Prey-predator Fishery Research Articles

Dynamics of a predator–prey fishery model with birth pulse, impulsive releasing and harvesting on prey

Dynamics of a predator–prey fishery model with birth pulse, impulsive releasing and harvesting on prey

Optimal harvest for predator–prey fishery models with variable price and marine protected area

In this paper, we propose a predator–prey fishery model with prey harvesting, variable price and marine protected area. We assume the price changes faster than other processes such as population growth and predation, and get a slow fast Ordinary Differential Equation (ODE) system. A simplified three-dimensional model is obtained by using approximate aggregation methods. The results show that there are two main scenarios, one is the depletion of fish stocks due to overfishing by fishermen, which is called a “catastrophic” equilibrium; and the other is a stable and sustainable fishery equilibrium. In order to avoid the extinction of fish, we consider establishing marine protected area (MPA) near fishing areas where fishermen only catch the prey. The results of the study provide the conditions for the establishment of MPA, allowing us to avoid the extinction of prey populations and establish sustainable fisheries. As another possibility, we consider increasing taxes to discourage overfishing by fishermen. In addition, when the tax revenue increases, the optimal harvest strategy is obtained. The optimal policy ensures the sustainable development of fisheries and maximizes the interests of fishermen.

Stability and Hopf bifurcation of an intraguild prey-predator fishery model with two delays and Michaelis-Menten type predator harvest.

In this paper, we have proposed and investigated an intraguild predator-prey system incorporating two delays and a harvesting mechanism based on the Michaelis-Menten principle, and it was assumed that the two species compete for a shared resource. Firstly, we examined the properties of the relevant characteristic equations to derive sufficient conditions for the asymptotical stability of equilibria in the delayed model and the existence of Hopf bifurcation. Using the normal form method and the central manifold theorem, we analyzed the stability and direction of periodic solutions arising from Hopf bifurcations. Our theoretical findings were subsequently validated through numerical simulations. Furthermore, we explored the impact of harvesting on the quantity of biological resources and examined the critical values associated with the two delays.

Study of dynamical behaviors of harvested stage-structured predator–prey fishery model with fear effect on prey under interval uncertainty

The lack of clarity in many parameters used in ecological models is due to the constant changes in nature. The fundamental predator–prey model discussed in this article takes into account several ecological parameters as parametric-functional interval numbers. It discusses the dynamic characteristics of the model within the context of an uncertain environment. To account for this uncertainty, the model incorporates the concepts of bionomic equilibria and maximum sustainable yield (MSY). The article explores the optimal harvesting policy by applying Pontryagin’s maximum principle in an imprecise and uncertain environment. The proposed imprecise model formulation includes two additional variables: the influence of fear on prey population growth rate and the harvesting of prey and juvenile predator populations. The system of interval differential equations governs the interactions between these imprecise species, allowing for mathematical analysis of the proposed system. The aim of this study is to evaluate the robustness of an imprecise prey–predator model within an unpredictable environment. The article concludes by presenting rigorous numerical simulations that support all the analytical findings.

An ecological‐economic fishery model: Maximizing the societal benefit through an integrated approach of fishing and ecotourism

One of the essential tasks of the modern fishery is to integrate fishery‐based tourism with it as a part of sustainable development. Here, we propose and study a two‐species fishery model where the ecological and economic concepts are integrated. The objectives are to demonstrate the effect of fishing tax and tourist entrance fees to stabilize fishery dynamics and maximize revenue generation. For this, we consider a predator–prey fishery model coupled with dynamic harvesting and time‐dependent fish price variation. The prey fish is commercially harvested, and the predatory fish is used for recreation purposes of the visitors with an entry fee. A fishing tax is levied on the fishers to restrict the overfishing of this renewable resource. We provide the local and global stability conditions of different equilibrium points of the system and unveil the broader dynamics through bifurcation analysis. Pontryagin's maximum principle shows the existence of an optimal fishing tax that maximizes overall revenue generation.

