shahnaz bathul - Academia.edu (original) (raw)

Papers by shahnaz bathul

Research paper thumbnail of Mathematical Foundations of Computer Science

Research paper thumbnail of A Comparative Study of Non Linear and Linear System of Three SpeciesEcological Model

The present paper is devoted to Numerical investigation of three species ecological model with a ... more The present paper is devoted to Numerical investigation of three species ecological model with a Prey ( 1 N ), a predator ( 2 N ) and a competitor ( ) 3 N to the both prey ( 1 N ) and Predator ( 2 N ). In addition to that, the species are provided with alternative food. The model is characterized by a set of first order non-linear ordinary differential equations. All the eight equilibrium points of the model are identified and study the comparison between linear system of equations and nonlinear system of equations using Numerical example with mat lab simulation.

Research paper thumbnail of Special functions and complex variables

Research paper thumbnail of Long-term Demographic Dynamics along a Mediterranean Urban Gradient: A Reflection on Population Growth and Socio-environmental Systems

International Journal of Ecological Economics and Statistics, 2018

Population trends were investigated over a time period between 1871 and 2016 at the municipal spa... more Population trends were investigated over a time period between 1871 and 2016 at the municipal spatial scale in Latium, central Italy, a region encompassing the Rome's metropolitan area. A multidimensional approach incorporating principal component analysis and hierarchical clustering was used to assess long-term demographic dynamics along the metropolitan gradient in Rome. Results of this study pointed out the importance of internal migrations along elevation (mountainous areas to lowlands) and urban gradients (inner Rome to countryside) that progressively consolidate the socioeconomic divide between accessible (and economically-dynamic) agricultural districts and peripheral (mountain) districts. Spatial divides in population density have grown over time, possibly driven by synergic processes of peri-urbanization and coastalization. Our study highlights the contribution of exploratory analysis of demographic dynamics to empirical knowledge of long-term socioeconomic transformati...

Research paper thumbnail of Effect of slip on peristaltic pumping of a hyperbolic tangent fluid in an inclined asymmetric channel

ABSTRACT The present paper deals with the study of the effect of slip on peristaltic flow of a Hy... more ABSTRACT The present paper deals with the study of the effect of slip on peristaltic flow of a Hyperbolic Tangent fluid in an inclined asymmetric channel under long wavelength and low Reynolds number assumptions. The non linear governing equations are solved using the regular perturbation method. Analysis has been carried out in the presence of velocity and slip conditions. Expressions for stream function, pressure gradient and pressure rise coefficients are derived. The effect of various parameters on the pumping phenomenon is discussed with the help of graphs.

Research paper thumbnail of Effect of internal-heating on weakly non-linear stability analysis of Rayleigh–Bénard convection under g-jitter

International Journal of Non-Linear Mechanics, 2013

In this paper, we study the combined effect of internal-heating and time-periodic gravity modulat... more In this paper, we study the combined effect of internal-heating and time-periodic gravity modulation on thermal instability in a viscous fluid layer, heated from below. The time-periodic gravity modulation, considered in this problem can be realized by vertically oscillating the fluid layer. A weak non-linear stability analysis has been performed by using power series expansion in terms of the amplitude of gravity modulation, which is assumed to be small. The Nusselt number has been obtained in terms of the amplitude of convection which is governed by the non-autonomous Ginzburg-Landau equation derived for the stationary mode of convection. Effects of various parameters such as internal Rayleigh number, Prandtl number, and amplitude and frequency of gravity modulation have been analysed on heat transport. It is found that the response of the convective system to the internal Rayleigh number is destabilizing. Further, it is found that the heat transport can be controlled by suitably adjusting the external parameters of the system.

Research paper thumbnail of Electrical Potential, Mass Transport and Velocity Distribution of Electro-osmotic Flow in a Nanochannel by Incorporating the Variation of Dielectric Constant of Aqueous Electrolyte Solution

Cornell University - arXiv, Feb 19, 2016

We consider a coupled system of Navier-Stokes, Maxwell-Stefan and Poisson-Boltzmann equations by ... more We consider a coupled system of Navier-Stokes, Maxwell-Stefan and Poisson-Boltzmann equations by incorporating the variation of dielectric constant, which governs the electro-osmotic flow in nanochannel, describing the evolution of the velocity, concentration and potential fields of dissolved constituents in an aqueous electrolyte solution. We apply the finite difference technique to solve one and two dimensional systems of these equations. The solutions give an extremely accurate prediction of the dielectric constant for a variety of salts and a wide range of concentrations.

