scholarly journals Mutualisim between Two Species and a Mortal Predator with Holling Type-II Response Function

2019 ◽  
Vol 8 (2) ◽  
pp. 4070-4086
Keyword(s):  
Steady State ◽  
Global Stability ◽  
General Model ◽  
Local Stability ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Type Ii ◽  
Mutualistic Interaction ◽  
Boundary Equilibrium ◽  
Type Ii Functional Response

We analyze the dynamics of a general model of three-species mutualistic interaction among two species and a mortal predator, which consumes the first mutual species in terms of Holling type-II functional response manner. Local stability around the existing equilibrium points is investigated by using perturbed method. Sufficient conditions for the global stability are obtained by means of employing Lyapunov’s method around boundary equilibrium points. The population stochasticity around the steady state of co-existence due to white noise is also computed. Finally, the numerical illustrations are carried out to support the study

scholarly journals Dynamic Behaviors of a Nonautonomous Discrete Predator-Prey System Incorporating a Prey Refuge and Holling Type II Functional Response

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Yumin Wu ◽  
Fengde Chen ◽  
Wanlin Chen ◽  
Yuhua Lin
Keyword(s):  
Global Stability ◽  
Numerical Simulations ◽  
Functional Response ◽  
Sufficient Conditions ◽  
Type Ii ◽  
Prey Refuge ◽  
Predator Species ◽  
Predator Prey ◽  
Prey System ◽  
Type Ii Functional Response

A nonautonomous discrete predator-prey system incorporating a prey refuge and Holling type II functional response is studied in this paper. A set of sufficient conditions which guarantee the persistence and global stability of the system are obtained, respectively. Our results show that if refuge is large enough then predator species will be driven to extinction due to the lack of enough food. Two examples together with their numerical simulations show the feasibility of the main results.

scholarly journals Local Stability Analysis On Lotka-Volterra Predator-Prey Models Functional Responses With Constant prey Refuge

2021 ◽  
Author(s):  
Chongming Li
Keyword(s):  
Stability Analysis ◽  
Functional Response ◽  
Local Stability ◽  
Equilibrium Points ◽  
Saddle Points ◽  
Stable Node ◽  
Type I ◽  
Type Ii ◽  
Functional Responses ◽  
Type Ii Functional Response

The dynamical behaviours of the predators and prey can be described by studying the local stability of the planar systems. Type I functional response shows that the rate of consumption per predator is proportional to prey’s density while type II functional response is related to the situation that predators would reach satiation as they consumed sufficient amount of prey. We seek out a method of using transformation to reduce the number of parameters of original models and then study the stability analysis of equilibrium points. Under suitable restrictions on the new parameters, we prove that the positive interior equilibrium is a stable node for the system of type I and type II functional responses. Moreover, in the case of type II functional response, the boundary equilibria can have more types of stability other than saddle points.

scholarly journals Dynamics of the Stage Structured Population Model, Predator Accompanied by Michaelis Menten Holling Type Functional Responseand Delay and Prey Taking Refuge

2021 ◽  
Vol 12 (2) ◽  
Author(s):  
N. Mohana Sorubha Sundari, Et. al.
Keyword(s):  
Population Model ◽  
Local Stability ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Type Ii ◽  
Structured Population Model ◽  
Predator Prey ◽  
Structured Population ◽  
Prey System ◽  
Two Stages

The current work considers predator prey system, prey taking refuge, predator reckoned with time delay and Michaelis Menten Holling type II response function undergoing two stages: juvenile and mature. From the characteristic equation, we derive conditions for the local stability of the system at the equilibrium points. Also, at the coexistence equilibrium point, the system is analyzed for the occurrence of Hopf bifurcation. Lyapunov function provides sufficient conditions for the global stability of the system. Numerical simulations are given to support the theory.

scholarly journals Local Stability Analysis On Lotka-Volterra Predator-Prey Models Functional Responses With Constant prey Refuge

