Supercritical Neimark–Sacker Bifurcation and Hybrid Control in a Discrete-Time Glycolytic Oscillator Model

We study the local dynamical properties, Neimark–Sacker bifurcation, and hybrid control in a glycolytic oscillator model in the interior of ℝ+2. It is proved that, for all parametric values, Pxy+α/β+α2,α is the unique positive equilibrium point of the glycolytic oscillator model. Further local dynamical properties along with different topological classifications about the equilibrium Pxy+α/β+α2,α have been investigated by employing the method of linearization. Existence of prime period and periodic points of the model under consideration are also investigated. It is proved that, about the fixed point Pxy+α/β+α2,α, the discrete-time glycolytic oscillator model undergoes no bifurcation, except Neimark–Sacker bifurcation. A further hybrid control strategy is applied to control Neimark–Sacker bifurcation in the discrete-time model. Finally, theoretical results are verified numerically.

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Impact of Dispersion on Dynamics of a Discrete Metapopulation Model

Open Systems & Information Dynamics ◽

10.1007/s11080-007-9063-1 ◽

2007 ◽

Vol 14 (04) ◽

pp. 379-396 ◽

Cited By ~ 1

Author(s):

Yu Huang ◽

Xingfu Zou

Keyword(s):

Periodic Orbit ◽

Discrete Time ◽

Positive Equilibrium ◽

Global Dynamics ◽

Metapopulation Model ◽

Time Model ◽

Discrete Time Model ◽

Global Extinction ◽

Dispersion Terms ◽

The Impact

We propose and analyze a discrete time model for metapopulation on two patches with local logistic dynamics. The model carries a delay in the dispersion terms, and our results on this model show that the impact of the dispersion on the global dynamics of the metapopulation is complicated and interesting: it can affect the existence of a positive equilibrium; it can either drive the metapopulation to global extinction, or prevent the metapopulation from going to global extinction and stabilize a positive equilibrium; it can also destabilize a positive equilibrium or a periodic orbit.

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A Discrete-Time Compartmental Epidemiological Model for COVID-19 with a Case Study for Portugal

Axioms ◽

10.3390/axioms10040314 ◽

2021 ◽

Vol 10 (4) ◽

pp. 314

Author(s):

Sandra Vaz ◽

Delfim F. M. Torres

Keyword(s):

Mathematical Model ◽

Discrete Time ◽

Continuous Time ◽

Local Stability ◽

Real Data ◽

Time Model ◽

Discrete Time Model ◽

Theoretical Results ◽

Disease Free

Recently, a continuous-time compartmental mathematical model for the spread of the Coronavirus disease 2019 (COVID-19) was presented with Portugal as case study, from 2 March to 4 May 2020, and the local stability of the Disease Free Equilibrium (DFE) was analysed. Here, we propose an analogous discrete-time model and, using a suitable Lyapunov function, we prove the global stability of the DFE point. Using COVID-19 real data, we show, through numerical simulations, the consistence of the obtained theoretical results.

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Qualitative analysis of a discrete-time phytoplankton–zooplankton model with Holling type-II response and toxicity

Advances in Difference Equations ◽

10.1186/s13662-021-03599-z ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Muhammad Salman Khan ◽

Maria Samreen ◽

Hassen Aydi ◽

Manuel De la Sen

Keyword(s):

Mathematical Models ◽

Discrete Time ◽

Hybrid Method ◽

Continuous Time ◽

Hybrid Control ◽

Numerical Study ◽

Equilibrium Points ◽

Control Technique ◽

Positive Equilibrium ◽

Time Model

AbstractThe interaction among phytoplankton and zooplankton is one of the most important processes in ecology. Discrete-time mathematical models are commonly used for describing the dynamical properties of phytoplankton and zooplankton interaction with nonoverlapping generations. In such type of generations a new age group swaps the older group after regular intervals of time. Keeping in observation the dynamical reliability for continuous-time mathematical models, we convert a continuous-time phytoplankton–zooplankton model into its discrete-time counterpart by applying a dynamically consistent nonstandard difference scheme. Moreover, we discuss boundedness conditions for every solution and prove the existence of a unique positive equilibrium point. We discuss the local stability of obtained system about all its equilibrium points and show the existence of Neimark–Sacker bifurcation about unique positive equilibrium under some mathematical conditions. To control the Neimark–Sacker bifurcation, we apply a generalized hybrid control technique. For explanation of our theoretical results and to compare the dynamics of obtained discrete-time model with its continuous counterpart, we provide some motivating numerical examples. Moreover, from numerical study we can see that the obtained system and its continuous-time counterpart are stable for the same values of parameters, and they are unstable for the same parametric values. Hence the dynamical consistency of our obtained system can be seen from numerical study. Finally, we compare the modified hybrid method with old hybrid method at the end of the paper.

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Analysis of Oscillatory Patterns of a Discrete-Time Rosenzweig–MacArthur Model

International Journal of Bifurcation and Chaos ◽

10.1142/s021812741850075x ◽

2018 ◽

Vol 28 (06) ◽

pp. 1850075

Author(s):

Xiaoqin P. Wu ◽

Liancheng Wang

Keyword(s):

Discrete Time ◽

Normal Forms ◽

Critical Values ◽

Positive Equilibrium ◽

Normal Form Theory ◽

Center Manifold Theorem ◽

Form Theory ◽

Positive Equilibrium Point ◽

Associated Species ◽

Theoretical Results

This article investigates the oscillatory patterns of the following discrete-time Rosenzweig–MacArthur model [Formula: see text] The system describes the evolution and interaction of the populations of two associated species (prey and predator) from generation to generation. We show that this system can exhibit co-dimension-1 bifurcations (flip and Neimark–Sacker bifurcations) as [Formula: see text] crosses some critical values and codimension-2 bifurcations (1:2, 1:3, and 1:4 resonances) for certain critical values of [Formula: see text] at the positive equilibrium point. The normal form theory and the center manifold theorem are used to obtain the normal forms. For codimension-2 bifurcations, the bifurcation diagrams are established by using these normal forms along the orbits of differential equations. Numerical simulations are presented to confirm the theoretical results.

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