CONSEQUENCES OF MULTIPLE ALLEE EFFECT IN AN OPEN ACCESS FISHERY MODEL

2015 ◽  
Vol 23 (supp01) ◽  
pp. S101-S121 ◽  
Author(s):  
EDUARDO GONZÁLEZ-OLIVARES ◽  
JOSÉ D. FLORES
Keyword(s):  
Open Access ◽  
Allee Effect ◽  
Fishing Effort ◽  
Positive Equilibrium ◽  
Allee Effects ◽  
Continuous Time Model ◽  
Time Model ◽  
Parsimony Principle ◽  
Positive Equilibrium Point ◽  
Fishery Model

This work deals with the dynamics of a bioeconomic continuous time model, where the combined action of the fishing effort exerted by men (as a predator) and multiple Allee effect or depensation on the growth rate of a self-regenerating resource (the prey) are considered. It has been recently established that a depensation phenomenon appears by diverse causes and new functions have been proposed to describe multiple Allee effects. One of these formalizations is here incorporated in the well-known Smith's model, one of the simplest models to open access fisheries. We prove that this new and complex expression is topologically equivalent to a simpler form. Then, we postulate that the parsimony principle must be used to describe this phenomenon. It is also shown that in the phase plane of biomass-effort on the proposed model, the origin is an attractor equilibrium for all parameters values as a consequence of the Allee effect. Moreover, there is a subset of the parameter values, for which two limit cycles exist surrounding the unique positive equilibrium point of the system, one of them being asymptotically stable (the non damped oscillatory tragedy of the commons); hence, multiestability exists, particularly three-stability.

Stability and Bifurcation in a Logistic Model with Allee Effect and Feedback Control

2020 ◽  
Vol 30 (15) ◽  
pp. 2050231
Author(s):  
Zhenliang Zhu ◽  
Mengxin He ◽  
Zhong Li ◽  
Fengde Chen
Keyword(s):  
Feedback Control ◽  
Logistic Model ◽  
Allee Effect ◽  
Control Variable ◽  
Positive Equilibrium ◽  
Threshold Condition ◽  
Local Asymptotic Stability ◽  
Linearized System ◽  
Positive Equilibrium Point ◽  
Two Parameters

This paper aims to study the dynamic behavior of a logistic model with feedback control and Allee effect. We prove the origin of the system is always an attractor. Further, if the feedback control variable and Allee effect are big enough, the species goes extinct. According to the analysis of the Jacobian matrix of the corresponding linearized system, we obtain the threshold condition for the local asymptotic stability of the positive equilibrium point. Also, we study the occurrence of saddle-node bifurcation, supercritical and subcritical Hopf bifurcations with the change of parameter. By calculating a universal unfolding near the cusp and choosing two parameters of the system, we can draw a conclusion that the system undergoes Bogdanov–Takens bifurcation of codimension-2. Numerical simulations are carried out to confirm the feasibility of the theoretical results. Our research can be regarded as a supplement to the existing literature on the dynamics of feedback control system, since there are few results on the bifurcation in the system so far.

DYNAMICS OF A DIFFUSIVE PREDATOR–PREY MODEL WITH ADDITIVE ALLEE EFFECT

2012 ◽  
Vol 05 (02) ◽  
pp. 1250023 ◽  
Author(s):  
YONGLI CAI ◽  
WEIMING WANG ◽  
JINFENG WANG
Keyword(s):  
Allee Effect ◽  
Dynamical Behavior ◽  
Positive Equilibrium ◽  
Global Asymptotical Stability ◽  
Allee Effects ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Prey Model ◽  
Holling Ii Functional Response

In this paper, we investigate the dynamics of a diffusive predator–prey model with Holling-II functional response and the additive Allee effect in prey. We show the local and global asymptotical stability of the positive equilibrium, and give the conditions of the existence of the Hopf bifurcation. By carrying out global qualitative and bifurcation analysis, it is shown that the weak and strong Allee effects in prey can induce different dynamical behavior in the predator–prey model. Furthermore, we use some numerical simulations to illustrate the dynamics of the model. The results may be helpful for controlling and managing the predator–prey system.

scholarly journals Supercritical Neimark–Sacker Bifurcation and Hybrid Control in a Discrete-Time Glycolytic Oscillator Model

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
A. Q. Khan ◽  
E. Abdullah ◽  
Tarek F. Ibrahim
Keyword(s):  
Discrete Time ◽  
Hybrid Control ◽  
Oscillator Model ◽  
Positive Equilibrium ◽  
Time Model ◽  
Discrete Time Model ◽  
Prime Period ◽  
Dynamical Properties ◽  
Positive Equilibrium Point ◽  
Theoretical Results

We study the local dynamical properties, Neimark–Sacker bifurcation, and hybrid control in a glycolytic oscillator model in the interior of ℝ+2. It is proved that, for all parametric values, Pxy+α/β+α2,α is the unique positive equilibrium point of the glycolytic oscillator model. Further local dynamical properties along with different topological classifications about the equilibrium Pxy+α/β+α2,α have been investigated by employing the method of linearization. Existence of prime period and periodic points of the model under consideration are also investigated. It is proved that, about the fixed point Pxy+α/β+α2,α, the discrete-time glycolytic oscillator model undergoes no bifurcation, except Neimark–Sacker bifurcation. A further hybrid control strategy is applied to control Neimark–Sacker bifurcation in the discrete-time model. Finally, theoretical results are verified numerically.