Complex dynamics of two prey–predator harvesting models with prey refuge and interval‐valued imprecise parameters

In view of the phenomena that species in nature are affected by the complexity of the ecosystem, some key biological parameters have certain inaccuracy or uncertainty and part of the prey hides in the shelter from predation. Moreover, different species exhibit different living habits, so it is of certain interest to investigate species‐related imprecise biological parameters. Besides, from the perspective of scientific and reasonable exploitation and utilization of fishery resources, two types of harvesting strategies are introduced into the system. Firstly, the dynamic properties of the system guided by continuous harvesting strategy are analyzed, which demonstrates the impact of refuge rate, imprecision indexes and harvesting efforts on the system dynamics. Meanwhile, the bio‐economic equilibrium is discussed, and the optimal harvesting strategy is obtained. Secondly, the complex dynamics of the system induced by weighted harvesting activity are analyzed. The predator‐extinction periodic solution and coexistent order‐1 periodic solution is discussed according to different weights and thresholds. Finally, the correctness of the obtained results and the feasibility of the harvesting strategy are verified through numerical simulation in MATLAB. The results of this study provide a reference for the scientific and reasonable exploitation and utilization of prey–predator fishery resources in the imprecise parameter environment.

Controlling the chaos and bifurcations of a discrete prey-predator model

<abstract><p>In this paper, we explore the existence of fixed points, local dynamics at fixed points, bifurcations and chaos of a discrete prey-predator fishery model with harvesting. More specifically, it is proved that, for all involved parameters, the model has trivial fixed point, but it has semitrivial and interior fixed points under definite parametric condition(s). We study the local behavior at fixed points by applying the theory of linear stability. Furthermore, it is shown that flip bifurcation does not occur at semitrivial and trivial fixed points, but that the model undergoes Neimark-Sacker bifurcation at interior fixed point. It is also proved that, at interior fixed point, the model undergoes the flip bifurcation. By using a feedback control strategy, the chaos control is also examined. Finally, to illustrate the theoretical findings, detailed numerical simulations are provided.</p></abstract>

Combined influences of environmental enrichment and harvesting mediate rich dynamics in an intraguild predation fishery system

Natural resources and its enrichment activities are often being associated with the paradoxical effects in ecology such as in predator–prey and fishery models. Several studies have shown that an increase in resources can cause the “Paradox of Enrichment” i.e., the destabilization of certain species population as the system is enriched. This situation is rather contradictory to our logical intuition. Parallel with this view, harvesting effort and the intensity of enrichment activities are among crucial factors in shaping the sustainability of fishery activities and natural resources. This is because an extreme harvesting effort could lead to resource depletion and threaten vulnerable marine ecosystems. In this work, we investigate the joint effects of enrichment and harvesting activities in an intraguild predator–prey fishery model that incorporates the Michaelis–Menten type of harvesting of predatory fishes. We study the existence and stability of the steady states in this system analytically and conduct numerical simulations, together with bifurcation analysis to verify our analytical findings. Our co-dimension one bifurcation analysis shows that the system can exhibit local bifurcations such as transcritical, saddle–node, subcritical, and supercritical Hopf bifurcations. Co-dimension two bifurcation analysis further demonstrates the existence of global dynamics through the appearance of Generalized Hopf (GH) and Zero-Hopf (ZH) bifurcations, which occur due to the interactions of distinct local bifurcations in this ecological system. The limit point bifurcation of cycles that emanates from the GH point has a bounded parameter space region in which tristability phenomenon (between stable steady states and stable limit cycles) occurs. In a scenario where the resources are enriched in the presence of harvesting activity, bistability emerges with the long-term outcomes lead to the exclusion of either prey or predator population, depending on the initial conditions of species. Based on these findings, we propose another interpretation to the “Paradox of Enrichment” as postulated in ecological theories: this predator–prey fishery system can be destabilized through an alternative stable states phenomenon, which results in the extinction of certain species as the ecosystem is further enriched.

