Prey Harvesting Research Articles

Dynamic analysis of a modified Leslie-Gower model with Michaelis-Menten prey harvesting and additive allee effect on predator population

In the current study, the dynamics of a Leslie-Gower model have been studied considering the Allee effect in the growth of the predator and the Michaelis-Menten type harvesting effect on the prey from biological and theoretical perspectives. Key findings highlight how these effects influence the existence of positive equilibria, with local stability properties and potential bifurcation behaviors examined using the center manifold theorem and linearization. Criteria for saddle-node bifurcation are established through Sotomayor’s theorem, while the direction and stability of Hopf bifurcation are explored via the first Lyapunov exponent. An optimal control model is developed to promote sustainable ecosystem development, utilizing the harvesting rate as a control parameter. Further, a few numerical simulations validate our significant findings using phase portrait diagrams.

Effects of extra resource and harvesting on the pattern formation for a predation system

We deal with a reaction–diffusion predation system with extra resource provided to predator and harvesting on prey. We first discuss the long-time behaviors of parabolic system, including the dissipativeness and persistence of positive solutions. It is shown that under certain constraints of harvesting, the quality of extra resource, the predation and transition rates of predator, the dissipativeness is compatible with persistence. Secondly, some properties of steady-state system are investigated, mainly including the existence of non-constant positive solutions, Turing and steady-state bifurcation phenomena. It is found that the extra resource, prey harvesting and diffusion have significant impacts on the pattern formations. Furthermore, some numerical simulations on Turing patterns and steady-state bifurcation solutions are performed to illustrate the theoretical analysis. We observe that when the quantity of extra resources is low, changes in the quality of extra resources can lead to significant changes in the spatial distribution of species, which is in sharp contrast to the case of high quantity of extra resource. Additionally, we conclude that different diffusion rates of predator can lead to different spatial patterns for the system.

Dynamics of a slow-fast Leslie-Gower predator-prey model with prey harvesting.

For the Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, the known results are on the saddle-node bifurcation and the Hopf bifurcation of codimensions 1, the Bogdanov-Takens bifurcations of codimensions 2 and 3, and on the cyclicity of singular slow-fast cycles. Here, we focus on the global dynamics of the model in the slow-fast setting and obtain much richer dynamical phenomena than the existing ones, such as global stability of an equilibrium; an unstable canard cycle exploding to a homoclinic loop; coexistence of a stable canard cycle and an inner unstable homoclinic loop; and, consequently, coexistence of two canard cycles: a canard explosion via canard cycles without a head, canard cycles with a short head and a beard and a relaxation oscillation with a short beard. This last one should be a new dynamical phenomenon. Numerical simulations are provided to illustrate these theoretical results.

COMPLEX DYNAMICAL BEHAVIORS OF A DISCRETE MODIFIED LESLIE–GOWER PREDATOR–PREY MODEL WITH PREY HARVESTING

Recently, the research on the modified Leslie–Gower model has become an appealing topic. Due to economic efficiency and the complexity of discrete models, we investigate a discrete modified Leslie–Gower predator–prey model with prey harvesting in this paper. The stability of fixed points and bifurcations of the interior fixed points are studied. According to bifurcation theory and normal forms, we derived the conditions of codimension 2 bifurcations occurred, including 1:1 strong resonance bifurcation and fold-flip bifurcation. These two bifurcations are unusual in bifurcation analysis on discrete systems. In addition, the continuation curves, bifurcation diagrams, and phase diagrams are used to demonstrate theoretical results. Our study shows the interesting dynamics of this model that are very different from the continuous one.

Dynamic Analysis of a Fractional-Modified Leslie-Gower Model with Predator Harvesting

This paper presents an investigation of a modified Leslie-Gower predator-prey model that incorporates fractional discrete-time Michaelis-Menten type prey harvesting. The analysis focuses on the topology of nonnegative interior fixed points, including their existence and stability dynamics. We derive conditions for the occurrence of flip and Neimark-Sacker bifurcations using the center manifold theorem and bifurcation theory. Numerical simulations, conducted with a computer package, are presented to demonstrate the consistency of the theoretical findings. Overall, our study sheds light on the complex dynamics that arise in this model and highlights the importance of considering fractional calculus in predator-prey systems with harvesting.

Spatiotemporal flow-induced instability of predator-prey model with Crowley-Martin functional response and prey harvesting.