2021 ◽  
Author(s):  
Chongming Li
Keyword(s):  
Stability Analysis ◽  
Functional Response ◽  
Local Stability ◽  
Equilibrium Points ◽  
Saddle Points ◽  
Stable Node ◽  
Type I ◽  
Type Ii ◽  
Functional Responses ◽  
Type Ii Functional Response

The dynamical behaviours of the predators and prey can be described by studying the local stability of the planar systems. Type I functional response shows that the rate of consumption per predator is proportional to prey’s density while type II functional response is related to the situation that predators would reach satiation as they consumed sufficient amount of prey. We seek out a method of using transformation to reduce the number of parameters of original models and then study the stability analysis of equilibrium points. Under suitable restrictions on the new parameters, we prove that the positive interior equilibrium is a stable node for the system of type I and type II functional responses. Moreover, in the case of type II functional response, the boundary equilibria can have more types of stability other than saddle points.

scholarly journals Global Stability and Optimal Control of Dengue with Two Coexisting Virus Serotypes

MATEMATIKA ◽  
2019 ◽  
Vol 35 (4) ◽  
pp. 149-170
Author(s):  
Afeez Abidemi ◽  
Rohanin Ahmad ◽  
Nur Arina Bazilah Aziz
Keyword(s):  
Optimal Control ◽  
Global Stability ◽  
Disease Transmission ◽  
Deterministic Model ◽  
Control Strategies ◽  
Equilibrium Points ◽  
Effective Control ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Boundary Equilibrium

This study presents a two-strain deterministic model which incorporates Dengvaxia vaccine and insecticide (adulticide) control strategies to forecast the dynamics of transmission and control of dengue in Madeira Island if there is a new outbreak with a different virus serotypes after the first outbreak in 2012. We construct suitable Lyapunov functions to investigate the global stability of the disease-free and boundary equilibrium points. Qualitative analysis of the model which incorporates time-varying controls with the specific goal of minimizing dengue disease transmission and the costs related to the control implementation by employing the optimal control theory is carried out. Three strategies, namely the use of Dengvaxia vaccine only, application of adulticide only, and the combination of Dengvaxia vaccine and adulticide are considered for the controls implementation. The necessary conditions are derived for the optimal control of dengue. We examine the impacts of the control strategies on the dynamics of infected humans and mosquito population by simulating the optimality system. The disease-freeequilibrium is found to be globally asymptotically stable whenever the basic reproduction numbers associated with virus serotypes 1 and j (j 2 {2, 3, 4}), respectively, satisfy R01,R0j 1, and the boundary equilibrium is globally asymptotically stable when the related R0i (i = 1, j) is above one. It is shown that the strategy based on the combination of Dengvaxia vaccine and adulticide helps in an effective control of dengue spread in the Island.

scholarly journals A Stage-Structure Rosenzweig-MacArthur Model with Effect of Prey Refuge

2020 ◽  
Vol 1 (1) ◽  
pp. 1-7
Author(s):  
Lazarus Kalvein Beay ◽  
Maryone Saija
Keyword(s):  
Equilibrium Point ◽  
Local Stability ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Structure Population ◽  
Prey Refuge ◽  
Behavioral System ◽  
Trivial Equilibrium ◽  
Coexistence Equilibrium

We proposed and analyzed a stage-structure Rosenzweig-MacArthur model incorporating a prey refuge.  It is assumed that the prey is a stage-structure population consisting of two compartments known as immature prey and mature prey. The model incorporates the functional response Holling type-II. In this work, we investigate all the biologically feasible equilibrium points, and it is shown that the system has three equilibrium points. Sufficient conditions for the local stability of the non-negative equilibrium point of the model are also derived. All points are conditionally locally asymptotically stable. By constructing Jacobian matrix and determined eigenvalues, we analyzed the local stability of the trivial equilibrium and non-predator equilibrium points. Specifically for coexistence equilibrium point, Routh-Hurwitz criterion used to analyze local stability. In addtion, we investigated the effect of immature prey refuge. Our mathematical analysis exhibits that immature prey refuge have played a crucial role in the behavioral system. When the effect of immature prey refuge (constant m) increases, it is can stabilize non-predator equilibrium point, where all the species can not exists together. And conversely, if contant m decreases, it is can stabilize coexistence equilibrium point then all the species can exists together. The work is completed with a numerical simulation to confirmed analitical results