Research paper thumbnail of The Study of Difference of Efficiency of a Bank in a Rural Area and a Bank in a Town

Queueing theory and the analysis of the efficiency of the queueing system of a bank in different ... more Queueing theory and the analysis of the efficiency of the queueing system of a bank in different locations. Starting with the historical back ground and important concepts of the queueing theory, we measured the effectiveness of banks in rural and town areas by using probability curves of arrivals and waiting times.

Research paper thumbnail of Thermal Instability and Heat Transport in a Micropolar Fluid in the presence of Suction-Injection-Combination and Internal Heat Generation

The effect of vertical throughflow or suction–injection– combination (SIC) on the thermal instabi... more The effect of vertical throughflow or suction–injection– combination (SIC) on the thermal instability and heat transport in a micropolar fluid layer in the presence of internal heat generation, between permeable horizontal boundaries, when heated uniformly from below, is examined analytically. Using normal mode analysis, the eigenvalue is obtained for free–free velocity and isothermal temperature boundary condition on the spin-vanishing boundaries. Heat transport in the system is studied using Ginzburg Landau equation derived from Lorenz equations. The influence of various micropolar fluid parameters, Peclet number and internal Rayleigh number has been analysed by plotting the graphs.

Research paper thumbnail of Bionomic equilibrium of a three species model with optimal harvesting for the first prey

This paper deals with the selective harvesting of a preypredator system based on catch-per-unit-e... more This paper deals with the selective harvesting of a preypredator system based on catch-per-unit-effort hypothesis. Here all the three species fulfill the logistic growth law. Using Routh-Hurwitz criteria the existence of positive equilibrium point and its local and global stability is studied. The extant of bionomic equilibrium has been established and the optimal harvesting policy is studied using Pontryagins maximum principle. Analysis of the effects of harvesting of one the prey species is worked out in this paper. The analytical results are illustrated using numerical examples. AMS Subject Classification: 34D23, 91B76, 49J15.

Research paper thumbnail of A Case Study of Bank Queueing Model

This paper deals with the Queueing theory and the analysis of queueing system by using probabilit... more This paper deals with the Queueing theory and the analysis of queueing system by using probability curves. Starting with the basis of the distributions and important concepts of queueing theory, probability curves of Gamma distribution are used to analyse the banking service.

Research paper thumbnail of Time Delayed Analysis of Two Prey One Predator Ecological Model

International Journal of Ecological Economics and Statistics, 2018

This paper is devoted to an analytical study of a three species ecological model in which a preda... more This paper is devoted to an analytical study of a three species ecological model in which a predator is preying on the other two species which are mutually helping each other. In addition to that, all the three species are provided with an alternate food. A time delay is proposed in the interaction between the prey1 and the predator. The model is characterized by a system of integro-differential equations. The local stability is established at the interior equilibrium points. A suitable Lyapunov function is constructed to discuss the global asymptotic stability of the system. Finally a suitable set of parameters are identified and numerical simulation is carried out for which the delay arguments further stabilizes the system.

Research paper thumbnail of A Two Species Amensalism Model with Time Delay

International Journal of Ecological Economics and Statistics, 2015

The Present paper investigates a two species amensalism model with time delay. The model is repre... more The Present paper investigates a two species amensalism model with time delay. The model is represented by a couple of integro-differential equations. All possible equilibrium points are identified and their stability is discussed and is supported by suitable numerical examples using Matlab simulation .Further, global stability of the system is established by constructing a suitable Lyapunov’s function. The dynamics delay effect is studied with suitable Numerical examples.

Research paper thumbnail of Stability Analysis of Discrete-Time Bidirectional Associative Neural Networks with Hysteresis

Artificial Intelligent Systems and Machine Learning, 2012

In this paper continuous hysteretic neuron model has been studied and discretized the model by us... more In this paper continuous hysteretic neuron model has been studied and discretized the model by using proper approximation. Sufficient condition for global exponential stability of a unique equilibrium is obtained. Motivated by the applications of bidirectional associative neural networks with hysteresis in artificial neural networks, we studied the global dynamics of bidirectional associative neural network with hysteresis. The hysteretic neural network model is envisaged to be efficient and robust for various applications such as medical image processing, military data processing, etc. Hysteretic feedback control phenomena also manage glucose vs. lactose utilization preference in Escherichia coli and ensure unidirectional cell-cycle progression in eukaryotes. The result improves the earlier publications due to the bidirectional associative memory and it removes restrictions on the neutral delays. Our result shows that after discretization of hysteretic neuron models, the network co...

