scholarly journals A Stochastic Model for Three Species

2018 ◽  
Vol 7 (4.10) ◽  
pp. 497
Author(s):  
Y. Suresh Kumar ◽  
N. Seshagiri Rao ◽  
B. V AppaRao
Keyword(s):  
Stochastic Process ◽  
Stochastic Model ◽  
Global Stability ◽  
Numerical Simulations ◽  
Local Stability ◽  
Equilibrium Points ◽  
Limited Resources

The present work is related to a three species ecosystem including a mutualism interaction between two species and a predator, where the predator is depending on both the mutual species. All three species in this model are considered in limited resources. The sustainability of the system (local stability) is discussed through the perturbed technique at the possible existing each equilibrium points. Using Lyapunov's technique the global stability of the system is also described. Further the nature of the system is observed by introducing the stochastic process to the species and the numerical simulations are studied to know the interaction among the species. 

More Dynamical Properties Revealed from a 3D Lorenz-like System

2014 ◽  
Vol 24 (10) ◽  
pp. 1450133 ◽  
Author(s):  
Haijun Wang ◽  
Xianyi Li
Keyword(s):  
Numerical Simulations ◽  
Local Stability ◽  
Equilibrium Points ◽  
Heteroclinic Orbits ◽  
Mathematical Work ◽  
Chaotic Attractors ◽  
Heteroclinic Cycles ◽  
Homoclinic And Heteroclinic Orbits ◽  
Dynamical Properties ◽  
Theoretical Results

After a 3D Lorenz-like system has been revisited, more rich hidden dynamics that was not found previously is clearly revealed. Some more precise mathematical work, such as for the complete distribution and the local stability and bifurcation of its equilibrium points, the existence of singularly degenerate heteroclinic cycles as well as homoclinic and heteroclinic orbits, and the dynamics at infinity, is carried out in this paper. In particular, another possible new mechanism behind the creation of chaotic attractors is presented. Based on this mechanism, some different structure types of chaotic attractors are numerically found in the case of small b > 0. All theoretical results obtained are further illustrated by numerical simulations. What we formulate in this paper is to not only show those dynamical properties hiding in this system, but also (more mainly) present a kind of way and means — both "locally" and "globally" and both "finitely" and "infinitely" — to comprehensively explore a given system.

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 A Mathematical Model to Study the Effectiveness of Some of the Strategies Adopted in Curtailing the Spread of COVID-19

2020 ◽  
Vol 2020 ◽  
pp. 1-6 ◽  
Author(s):  
Isa Abdullahi Baba ◽  
Bashir Abdullahi Baba ◽  
Parvaneh Esmaili
Keyword(s):  
Mathematical Model ◽  
Stability Analysis ◽  
Numerical Simulations ◽  
Government Policy ◽  
Local Stability ◽  
Equilibrium Points ◽  
Compartmental Models ◽  
Effective Strategy ◽  
Local Stability Analysis ◽  
The Government

In this paper, we developed a model that suggests the use of robots in identifying COVID-19-positive patients and which studied the effectiveness of the government policy of prohibiting migration of individuals into their countries especially from those countries that were known to have COVID-19 epidemic. Two compartmental models consisting of two equations each were constructed. The models studied the use of robots for the identification of COVID-19-positive patients. The effect of migration ban strategy was also studied. Four biologically meaningful equilibrium points were found. Their local stability analysis was also carried out. Numerical simulations were carried out, and the most effective strategy to curtail the spread of the disease was shown.

scholarly journals A Mathematical Modeling of Tuberculosis Dynamics with Hygiene Consciousness as a Control Strategy

2020 ◽  
pp. 38-48
Author(s):  
Phineas Z. Mawira ◽  
David M. Malonza
Keyword(s):  
Stability Analysis ◽  
Global Stability ◽  
Control Strategy ◽  
Local Stability ◽  
Equilibrium Points ◽  
Asymptotically Stable ◽  
Linear Ordinary Differential Equations ◽  
Global Stability Analysis ◽  
Local Stability Analysis ◽  
Disease Free

Tuberculosis, an airborne infectious disease, remains a major threat to public health in Kenya. In this study, we derived a system of non-linear ordinary differential equations from the SLICR mathematical model of TB to study the effects of hygiene consciousness as a control strategy against TB in Kenya. The effective basic reproduction number (R0) of the model was determined by the next generation matrix approach. We established and analyzed the equilibrium points. Using the Routh-Hurwitz criterion for local stability analysis and comparison theorem for global stability analysis, the disease-free equilibrium (DFE) was found to be locally asymptotically stable given that R0 < 1.  Also by using the Routh-Hurwitz criterion for local stability analysis and Lyapunov function and LaSalle’s invariance principle for global stability analysis, the endemic equilibrium (EE) point was found to be locally asymptotically stable given that R0 > 1. Using MATLAB ode45 solver, we simulated the model numerically and the results suggest that hygiene consciousness can helpin controlling TB disease if incorporated effectively.