An Age-Structured Open-Access Fishery Model

1994 ◽  
Vol 51 (4) ◽  
pp. 900-912 ◽  
Author(s):  
Richard McGarvey
Keyword(s):  
Open Access ◽  
Density Dependence ◽  
Dynamic Model ◽  
Fishing Effort ◽  
Steady States ◽  
Volterra Model ◽  
Fish Stock ◽  
Placopecten Magellanicus ◽  
Recruitment Variability ◽  
Fishery Model

A dynamic model for open-access fisheries is presented. In addition to density dependence in recruitment and fishing effort changing in proportion to the level of profit fishermen earn which characterizes previous open-access models, it incorporates full age structure for the fish stock, lognormal environmental recruitment variability, and gear selectivity. The predator–prey cycling solution of the original Schaefer dynamic model, and subsequent open-access models, persists for these model extensions. Density dependence in recruitment induces greater global stability. Environmental recruitment variability, common in marine populations, is destabilizing in the neighborhood of the open-access equilibrium. These two influences, combined in the open-access fishery model, generate robust long-lasting irregular cycles of stock and effort. Volterra proved for the original Lotka–Volterra model that the time averages of the variables over one cycle were exactly equal to their equilibrium steady states. This is shown to extend as a good approximation for the model presented here. Approximating model steady states of effort and catch by the corresponding averages from data time series underlies a new algorithm of parameter evaluation, applied here to an open-access model of the Georges Bank sea scallop (Placopecten magellanicus) fishery.

scholarly journals A New Global Stable Conclusion on a Diffusive Leslie-Gower Predator-Prey System with Additive Allee Effect

2019 ◽  
Vol 2019 ◽  
pp. 1-5
Author(s):  
Liu Yang
Keyword(s):  
Allee Effect ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Comparison Method ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Homogeneous Neumann Boundary Condition ◽  
Lyapunov Function Method ◽  
Positive Equilibrium Point

In this work, a diffusive Leslie-Gower predator-prey model with additive Allee effect on prey under a homogeneous Neumann boundary condition is reconsidered. We establish new sufficient conditions for the global stability of the unique positive equilibrium point of the system by using the comparison method rather than the Lyapunov function method. It is shown that our result supplements and complements one of the main results of Yang and Zhong, 2015. Furthermore, numerical simulations are performed to consolidate the analytic finding.

Dynamics of a prey predator disease model with Holling type functional response and stochastic perturbation

2020 ◽  
pp. 471-488
Author(s):  
M. N. Srinivas ◽  
G. Basava Kumar ◽  
V. Madhusudanan
Keyword(s):  
Equilibrium Point ◽  
Disease Model ◽  
Stochastic Perturbation ◽  
Positive Equilibrium ◽  
Type Ii ◽  
Asymptotically Stable ◽  
Unique Positive Solution ◽  
Stochastic Nature ◽  
Research Article ◽  
Positive Equilibrium Point

The present research article constitutes Holling type II and IV diseased prey predator ecosystem and classified into two categories namely susceptible and infected predators.We show that the system has a unique positive solution. The deterministic and stochastic nature of the dynamics of the system is investigated. We check the existence of all possible steady states with local stability. By using Routh-Hurwitz criterion we showed that the positive equilibrium point $E_{7}$ is locally asymptotically stable if $x^{*} > \sqrt{m_{1}}$ .Moreover condition of the global stability of positive equilibrium point $E_{7}$ are also entrenched with help of Lyupunov theorem. Some Numerical simulations are carried out to illustrate our analytical findings.

scholarly journals The influence of density in population dynamics with strong and weak Allee effect

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
Kamrun Nahar Keya ◽  
Md. Kamrujjaman ◽  
Md. Shafiqul Islam
Keyword(s):  
Population Dynamics ◽  
Allee Effect ◽  
Global Bifurcation ◽  
Reaction Diffusion ◽  
Logistic Growth ◽  
Allee Effects ◽  
Reaction Diffusion Model ◽  
Positive Steady States ◽  
The Stability ◽  
The Impact

AbstractIn this paper, we consider a reaction–diffusion model in population dynamics and study the impact of different types of Allee effects with logistic growth in the heterogeneous closed region. For strong Allee effects, usually, species unconditionally die out and an extinction-survival situation occurs when the effect is weak according to the resource and sparse functions. In particular, we study the impact of the multiplicative Allee effect in classical diffusion when the sparsity is either positive or negative. Negative sparsity implies a weak Allee effect, and the population survives in some domain and diverges otherwise. Positive sparsity gives a strong Allee effect, and the population extinct without any condition. The influence of Allee effects on the existence and persistence of positive steady states as well as global bifurcation diagrams is presented. The method of sub-super solutions is used for analyzing equations. The stability conditions and the region of positive solutions (multiple solutions may exist) are presented. When the diffusion is absent, we consider the model with and without harvesting, which are initial value problems (IVPs) and study the local stability analysis and present bifurcation analysis. We present a number of numerical examples to verify analytical results.

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