Research paper thumbnail of Stochastic and Diffusion Analysis of a Modified Leslie-Gower Model with Holling Type Functional Responses

International Journal of Ecological Economics and Statistics, 2019

The present work is related to the modified Leslie-Gower prey-predator model with diffusion effec... more The present work is related to the modified Leslie-Gower prey-predator model with diffusion effect. Conditions for the occurrence of stability at the interior equilibrium point are addressed using appropriate techniques for the reduced linearized system. Comparison between the deterministic and stochastic model are carried out. The effects of diffusion on the proposed system are analyzed. Simulations are carried out using suitable parameter values to strengthen the validity of the system.

Research paper thumbnail of Dynamics of a Prey and Two Predators System with Time Delay

International journal of pure and applied mathematics, 2018

The aim of this paper is to study the prey-predator system with delay effects. Initially, the pos... more The aim of this paper is to study the prey-predator system with delay effects. Initially, the positive equilibrium point of the proposed system is derived and its local stability is discussed using Routh-Hurwitz criterion. A well suited Lyapunov function describes the global asymptotic stability of the system. To preserve the stability of the system without violating its properties the length of time delay is estimated and some important conclusions are made at the end.

Research paper thumbnail of Least Square and Gauss Jordan Methods applied to fit the Dielectric data Vs. Concentration of Ionic Liquids

Ionic liquids are not only prominent for life but also important in chemical and biological indus... more Ionic liquids are not only prominent for life but also important in chemical and biological industry. Starting from DebyeHuckel theory of electrolytes, a rich progress in theoretical and experimental studies was made over the past hundred years. K.S.Pitzer developed a set of equations for six coefficients on the basis of virial theorem and Statistical thermodynamics , starting from the Debye-Huckel model .He adopted calculations of Guggenheim ,with the modification of Scatchard ,Hammer and Bronsted to incorporate specific interactions, for electrolytic state ,in which terms related to short range forces between ions were partly accounted. However the very significant lowering of Dielectric constant of solvent continuum on addition of electrolyte, as modeled by Glueckauff was not considered in their work. We report a better & more general method of possible correction starting with the dielectric data of Brahmajirao, P.S.K.M.Rao and Sastry by using Least Square and Gauss Jordan Techn...

Research paper thumbnail of Creeping Flow from a Circular Orifice in a Plane Wall

C reeping motion of a viscous fluid through a circular hole in a plane when the boundary conditio... more C reeping motion of a viscous fluid through a circular hole in a plane when the boundary conditions on the plane are prescribed is studied and integral solution for the problem presented. Infinite series representation and asymptotic expansion are developed to understand the far field behavior of the fluid flow and to facilitate numerical work.

Research paper thumbnail of The effect of probabilities of arrivals with time in a bank

This paper deals with queueing theory and analysis of probability curves of pure birth model. Sta... more This paper deals with queueing theory and analysis of probability curves of pure birth model. Starting with historical back grounds and important concepts of queueing theory, we obtained a relation to find time "t" where we get highest probability for arrivals which follow Poission probability distribution.

Research paper thumbnail of An Optimal Harvesting Policy for the Two Prey Species in a Three Species Ecological Model

International Journal of Ecology & Development, 2017

In this study, a system of non-linear ordinary differential equations emerging from a three speci... more In this study, a system of non-linear ordinary differential equations emerging from a three species ecological model with one predator feeding on two prey species is considered for stability analysis and optimal harvesting. Here we harvested both the prey species based on catch-per-unit-effort hypothesis and all the three species grows logistically. The positive equilibrium point of the proposed system has been identified and its local stability is analyzed using Routh-Hurwitz criteria. A suitable Lyapunov function is constructed and the global stability of the system is studied. The conditions for the occurrence of local and global stability are also provided. Also the bioeconomic equilibrium has been derived and by virtue of Pontryagin’s maximum principle, the optimal harvesting policy and its effects are discussed. Numerical simulation provides justification to the theoretical results.

