scholarly journals Student Group Dynamic Model Based on Understanding in Mathematics Subjects

2021 ◽  
Vol 7 (1) ◽  
pp. 41-50
Author(s):  
M. Ivan Ariful Fathoni ◽  
Anisa Fitri ◽  
Hanifahtul Husnah
Keyword(s):  
Equilibrium Point ◽  
Subject Matter ◽  
Equilibrium Points ◽  
Point Of View ◽  
Dimensional System ◽  
Student Group ◽  
Group Dynamic ◽  
Trivial Equilibrium ◽  
Two Dimensional System ◽  
The Stability

This study discusses the interaction of students with a mathematical modeling point of view. This interaction involves students who understand and do not understand mathematics subject matter. The interaction process between groups is modeled in a two-dimensional system of differential equations. Variable A is the percentage of students who understand the material, and variable B is the percentage of students who do not understand the material. The dynamic analysis results obtained by one trivial equilibrium point and three non-trivial equilibrium points exist with several conditions. Based on the stability analysis of the non-trivial equilibrium point, it is found that the conditions without students do not understand mathematics subject matter. This condition is the goal of this study, which involves interaction between students; it can increase the learning process's success.

Impact of Additive Allee Effect on the Dynamics of an Intraguild Predation Model with Specialist Predator

2020 ◽  
Vol 30 (16) ◽  
pp. 2050239
Author(s):  
Udai Kumar ◽  
Partha Sarathi Mandal
Keyword(s):  
System Dynamics ◽  
Equilibrium Point ◽  
Intraguild Predation ◽  
Allee Effect ◽  
Equilibrium Points ◽  
Dimensional System ◽  
Global Dynamics ◽  
Interior Equilibrium ◽  
Trivial Equilibrium ◽  
The Impact

Many important factors in ecological communities are related to the interplay between predation and competition. Intraguild predation or IGP is a mixture of predation and competition which is a very basic three-dimensional system in food webs where two species are related to predator–prey relationship and are also competing for a shared prey. On the other hand, Allee effect is also a very important ecological factor which causes significant changes to the system dynamics. In this work, we consider a intraguild predation model in which predator is specialist, the growth of shared prey population is subjected to additive Allee effect and there is Holling-Type III functional response between IG prey and IG predator. We analyze the impact of Allee effect on the global dynamics of the system with the prior knowledge of the dynamics of the model without Allee effect. Our theoretical and numerical analyses suggest that: (1) Trivial equilibrium point is always locally asymptotically stable and it may be globally stable also. Hence, all the populations may go to extinction depending upon initial conditions; (2) Bistability is observed between unique interior equilibrium point and trivial equilibrium point or between boundary equilibrium point and trivial equilibrium point; (3) Multiple interior equilibrium points exist under certain parameters range. We also provide here a comprehensive study of bifurcation analysis by considering Allee effect as one of the bifurcation parameters. We observed that Allee effect can generate all possible bifurcations such as transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, Bogdanov–Taken bifurcation and Bautin bifurcation. Finally, we compared our model with the IGP model without Allee effect for better understanding the impact of Allee effect on the system dynamics.

scholarly journals A Stage-Structure Rosenzweig-MacArthur Model with Effect of Prey Refuge

2020 ◽  
Vol 1 (1) ◽  
pp. 1-7
Author(s):  
Lazarus Kalvein Beay ◽  
Maryone Saija
Keyword(s):  
Equilibrium Point ◽  
Local Stability ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Structure Population ◽  
Prey Refuge ◽  
Behavioral System ◽  
Trivial Equilibrium ◽  
Coexistence Equilibrium

