Locating the equilibrium points of a predator–prey model by means of affine state feedback

2008 ◽  
Vol 345 (5) ◽  
pp. 489-498 ◽  
Author(s):  
Wieslaw Krajewski ◽  
Umberto Viaro
Keyword(s):  
State Feedback ◽  
Equilibrium Points ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

scholarly journals Dynamics of an Eco-Epidemic Predator–Prey Model Involving Fractional Derivatives with Power-Law and Mittag–Leffler Kernel

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 785
Author(s):  
Hasan S. Panigoro ◽  
Agus Suryanto ◽  
Wuryansari Muharini Kusumawinahyu ◽  
Isnani Darti
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Derivative ◽  
Equilibrium Point ◽  
Power Law ◽  
Equilibrium Points ◽  
Essential Difference ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Infected Prey

In this paper, we consider a fractional-order eco-epidemic model based on the Rosenzweig–MacArthur predator–prey model. The model is derived by assuming that the prey may be infected by a disease. In order to take the memory effect into account, we apply two fractional differential operators, namely the Caputo fractional derivative (operator with power-law kernel) and the Atangana–Baleanu fractional derivative in the Caputo (ABC) sense (operator with Mittag–Leffler kernel). We take the same order of the fractional derivative in all equations for both senses to maintain the symmetry aspect. The existence and uniqueness of solutions of both eco-epidemic models (i.e., in the Caputo sense and in ABC sense) are established. Both models have the same equilibrium points, namely the trivial (origin) equilibrium point, the extinction of infected prey and predator point, the infected prey free point, the predator-free point and the co-existence point. For a model in the Caputo sense, we also show the non-negativity and boundedness of solution, perform the local and global stability analysis and establish the conditions for the existence of Hopf bifurcation. It is found that the trivial equilibrium point is a saddle point while other equilibrium points are conditionally asymptotically stable. The numerical simulations show that the solutions of the model in the Caputo sense strongly agree with analytical results. Furthermore, it is indicated numerically that the model in the ABC sense has quite similar dynamics as the model in the Caputo sense. The essential difference between the two models is the convergence rate to reach the stable equilibrium point. When a Hopf bifurcation occurs, the bifurcation points and the diameter of the limit cycles of both models are different. Moreover, we also observe a bistability phenomenon which disappears via Hopf bifurcation.

scholarly journals Examination of Stability of the Mathematical Predator-Prey Model by Observing the Distance between Predator and Prey

2019 ◽  
Vol 8 (6S3) ◽  
pp. 65-70
Keyword(s):  
Growth Model ◽  
Equilibrium Points ◽  
Predation Rate ◽  
Logistic Growth ◽  
Asymptotically Stable ◽  
Defense Behavior ◽  
Long Distance ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

Maintaining distance is one of the strategies that can be applied by prey to defend themselves or to avoid predatory attacks. This defense behavior can affect predation rates. The distance or difference in the number of prey and predator populations will affect the level of balanced ecosystem. The distance is also affecting predation rate, when there’s a long distance between prey and predator thus the predation rate decreases and vice versa. The purpose of this thesis is to analyze the stability of the mathematical equilibrium on predator-prey model by observing the distance. There are two types of model being observed, type one uses exponential growth model and type two is using a logistic growth model. The analytics results obtain three equilibrium points, namely the unstable extinction equilibrium point, and the asymptotically stable predator extinction with certain conditions and asymptotically stable coexistence with certain conditions. Then numerical simulation is conducted to support the analytical results.

scholarly journals Analysis of The Rosenzweig-MacArthur Predator-Prey Model with Anti-Predator Behavior

CAUCHY ◽  
2021 ◽  
Vol 6 (4) ◽  
pp. 260-269
Author(s):  
Ismail Djakaria ◽  
Muhammad Bachtiar Gaib ◽  
Resmawan Resmawan
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Point ◽  
Equilibrium Points ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Existence And Stability ◽  
The Stability ◽  
Population Equilibrium ◽  
Predator Behavior

