Dynamic behaviors of a Lotka-Volterra type predator-prey system with Allee effect on the predator species and density dependent birth rate on the prey species

Chen, Fengde; Guan, Xinyu; Huang, Xiaoyan; Deng, Huan

doi:10.1515/math-2019-0082

Cited by 13 publications

(3 citation statements)

References 34 publications

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“…Literature [14] introduced Markov transforms into the predator-feeder model to explore the stochastic bifurcation and dynamic equilibrium in the system, finding quantitative conditions for the smooth distribution and proving that it is more sensitive to the rate of leapfrogging. Literature [15] discusses the equilibrium point of the Lotka-Volterra predator-eater system with Allee's influence, and an iterative method is used to derive the conditions for the infinite time-lag system to be in a positive equilibrium global attractor point, which is verified by numerical simulations. Literature [16] analyzes the third-order predator system with a time lag, proposes that the stability of the distributional solution is an important catch for resolving the global of the system, and discusses the convergence of the distributional solution under different parameter value conditions.…”

Section: Introductionmentioning

confidence: 99%

A study of periodic solutions of several types of nonlinear models in biomathematics

2024

Applied Mathematics and Nonlinear Sciences

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Biomathematics is a cross-discipline formed by the interpenetration of mathematics with life sciences, biology, and other disciplines, and biomathematical models provide an effective tool for solving problems in the above application areas. Our aim in this paper is to combine mathematical analytical tools and numerical simulation methods to investigate the existence and steady state of periodic solutions in different nonlinear models. Time lags with both discrete and distributed characteristics are introduced into the Lotka-Volterra predator-feeder system, and based on the discussion of the central manifold theorem and canonical type theory, it is proved that the branching periodic solution exists when the discrete time lag parameter τ > τ 0. In the SEIRS infectious disease model with nonlinear incidence term and vertical transmission, the global stability of the disease-free equilibrium point and the local asymptotic stability of the endemic equilibrium point are analyzed through the computation and discussion of the fundamental regeneration number R 0 (p, q). A class of convergence-growth models with nonlinear sensitivity functions is studied, and the global boundedness of classical solutions and their conditions are demonstrated based on global dynamics. A mathematical generalization of the muscular vascular model is made by introducing a centralized parameter, the relationship between periodic solutions and chaotic phenomena is explored utilizing a systematic equivalence transformation, and the equation of the homoscedastic orbitals is deduced to be z 2 = x 2 ( A - 1 2 x 2 ) {z^2} = {x^2}\left( {A - {1 \over 2}{x^2}} \right) .

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Section: Introductionmentioning

confidence: 99%

A study of periodic solutions of several types of nonlinear models in biomathematics

2024

Applied Mathematics and Nonlinear Sciences

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“…But in a real ecological system, a birth rate of the prey species is dependent on the density of prey. In [11], the authors considered density dependent birth rate of the prey species and discussed the dynamical behaviour of the predator-prey system. Aforesaid studies are mainly confined on continuous predator-prey models with two variables.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of discrete predator- prey system with fear effect and density dependent birth rate of the prey species

Mukherjee¹

2022

Math Appl Sci Eng

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This paper analyses a discrete predator-prey system with fear effect and density dependent birth rate of the prey species. The fixed points of the system are determined and their stability is examined. The criterion for Neimark-Sacker bifurcation and flip bifurcation is developed. The chaotic orbit at an unstable fixed point can be stabilized by applying the state feedback control method. Numerically, we illustrate our analytical findings and observe the complex behaviour of the system that leads to stable state to chaotic one.

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“…The predator-prey relationship has been highly valued by scholars because of its widespread existence . Such topics as the influence of the stage structure [6][7][8][11][12][13][14][15], the influence of refuge [17,18,20,21], the influence of mutual interferences [10], the fear effect of the prey species [1][2][3][4][5], the influence of cannibalism [16], and the influence of the Allee effect [19] are extensively studied by scholars.…”

Section: Introductionmentioning

confidence: 99%

Impact of Michaelis–Menten type harvesting in a Lotka–Volterra predator–prey system incorporating fear effect

Lai

et al. 2020

Adv Differ Equ

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We propose and study a Lotka–Volterra predator–prey system incorporating both Michaelis–Menten-type prey harvesting and fear effect. By qualitative analysis of the eigenvalues of the Jacobian matrix we study the stability of equilibrium states. By applying the differential inequality theory we obtain sufficient conditions that ensure the global attractivity of the trivial equilibrium. By applying Dulac criterion we obtain sufficient conditions that ensure the global asymptotic stability of the positive equilibrium. Our study indicates that the catchability coefficient plays a crucial role on the dynamic behavior of the system; for example, the catchability coefficient is the Hopf bifurcation parameter. Furthermore, for our model in which harvesting is of Michaelis–Menten type, the catchability coefficient is within a certain range; increasing the capture rate does not change the final number of prey population, but reduces the predator population. Meanwhile, the fear effect of the prey species has no influence on the dynamic behavior of the system, but it can affect the time when the number of prey species reaches stability. Numeric simulations support our findings.

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Dynamic behaviors of a Lotka-Volterra type predator-prey system with Allee effect on the predator species and density dependent birth rate on the prey species

Cited by 13 publications

References 34 publications

A study of periodic solutions of several types of nonlinear models in biomathematics

A study of periodic solutions of several types of nonlinear models in biomathematics

Dynamics of discrete predator- prey system with fear effect and density dependent birth rate of the prey species

Impact of Michaelis–Menten type harvesting in a Lotka–Volterra predator–prey system incorporating fear effect

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