The Study of a Predator-Prey Model with Fear Effect Based on State-Dependent Harvesting Strategy

In presence of predator population, the prey population may significantly change their behavior. Fear for predator population enhances the survival probability of prey population, and it can greatly reduce the reproduction of prey population. In this study, we propose a predator-prey fishery model introducing the cost of fear into prey reproduction with Holling type-II functional response and prey-dependent harvesting and investigate the global dynamics of the proposed model. For the system without harvest, it is shown that the level of fear may alter the stability of the positive equilibrium, and an expression of fear critical level is characterized. For the harvest system, the existence of the semitrivial order-1 periodic solution and positive order- q ( q ≥ 1 ) periodic solution is discussed by the construction of a Poincaré map on the phase set, and the threshold conditions are given, which can not only transform state-dependent harvesting into a cycle one but also provide a possibility to determine the harvest frequency. In addition, to ensure a certain robustness of the adopted harvest policy, the threshold condition for the stability of the order- q periodic solution is given. Meanwhile, to achieve a good economic profit, an optimization problem is formulated and the optimum harvest level is obtained. Mathematical findings have been validated in numerical simulation by MATLAB. Different effects of different harvest levels and different fear levels have been demonstrated by depicting figures in numerical simulation using MATLAB.

THE STABILITY OF THE CRITICAL POINTS OF THE GENERALIZED GAUSE TYPE PREDATOR-PREY FISHERY MODELS WITH PROPORTIONAL HARVESTING AND TIME DELAY

In the marine ecosystem, the time delay or lag may occur in the predator response function, which measures the rate of capture of prey by a predator. This is because, when the growth of the prey population is null at the time delay period, the predator’s growth is affected by its population and prey population densities only after the time delay period. Therefore, the generalized Gause type predator-prey fishery models with a selective proportional harvesting rate of fish and time lag in the Holling type II predator response function are proposed to simulate and solve the population dynamical problem. From the mathematical analysis of the models, a certain dimension of time delays in the predator response or reaction function can change originally stable non-trivial critical points to unstable ones. This is due to the existence of the Hopf bifurcation that measures the critical values of the time lag, which will affect the stabilities of the non-trivial critical points of the models. Therefore, the effects of increasing and decreasing the values of selective proportional harvesting rate terms of prey and predator on the stabilities of the non-trivial critical points of the fishery models were analysed. Results have shown that, by increasing the values of the total proportion of prey and predator harvesting denoted by qx Ex and qy Ey respectively, within the range 0.3102 ≤ qx Ex ≤ 0.9984 and 0.5049 ≤ qy Ey ≤ 0.5363, the originally unstable non-trivial critical points of the fishery models can be stable.

Dynamical behaviours of prey-predator fishery model with two reserved area for prey in the presence of toxicity and response function Holling type IV

This paper proposes a model with two type of preys and one predator of fishery with Holling type IV function response. The effect of harvesting was incorporated to both populations and thoroughly analysed. We study the dynamics as the prey-predator system of fishing in two fishing zones: a free zone and a reserved zone. The equilibrium points are calculated and the local and global stability conditions of the system are obtained. The local stability conditions were obtained by the Routh-Hurwitz criterion. In addition, the global stability of the coexistence equilibrium point is proved by defining an appropriate Lyapunov function. The optimal harvesting policy is discussed by using the Pontryagin’s Maximal Principle. Finally, numerical simulations are carried to verify the analytical results.

Dynamical behaviors and optimal harvesting of an intraguild prey-predator fishery model with Michaelis-Menten type predator harvesting

