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We investigate some qualitative behavior of the solutions of the difference equation xn+ = axn - bxn/(cxx - dxn-1), n = 0,1,..., where the initial conditions x-1, x0 are arbitrary real numbers and a, b, c, d are positive constants.

This research paper investigates discrete predator-prey dynamics with two logistic maps. The study extensively examines various aspects of the system's behavior. Firstly, it thoroughly investigates the existence and stability of fixed points within the system. We explores the emergence of transcritical bifurcations, period-doubling bifurcations, and Neimark-Sacker bifurcations that arise from coexisting positive fixed points. By employing central bifurcation theory and bifurcation theory techniques. Chaotic behavior is analyzed using Marotto's approach. The OGY feedback control method is implemented to control chaos. Theoretical findings are validated through numerical simulations., (© 2024. The Author(s).)

In this study, we investigate the dynamics of a discrete-time with predator-prey system with a Holling-III type functional response model. The center manifold theorem and bifurcation theory are used to create existence conditions for flip bifurcations and Neimark-Sacker bifurcations. Bifurcation diagrams, maximum Lyapunov exponents, and phase portraits are examples of numerical simulations that not only show the soundness of theoretical analysis but also show complicated dynamical behaviors and biological processes. From the point of view of biology, this implies that the tiny integral step size can steady the system into locally stable coexistence. Yet, the large integral step size may lead to instability in the system, producing more intricate and richer dynamics. This also means that when the intrinsic death rate of the predator is high, this leads to a chaotic growth rate of the prey. The model has bifurcation features that are similar to those seen in logistic models. In addition, there is a bidirectional Neimark-Sacker bifurcation for both prey and predator, and therefore we obtain a direct correlation in symbiosis. This means that the higher the growth rate of the prey, the greater the growth rate of the predator. Therefore, the operation of predation has increased. The opposite is also true. Finally, the OGY approach is used to control chaos in the predator and prey model. which led to a new concept which we call bifurcation phase of control chaos., (© 2022. The Author(s).)

In this work, by using both the comparison technique with first-order differential inequalities and the Riccati transformation, we extend this development to a class of third-order neutral differential equations of the mixed type. We present new criteria for oscillation of all solutions, which improve and extend some existing ones in the literature. In addition, we provide an example to illustrate our results.

COVID-19 has become a world wide pandemic since its first appearance at the end of the year 2019. Although some vaccines have already been announced, a new mutant version has been reported in UK. We certainly should be more careful and make further investigations to the virus spread and dynamics. This work investigates dynamics in Lotka-Volterra based Models of COVID-19. The proposed models involve fractional derivatives which provide more adequacy and realistic description of the natural phenomena arising from such models. Existence and boundedness of non-negative solution of the fractional model is proved. Local stability is also discussed based on Matignon's stability conditions. Numerical results show that the fractional parameter has effect on flattening the curves of the coexistence steady state. This interesting foundation might be used among the public health strategies to control the spread of COVID-19 and its mutated versions., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2021 The Author(s).)