Ecological systems can generate striking large-scale spatial patterns through local interactions and migration. In the presence of diffusion and advection, this work examines the formation of flow-induced patterns in a predator-prey system with a Crowley-Martin functional response and prey harvesting, where the advection reflects the unidirectional flow of each species migration (or flow). Primarily, the impact of diffusion and advection rates on the stability and the associated Turing and flow-induced patterns are investigated. The theoretical implication of flow-induced instability caused by population migration, mainly the relative migrations between prey and predator, is examined, and it also shows that Turing instability is the particular condition of flow-induced instability. The influence of the relative flow of both species and prey-harvesting effort on the emerging pattern is reported. Advection impacts a wide range of spatiotemporal patterns, including bands, spots, and a mixture of bands and spots in both harvested and unharvested dynamics. We also observe the diagonally bend-type banded patterns and straight-type banded patterns due to positive and negative relative flows, respectively. Here, the increasing relative flow increases the band length. The growing harvesting effort also decreases the band length, producing a thin band and a mixture of spots and bands due to the negative and positive relative flows, respectively. One exciting result observed here is that harvesting effort drives the flow-Turing and flow-Turing-Hopf instability into pure-flow instability.

Spatio-temporal dynamics in a delayed prey–predator model with nonlinear prey refuge and harvesting

In the current study, we introduce a model for the temporal and spatial interactions between prey and predator. The model incorporates a nonlinear refuge mechanism for prey, along with linear harvesting of prey and nonlinear harvesting for predators. Initially, we examined the well-posed nature of the model by analyzing the presence of all feasible equilibria and investigating the corresponding dynamics. Following that, we delve into the dynamics of the temporal model, focusing specifically on aspects such as uniform boundedness, permanence, and stability of viable equilibria. We demonstrate analytically that the proposed model experiences transcritical, saddle–node, Hopf, and Bogdanov–Takens bifurcations. It shows a variety of intricate dynamics involving Generalized Hopf and double Hopf. Then, discrete-time delay effects arising from the gestation of predator species have been incorporated into the temporal system. Hopf bifurcation for the delay parameter was detected in this investigation. Subsequently, we established conditions for self-diffusion instability and Turing instability in the spatiotemporal model, both with and without delay, employing the homogeneous Neumann boundary condition. Moreover, the discussion of sensitivity analysis (PRCC) serves to illustrate how crucial parameters influence the dynamics of the system. In addition, we conduct numerical simulations aiming to corroborate and validate the analytical results obtained.

Bifurcation analysis of a Leslie-type predator–prey system with prey harvesting and group defense

In this paper, we investigate a Leslie-type predator–prey model that incorporates prey harvesting and group defense, leading to a modified functional response. Our analysis focuses on the existence and stability of the system’s equilibria, which are essential for the coexistence of predator and prey populations and the maintenance of ecological balance. We identify the maximum sustainable yield, a critical factor for achieving this balance. Through a thorough examination of positive equilibrium stability, we determine the conditions and initial values that promote the survival of both species. We delve into the system’s dynamics by analyzing saddle-node and Hopf bifurcations, which are crucial for understanding the system transitions between various states. To evaluate the stability of the Hopf bifurcation, we calculate the first Lyapunov exponent and offer a quantitative assessment of the system’s stability. Furthermore, we explore the Bogdanov–Takens (BT) bifurcation, a co-dimension 2 scenario, by employing a universal unfolding technique near the cusp point. This method simplifies the complex dynamics and reveals the conditions that trigger such bifurcations. To substantiate our theoretical findings, we conduct numerical simulations, which serve as a practical validation of the model predictions. These simulations not only confirm the theoretical results but also showcase the potential of the model for predicting real-world ecological scenarios. This in-depth analysis contributes to a nuanced understanding of the dynamics within predator–prey interactions and advances the field of ecological modeling.

Response of Prey-Predator-Scavenger Interactions with the Inclusion of Prey Harvesting

Response of Prey-Predator-Scavenger Interactions with the Inclusion of Prey Harvesting

Bifurcation Analysis of a Holling–Tanner Model with Generalist Predator and Constant-Yield Harvesting

In this paper, we introduce constant-yield prey harvesting into the Holling–Tanner model with generalist predator. We prove that the unique positive equilibrium is a cusp of codimension 4. As the parameter values change, the system exhibits degenerate Bogdanov–Takens bifurcation of codimension 4. Using the resultant elimination method, we show that the positive equilibrium is a weak focus of order 2, and the system undergoes degenerate Hopf bifurcation of codimension 2 and has two limit cycles. By numerical simulations, we demonstrate that the system exhibits homoclinic bifurcation and saddle–node bifurcation of limit cycles as the parameters are varied. The main results show that constant-yield prey harvesting and generalist predator can lead to complex dynamic behavior of the model.

