Dynamical analysis of a two prey-one predator system with quadratic self interaction

Aybar, Ilknur Kusbeyzi; Aybar, Orhan Özgür; Dukarić, Maša; Ferčec, Brigita

doi:10.1016/j.amc.2018.03.123

Cited by 4 publications

(5 citation statements)

References 30 publications

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“…The authors evaluated the predator-prey relationship in three species in three dimensions by using an ordinary differential equation. For the calculation, the subjective population sizes of two prey species and one predator creature that share their habitats were considered (Aybar et al 2018). Prey cooperation benefits both prey populations in many ecosystems.…”

Section: Introductionmentioning

confidence: 99%

Modelling the Multiteam Prey–predator Dynamics Using the Delay Differential Equation

Raj

Kumar

2023

MJS

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In nature, many species form teams and move in herds from one place to another. This helps them in reducing the risk of predation. Time delay caused by the age structure, maturation period, and feeding time is a major factor in real-time prey–predator dynamics that result in periodic solutions and the bifurcation phenomenon. This study analysed the behaviour of teamed-up prey populations against predation by using a mathematical model. The following variables were considered: the prey population Pr1, the prey population Pr2, and the predator population Pr3. The interior equilibrium point was calculated. A local satiability analysis was performed to ensure a feasible interior equilibrium. The effect of the delay parameter on the dynamics was examined. A Hopf bifurcation was noted when the delay parameter crossed the critical value. Direction analysis was performed using the centre manifold theorem. The graphs of analytical results were plotted using MATLAB.

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Section: Introductionmentioning

confidence: 99%

Modelling the Multiteam Prey–predator Dynamics Using the Delay Differential Equation

Raj

Kumar

2023

MJS

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“…Hence, Hopf bifurcation scheme has been extensively used to acquire more information about periodic solution properties near an equilibrium point of a nonlinear system [21]. In [2], I.K. Aybar et al discussed the two prey-one predator system consisting of quadratic self interaction in the prey equations.…”

Section: Introductionmentioning

confidence: 99%

“…The detailed biological meanings of parameters are shown in Table 1. The authors of [2] investigated the the singular points' stability of system (1.1), and Quadratic self interaction rate of prey-B by means of numerical simulation, they showed that solutions trajectories of the system can be approached to the stable singular points under given conditions. They also introduced an approach for examining the existence of Hopf bifurcation.…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by the recent work of I.K. Aybar et al [2] with considering the fact that the more realistic models should consist of delay differential equations without instantaneous feedbacks, we focus on the following time delayed predator-prey system with a single delay consisting of three species which represent the population densities of two prey and one predator species:…”

Section: Introductionmentioning

confidence: 99%

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Exploring The Influence of Time-Delay On The Dynamics of a Two Prey-One Predator System With Quadratic Self-Interaction

Surosh¹,

Ghaziani²,

Alidousti³

2022

DSA

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This work is devoted to studying the effects of time-delay on the dynamics of a two prey-one predator system with quadratic self-interaction. The essential dynamical structures of the delayed system are analyzed by means of local stability analysis and bifurcation theory.Taking the time lag τ as a free parameter, the necessary conditions for the existence of the Hopf bifurcation around the interior equilibrium of the system has been derived both analytically and numerically. It is observed that a Hopf bifurcation occurs when the bifurcation parameter crosses a certain threshold value and it is found that the dynamics of the system can be effected significantly by the time delay which has both stabilizing and destabilizing impacts depending on the magnitude of the delay. Moreover, we derived the explicit formulas in order to determine the direction of the Hopf bifurcation and examine the nature of the bifurcating periodic solution by using the normal form method and the center manifold theorem. Eventually, some numerical simulations are given to verify the derived theoretical analysis.

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“…Various generalized predator-prey models that involve quadratic functions which exhibit logistic behaviour [1][2][3], cubic functions which show different rates of reproduction [4,5], Holling type II functions which state constant consumption [6,7] and Beddington-DeAngelis functions which indicate mutual interference and extinction [8,9] have been shown to overcome some of the biological problems of the original Lotka-Volterra model [10,11]. Population carrying capacities are introduced into these generalizations by adding the proposed functions to the self-interaction and the coupling terms [12].…”

Section: Introductionmentioning

confidence: 99%

Stability and Hopf Bifurcation in Three-Dimensional Predator-Prey Models with Allee Effect

Aybar

2019

Sakarya University Journal of Science

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In this study, we perform the stability and Hopf bifurcation analysis for two population models with Allee effect. The population models within the scope of this study are the one prey-two predator model with Allee growth in the prey and the two prey-one predator model with Allee growth in the preys. Our procedure for investigating each model is as follows. First, we investigate the singular points where the system is stable. We provide the necessary parameter conditions for the system to be stable at the singular points. Then, we look for Hopf bifurcation at each singular point where a family of limit cycles cycle or oscillate. We provide the parameter conditions for Hopf bifurcation to occur. We apply the algebraic invariants method to fully examine the system. We investigate the algebraic properties of the system by finding all algebraic invariants of degree two and three. We give the conditions for the system to have a first integral.

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Dynamical analysis of a two prey-one predator system with quadratic self interaction

Cited by 4 publications

References 30 publications

Modelling the Multiteam Prey–predator Dynamics Using the Delay Differential Equation

Modelling the Multiteam Prey–predator Dynamics Using the Delay Differential Equation

Exploring The Influence of Time-Delay On The Dynamics of a Two Prey-One Predator System With Quadratic Self-Interaction

Stability and Hopf Bifurcation in Three-Dimensional Predator-Prey Models with Allee Effect

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