Search

In this article, we compare the two ecological services known as yield and resilience, for a tri-trophic food chain model consisting of a prey, an intermediate predator and a top predator. For this comparison process, we use both analytical and numerical techniques. It is shown that a variety of patterns are possible based on the intensity of efforts distributed among different trophic levels. Thus we may suggest that fishing down the food chain, as suggested by Pauly et al. (1998) is not bound to happen. Our analysis also shows that balancing the harvest between prey, intermediate predator and top predator could give more yield and stabilizing the ecosystem, than the selective harvesting of any one species. This balanced harvesting may not be a win–win situation for yield and resilience, but it could be a most favourable strategy to balance them. This research would help to correlate resilience with yield and determines the desirable selection of two policies, resilience maximizing yield or maximum sustainable yield to safeguard ecological communities.

The coronavirus disease 2019 (COVID-19) outbreak, due to SARS-CoV-2 (severe acute respiratory syndrome coronavirus 2), originated in Wuhan, China and is now a global pandemic. The unavailability of vaccines, delays in diagnosis of the disease, and lack of proper treatment resources are the leading causes of the rapid spread of COVID-19. The world is now facing a rapid loss of human lives and socioeconomic status. As a mathematical model can provide some real pictures of the disease spread, enabling better prevention measures. In this study, we propose and analyze a mathematical model to describe the COVID-19 pandemic. We have derived the threshold parameter basic reproduction number, and a detailed sensitivity analysis of this most crucial threshold parameter has been performed to determine the most sensitive indices. Finally, the model is applied to describe COVID-19 scenarios in India, the second-largest populated country in the world, and some of its vulnerable states. We also have short-term forecasting of COVID-19, and we have observed that controlling only one model parameter can significantly reduce the disease's vulnerability.

This paper describes a prey–predator type ecological model with infection in the prey populations. We also consider here a nonlinear functional response for disease transmission and a constant amount of refuge for the sound prey populations. The dynamical behavior of the mathematical model is described from the point of view of stability and bifurcation. A geometric method is also applied to establish the global asymptotic stability at the co-existence equilibrium point. Some computer simulation works have been presented to illustrate the theoretical results.

We propose and analyze a mathematical model for tuberculosis (TB) transmission to study the role of exogenous reinfection and endogenous reactivation. The model exhibits two equilibria: a disease free and an endemic equilibria. We observe that the TB model exhibits transcritical bifurcation when basic reproduction number R 0 = 1 . Our results demonstrate that the disease transmission rate β and exogenous reinfection rate α plays an important role to change the qualitative dynamics of TB. The disease transmission rate β give rises to the possibility of backward bifurcation for R 0 1 , and hence the existence of multiple endemic equilibria one of which is stable and another one is unstable. Our analysis suggests that R 0 1 may not be sufficient to completely eliminate the disease. We also investigate that our TB transmission model undergoes Hopf-bifurcation with respect to the contact rate β and the exogenous reinfection rate α . We conducted some numerical simulations to support our analytical findings.

In this paper, we proposed and analyzed a susceptible-infected-recovered (SIR) type epidemic model to investigate the effect of transport-related infectious diseases namely tuberculosis, measles, rubella, influenza, sexually transmitted diseases, etc. The existence and stability criteria of both the diseases include free equilibrium point and endemic equilibrium point which are established and the threshold parametric condition for which the system passes through a transcritical bifurcation is also obtained. Optimal control strategy for control parameters is formulated and solved both theoretically and numerically. Lastly, we not only illustrate our theoretical results through graphical illustrations but also computer simulation is used to show that our model would be a good model to study the SARS epidemic in 2003.

The world fisheries are mainly focused on the yield from fishing, and hence maximum sustainable yield (MSY) policy has become the most popular management tool. But this aim of maximum productivity, particularly in the multitrophic system, impacts the resilience, considering the potential of the system to absorb the disturbances applied to the system. Due to harvesting effort used on different individual trophic levels, the trade-offs and synergies between yield and resilience of a tri-trophic food chain system have been studied. In Holling type II functional response, the enrichment of carrying capacity can destabilize the system and may adversely affect the system resilience. In the case of prey harvesting, for both Holling type I and II responses, the existence of MSY depends on some parametric restrictions, but resilience maximizing yield (RMY) always exists. For predator harvesting, only RMY may be a favorable harvesting strategy as MSY does not exist. But for the top predator harvesting with both for Holling type I and II functional responses, both the pair MSY and RMY exist. Either MSY, or RMY, or a balanced approach between them, maybe implemented depending upon the requirements of more yield, or more resilience, or balancing them.