Dynamics of a discrete one‐predator two‐prey system with Michaelis–Menten‐type prey harvesting and prey refuge

In this paper, we propose and study a discretized one‐predator two‐prey system along with prey refuge and Michaelis–Menten‐type prey harvesting. The interaction among the species is considered as Holling type III functional response. Firstly, existence and local stability of all the fixed points are derived under certain parametric conditions. Furthermore, a special consideration is made to global asymptotic stability of the interior fixed point. Then, we have shown that the system undergoes different types of bifurcations including transcritical bifurcation, flip bifurcation, and Neimark–Sacker bifurcation by using center manifold theorem, bifurcation theory, and normal form method. Also, Feigenbaum's constant of the system is calculated. It is observed that both harvesting and refuge have a stabilizing effect on the system, and the stabilizing effect of harvesting dominates the stabilizing effect of refuge. Of most interest is the finding of coexisting attractors and multistability. In particular, optimal harvesting policy has been obtained by extension of Pontryagin's maximum principle to discrete system. Finally, some intriguing numerical simulations are provided to verify our analytic findings and rich dynamics of the three species system.

DYNAMICS OF DELAYED AUTONOMOUS MODEL WITH CARRY-OVER EFFECTS, INCLUDING PREY HARVESTING

This paper proposes a two-dimensional modified Leslie–Gower predator–prey model with carry-over effects, fear, prey protection and additional food, including prey harvesting. For this, it assumes the carry-over effect with delay incorporated in the birth rate of the prey population and gestation delay in Leslie–Gower term of a predator, and supposes that the predator consumes the prey according to Holling type II functional response. This dynamical system is split into two groups, namely non-delayed and delayed systems, respectively. In the non-delayed system, first, we discuss positivity, boundedness solutions, and the existence of equilibria, especially finding the coexistence of interior equilibrium points under certain conditions. After that, local and global stabilities are systematically analyzed. In addition, we obtain the conditions of Hopf-bifurcation in terms of the growth rate of prey species, prey protection and preference rate of predator for the non-delayed system. Whereas in the delayed system, both the delays have an essential role in governing the dynamical system discussed in three cases, and each case found the Hopf-bifurcation parameters for which the system changes stability behavior into instability. At last, the numerical simulations have been carried out to verify the theoretical findings.

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.

Existence and Uniqueness of a Canard Cycle with Cyclicity at Most Two in a Singularly Perturbed Leslie–Gower Predator–Prey Model with Prey Harvesting

Yao and Huzak [2022] proved that the cyclicity of canard cycles in a singularly perturbed Leslie–Gower predator–prey model with prey harvesting is at most two in a region of parameters. In this paper, we further show that there exists only one canard cycle with cyclicity at most two under explicit parameters conditions.

DAMPAK KEKERINGAN TERHADAP KESTABILAN SISTEM MANGSA PEMANGSA DENGAN FUNGSI RESPON HOLLING TIPE III DAN HASIL PEMANENAN KONSTAN PADA MANGSA

ABSTRACT: The dynamic relationship between prey populations and predator populations can be represent in the mathematical model. This research was developed from the mathematical model of predator-prey which was first introduced by Lotka-Volterra, namely by increasing the realism of the model through apply the logistic model to the prey and involving Holling type-III response function. The effects of drought on both populations and constant harvesting of prey are also included in the mathematical model. After the mathematical model is formed, a nondimensional model is then carried out to create a shape from the model that was built previously. This study aims to analyze the impact of drought on the prey-prey system with a type III Holling functional response and constant-yield prey harvesting. There is at least one equilibrium point and there are at most three equilibrium points in the model. Numerical simulations are carried out on the model to see the phase portrait. The simulation results show that if the rate of drought is greater than the intrinsic growth rate of prey then both populations will go towards extinction. However, on the other hand, if the intrinsic growth rate of the prey is greater than the rate of drought, then the dynamics relationship between the predator and prey populations depends on the pattern of constant-yield prey harvesting. Thus, a stable dynamic relationship occurs on the constant-yield prey harvesting at the interval 0<h<=K(r-a1)2

