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(searched for: doi:10.53391/mmnsa.2021.01.009)
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Zhuo Ba, Xianyi Li
Electronic Research Archive, Volume 31, pp 1405-1438; https://doi.org/10.3934/era.2023072

Abstract:
In this paper, a discrete predator-prey model incorporating Allee effect and cannibalism is derived from its continuous version by semidiscretization method. Not only the existence and local stability of fixed points of the discret system are investigated, but more important, the sufficient conditions for the occurrence of its period-doubling bifurcation and Neimark-Sacker bifurcation are obtained using the center manifold theorem and local bifurcation theory. Finally some numerical simulations are given to illustrate the existence of Neimark-Sacker bifurcation. The outcome of the study reveals that this discrete system undergoes various bifurcations including period-doubling bifurcation and Neimark-Sacker bifurcation.
, Ibraheem M. Alsulami, Umbreen Sadiq
Published: 25 September 2022
Journal: Complexity
Complexity, Volume 2022, pp 1-14; https://doi.org/10.1155/2022/6921934

Abstract:
The local dynamics, chaos, and bifurcations of a discrete Brusselator system are investigated. It is shown that a discrete Brusselator system has an interior fixed point P1,r if r>0. Then, by linear stability theory, local dynamical characteristics are explored at interior fixed point P1,r. Furthermore, for the discrete Brusselator system, the existence of periodic points is investigated. The existence of bifurcations around an interior fixed point is also investigated and proved that the discrete Brusselator model undergoes hopf and flip bifurcations if r,hℋℬ|P1,r=r,h,h=2r and r,hℱℬ|P1,r=r,h,h=4/2rr24r, respectively. The next feedback control method is utilized to stabilize the chaos that exists in the discrete Brusselator system. Finally, obtained results are verified numerically.
, Kasey Cooper, Ava Dreher, Caroline McCrorey
Mathematical Biosciences and Engineering, Volume 20, pp 3355-3378; https://doi.org/10.3934/mbe.2023158

Abstract:
Cannibalism, or intraspecific predation, is the act of an organism consuming another organism of the same species. In predator-prey relationships, there is experimental evidence to support the existence of cannibalism among juvenile prey. In this work, we propose a stage-structured predator-prey system where cannibalism occurs only in the juvenile prey population. We show that cannibalism has both a stabilizing and destabilizing effect depending on the choice of parameters. We perform stability analysis of the system and also show that the system experiences a supercritical Hopf, saddle-node, Bogdanov-Takens and cusp bifurcation. We perform numerical experiments to further support our theoretical findings. We discuss the ecological implications of our results.
Qiulin Huang, , Chuanjun Dai, Zengling Ma, Qi Wang, Min Zhao
Mathematical Biosciences and Engineering, Volume 20, pp 930-954; https://doi.org/10.3934/mbe.2023043

Abstract:
Within the framework of physical and ecological integrated control of cyanobacteria bloom, because the outbreak of cyanobacteria bloom can form cyanobacteria clustering phenomenon, so a new aquatic ecological model with clustering behavior is proposed to describe the dynamic relationship between cyanobacteria and potential grazers. The biggest advantage of the model is that it depicts physical spraying treatment technology into the existence pattern of cyanobacteria, then integrates the physical and ecological integrated control with the aggregation of cyanobacteria. Mathematical theory works mainly investigate some key threshold conditions to induce Transcritical bifurcation and Hopf bifurcation of the model $ (2.1) $, which can force cyanobacteria and potential grazers to form steady-state coexistence mode and periodic oscillation coexistence mode respectively. Numerical simulation works not only explore the influence of clustering on the dynamic relationship between cyanobacteria and potential grazers, but also dynamically show the evolution process of Transcritical bifurcation and Hopf bifurcation, which can be clearly seen that the density of cyanobacteria decreases gradually with the evolution of bifurcation dynamics. Furthermore, it should be worth explaining that the most important role of physical spraying treatment technology can break up clumps of cyanobacteria in the process of controlling cyanobacteria bloom, but cannot change the dynamic essential characteristics of cyanobacteria and potential grazers represented by the model $ (2.1) $, this result implies that the physical spraying treatment technology cannot fundamentally eliminate cyanobacteria bloom. In a word, it is hoped that the results of this paper can provide some theoretical support for the physical and ecological integrated control of cyanobacteria bloom.
Yi Tian
Mathematical Modelling and Control, Volume 2, pp 75-80; https://doi.org/10.3934/mmc.2022009

Abstract:
Fractal ordinary differential equations are successfully established by He's fractal derivative in a fractal space, and their variational principles are obtained by semi-inverse transform method.Taylor series method is used to solve the given fractal equations with initial boundary value conditions, and sometimes Ying Buzu algorithm play an important role in this process. Examples show the Taylor series method and Ying Buzu algorithm are powerful and simple tools.
Ming-Zhen Xin, Bin-Guo Wang, Yashi Wang
Mathematical Biosciences and Engineering, Volume 19, pp 9125-9146; https://doi.org/10.3934/mbe.2022424

Abstract:
Influenza is a respiratory infection caused influenza virus. To evaluate the effect of environment noise on the transmission of influenza, our study focuses on a stochastic influenza virus model with disease resistance. We first prove the existence and uniqueness of the global solution to the model. Then we obtain the existence of a stationary distribution to the positive solutions by stochastic Lyapunov function method. Moreover, certain sufficient conditions are provided for the extinction of the influenza virus flu. Finally, several numerical simulations are revealed to illustrate our theoretical results. Conclusively, according to the results of numerical models, increasing disease resistance is favorable to disease control. Furthermore, a simple example demonstrates that white noise is favorable to the disease's extinction.
Yousef Alnafisah, Moustafa El-Shahed
Aims Mathematics, Volume 7, pp 11905-11918; https://doi.org/10.3934/math.2022664

Abstract:
In this paper, a deterministic and stochastic model for hepatitis C with different types of virus genomes is proposed and analyzed. Some sufficient conditions are obtained to ensure the stability of the deterministic equilibrium points. We perform a stochastic extension of the deterministic model to study the fluctuation between environmental factors. Firstly, the existence of a unique global positive solution for the stochastic model is investigated. Secondly, sufficient conditions for the extinction of the hepatitis C virus from the stochastic system are obtained. Theoretical and numerical results show that the smaller white noise can ensure the persistence of susceptible and infected populations while the larger white noise can lead to the extinction of disease. By introducing the basic reproduction number $ R_0 $ and the stochastic basic reproduction number $ R_0^s $, the conditions that cause the disease to die out are indicated. The importance of environmental noise in the propagation of hepatitis C viruses is highlighted by these findings.
, Billel Semmar, Kamal Al Nasr
Published: 1 January 2022
Nonlinear Engineering, Volume 11, pp 100-111; https://doi.org/10.1515/nleng-2022-0013

Abstract:
In this article, a prey–predator system is considered in Caputo-conformable fractional-order derivatives. First, a discretization process, making use of the piecewise-constant approximation, is performed to secure discrete-time versions of the two fractional-order systems. Local dynamic behaviors of the two discretized fractional-order systems are investigated. Numerical simulations are executed to assert the outcome of the current work. Finally, a discussion is conducted to compare the impacts of the Caputo and conformable fractional derivatives on the discretized model.
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