In recent years, the important challenging issues are biological invasions to biodiversity, overexploitation and extinction of several species. Invasive species often alter the workings of ecosystems around the globe. In the present research, our aim is to study the possible impacts of invasive species on the sustainable use of native exploited species. To address this issue, cleaning operation is introduced to protect biodiversity and recover stocks. It is found that presence of invasive species reduces the maximum sustainable yield (MSY) of native species. In case of prey–predator system, prey harvesting at MSY level causes the extinction of predator species, but extinction effort increases with the cleaning effort. It is also observed that when independent efforts are applied on both the prey and predator species, global maximum sustainable total yield (MSTY) exists and it increases as the cleaning effort increases. In all the cases it is found that appropriate cleaning effort may reduce the impacts of biological invasions on the sustainable use of resources.

This paper investigates some economic consequences of implementing ecotourism which is inhabited by two interacting stocks: a stock of prey (fish) and a stock of predator (marine mammals, e.g., whale, seals, shark, etc.). The prey species is targeted for commercial fishing while the predator species is not subject to fishing but is a potential basis to implement the eco-tourism. We specifically address two key issues. First, we examine the consequences of entrance fee on the total benefit (fishing + eco-tourism) depending on the sensitivity of the number of tourists to the predator abundance level. It is found that total benefit always increases with entrance fee until the entrance fee reaches its optimal value. Second, we examine the effects on interacting species from attempts to reach MSY in prey species. It is found that total benefit always higher than the benefit coming from fishing only even if the fishing is done at MSY level.

Balancing the representation of marine ecosystems and threatened resources is critical for optimizing total rent (fishing+ecotourism), achieving the convention on sustainable use of resources. Comprehensive marine zoning provides a useful tool with which these essential concepts are addressed. A bioeconomic model is constructed, allowing three different aims to reserve, fishing, ecotourism, and total optimized rent, resource abundance, and the number of tourist changes. We provide ecological justification for the introduction of three different zones in the dynamic resource model. We obtain the conditions of optimizing total rent if the resource has no movement rate and fast movement rate among three different zones. Apart from the analytical results for the models under consideration, we perform extensive numerical simulations to construct the relevant, optimized benefit, resource abundance, and some tourists change diagrams. Our analytical and supportive numerical findings reveal that reserve is always good for fishery's sustainable management in the presence of fishing and ecotourism. Finally, we discuss the ecological implication of the current investigation in the conclusion section.

A prey-predator model system is developed; specifically the disease is considered into the prey population. Here the prey population is taken as pest and the predators consume the selected pest. Moreover, we assume that the prey species is infected with a viral disease forming into susceptible and two-stage infected classes, and the early stage of infected prey is more vulnerable to predation by the predator. Also, it is assumed that the later stage of infected pests is not eaten by the predator. Different equilibria of the system are investigated and their stability analysis and Hopf bifurcation of the system around the interior equilibriums are discussed. A modified model has been constructed by considering some alternative source of food for the predator population and the dynamical behavior of the modified model has been investigated. We have demonstrated the analytical results by numerical analysis by taking some simulated set of parameter values.

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.

This paper describes a prey–predator fishery model incorporating partial closure for the populations. It is assumed that the predator population partially dependent on a logistically growing resource with Beddington–De Angelis type functional response. The proposed system also reflects the dynamic interaction between the net economic revenue and the fishing effort used to harvest the populations. The steady states of the system are determined and the dynamic behavior of the system is discussed. The existence of Hopf bifurcation phenomenon is examined at the interior equilibrium point of the proposed system. We have adopted partial closure for the populations as a controlling instrument to regulate the harvesting of the populations. A dynamic framework towards the optimal utilization of the resource is developed using Pontryagin's 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 a sustainable ecosystem. Results are analyzed with the help of graphical illustrations.