We proposed and analyzed a stage-structure Rosenzweig-MacArthur model incorporating a prey refuge.  It is assumed that the prey is a stage-structure population consisting of two compartments known as immature prey and mature prey. The model incorporates the functional response Holling type-II. In this work, we investigate all the biologically feasible equilibrium points, and it is shown that the system has three equilibrium points. Sufficient conditions for the local stability of the non-negative equilibrium point of the model are also derived. All points are conditionally locally asymptotically stable. By constructing Jacobian matrix and determined eigenvalues, we analyzed the local stability of the trivial equilibrium and non-predator equilibrium points. Specifically for coexistence equilibrium point, Routh-Hurwitz criterion used to analyze local stability. In addtion, we investigated the effect of immature prey refuge. Our mathematical analysis exhibits that immature prey refuge have played a crucial role in the behavioral system. When the effect of immature prey refuge (constant m) increases, it is can stabilize non-predator equilibrium point, where all the species can not exists together. And conversely, if contant m decreases, it is can stabilize coexistence equilibrium point then all the species can exists together. The work is completed with a numerical simulation to confirmed analitical results

scholarly journals Mathematical Modelling of Deforestation Due to Population Density and Industrialization

Jurnal Varian ◽  
2021 ◽  
Vol 5 (1) ◽  
pp. 9-16
Author(s):  
Didiharyono D. ◽  
Irwan Kasse
Keyword(s):  
Numerical Simulation ◽  
Population Density ◽  
Mathematical Modelling ◽  
Equilibrium Point ◽  
Equilibrium Points ◽  
Stability Criteria ◽  
Hurwitz Stability ◽  
Stable Conditions ◽  
Linear Differential ◽  
The Stability

The focus of the study in this paper is to model deforestation due to population density and industrialization. To begin with, it is formulated into a mathematical modelling which is a system of non-linear differential equations. Then, analyze the stability of the system based on the Routh-Hurwitz stability criteria. Furthermore, a numerical simulation is performed to determine the shift of a system. The results of the analysis to shown that there are seven non-negative equilibrium points, which in general consist equilibrium point of disturbance-free and equilibrium points of disturbances. Equilibrium point TE7(x, y, z) analyzed to shown asymptotically stable conditions based on the Routh-Hurwitz stability criteria. The numerical simulation results show that if the stability conditions of a system have been met, the system movement always occurs around the equilibrium point.

Dynamics of an ecological system

2019 ◽  
Vol 10 (4) ◽  
pp. 355-376
Author(s):  
Shashi Kant
Keyword(s):  
Saddle Point ◽  
Numerical Simulations ◽  
Equilibrium Point ◽  
Local Stability ◽  
Deterministic Model ◽  
Equilibrium Points ◽  
Prey Refuge ◽  
Trivial Equilibrium ◽  
Stability Results ◽  
Predator System

AbstractIn this paper, we investigate the deterministic and stochastic prey-predator system with refuge. The basic local stability results for the deterministic model have been performed. It is found that all the equilibrium points except the positive coexisting equilibrium point of the deterministic model are independent of the prey refuge. The trivial equilibrium point, predator free equilibrium point and prey free equilibrium point are always unstable (saddle point). The existence and local stability of the coexisting equilibrium point is related to the prey refuge. The permanence and extinction conditions of the proposed biological model have been studied rigourously. It is observed that the stochastic effect may be seen in the form of decaying of the species. The numerical simulations for different values of the refuge values have also been included for understanding the behavior of the model graphically.

scholarly journals Equilibrium Further Studied for Combined System of Cournot and Bertrand: A Differential Approach

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Bingyuan Gao ◽  
Yueping Du
Keyword(s):  
General Equilibrium ◽  
Equilibrium Point ◽  
Price Competition ◽  
Dynamic Equilibrium ◽  
Equilibrium Points ◽  
Cournot Model ◽  
Combined System ◽  
Quantity Competition ◽  
Bertrand Model ◽  
The Stability

In general, quantity competition and price competition exist simultaneously in a dynamic economy system. Whether it is quantity competition or price competition, when there are more than three companies in one market, the equilibrium points will become chaotic and are very difficult to be derived. This paper considers generally dynamic equilibrium points of combination of the Bertrand model and Cournot model. We analyze general equilibrium points of the Bertrand model and Cournot model, respectively. A general equilibrium point of the combination of the Cournot model and Bertrand model is further investigated in two cases. The theory of spatial agglomeration and intermediate value theorem are introduced. In addition, the stability of equilibrium points is further illustrated on celestial bodies motion. The results show that at least a general equilibrium point exists in combination of Cournot and Bertrand. Numerical simulations are given to support the research results.