The present paper discusses the dynamics and optimal harvesting of an intraguild prey-predator fishery model by incorporating the nonlinear Michaelis-Menten type of harvesting in predator. To our knowledge, there is limited literature working on Michaelis-Menten type of harvesting in a three species intraguild model with variable carrying capacity. The carrying capacity is proportional to the density of biotic resource. The existence of the possible equilibria has been studied along with the stability criteria. We consider the impact of predator fish harvesting as the bifurcation parameter to analyze the long time behavior of the proposed system. From the economic perspective, bionomic equilibrium of the system is studied and optimal harvesting policy is derived with the assistance of Pontryagin Maximum Principle. Finally, numerical results are presented to verify our analytical results. It is shown that in the bifurcation diagrams, the system can exhibit transcritical and Hopf bifurcations in the neighborhood of coexistence equilibrium at low and relatively high level of predator harvesting respectively. Interestingly, the system enters to a bistable region where both the coexistence and predator-free equilibria can be stable depending on the initial values. This bistable behavior might be novel to the existing literature that studied intraguild models with variable carrying capacity. The objective of this study is to derive the optimal threshold for the predator harvesting that gives maximum financial profit while sustaining the fishery resources.

Dynamical behaviours of prey-predator fishery model with two reserved area for prey in the presence of toxicity and response function Holling type IV

This paper proposes a model with two type of preys and one predator of fishery with Holling type IV function response. The effect of harvesting was incorporated to both populations and thoroughly analysed. We study the dynamics as the prey-predator system of fishing in two fishing zones: a free zone and a reserved zone. The equilibrium points are calculated and the local and global stability conditions of the system are obtained. The local stability conditions were obtained by the Routh-Hurwitz criterion. In addition, the global stability of the coexistence equilibrium point is proved by defining an appropriate Lyapunov function. The optimal harvesting policy is discussed by using the Pontryagin’s Maximal Principle. Finally, numerical simulations are carried to verify the analytical results.

Harvesting of a prey-predator model fishery in the presence of competition and toxicity with two effort function

In this paper, we propose a predator-prey model with harvesting and reserved area for prey with the presence of competition and toxicity with two effort functions. First, we prove the boundedness of the solutions. Then, the existence is studied, as well as the local and global stability of the equilibria. Lyapunov proved this last with certain conditions. The optimal harvesting policy is discussed using the Maximum Principle of Pantryagin. Finally, we ensure our results by numerical simulations.

Optimal Harvesting of a Prey-Predator Fishery: An Overlapping Generations Analysis

Optimal Harvesting of a Prey-Predator Fishery: An Overlapping Generations Analysis

A Proposed Bio-Economic Model for the Prey-Predator Fishery Model with Harvesting: Toward Modeling Dynamics in Applied Mathematics

This study proposed and analyzed a mathematical model. The proposed model was used for studying the two-prey predator system dynamics in relation to the fishery model. One of the parameters that were considered involved the possible impact of harvesting on the target populations. In particular, ecological dynamics were studied relative to tilapia, cichlid, and the Nile perch as pre-predator systems. Through system non-dimensionalization, the study computed equilibrium points before obtaining the conditions for global and local instability. To achieve the local instability condition, the investigation used Routh-Hurwitz Criterion and eigenvalues approach. With appropriate Lyapunov function defined, the coexistence equilibrium was proved. In turn, there was the analysis of the bio-economic equilibrium before conducting numerical simulations to verify the analytical outcomes that the study obtained. From the numerical results, the study established that if tilapia and cichlid fishes are not overharvested, the three species are likely to coexist. The populations were observed to play a crucial role to the Nile perch population’s growth rate. Hence, the study’s proposed mathematical model proved informative in such a way that it paved the way for the fishery control management departments to avoid overharvesting tilapia and cichlid fishes.

Optimal harvesting of prey-predator fishery modeling in a two patch environment and harvesting in unprotected area

This article deals with the dynamics of prey and predator populations in a two patch environment, a protected area and an unprotected area for fishing. The prey disperses between the two patches and migrates easily. There are two predators, one is in the protected area and another is in the unprotected area. The predators cannot migrate. Both prey and predator in unprotected area are harvested with constant efforts. The dynamical behavior of the populations is stated as a system of differential equations. The existence of a positive equilibrium point and its stability are investigated. We discuss the local stability of the positive equilibrium point. The stable equilibrium point is then associated with optimal harvesting problems. Based on the analysis, we found that there exist a stable positive equilibrium point when there is no harvesting. For model with constant efforts for both prey and predator, we found that over fishing will maximize the profit but the predator in the unprotected area will be extinct. With the help of Pontryagin’s maximum principle in maximizing the present value of revenues, we found the extremal of the efforts that maximize the present value of revenues. This means that both prey and predator in the protected area as well as the prey and predator in the unprotected area are possibly coexist although the prey and the predator in the unprotected area are harvested with constant efforts. Some numerical simulations area given to confirm the result of analysis.