A prey-predator system with nonlinear incidence rate of the disease in prey is described.The dynamical behavior of the system is investigated at each equilibrium point.Sufficient conditions are derived for the uniformly strong persistence of the system.Results are obtained to extract the criteria for disease persistence and extinction. A prey-predator system with disease in prey is proposed. The proposed system is an extension of the model analyzed by Bhattacharyya and Mukhopadhyay (2011) which did not consider the density of fish population as a dynamic variable which significantly influence the dynamics of the system. The coexistence equilibria of the system is determined and the dynamic behavior of the system is investigated around coexistence equilibria. Incidence rate of the disease is considered as a bifurcation parameter to examine the occurrence of Hopf bifurcation in the neighborhood of the co-existing equilibria. Sufficient conditions are derived for the global stability of the system around coexistence equilibria. Uniform strong persistence of the system is discussed in order to ensure long-term survival of the species. The obtained results are useful to extract the criteria for disease extinction and persistence. Finally, some numerical simulations are given to verify the analytical results, and the system is analyzed through graphical illustrations.

This paper describes the dynamics of an infectious disease transmission modified Leslie–Gower type eco-epidemiological system in both deterministic and stochastic fluctuating environments with harvesting. The dynamics of the deterministic system is extensively investigated around coexistence equilibria. Sufficient conditions are derived for local and global stability of the system. The existence of Hopf bifurcation phenomenon is examined around interior equilibria of the system. Subsequently, we use normal form method and center manifold theorem to examine the nature of the Hopf bifurcation. The obtained results are useful to extract the criteria for disease extinction and control perspective. Later, a white noise term is incorporated to the system to describe the dynamics of the system in stochastic fluctuating environment. Sufficient conditions are derived for the mean square stability (MSS) of the system which can be used to evaluate necessary conditions for the asymptotic MSS and a threshold condition between asymptotic MSS and unstable system. Finally, some numerical simulations are carried out, and graphical illustrations are given in support of the analytical results obtained in both deterministic and stochastic systems.

In this present study, we propose and analyze a mathematical model of tuberculosis (TB) transmission considering social awareness effects during an epidemic. Possible equilibrium points of the model are investigated, and their stability criterion is discussed. Basic reproduction number R0 of the model is obtained through the next-generation matrix method. It has been shown that the infection-free equilibrium is locally stable when R0 1. The global asymptotic stability of the endemic equilibrium P* is verified by constructing a suitable Lyapunov function. The possibility of two endemic equilibria when R0 R 0 = 1 also verified using center manifold theory. The media awareness parameters influence the occurrence of backward bifurcation. An optimal control problem is framed considering a media intervention parameter u(t) as a control variable. The existence and characterization of the optimal solution to the problem solved analytically. Optimal media control strategy with accessible media intervention cost gradually reduce the prevalence of the disease. In addition to our analytical results, several numerical simulations are also performed to make the analysis more significant. A short discussion on the media guided transmission characteristic of the disease, obtained from our investigation is conducted at last.

In this paper, we consider a two-predator one-prey system to determine the feedback of exploitation in individual as well as joint population levels. As balancing yield with resilience is highly essential for the conservation of species in the marine ecosystem, here we measure both the maximum sustainable yield (MSY) and resilience simultaneously. Then we investigate both the trade-offs and synergies among maximum yield, conservation, and resilience that emerge from different harvesting plans. It is found that for single species harvesting, a prey species-oriented system is capable of producing more yield in compare to any predator-oriented system but for resilience, a prey species-oriented system is far behind the others. In the case of joint harvesting of all the species, it is observed that the first predator-oriented system has a better ability to absorb the disturbances than the other cases. The correlation between yield and resilience at the MSY level is studied in all the cases. It is further observed that the increase of intraspecific competition in the predator decreases the risk of sustainability. In this way, this study may be helpful for fishery management to fulfill their goals without affecting the ecosystem's health in the long run.

