community. Lower temperature inhibits the virus replication and thus infection does not persist into the system. On the other hand, virus production per insect increases with temperature increase within certain limit, and this increase of virus production brings some unforeseen consequences in the infection dynamics in pest control. We deal the problem by applying non-vulnerability concept on the mathematical model of viral infection that linked up with the temperature-viral developmental model. Using Lyapunov-like function, we determine a range of temperature and show that the system would become endemic and remain in stable situation if temperature could be maintained in this prescribed range, whereas that range depends on other parameters of the system. © 2007 Elsevier Ltd. All rights reserved.

Journal of Theoretical Biology

Effect of temperature on viral infection and its control: A mathematical approach

The interrelationship of latency period in viral infection and overall infection process in host community are of critical importance in context of pest control programme. Both of them regulate the overall system stability as they are dynamically linked to predation by natural enemies in the system. The present paper deals with the role of latency period in viral infection through mathematical modeling and analysis. We propose a four dimensional mathematical model with delayed infection in pest community. It is shown that there exists a certain value of delay, say T* such that for T &gt; T* the system exhibits global stability towards disease-free equilibrium. But for T &lt; T* system exhibits stability switches of the endemic equilibrium. Finally, some conclusive remarks have been drawn through numerical simulations of the model. © 2007 Elsevier Inc. All rights reserved.

Mathematical Biosciences

Role of latency period in viral infection: A pest control model

The object of the present paper is to study an integrated pest management (IPM) problem in an agroecosystem (paddy-fish culture) through mathematical modeling and analysis, where release of sterile males and spraying of pesticide have been used as control measures for pest population. Using optimal analysis of the model, we have shown that restricted and proper use of control measures might enhance the crop production of the system in an economically viable way. The paper also considers the vulnerability of the underlined ecosystem due to the effect of temperature on the pest growth. Using Liapunov-like function, we have found out a suitable range of temperature, where this IPM strategy remains effective. Some important remarks have finally been made on the basis of numerical simulation. © 2006 Elsevier Inc. All rights reserved.

An improved integrated pest management model under 2-control parameters (sterile male and pesticide)

This paper deals with the mathematical modeling of pest management under viral infection (i.e. using viral pesticide) and analysis of its essential mathematical features. As the viral infection induces host lysis which releases more virus into the environment, on the average 'κ' viruses per host, κ∈(1,∞), the 'virus replication parameter' is chosen as the main parameter on which the dynamics of the infection depends. We prove that there exists a threshold value κ0 beyond which the endemic equilibrium bifurcates from the free disease one. Still for increasing κ values, the endemic equilibrium bifurcates towards a periodic solution. We further analyse the orbital stability of the periodic orbits arising from bifurcation by applying Poor's condition. A concluding discussion with numerical simulation of the model is then presented. © 2005 Elsevier Ltd. All rights reserved.

Pest control through viral disease: Mathematical modeling and analysis

The paper deals with the effect of pesticide on the pest affecting a paddy field and also on the fishes living in the soil water. An objective function is formed in terms of projected loss of individual fish, the projected profit of killing a pest and the cost of spraying insecticide for a finite time period. Next the corresponding pest management model is formed by considering the growth equations of paddy, pest, and fish of the soil water. The model is analyzed by control theoretic means to determine the threshold limits of pesticide to be used. Lastly, the optimal value of pesticide is obtained. It is shown that the optimal value lies within the threshold limits. Finally corresponding to this optimal threshold value of pesticide, the optimal values of the biomass of paddy and that of the fish of the soil water are also obtained. The results are finally verified numerically by special choice of parameters compatible with the model. © World Scientific Publishing Company.

Journal of Biological Systems

A new approach to pest management problem

We construct representations of certain direct limit Lie groups G=lim Gn via direct limits of Zuckerman derived functor modules of the groups Gn. We show such direct limits exist when the degree of cohomology can be held constant, and discuss some examples for the groups Sp(p,<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∞</mo></math>) and SO(2p,<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∞</mo></math>), relating to the discrete series and ladder representations. We show that our examples belong to the "admissible" class of Ol'shanskii, and also discuss the globalizations of the Harish-Chandra modules obtained by the derived functor construction. The representations constructed here are the first ones in cohomology of non-zero degree for direct limits of non-compact Lie groups.&nbsp;