Impact of Harvesting on a Bioeconomic Predator-Prey Fishery Model Subject to Environmental Toxicant.

The present paper studies a predator-prey fishery model which incorporates the independent harvesting strategies and nonlinear impact of an anthropogenic toxicant. Both fish populations are harvested with different harvesting efforts, and the cases for the presence and non-presence of harvesting effort are discussed. The prey fish population is assumed to be infected by the toxicant directly which causes indirect infection to predator fish population through the feeding process. Each equilibrium of the proposed system is examined by analyzing the respective local stability properties. Dynamical behavior and bifurcations are studied with the assistance of threshold conditions influencing the persistence and extinction of both predator and prey. Bionomic equilibrium solutions for three possible cases are investigated with certain restrictions. Optimal harvesting policy is explored by utilizing the Pontryagin's Maximum Principle to optimize the profit while maintaining the sustainability of the marine ecosystem. Bifurcation analysis showed that the harvesting parameters are the key elements causing fishery extinction. Numerical simulations of bionomic and optimal equilibrium solutions showed that the presence of toxicant has a detrimental effect on the fish populations.

Harvesting in a toxicated intraguild predator–prey fishery model with variable carrying capacity

Latterly, variable carrying capacity in the predator–prey population models has been widely studied to provide more realistic understanding about population dynamics. In the present paper, we discuss the bio-economic harvesting of an intraguild fishery model where the logistic carrying capacities of both predator and prey fish populations are variable based on the shared biotic resource. Distinct from most fishery models, we incorporate the independent harvesting strategies on the fisheries with different harvesting efforts by assuming them to have different economic values. In our model, the prey fish population is assumed to be infected directly by an anthropogenic toxicant from the environment while the predator is infected indirectly when they feed on prey. We investigate the local stability of trivial equilibria with Routh–Hurwitz criterion and the global stability of non-trivial equilibrium by constructing a Lyapunov function. Bifurcation and numerical analyses are presented to show distinctive result that a Hopf bifurcation occurs with respect to harvesting parameter instead of resource enrichment parameter as in the literature. Bionomic equilibrium is explored with several possibilities and restrictions. The nontrivial bionomic equilibrium is found to depend critically on the resource density. Finally, the optimal harvesting policy of the three dimensional model is derived by utilizing Pontryagin Maximum Principle. From this study, harvesting parameter is a crucial factor for the stability of the intraguild system. The objective is to obtain the optimal sustainable yield that provides maximum monetary profit while conserving the marine ecosystem.

Modeling Dynamics of Prey-Predator Fishery Model with Harvesting: A Bioeconomic Model

A mathematical model is proposed and analysed to study the dynamics of two-prey one predator system of fishery model with Holling type II function response. The effect of harvesting was incorporated to both populations and thoroughly analysed. We study the ecological dynamics of the Nile perch, cichlid, and tilapia fishes as prey-predator system of lake Victoria fishery in Tanzania. In both cases, by nondimensionalization of the system, the equilibrium points are computed and conditions for local and global stability of the system are obtained. Condition for local stability was obtained by eigenvalue approach and Routh-Hurwitz Criterion. Moreover, the global stability of the coexistence equilibrium point is proved by defining appropriate Lyapunov function. Bioeconomic equilibrium is analysed and numerical simulations are also carried out to verify the analytical results. The numerical results indicate that the three species would coexist if cichlid and tilapia fishes will not be overharvested as these populations contribute to the growth rates of Nile perch population. The fishery control management should be exercised to avoid overharvesting of cichlid and tilapia fishes.