scholarly journals A Stochastic Model for Three Species

2018 ◽  
Vol 7 (4.10) ◽  
pp. 497
Author(s):  
Y. Suresh Kumar ◽  
N. Seshagiri Rao ◽  
B. V AppaRao
Keyword(s):  
Stochastic Process ◽  
Stochastic Model ◽  
Global Stability ◽  
Numerical Simulations ◽  
Local Stability ◽  
Equilibrium Points ◽  
Limited Resources

The present work is related to a three species ecosystem including a mutualism interaction between two species and a predator, where the predator is depending on both the mutual species. All three species in this model are considered in limited resources. The sustainability of the system (local stability) is discussed through the perturbed technique at the possible existing each equilibrium points. Using Lyapunov's technique the global stability of the system is also described. Further the nature of the system is observed by introducing the stochastic process to the species and the numerical simulations are studied to know the interaction among the species. 

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

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Charles Raymond ◽  
Alfred Hugo ◽  
Monica Kung’aro
Keyword(s):  
Global Stability ◽  
Lake Victoria ◽  
Equilibrium Points ◽  
Nile Perch ◽  
Type Ii ◽  
Bioeconomic Model ◽  
Eigenvalue Approach ◽  
Fishery Model ◽  
Predator System ◽  
Control Management

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.

scholarly journals On the stabilization of internally coupled map lattice systems

2004 ◽  
Vol 2004 (2) ◽  
pp. 345-356 ◽  
Author(s):  
Weihong Huang
Keyword(s):  
Steady State ◽  
Periodic Orbit ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Coupled Map Lattice ◽  
Coupling Constants ◽  
Lattice Systems ◽  
Adjustment Mechanism ◽  
Adaptive Adjustment ◽  
Effectiveness And Efficiency

The adaptive adjustment mechanism is applied to the stabilization of an internally coupled map lattice system defined byxi,t+1=G((1−αi−βi)xi,t+αixi+1,t+βixi−1,t), wheref:ℝ→ℝis a nonlinear map, andαandβare nonnegative coupling constants that satisfy the constraintαi+βi<1, for allx∈ℝ,i=1,2,…,n. Sufficient conditions and ranges of adjustment parameters that guarantee the local stability of a generic steady state have been provided. Numerical simulations have demonstrated the effectiveness and efficiency for this mechanism to stabilize the system to a generic unstable steady state or a periodic orbit.

GLOBAL STABILITY OF A CHEMOSTAT MODEL WITH DELAYED RESPONSE IN GROWTH AND A LETHAL EXTERNAL INHIBITOR

2008 ◽  
Vol 01 (04) ◽  
pp. 503-520 ◽  
Author(s):  
ZHIQI LU ◽  
JINGJING WU
Keyword(s):  
Global Stability ◽  
Local Stability ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Delayed Response ◽  
Qualitative Properties ◽  
Chemostat Model ◽  
Stability And Instability ◽  
Discrete Delays ◽  
Globally Stable

A competition model between two species with a lethal inhibitor in a chemostat is analyzed. Discrete delays are used to describe the nutrient conversion process. The proved qualitative properties of the solution are positivity, boundedness. By analyzing the local stability of equilibria, it is found that the conditions for stability and instability of the boundary equilibria are similar to those in [9]. In addition, the global asymptotic behavior of the system is discussed and the sufficient conditions for the global stability of the boundary equilibria are obtained. Moreover, by numerical simulation, it is interesting to find that the positive equilibrium may be globally stable.

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