Research paper thumbnail of Mathematical Foundations of Computer Science

Research paper thumbnail of A Comparative Study of Non Linear and Linear System of Three SpeciesEcological Model

The present paper is devoted to Numerical investigation of three species ecological model with a ... more The present paper is devoted to Numerical investigation of three species ecological model with a Prey ( 1 N ), a predator ( 2 N ) and a competitor ( ) 3 N to the both prey ( 1 N ) and Predator ( 2 N ). In addition to that, the species are provided with alternative food. The model is characterized by a set of first order non-linear ordinary differential equations. All the eight equilibrium points of the model are identified and study the comparison between linear system of equations and nonlinear system of equations using Numerical example with mat lab simulation.

Research paper thumbnail of Special functions and complex variables

Research paper thumbnail of Long-term Demographic Dynamics along a Mediterranean Urban Gradient: A Reflection on Population Growth and Socio-environmental Systems

International Journal of Ecological Economics and Statistics, 2018

Population trends were investigated over a time period between 1871 and 2016 at the municipal spa... more Population trends were investigated over a time period between 1871 and 2016 at the municipal spatial scale in Latium, central Italy, a region encompassing the Rome's metropolitan area. A multidimensional approach incorporating principal component analysis and hierarchical clustering was used to assess long-term demographic dynamics along the metropolitan gradient in Rome. Results of this study pointed out the importance of internal migrations along elevation (mountainous areas to lowlands) and urban gradients (inner Rome to countryside) that progressively consolidate the socioeconomic divide between accessible (and economically-dynamic) agricultural districts and peripheral (mountain) districts. Spatial divides in population density have grown over time, possibly driven by synergic processes of peri-urbanization and coastalization. Our study highlights the contribution of exploratory analysis of demographic dynamics to empirical knowledge of long-term socioeconomic transformati...

Research paper thumbnail of Effect of slip on peristaltic pumping of a hyperbolic tangent fluid in an inclined asymmetric channel

ABSTRACT The present paper deals with the study of the effect of slip on peristaltic flow of a Hy... more ABSTRACT The present paper deals with the study of the effect of slip on peristaltic flow of a Hyperbolic Tangent fluid in an inclined asymmetric channel under long wavelength and low Reynolds number assumptions. The non linear governing equations are solved using the regular perturbation method. Analysis has been carried out in the presence of velocity and slip conditions. Expressions for stream function, pressure gradient and pressure rise coefficients are derived. The effect of various parameters on the pumping phenomenon is discussed with the help of graphs.

Research paper thumbnail of Effect of internal-heating on weakly non-linear stability analysis of Rayleigh–Bénard convection under g-jitter

International Journal of Non-Linear Mechanics, 2013

In this paper, we study the combined effect of internal-heating and time-periodic gravity modulat... more In this paper, we study the combined effect of internal-heating and time-periodic gravity modulation on thermal instability in a viscous fluid layer, heated from below. The time-periodic gravity modulation, considered in this problem can be realized by vertically oscillating the fluid layer. A weak non-linear stability analysis has been performed by using power series expansion in terms of the amplitude of gravity modulation, which is assumed to be small. The Nusselt number has been obtained in terms of the amplitude of convection which is governed by the non-autonomous Ginzburg-Landau equation derived for the stationary mode of convection. Effects of various parameters such as internal Rayleigh number, Prandtl number, and amplitude and frequency of gravity modulation have been analysed on heat transport. It is found that the response of the convective system to the internal Rayleigh number is destabilizing. Further, it is found that the heat transport can be controlled by suitably adjusting the external parameters of the system.

Research paper thumbnail of Electrical Potential, Mass Transport and Velocity Distribution of Electro-osmotic Flow in a Nanochannel by Incorporating the Variation of Dielectric Constant of Aqueous Electrolyte Solution

Cornell University - arXiv, Feb 19, 2016

We consider a coupled system of Navier-Stokes, Maxwell-Stefan and Poisson-Boltzmann equations by ... more We consider a coupled system of Navier-Stokes, Maxwell-Stefan and Poisson-Boltzmann equations by incorporating the variation of dielectric constant, which governs the electro-osmotic flow in nanochannel, describing the evolution of the velocity, concentration and potential fields of dissolved constituents in an aqueous electrolyte solution. We apply the finite difference technique to solve one and two dimensional systems of these equations. The solutions give an extremely accurate prediction of the dielectric constant for a variety of salts and a wide range of concentrations.