This paper illustrates the effects of exploitation for yield maximization in prey–predator system where two predators are competing over same resource. Harvesting the prey (bottom trophic level) for yield maximization causes the extinction of the predators, but exploitation of either predator at MSY level becomes biologically sustainable policy. We show that harvesting multiple trophic levels including the prey species for total yield maximization may be sustainable under equal harvesting effort, but predator species goes to extinction under independent harvesting efforts. We also establish that neither predator goes to extinction if both predator trophic levels (i.e., excluding the prey species) are exploited under independent effort at maximum level; though either predator may be driven to extinction under combined harvesting effort.

This paper deals with the surplus production models of Verhulst-Schaefer and Gompertz-Fox that are applied to the European Hake fishery to develop static as well as dynamic frameworks to investigate the sustainability properties of the stock, optimal utilization of the resource and the management of the fishery. To illustrate the approach through which long-run sustainability can achieve an optimum exploitation of the fishery, a conventional economic model is combined with a biological population growth model to develop a deterministic bioeconomic model. The parameters of the bioeconomic model are estimated using empirical data of catch and effort of the European Hake fishery. Standard reference points are analyzed and tax policies are introduced to attain the standard reference points. In order to achieve maximum profit from the fishery, optimal steady state solutions are determined for separate discount rates.

This paper describes a multiple delayed modified Leslie–Gower type predator–prey system with a strong Allee effect in prey population growth. Non-selective effort is used to harvest the population. The dynamical characteristics of the delay induced system are rigorously studied using mathematical tools. The existence of coexistence equilibria is ensured, and the dynamic behavior of the system is investigated around coexistence equilibria. Uniform strong persistence and permanence of the system are discussed in order to ensure long-term survival of the species. The stability of the delay preserved system is investigated. Sufficient conditions are derived for local and global stability of the system. The existence of Hopf bifurcation phenomenon is examined around interior equilibria of the system. Subsequently, we use normal form method and center manifold theorem to examine the nature of the Hopf bifurcation. Finally, numerical simulations are carried out to validate the analytical findings.

This paper examines the application of maximum sustainable yield (MSY) policy in ecosystem and indicates when the ecosystem based fisheries management approach is required for conservation purpose. To describe the possible impacts of applying global MSY policy in an ecosystem, we have considered both specialist and generalist prey–predator models with different fishing efforts. It is found that harvesting both prey and predator species in specialist prey–predator systems, to achieve global maximum sustainable total yield (MSTY) under independent efforts, will cause the extinction of the predator species. In contrast, the global MSTY may exist in a generalist prey–predator system. If global MSTY does not exist, then it is due to the extinction of predator species. Hence, the prey species never goes to extinction under independent efforts and this scenario is quite different from the one found under combined harvesting effort.

This paper deals with a prey–predator system in the presence of some alternative food to predator and harvesting of prey species. Instead of the usual harvesting function based on catch-per-unit-effort (CPUE) hypothesis, we have considered an alternative functional form to overcome some of the unrealistic features of the earlier function. In absence of exploitation, it is observed that alternative source of food to the predator has a negative effect on the growth of the prey species. Moreover, in the presence of alternative food maximum sustainable yield (MSY) level of prey exploitation may not prevent the extinction of the predator species. Hence, to protect the predator species from extinction a control instrument, tax is imposed on the landed fish and the fishing effort is taken as dynamic variable. Optimal tax policy is obtained using Pontryagin's maximum principle. Some numerical simulations are given to verify some of our analytical results and optimal and suboptimal paths are obtained.

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 Pontryagin's 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.

In this paper, we make a systematic analysis of the dynamics of a predator–prey system with type-II functional response, in which the predator growth rate is affected by the presence of a super predator. The main aim of this research is to study the consequences of the presence of a super predator on the system dynamics. The existence and stability of the different possible equilibrium points are studied, and we conclude that the maximum consumption rate of a super predator plays a key role in determining the eventual state of the ecosystem. A detailed bifurcation analysis is carried out through numerical simulations, and we observe that theoretically it is possible to control the dynamics of the system by manipulating the consumption rate of the super predator.

In this paper, we propose and analyze an ecological system consisting of pest and its natural enemy as predator. Here we also consider the role of infection to the pest population and the presence of some alternative source of food to the predator population. We analyze the dynamics of this system in a systemic manner, study the dependence of the dynamics on some vital parameters and discuss the global behavior and controllability of the proposed system. The investigation also includes the use of pesticide control to the system and finally we use Pontryagin’s maximum principle to derive the optimal pest control strategy. We also illustrate some of the key findings using numerical simulations.

