Soovoojeet Jana

Global dynamics and bifurcation in a stage structured prey–predator fishery model with harvesting

Abstract This paper describes a prey–predator model with stage structure for predator and selective harvesting effort on predator population. The Holling type II functional response function is taken into consideration. All the equilibria of the proposed system are determined and the behavior of the system is investigated near them. Local stability of the system is analyzed. Geometric approach is used to derive the sufficient conditions for global stability of the system. The occurrence of Hopf bifurcation of the model system in the neighborhood of the co-existing equilibrium point is shown through considering maximal relative increase of predation as bifurcation parameter. Fishing effort used to harvest predator population is considered as a control to develop a dynamic framework to investigate the optimal utilization of the resource, sustainability properties of the stock and the resource rent earned from the resource. Pontryagin’s maximum principle is used to characterize the optimal control. The optimal system is derived and then solved numerically using an iterative method with Runge–Kutta fourth-order scheme. Simulation results show that the optimal control scheme can achieve sustainable ecosystem.

A theoretical study on mathematical modelling of an infectious disease with application of optimal control

In this paper, we propose and analyze an epidemic problem which can be controlled by vaccination as well as treatment. In the first part of our analysis we study the dynamical behavior of the system with fixed control for both vaccination and treatment. Basic reproduction number is obtained in all possible cases and it is observed that the simultaneous use of vaccination and treatment control is the most favorable case to prevent the disease from being epidemic. In the second part, we take the controls as time dependent and obtain the optimal control strategy to minimize both the infected populations and the associated costs. All the analytical results are verified by simulation works. Some important conclusions are given at the end of the paper.

Dynamics of pest and its predator model with disease in the pest and optimal use of pesticide.

In this paper, we propose and analyze a prey-predator system. Here the prey population is taken as pest and the predators are those eat the pests. Moreover we assume that the prey species is infected with a viral disease forming into susceptible and infected classes and infected prey is more vulnerable to predation by the predator. The dynamical behavior of this system both analytically and numerically is investigated from the point of view of stability and bifurcation. Then we explicitly introduce a control variable for pest control into the analysis by considering the associated control cost. In the nonconstant control case, we use Pontrygins Maximum principle to derive necessary conditions for the optimal control of the pest. Then we demonstrated the analytical results by numerical analysis and characterized the effects of the parameter values on optimal strategy.

Original article: Global stability and bifurcation of time delayed prey-predator system incorporating prey refuge

This paper describes a prey-predator model with Holling type II functional response incorporating prey refuge. The equilibria of the proposed system are determined and the behavior of the system is investigated around equilibria. Density-dependent mortality rate for the predator is considered as bifurcation parameter to examine the occurrence of Hopf bifurcation in the neighborhood of the co-existing equilibrium point. Discrete-type gestational delay of predators is also incorporated on the system. The dynamics of the delay induced prey-predator system is analyzed. Delay preserving stability and direction of the system is studied. Global stability of the delay preserving system is shown. Finally, some numerical simulations are given to verify the analytical results, and the system is analyzed through graphical illustrations.

Stability and bifurcation analysis of a stage structured predator prey model with time delay

Abstract In this paper we proposed and analyzed a prey predator system with stage-structured for the predator population. A time delay is incorporated due to the gestation for the matured predator. All the possible non-negative equilibria are obtained and their local as well as global behavior are studied. Choosing delay as a bifurcation parameter, the existence of the Hopf bifurcation of the system has been investigated. Moreover, we use the normal form method and the center manifold theorem to examine the direction of the Hopf bifurcation and the nature of the bifurcating periodic solution. Some numerical simulations are given to support the analytical results. Some interesting conclusions are obtained from our analysis and it is given at the end of the paper.

Complex Dynamics of an SIR Epidemic Model with Saturated Incidence Rate and Treatment

This paper describes a traditional SIR type epidemic model with saturated infection rate and treatment function. The dynamics of the model is studied from the point of view of stability and bifurcation. Basic reproduction number is obtained and it is shown that the model system may possess a backward bifurcation. The global asymptotic stability of the endemic equilibrium is studied with the help of a geometric approach. Optimal control problem is formulated and solved. Some numerical simulation works are carried out to validate our analytical results.

Mathematical analysis of an epidemic model with isolation and optimal controls

ABSTRACT In this article, we consider a susceptible–infectious–recovered (SIR) type epidemic model with some isolation to the susceptible, treatment for infectives, and vaccination to the newly recruited individuals. The basic reproduction number is obtained, and both the existence and stability of the disease-free and endemic equilibrium are discussed. We study the influence of isolation on infected individuals. We also study the existence of optimal control for both vaccination and treatment to minimize both the infected population and the costs required to control the disease. All the theoretical results are verified through numerical simulations and some of the key findings are given at the end of the article.

Dynamical Behavior of an Epidemic Model in a Fuzzy Transmission

In this paper, we have formulated a simple SIS type epidemic model in the presence of treatment control, and we have discussed the dynamical behavior of the system. The system is modified by considering both the disease transmission rate and the treatment function as fuzzy numbers, and also the fuzzy expected value of the infected individuals is calculated. Furthermore, the fuzzy basic reproduction number is investigated and a threshold condition of pathogen is obtained at which the system undergoes a transcritical bifurcation.

A Theoretical Approach on Controlling Agricultural Pest by Biological Controls

Abstract In this paper we propose and analyze a prey-predator type dynamical system for pest control where prey population is treated as the pest. We consider two classes for the pest namely susceptible pest and infected pest and the predator population is the natural enemy of the pest. We also consider average delay for both the predation rate i.e. predation to the susceptible pest and infected pest. Considering a subsystem of original system in the absence of infection, we analyze the existence of all possible non-negative equilibria and their stability criteria for both the subsystem as well as the original system. We present the conditions for transcritical bifurcation and Hopf bifurcation in the disease free system. The theoretical evaluations are demonstrated through numerical simulations.

Application of various control strategies to Japanese encephalitic: A mathematical study with human, pig and mosquito

Japanese encephalitis (JE) is a public health problem that threats the entire world today. Japanese Encephalitis virus (JEV) mostly became a threat due to the significant number of increase of susceptible mosquito vectors and vertebrate hosts in Asia by which around 70,000 cases and 10,000 deaths per year took place in children below 15 years of age. In this paper, a mathematical model of JE due to JEV from the vector source (infected mosquito) and two vertebrate hosts (infected human and infected pig) is formulated. The disease can be controlled by applying several control measures such as vaccination, medicine and insecticide to the JE infection causing species. The model has been formulated as an optimal control problem and has been solved using Pontryagins maximum principle. Also, the stability of the system has been studied with the help of basic reproduction number for disease free and endemic equilibrium. The results of fixed control for endemic equilibrium is presented numerically and depicted graphically. The effects of different control strategies on human, pig and mosquito has been analyzed using Runge-Kutta 4th order forward and backward techniques and presented thereafter graphically.