Research paper thumbnail of The Study of Difference of Efficiency of a Bank in a Rural Area and a Bank in a Town

Queueing theory and the analysis of the efficiency of the queueing system of a bank in different ... more Queueing theory and the analysis of the efficiency of the queueing system of a bank in different locations. Starting with the historical back ground and important concepts of the queueing theory, we measured the effectiveness of banks in rural and town areas by using probability curves of arrivals and waiting times.

Research paper thumbnail of Thermal Instability and Heat Transport in a Micropolar Fluid in the presence of Suction-Injection-Combination and Internal Heat Generation

The effect of vertical throughflow or suction–injection– combination (SIC) on the thermal instabi... more The effect of vertical throughflow or suction–injection– combination (SIC) on the thermal instability and heat transport in a micropolar fluid layer in the presence of internal heat generation, between permeable horizontal boundaries, when heated uniformly from below, is examined analytically. Using normal mode analysis, the eigenvalue is obtained for free–free velocity and isothermal temperature boundary condition on the spin-vanishing boundaries. Heat transport in the system is studied using Ginzburg Landau equation derived from Lorenz equations. The influence of various micropolar fluid parameters, Peclet number and internal Rayleigh number has been analysed by plotting the graphs.

Research paper thumbnail of Bionomic equilibrium of a three species model with optimal harvesting for the first prey

This paper deals with the selective harvesting of a preypredator system based on catch-per-unit-e... more This paper deals with the selective harvesting of a preypredator system based on catch-per-unit-effort hypothesis. Here all the three species fulfill the logistic growth law. Using Routh-Hurwitz criteria the existence of positive equilibrium point and its local and global stability is studied. The extant of bionomic equilibrium has been established and the optimal harvesting policy is studied using Pontryagins maximum principle. Analysis of the effects of harvesting of one the prey species is worked out in this paper. The analytical results are illustrated using numerical examples. AMS Subject Classification: 34D23, 91B76, 49J15.

Research paper thumbnail of A Case Study of Bank Queueing Model

This paper deals with the Queueing theory and the analysis of queueing system by using probabilit... more This paper deals with the Queueing theory and the analysis of queueing system by using probability curves. Starting with the basis of the distributions and important concepts of queueing theory, probability curves of Gamma distribution are used to analyse the banking service.

Research paper thumbnail of Time Delayed Analysis of Two Prey One Predator Ecological Model

International Journal of Ecological Economics and Statistics, 2018

This paper is devoted to an analytical study of a three species ecological model in which a preda... more This paper is devoted to an analytical study of a three species ecological model in which a predator is preying on the other two species which are mutually helping each other. In addition to that, all the three species are provided with an alternate food. A time delay is proposed in the interaction between the prey1 and the predator. The model is characterized by a system of integro-differential equations. The local stability is established at the interior equilibrium points. A suitable Lyapunov function is constructed to discuss the global asymptotic stability of the system. Finally a suitable set of parameters are identified and numerical simulation is carried out for which the delay arguments further stabilizes the system.

Research paper thumbnail of A Two Species Amensalism Model with Time Delay

International Journal of Ecological Economics and Statistics, 2015

The Present paper investigates a two species amensalism model with time delay. The model is repre... more The Present paper investigates a two species amensalism model with time delay. The model is represented by a couple of integro-differential equations. All possible equilibrium points are identified and their stability is discussed and is supported by suitable numerical examples using Matlab simulation .Further, global stability of the system is established by constructing a suitable Lyapunov’s function. The dynamics delay effect is studied with suitable Numerical examples.

Research paper thumbnail of Stability Analysis of Discrete-Time Bidirectional Associative Neural Networks with Hysteresis

Artificial Intelligent Systems and Machine Learning, 2012

In this paper continuous hysteretic neuron model has been studied and discretized the model by us... more In this paper continuous hysteretic neuron model has been studied and discretized the model by using proper approximation. Sufficient condition for global exponential stability of a unique equilibrium is obtained. Motivated by the applications of bidirectional associative neural networks with hysteresis in artificial neural networks, we studied the global dynamics of bidirectional associative neural network with hysteresis. The hysteretic neural network model is envisaged to be efficient and robust for various applications such as medical image processing, military data processing, etc. Hysteretic feedback control phenomena also manage glucose vs. lactose utilization preference in Escherichia coli and ensure unidirectional cell-cycle progression in eukaryotes. The result improves the earlier publications due to the bidirectional associative memory and it removes restrictions on the neutral delays. Our result shows that after discretization of hysteretic neuron models, the network co...