This paper discusses the analysis of the Rosenzweig-MacArthur predator-prey model with anti-predator behavior. The analysis is started by determining the equilibrium points, existence, and conditions of the stability. Identifying the type of Hopf bifurcation by using the divergence criterion. It has shown that the model has three equilibrium points, i.e., the extinction of population equilibrium point (E0), the non-predatory equilibrium point (E1), and the co-existence equilibrium point (E2). The existence and stability of each equilibrium point can be shown by satisfying several conditions of parameters. The divergence criterion indicates the existence of the supercritical Hopf-bifurcation around the equilibrium point E2. Finally, our model's dynamics population is confirmed by our numerical simulations by using the 4th-order Runge-Kutta methods.

A Michaelis–Menten Predator–Prey Model with Strong Allee Effect and Disease in Prey Incorporating Prey Refuge

2018 ◽  
Vol 28 (06) ◽  
pp. 1850073 ◽  
Author(s):  
Sangeeta Saha ◽  
Alakes Maiti ◽  
G. P. Samanta
Keyword(s):  
Allee Effect ◽  
Equilibrium Points ◽  
Prey Refuge ◽  
Susceptible Population ◽  
Bifurcation Theorem ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Lasalle Theorem ◽  
Transcritical Bifurcations

Here, we have proposed a predator–prey model with Michaelis–Menten functional response and divided the prey population in two subpopulations: susceptible and infected prey. Refuge has been incorporated in infected preys, i.e. not the whole but only a fraction of the infected is available to the predator for consumption. Moreover, multiplicative Allee effect has been introduced only in susceptible population to make our model more realistic to environment. Boundedness and positivity have been checked to ensure that the eco-epidemiological model is well-behaved. Stability has been analyzed for all the equilibrium points. Routh–Hurwitz criterion provides the conditions for local stability while on the other hand, Bendixson–Dulac theorem and Lyapunov LaSalle theorem guarantee the global stability of the equilibrium points. Also, the analytical results have been verified numerically by using MATLAB. We have obtained the conditions for the existence of limit cycle in the system through Hopf Bifurcation theorem making the refuge parameter as the bifurcating parameter. In addition, the existence of transcritical bifurcations and saddle-node bifurcation have also been observed by making different parameters as bifurcating parameters around the critical points.

scholarly journals Model matematis predator-prey tanaman padi, hama penggerek batang, tikus, dan wereng batang coklat di Karawang

2018 ◽  
Vol 13 (1) ◽  
pp. 52-62
Author(s):  
Tesa Nur Padilah ◽  
Betha Nurina Sari ◽  
Hannie Hannie
Keyword(s):  
Mathematical Model ◽  
Equilibrium Points ◽  
Brown Planthopper ◽  
Stem Borer ◽  
Behavioral Analysis ◽  
Predator Prey ◽  
Rice Plants ◽  
Predator Prey Model ◽  
Prey Model ◽  
Plant Pest