The Qualitative Analysis of Two Populations Commensalisms Model

2012 ◽  
Vol 524-527 ◽  
pp. 3705-3708
Author(s):  
Guang Cai Sun
Keyword(s):  
Qualitative Analysis ◽  
Equilibrium Point ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Mathematics Model ◽  
Stable Equilibrium Point ◽  
The Stability ◽  
Two Populations

This paper deals with the mathematics model of two populations Commensalisms symbiosis and the stability of all equilibrium points the system. It has given the conclusion that there is only one stable equilibrium point the system. This paper also elucidates the biology meaning of the model and its equilibrium points.

scholarly journals A Four-Zone Model and Nonlinear Dynamic Analysis of Solution Multiplicity of Buoyancy Ventilation in Underground Building

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-24
Author(s):  
Yuxing Wang ◽  
Chunyu Wei
Keyword(s):  
Differential Equations ◽  
Equilibrium Point ◽  
Equilibrium Points ◽  
Underground Structure ◽  
Unstable Equilibrium ◽  
Phase Portraits ◽  
Zone Model ◽  
Unstable Equilibrium Point ◽  
Index 2 ◽  
The Stability

The solution multiplicity of natural ventilation in buildings is very important to personnel safety and ventilation design. In this paper, a four-zone model of buoyancy ventilation in typical underground building is proposed. The underground structure is divided to four zones, a differential equation is established in each zone, and therefore, there are four differential equations in the underground structure. By solving and analyzing the equilibrium points and characteristic roots of the differential equations, we analyze the stability of three scenarios and obtain the criterions to determine the stability and existence of solutions for two scenarios. According to these criterions, the multiple steady states of buoyancy ventilation in any four-zone underground buildings for different stack height ratios and the strength ratios of the heat sources can be obtained. These criteria can be used to design buoyancy ventilation or natural exhaust ventilation systems in underground buildings. Compared with the two-zone model in (Liu et al. 2020), the results of the proposed four-zone model are more consistent with CFD results in (Liu et al. 2018). In addition, the results of proposed four-zone model are more specific and more detailed in the unstable equilibrium point interval. We find that the unstable equilibrium point interval is divided into two different subintervals corresponding to the saddle point of index 2 and the saddle focal equilibrium point of index 2, respectively. Finally, the phase portraits and vector field diagrams for the two scenarios are given.

STABILITY BOUNDARY CHARACTERIZATION OF NONLINEAR AUTONOMOUS DYNAMICAL SYSTEMS IN THE PRESENCE OF A SUPERCRITICAL HOPF EQUILIBRIUM POINT

2013 ◽  
Vol 23 (12) ◽  
pp. 1350196 ◽  
Author(s):  
JOSAPHAT R. R. GOUVEIA ◽  
FABÍOLO MORAES AMARAL ◽  
LUÍS F. C. ALBERTO
Keyword(s):  
Dynamical Systems ◽  
Equilibrium Point ◽  
Equilibrium Points ◽  
Stability Boundary ◽  
Complete Characterization ◽  
Center Manifolds ◽  
The Stability ◽  
Hyperbolic Equilibrium ◽  
Autonomous Dynamical Systems

A complete characterization of the boundary of the stability region (or area of attraction) of nonlinear autonomous dynamical systems is developed admitting the existence of a particular type of nonhyperbolic equilibrium point on the stability boundary, the supercritical Hopf equilibrium point. Under a condition of transversality, it is shown that the stability boundary is comprised of all stable manifolds of the hyperbolic equilibrium points lying on the stability boundary union with the center-stable and\or center manifolds of the type-k, k ≥ 1, supercritical Hopf equilibrium points on the stability boundary.

On the stability of steady-states of a two-dimensional system of ferromagnetic nanowires

2017 ◽  
Vol 23 (2) ◽  
Author(s):  
Sharad Dwivedi ◽  
Shruti Dubey
Keyword(s):  
Finite Number ◽  
Steady States ◽  
Dimensional System ◽  
Two Dimensional ◽  
Ferromagnetic Nanowires ◽  
Two Dimensional System ◽  
The Stability

AbstractWe investigate the stability features of steady-states of a two-dimensional system of ferromagnetic nanowires. We constitute a system with the finite number of nanowires arranged on the

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