Though the maximum sustainable yield (MSY) approach has been legally adopted for the management of world fisheries, it does not provide any guarantee against from species extinction in multispecies communities. In the present article, we describe the appropriateness of the MSY policy in a Holling–Tanner prey–predator system with different types of functional responses. It is observed that for both type I and type II functional responses, harvesting of either prey or predator species at the MSY level is a sustainable fishing policy. In the case of combined harvesting, both the species coexist at the maximum sustainable total yield (MSTY) level if the biotic potential of the prey species is greater than a threshold value. Further, increase of the biotic potential beyond the threshold value affects the persistence of the system.

This article investigates the effects of reaching the maximum sustainable yield ( MSY ) in prey–predator systems where the prey population follows logistic law of growth. Two different models are proposed: (i) first model involves linear prey–predator interaction and intraspecific competition among predator populations, and (ii) the second one is a ratio-dependent prey–predator system. In the first model, our results suggest that the introduction of intraspecific competition among predator population has important consequences for the fishing to reach MSY from prey species and maximum sustainable total yield (MSTY) for combined harvesting of both prey and predator species. On the otherhand, in the second model, our results suggest that though the harvesting of prey species at MSY level shall be guaranteed the coexistence of both the species, but the combined harvesting of both the species at MSTY level may cause extinction of the predator species. However, for both the models, predator harvesting at MSY level may be a sustainable fishing policy. Therefore, based on our results we can conclude that MSY (or MSTY) policy in prey–predator systems in nature are not likely to fit requirements of Conservation of Biological Diversity ( CBD, 1992 ) in all cases.

In the present paper, we study a dynamic reaction model in which (i) the predator is provided with an alternative food in addition to the prey species, (ii) the predator is harvested, and (iii) a tax is imposed to regulate the system. The existence of possible steady states along with their local as well as global stability is discussed for both the exploited and unexploited systems. Boundedness of the system is also discussed. It is seen that the system undergoes a Hopf bifurcation by the addition of alternative prey and the criteria for the Hopf-bifurcation is also discussed. Optimal tax policy is discussed using Pontryagin’s maximal principle. Finally, some numerical simulations are given to show the consistency with theoretical analysis.

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 Pontrygin's 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.

In the present paper, we develop a simple two species prey-predator model in which the predator is partially coupled with alternative prey. The aim is to study the consequences of providing additional food to the predator as well as the effects of harvesting efforts applied to both the species. It is observed that the provision of alternative food to predator is not always beneficial to the system. A complete picture of the long run dynamics of the system is discussed based on the effort pair as control parameters. Optimal augmentations of prey and predator biomass at final time have been investigated by optimal control theory. Also the short and large time effects of the application of optimal control have been discussed. Finally, some numerical illustrations are given to verify our analytical results with the help of different sets of parameters.

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.

This paper describes a prey–predator fishery 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. We have incorporated state feedback controller to stabilize the model system in the case of positive economic interest. Fishing effort used to harvest the adult prey and predator populations is used 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.

In this paper, we analyze the dynamical behaviour of a bioeconomic model system using differential algebraic equations. The system describes a prey–predator fishery with prey dispersal in a two-patch environment, one of which is a free fishing zone and other is a protected zone. It is observed that a singularity-induced bifurcation phenomenon appears when a variation of the economic interest of harvesting is taken into account. We have incorporated a state feedback controller to stabilize the model system in the case of positive economic interest. A discrete-type gestational delay of predators is incorporated, and its effect on the dynamical behaviour of the model is analyzed. The occurrence of Hopf bifurcation of the proposed model with positive economic profit is shown in the neighbourhood of the coexisting equilibrium point through considering the delay as a bifurcation parameter. Finally, some numerical simulations are given to verify the analytical results, and the system is analyzed through graphical illustrations.

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 Pontryagin's 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.

An SIR epidemic model with time delay, information variable and saturated incidence rate, where the susceptibles are assumed to satisfy the logistic equation and the incidence term, is of saturated form with the susceptibles. This model exhibits two bifurcations, one is transcritical bifurcation and the other is Hopf bifurcation. The local and global stability of endemic equilibrium is also discussed. Finally, numerical simulations are carried out to explain the mathematical conclusions.