Karawang merupakan salah satu pusat penanaman padi di Pulau Jawa. Keberhasilan panen dapat terganggu oleh adanya organisme pengganggu tumbuhan (OPT) sehingga dapat mengancam target swasembada beras. Hubungan antara tanaman padi dengan OPT dapat dibentuk menjadi suatu model matematis yaitu model predator-prey. Untuk itu, penelitian ini bertujuan untuk menganalisis model matematis predator-prey tanaman padi dan OPT. Predator (pemangsa) adalah makhluk hidup yang memakan mangsa (prey). Model predator-prey antara tanaman padi dengan OPT yang dibahas adalah model tiga predator yaitu hama penggerek batang, tikus, dan wereng batang coklat dengan prey yaitu padi. Pertumbuhan padi mengikuti model pertumbuhan logistik. Model yang diturunkan berbentuk sistem persamaan diferensial nonlinier. Pada model diperoleh lima titik ekuilibrium. Analisis perilaku model dilakukan pada tiga titik ekuilibrium dan ketiganya bersifat stabil asimtotik. Simulasi model dengan menggunakan software Maple 13 sejalan dengan analisis perilaku model. Faktor-faktor yang berpengaruh agar populasi hama penggerek batang, tikus, dan wereng batang coklat dapat menurun bahkan hilang dari populasi yaitu tingkat kematian alami serta tingkat interaksi padi terhadap hama-hama tersebut. Predator-prey mathematical model of rice plants, stem borer, rat, and brown planthopper in Karawang AbstractKarawang was one of the center of rice planting in Java Island. The success of the harvest may be disrupted by the presence of plant pest organisms that may threaten the rice self-sufficiency target. The relationship between rice plants and pests can be formed into a mathematical model, that was a predator-prey model. Therefore, this research aimed to analyze the mathematical model of predator-prey between rice plants and plant pest organisme. Predators were living things that eat prey. The predator-prey model between rice plants and pests discussed was a three predator model of stem borer, rat, and brown stem rhizome with the prey, that was rice. Rice growth follows the logistic growth model. The derived model was an nonlinear differential equation system. In this model obtained five equilibrium points. Model behavioral analysis was performed on three equilibrium points and they were stable asymptotically. Simulations of the model using Maple 13 software were in good agreement with behavioral analysis model. Factors that influence the stem borer, rat, and brown planthopper population could decrease even disapear from the population were the natural death rate and the interaction rate of rice to the pests.

Stability analysis of a delayed predator–prey model with nonlinear harvesting efforts using imprecise biological parameters

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Amit K. Pal
Keyword(s):  
Equilibrium Points ◽  
Stable Limit Cycle ◽  
Limit Cycle Oscillation ◽  
Stable Limit ◽  
Biological Parameters ◽  
Delay Model ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Nonlinear Harvesting

Abstract In this paper, the dynamical behaviors of a delayed predator–prey model (PPM) with nonlinear harvesting efforts by using imprecise biological parameters are studied. A method is proposed to handle these imprecise parameters by using a parametric form of interval numbers. The proposed PPM is presented with Crowley–Martin type of predation and Michaelis–Menten type prey harvesting. The existence of various equilibrium points and the stability of the system at these equilibrium points are investigated. Analytical study reveals that the delay model exhibits a stable limit cycle oscillation. Computer simulations are carried out to illustrate the main analytical findings.

scholarly journals Predator-Prey Model with Refuge, Fear and Z-Control

2021 ◽  
Vol 26 (1) ◽  
pp. 40-57
Author(s):  
Ibrahim M. Elmojtaba ◽  
Kawkab Al-Amri ◽  
Qamar J.A. Khan
Keyword(s):  
Numerical Simulations ◽  
Equilibrium Points ◽  
Predator Population ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Long Run ◽  
Constant State

In this paper, we consider a predator-prey model incorporating fear and refuge.  Our results show that the predator-free equilibrium is globally asymptotically stable if the ratio between the death rate of predators and the conversion rate of prey into predator is greater than the value of prey in refuge at equilibrium.  We also show that the co-existence equilibrium points are locally asymptotically stable if the value of the prey outside refuge is greater than half of the carrying capacity.  Numerical simulations show that when the intensity of fear increases, the fraction of the prey inside refuge increases; however, it has no effect on the fraction of the prey outside refuge, in the long run. It is shown that the intensity of fear harms predator population size. Numerical simulations show that the application of Z-control will force the system to reach any desired state within a limited time, whether the desired state is a constant state or a periodic state. Our results show that when the refuge size is taken to be a non-constant function of the prey outside refuge, the systems change their dynamics. Namely, when it is a linear function or an exponential function, the system always reaches the predator-free equilibrium.  However, when it is taken as a logistic equation, the system reaches the co-existence equilibrium after long term oscillations.

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