Research paper thumbnail of Stochastic and Diffusion Analysis of a Modified Leslie-Gower Model with Holling Type Functional Responses

International Journal of Ecological Economics and Statistics, 2019

The present work is related to the modified Leslie-Gower prey-predator model with diffusion effec... more The present work is related to the modified Leslie-Gower prey-predator model with diffusion effect. Conditions for the occurrence of stability at the interior equilibrium point are addressed using appropriate techniques for the reduced linearized system. Comparison between the deterministic and stochastic model are carried out. The effects of diffusion on the proposed system are analyzed. Simulations are carried out using suitable parameter values to strengthen the validity of the system.

Research paper thumbnail of Dynamics of a Prey and Two Predators System with Time Delay

International journal of pure and applied mathematics, 2018

The aim of this paper is to study the prey-predator system with delay effects. Initially, the pos... more The aim of this paper is to study the prey-predator system with delay effects. Initially, the positive equilibrium point of the proposed system is derived and its local stability is discussed using Routh-Hurwitz criterion. A well suited Lyapunov function describes the global asymptotic stability of the system. To preserve the stability of the system without violating its properties the length of time delay is estimated and some important conclusions are made at the end.

Research paper thumbnail of Least Square and Gauss Jordan Methods applied to fit the Dielectric data Vs. Concentration of Ionic Liquids

Ionic liquids are not only prominent for life but also important in chemical and biological indus... more Ionic liquids are not only prominent for life but also important in chemical and biological industry. Starting from DebyeHuckel theory of electrolytes, a rich progress in theoretical and experimental studies was made over the past hundred years. K.S.Pitzer developed a set of equations for six coefficients on the basis of virial theorem and Statistical thermodynamics , starting from the Debye-Huckel model .He adopted calculations of Guggenheim ,with the modification of Scatchard ,Hammer and Bronsted to incorporate specific interactions, for electrolytic state ,in which terms related to short range forces between ions were partly accounted. However the very significant lowering of Dielectric constant of solvent continuum on addition of electrolyte, as modeled by Glueckauff was not considered in their work. We report a better & more general method of possible correction starting with the dielectric data of Brahmajirao, P.S.K.M.Rao and Sastry by using Least Square and Gauss Jordan Techn...

Research paper thumbnail of Creeping Flow from a Circular Orifice in a Plane Wall

C reeping motion of a viscous fluid through a circular hole in a plane when the boundary conditio... more C reeping motion of a viscous fluid through a circular hole in a plane when the boundary conditions on the plane are prescribed is studied and integral solution for the problem presented. Infinite series representation and asymptotic expansion are developed to understand the far field behavior of the fluid flow and to facilitate numerical work.

Research paper thumbnail of The effect of probabilities of arrivals with time in a bank

This paper deals with queueing theory and analysis of probability curves of pure birth model. Sta... more This paper deals with queueing theory and analysis of probability curves of pure birth model. Starting with historical back grounds and important concepts of queueing theory, we obtained a relation to find time "t" where we get highest probability for arrivals which follow Poission probability distribution.

Research paper thumbnail of An Optimal Harvesting Policy for the Two Prey Species in a Three Species Ecological Model

International Journal of Ecology & Development, 2017

In this study, a system of non-linear ordinary differential equations emerging from a three speci... more In this study, a system of non-linear ordinary differential equations emerging from a three species ecological model with one predator feeding on two prey species is considered for stability analysis and optimal harvesting. Here we harvested both the prey species based on catch-per-unit-effort hypothesis and all the three species grows logistically. The positive equilibrium point of the proposed system has been identified and its local stability is analyzed using Routh-Hurwitz criteria. A suitable Lyapunov function is constructed and the global stability of the system is studied. The conditions for the occurrence of local and global stability are also provided. Also the bioeconomic equilibrium has been derived and by virtue of Pontryagin’s maximum principle, the optimal harvesting policy and its effects are discussed. Numerical simulation provides justification to the theoretical results.