Kunal Chakraborty

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.

Optimal control of harvest and bifurcation of a prey―predator model with stage structure

Abstract This paper describes a prey–predator model with stage structure for prey. The adult prey and predator populations are harvested in the proposed system. The dynamic behavior of the model system is discussed. It is observed that singularity induced bifurcation phenomenon is appeared when variation of the economic interest of harvesting is taken into account. State feedback controller is incorporated to stabilize the model system in case of positive economic interest. Harvesting of prey and predator population are used as controls 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. The Pontryagin’s maximum principle is used to characterize the optimal controls. The optimality 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.

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.

Combined harvesting of a stage structured prey–predator model incorporating cannibalism in competitive environment

In this paper, we propose a prey-predator system with stage structure for predator. The proposed system incorporates cannibalism for predator populations in a competitive environment. The combined fishing effort is considered as control used to harvest the populations. The steady states of the system are determined and the dynamical behavior of the system is discussed. Local stability of the system is analyzed and sufficient conditions are derived for the global stability of the system at the positive equilibrium point. The existence of the Hopf bifurcation phenomenon is examined at the positive equilibrium point of the proposed system. We consider harvesting effort as a control parameter and subsequently, characterize the optimal control parameter in order to formulate the optimal control problem under the dynamic framework towards optimal utilization of the resource. Moreover, the optimal system is solved numerically to investigate the sustainability of the ecosystem using an iterative method with a Runge-Kutta fourth-order scheme. Simulation results show that the optimal control scheme can achieve sustainable ecosystem. Results are analyzed with the help of graphical illustrations.

On non-selective harvesting of a multispecies fishery incorporating partial closure for the populations

HighlightsWe describe a prey-predator fishery model incorporating partial closure for the populations.The criterion for the existence of interior equilibrium point, local and global stability of the system are derived.The existence of bionomic equilibrium is discussed.The optimal system is derived and solved numerically using an iterative method with Runge-Kutta fourth-order scheme.Numerical simulations are carried out to verify the analytically proved results. A prey-predator type fishery model incorporating partial closure for the populations is described in this paper. The proposed model deals with a problem of non-selective harvesting of a prey-predator system in which both the prey and the predator species obey logistic law of growth. The predator dependent Beddington DeAngelis type functional response is taken into consideration. Dynamic behavior of the system is analyzed. Partial closure for the populations is considered as a controlling instrument to regulate the harvesting of the populations. A dynamic framework towards the optimal utilization of the resource is developed using Pontryagins maximum principle. The optimal system is numerically solved using an iterative method with Runge-Kutta fourth order scheme. Simulation results show that the optimal control scheme can achieve sustainable ecosystem. Results are analyzed with the help of graphical illustrations.

REGULATION OF A PREY–PREDATOR FISHERY INCORPORATING PREY REFUGE BY TAXATION: A DYNAMIC REACTION MODEL

This paper, describes a prey–predator fishery model incorporating prey refuge. The proposed model reflecting the dynamic interaction between the net economic revenue and the fishing effort used to harvest the prey species in the presence of predation and a suitable tax. The steady states of the system are determined and the dynamic behavior of the model system is discussed. The occurrence of Hopf bifurcation of the proposed model system is examined through considering density-dependent mortality for the predator as bifurcation parameter. The optimal taxation policy is formulated and solved with the help of Pontryagins maximal principle. The objective of the paper is to maximize the monetary social benefit as well as prevent the predator species from extinction, keeping the ecological balance. Results are illustrated with the help of numerical examples.

A comparison of CO2 dynamics and air-water fluxes in a river-dominated estuary and a mangrove-dominated marine estuary

The fugacity of CO2 [fCO2 (water)] and air-water CO2 flux were compared between a river-dominated anthropogenically disturbed open estuary, the Hugli and a comparatively pristine mangrove dominated semi-closed marine estuary, the Matla on the east coast of India. Annual mean salinity of the Hugli Estuary (≈ 7.1) was much less compared to the Matla Estuary (≈ 20.0). All the stations of the Hugli Estuary were highly supersaturated with CO2 (annual mean ~ 2200 µatm) whereas, the Matla was marginally oversaturated (annual mean ~ 530 µatm). During the post-monsoon season, the outer station of the Matla Estuary was under saturated with respect to CO2 and acted as a sink. The annual mean CO2 emission from the Hugli Estuary (32.4 mol C m-2 yr-1) was 14 times higher than the Matla Estuary (2.3 mol C m-2 yr-1). CO2 efflux rate from the Hugli Estuary has increased drastically in the last decade, which is attributed to increased runoff from the river-dominated estuary.

Economic perspective of marine reserves in fisheries: A bioeconomic model

The present paper describes a prey-predator type fishery model with prey dispersal in a two-patch environment, one of which is a free fishing zone and other is protected zone. The objective of the paper is to maximize the net economic revenue earn from the fishery through implementing the sustainable properties of the fishery to keep the ecological balance. Biological measures are introduced to increase the understanding of the mechanisms in the bioeconomic system. The importance of marine reserve is analyzed through the obtained results of the numerical simulations of proposed model system. The results depict that reserves will be most effective when coupled with harvesting controls in adjacent fisheries. The paper also incorporates the induced cost and premium from establishing a marine protected area in a fishery. It is found that premium of marine protected area (MPA) increases with the increasing size of the reserve. Results are analyzed with the help of graphical illustrations.

Modeling and analysis of a marine plankton system with nutrient recycling and diffusion

This article describes a nutrient-phytoplankton-zooplankton system with nutrient recycling in the presence of toxicity. We have studied the dynamical behavior of the system with delayed nutrient recycling in the first part of the article. Uniform persistent of the system is examined. In the second part of the article, we have incorporated diffusion of the plankton population to the system and dynamical behavior of the system is analyzed with instantaneous nutrient recycling. The condition of the diffusion driven instability is obtained. The conditions for the occurrence of Hopf and Turing bifurcation critical line in a spatial domain are derived. Variation of the system with small periodicity of diffusive coefficient has been studied. Stability condition of the plankton system subject to the periodic diffusion coefficient of the zooplankton is derived. It is observed that nutrient-phytoplankton-zooplankton interactions are very complex and situation specific. Moreover, we have obtained different exciting results, ranging from stable situation to cyclic oscillatory behavior may occur under different favorable conditions, which may give some insights for predictive management.

An ecological perspective on marine reserves in prey–predator dynamics

This paper describes a prey–predator type fishery model with prey dispersal in a two-patch environment, one of which is a free fishing zone and other is a protected zone. The existence of possible steady states, along with their local stability, is discussed. A geometric approach is used to derive the sufficient conditions for global stability of the system at the positive equilibrium. Relative size of the reserve is considered as control in order to study optimal sustainable yield policy. Subsequently, the optimal system is derived and then solved numerically using an iterative method with Runge–Kutta fourth-order scheme. Numerical simulations are carried out to illustrate the importance of marine reserve in fisheries management. It is noted that the marine protected area enables us to protect and restore multi-species ecosystem. The results illustrate that dynamics of the system is extremely interesting if simultaneous effects of a regulatory mechanism like marine reserve is coupled with harvesting effort. It is observed that the migration of the resource, from protected area to unprotected area and vice versa, is playing an important role towards the standing stock assessment in both the areas which ultimately control the harvesting efficiency and enhance the fishing stock up to some extent.