Complexity Analysis of a Modified Predator-Prey System with Beddington–DeAngelis Functional Response and Allee-Like Effect on Predator

Wang, Shuangte; Yu, Hengguo

doi:10.1155/2021/5618190

Cited by 7 publications

(3 citation statements)

References 26 publications

(49 reference statements)

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“…For instance, Agrawal et al [17] have considered Hollings-II functional response [18] to explain the tumour-immune interaction. In the present study, Holling's type-III [19] and Beddington-DeAngelis (BD) functional responses [20] are used to defne the interaction. Holling's type-III functional response motivated by Dawes and Souza [18] is expressed for tumour-immune interaction as…”

Section: Introductionmentioning

confidence: 99%

Mathematical and Numerical Modelling of Interference of Immune Cells in the Tumour Environment

Sinha

Singh

Köksal

2023

Discrete Dynamics in Nature and Society

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In this article, the behaviour of tumour growth and its interaction with the immune system have been studied using a mathematical model in the form of partial differential equations. However, the development of tumours and how they interact with the immune system make up an extremely complex and little-understood system. A new mathematical model has been proposed to gain insight into the role of immune response in the tumour microenvironment when no treatment is applied. The resulting model is a set of partial differential equations made up of four variables: the population density of tumour cells, two different types of immune cells (CD4+ helper T cells and CD8+ cytotoxic T cells), and nutrition content. Such kinds of systems also occur frequently in science and engineering. The interaction of tumour and immune cells is exemplified by predator-prey models in ecology, in which tumour cells act as prey and immune cells act as predators. The tumour-immune cell interaction is expressed via Holling’s Type-III and Beddington-DeAngelis functional responses. The combination of finite volume and finite element method is used to approximate the system numerically because these approximations are more suitable for time-dependent systems having diffusion. Finally, numerical simulations show that the methods perform well and depict the behaviour of the model.

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Section: Introductionmentioning

confidence: 99%

Mathematical and Numerical Modelling of Interference of Immune Cells in the Tumour Environment

Sinha

Singh

Köksal

2023

Discrete Dynamics in Nature and Society

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“…Many researchers have studied the bifurcation and stability analysis of continuous time models with Allee effect in [4,30,34]. In the literature, a discrete-time models with Allee effect on prey population have been studied intensively, however, the research with Allee effect on predator population is relatively rare [3,26,31,32]. So, we introduce a discrete-time system with Allee effect on predator population.…”

Section: Introductionmentioning

confidence: 99%

On the analysis of stability, bifurcation, and chaos control of discrete-time predator-prey model with Allee effect on predator

Işık

Kangalgi̇l

2022

Hacettepe Journal of Mathematics and Statistics

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In this paper, a discrete predator-prey model with Allee effect which is obtained by the forward Euler method has been investigated. The local stability conditions of the model at the fixed point have been discussed. In addition, it is shown that the model undergoes Neimark-Sacker bifurcation by using bifurcation theory. Then, the direction of Neimark-Sacker bifurcation has been given. The OGY method is applied in order to control chaos in considered model due to emergence of Neimark-Sacker bifurcation. Some numerical simulations such as phase portraits and bifurcation figures have been presented to support the theoretical results. Also, the chaotic features are justified numerically by computing Lyapunov exponents. Because of consistency with the biological facts, the parameter values have been taken from literature [Controlling chaos and Neimark-Sacker bifurcation discrete-time predator-prey system, Hacet.

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“…have been introduced. Allee effect is one of the key ecological variables that can significantly alter the preypredator system [7,8]. When simulating the interactions between predators and prey, it is crucial to take into account the increase of the prey population to its carrying capacity in the absence of predators.…”

Section: Introductionmentioning

confidence: 99%

Güçlü Allee Etkili Av-Avcı Modelinin Kararlılığı, Neimark-Sacker Çatallanma Analizi ve Kaos Kontrol

Elmacı

Kangalgi̇l

2022

Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi

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In this study, the dynamical behaviors of a prey–predator model with multiple strong Allee effect are investigated. The fixed points of the model are examined for existence and topological classification. By selecting as the bifurcation parameter $\beta$, it is demonstrated that the model can experience a Neimark-Sacker bifurcation at the unique positive fixed point. Bifurcation theory is used to present the Neimark-Sacker bifurcation conditions of existence and the direction of the bifurcation. Additionally, some numerical simulations are provided to back up the analytical result. Following that, the model's bifurcation diagram and the triangle-shaped stability zone are provided.

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Complexity Analysis of a Modified Predator-Prey System with Beddington–DeAngelis Functional Response and Allee-Like Effect on Predator

Cited by 7 publications

References 26 publications

Mathematical and Numerical Modelling of Interference of Immune Cells in the Tumour Environment

Mathematical and Numerical Modelling of Interference of Immune Cells in the Tumour Environment

On the analysis of stability, bifurcation, and chaos control of discrete-time predator-prey model with Allee effect on predator

Güçlü Allee Etkili Av-Avcı Modelinin Kararlılığı, Neimark-Sacker Çatallanma Analizi ve Kaos Kontrol

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