In this paper, we analyze the dynamics of a delayed predator–prey system in the presence of harvesting. This is a modified version of the Leslie–Gower and Holling-type II scheme. The main result is given in terms of local stability, global stability, influence of harvesting and bifurcation. Direction of Hopf bifurcation and the stability of bifurcating periodic solutions are also studied by using the normal form method and center manifold theorem.

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.

Within the framework of a general equilibrium model we study the long-run dynamics of a prey-predator model in the presence of an alternative prey. Our results show that sustainability, i.e. a positive value of the population in the long run, essentially depends on individual harvesting efforts and digesting factors relative to alternative prey. A detailed bifurcation analysis evidences the richness of possible long-run dynamics. Our model clearly shows that the role of an alternative prey must be taken into consideration when studying prey-predator dynamics.

This study examines the impact of the creation of marine protected areas (MPAs), from both economic and biological perspectives. In particular, we examine the effects of protected patches and harvesting on resource populations. We conclude that protected patches are an effective means of conserving resource populations, even though extinction cannot be prevented in all cases. We discuss the dynamic optimization of a harvest policy by choosing E(t), the harvesting effort, as the dynamic variable. We also discuss the optimal equilibrium harvest policy and explain the biological and bioeconomic interpretations of the results.

In this paper, we have considered a prey–predator-type fishery model with Beddington–DeAngelis functional response and selective harvesting of predator species. We have established that when the time delay is zero, the interior equilibrium is globally asymptotically stable provided it is locally asymptotically stable. It is also shown that the time delay can cause a stable equilibrium to become unstable and even a switching of stabilities. Lastly, some numerical simulations are carried out.

In recent years overexploitation and collapse of several biological resources have been seen in fisheries, forestry and wildlife. This is due to unconventional and indiscrete harvesting of the resources. Consequently there is much current concern to find principles for the control and management of resources, particularly in those circumstances where both prey and predator species are being exploited. This work illustrates the use of harvesting efforts as a control to obtain strategies for the control of a prey–predator system with Holling type III functional response.

The dynamics of a prey-predator system, where predator population has two stages, juvenile and adult with harvesting are modelled by a system of delay differential equation. Our analysis shows that, both the delay and harvesting effort may play a significant role on the stability of the system. Numerical simulations are given to illustrate the results.

The excessive and unsustainable exploitation of marine resources has to led to the promotion of marine reserve as a fisheries management tool. In this paper we study a prey–predator system in a two-patch environment: one accessible to both prey and predators (patch 1) and the other one being a refuge for the prey (patch 2). The prey refuge (patch 2) constitutes a reserve zone of prey and fishing is not permitted, while the unreserved zone area is an open-access fishery zone. The existence of possible steady states, along with their local and global stability, is discussed. We then examine the possibilities of the existence of bionomic equilibrium. An optimal harvesting policy is given using Pontryagin’s maximum principle.

This paper deals with the problem of a ratio-dependent prey-predator model with combined harvesting. The existence of steady states and their stability are studied using eigenvalue analysis. Boundedness of the exploited system is examined. We derive conditions for persistence and global stability of the system. The possibility of existence of bionomic equilibria has been considered. The problem of optimal harvest policy is then solved by using Pontryagin’s maximal principle.

This paper deals with a prey–predator model incorporating a prey refuge and harvesting of the predator species. A mathematical analysis shows that prey refuge plays a crucial role for the survival of the species and that the harvesting effort on the predator may be used as a control to prevent the cyclic behaviour of the system. The optimal harvest policy is studied, and the solution is derived in the equilibrium case by using Pontryagin's maximal principle. Some numerical simulations are taken to illustrate the established results.

In this paper, we have considered a prey-predator model with Holling type of predation and harvesting of mature predator species. Our study shows that, using the harvesting effort as control, it is possible to break the cyclic behavior of the system and drive it to a required steady state. It is also shown that the time delay can cause a stable equilibrium to become unstable and even switching of stabilities. We have also discussed the effects of bycatch of the immature species. Lastly, some numerical simulations are carried out.