FILIPPOV TYPE PREY–PREDATOR SYSTEM FOR SELECTIVE HARVESTING OF PREY

In this paper, the dynamics of a Filippov system with logistically growing prey and Holling Type II predation is explored. The selective nonlinear harvesting of prey is carried out when the ratio of prey to predator is above a specified threshold value. In order to prevent the extinction of both the species and sustainable economic gains, no harvesting is allowed when the ratio is below a threshold value. The equilibrium states and their stability for the individual smooth are analyzed. Filippov convex method is applied to determine the dynamics on the boundary. The sliding mode dynamics is scrutinized, and it is observed that there is no escaping region. The existence of boundary points, tangent points and pseudo-equilibrium points are established. The conditions are obtained for the visibility of the tangent points. The complex behavior within the sliding region is studied by experimenting with different initial conditions. The increase in harvesting effort may lead to extinction of both the species through the sliding region of the boundary. The change in the stability behavior of both the interior equilibrium states is investigated with respect to model parameters. Border cross bifurcates with threshold value as the bifurcation parameter is shown for both the interior equilibrium states.

Global Hopf Bifurcation of a Diffusive Modified Leslie–Gower Predator–Prey Model with Delay and Michaelis–Menten Type Prey Harvesting

Global Hopf Bifurcation of a Diffusive Modified Leslie–Gower Predator–Prey Model with Delay and Michaelis–Menten Type Prey Harvesting

Bifurcation Patterns in a Discrete Predator–Prey Model Incorporating Ratio-Dependent Functional Response and Prey Harvesting

Bifurcation Patterns in a Discrete Predator–Prey Model Incorporating Ratio-Dependent Functional Response and Prey Harvesting

МОДЕЛИРОВАНИЕ ДИНАМИКИ ВЗАИМОДЕЙСТВУЮЩИХ ПОПУЛЯЦИЙ ТИПА «ХИЩНИК–ЖЕРТВА» ПРИ ПОСТОЯННОЙ МИГРАЦИИ ОСОБЕЙ С СОПРЕДЕЛЬНЫХ ТЕРРИТОРИЙ

The article deals with the research of the dynamics of a local predator-prey community with constant migration of individuals from neighboring territories. We studied several models with a constant inflow of individuals into both predator and prey populations. It is shown that changes in the overall dynamics are significantly influenced by the number of predator migrants: their large influx leads to the rapid and almost complete extinction of preys. The model considering only a constant prey inflow is successfully applied to modeling of various processes based on the principles of predator-prey population interactions, for example, when studying the consumption of difficult-to-renewable or nonrenewable natural resources. The study of the model provides allows getting estimations of the resources use efficiency and and the degree of modernization of the consumer sector. It is shown that two-dimensional models - modifications of the basic Volterra and Bazykin models, lead to structurally stable fluctuation regimes corresponding to the focus and limit cycle. We found that these models also contain fast-slow periodic dynamics, corresponding to a maximum limit cycle with strong spikes and smoother declines in the population size. The emergence of such regimes corresponds to the alternation of periods of active extinctions of prey, accompanied by an increase in the number of predators, and periods of long-term restoration of the prey, during which there is no harvesting of prey. We used methods of dynamic systems analysis to study the models. The construction of two-dimensional parametric portraits shows that, to get stable dynamics of the basic models of two interacting biological species, no complication is required, for example, by adding nonlinear terms. Stable regimes are also observed in simpler models, such as the original Lotka-Volterra or Bazikin models, where the recovery rate of prey population is a constant value.

The study on the complex nature of a predator-prey model with fractional-order derivatives incorporating refuge and nonlinear prey harvesting

<abstract> <p>The main objective of our research was to explore and develop a fractional-order derivative within the predator-prey framework. The framework includes prey refuge and selective nonlinear harvesting, where the harvesting progressively approaches a threshold value as the density of the harvested population advances. For memory effect, a non-integer order derivative is better than an integer-order derivative. The solutions to the fractional framework were shown to be existence, uniqueness, non-negativity, and boundedness. Matignon's condition was used for analysing local stability, and a suitable Lyapunov function provided global stability. While discussing the Hopf bifurcation's existence condition, we explored derivative order and refuge as bifurcation parameters. We aimed at redefining the predator-prey framework to incorporate fractional order, refuge, and harvesting. This kind of nonlinear harvesting is more realistic and reasonable than the model with constant yield harvesting and constant effort harvesting. The Adams-Bashforth-Moulton PECE algorithm in MATLAB software was used to simulate the proposed outcomes, investigate the impact on various factors, and analyse harvesting's effect on non-integer order predator-prey interactions.</p> </abstract>