Optimal control application to the epidemiology of HBV and HCV co-infection

2021 ◽  
Author(s):  
Muhammad Naeem Jan ◽  
Gul Zaman ◽  
Nigar Ali ◽  
Imtiaz Ahmad ◽  
Zahir Shah
Keyword(s):  
Optimal Control ◽  
Global Stability ◽  
Local Stability ◽  
Equilibrium Points ◽  
Optimality System ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Before And After ◽  
Control Versus ◽  
Order Four

It is very important to note that a mathematical model plays a key role in different infectious diseases. Here, we study the dynamical behaviors of both hepatitis B virus (HBV) and hepatitis C virus (HCV) with their co-infection. Actually, the purpose of this work is to show how the bi-therapy is effective and include an inhibitor for HCV infection with some treatments, which are frequently used against HBV. Local stability, global stability and its prevention from the community are studied. Mathematical models and optimality system of nonlinear DE are solved numerically by RK4. We use linearization, Lyapunov function and Pontryagin’s maximum principle for local stability, global stability and optimal control, respectively. Stability curves and basic reproductive number are plotted with and without control versus different values of parameters. This study shows that the infection will spread without control and can cover with treatment. The intensity of HBV/HCV co-infection is studied before and after optimal treatment. This represents a short drop after treatment. First, we formulate the model then find its equilibrium points for both. The models possess four distinct equilibria: HBV and HCV free, and endemic. For the proposed problem dynamics, we show the local as well as the global stability of the HBV and HCV. With the help of optimal control theory, we increase uninfected individuals and decrease the infected individuals. Three time-dependent variables are also used, namely, vaccination, treatment and isolation. Finally, optimal control is classified into optimality system, which we can solve with Runge–Kutta-order four method for different values of parameters. Finally, we will conclude the results for implementation to minimize the infected individuals.

Stability analysis in prey–predator system with mixed functional response

2016 ◽  
Vol 13 (4) ◽  
pp. 364-369
Author(s):  
V. Madhusudanan ◽  
S. Vijaya
Keyword(s):  
Global Stability ◽  
Numerical Simulations ◽  
Functional Response ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Type I ◽  
Content Type ◽  
Local And Global Stability ◽  
Interior Equilibrium ◽  
Predator System

Purpose This paper aims to propose and analyse a two-prey–one-predator system with mixed functional response. Design/methodology/approach The predator exhibits Holling type IV functional response to one prey and Holling type I response to other. The occurrence of various positive equilibrium points with feasibility conditions are determined. The local and global stability of interior equilibrium points are examined. The boundedness of system is analysed. The sufficient conditions for persistence of the system is obtained by using Bendixson–Dulac criteria. Numerical simulations are carried out to illustrate the analytical findings. Findings The authors have derived the local and global stability condition of interior equilibrium of the system. Originality/value The authors observe that the critical values of some system parameter undergo Hopf bifurcation around some selective equilibrium. Hence, numerical simulations reveal the condition for the system to be stable and oscillatory.

scholarly journals Mathematical Modeling of Typhoid Fever Disease Incorporating Unprotected Humans in the Spread Dynamics

2019 ◽  
pp. 1-11
Author(s):  
Julia Wanjiku Karunditu ◽  
George Kimathi ◽  
Shaibu Osman
Keyword(s):  
Typhoid Fever ◽  
Global Stability ◽  
Equilibrium Point ◽  
Local Stability ◽  
Jacobian Matrix ◽  
Equilibrium Points ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Spread Dynamics ◽  
Disease Free

A deterministic mathematical model of typhoid fever incorporating unprotected humans is formulated in this study and employed to study local and global stability of equilibrium points. The model incorporating Susceptible, unprotected, Infectious and Recovered humans which are analyzed mathematically and also result into a system of ordinary differential equations which are used for interpretations and comparison to the qualitative solutions in studying the spread dynamics of typhoid fever. Jacobian matrix was considered in the study of local stability of disease free equilibrium point and Castillo-Chavez approach used to determine global stability of disease free equilibrium point. Lyapunov function was used to study global stability of endemic equilibrium point. Both equilibrium points (DFE and EE) were found to be local and globally asymptotically stable. This means that the disease will be dependent on numbers of unprotected humans and other factors who contributes positively to the transmission dynamics.

scholarly journals Analysis of the Dynamics of SI-SI-SEIR Avian Influenza A(H7N9) Epidemic Model with Re-infection

2020 ◽  
pp. 43-73
Author(s):  
Oluwafemi I. Bada ◽  
Abayomi S. Oke ◽  
Winfred N. Mutuku ◽  
Patrick O. Aye
Keyword(s):  
Mathematical Model ◽  
Avian Influenza ◽  
Global Stability ◽  
Numerical Simulations ◽  
Epidemic Model ◽  
Influenza A ◽  
Equilibrium Points ◽  
Local And Global Stability ◽  
Continuous Contact ◽  
Non Linear

The spread of Avian influenza in Asia, Europe and Africa ever since its emergence in 2003, has been endemic in many countries. In this study, a non-linear SI-SI-SEIR Mathematical model with re-infection as a result of continuous contact with both infected poultry from farm and market is proposed. Local and global stability of the three equilibrium points are established and numerical simulations are used to validate the results.

A competition model with herd behaviour and allee effect

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2529-2542
Author(s):  
Prosenjit Sen ◽  
Alakes Maiti ◽  
G.P. Samanta
Keyword(s):  
Time Delay ◽  
Numerical Simulations ◽  
Discrete Time ◽  
Local Stability ◽  
Allee Effect ◽  
Equilibrium Points ◽  
Uniform Boundedness ◽  
Competition Model ◽  
Discrete Time Delay ◽  
Herd Behaviour

In this work we have studied the deterministic behaviours of a competition model with herd behaviour and Allee effect. The uniform boundedness of the system has been studied. Criteria for local stability at equilibrium points are derived. The effect of discrete time-delay on the model is investigated. We have carried out numerical simulations to validate the analytical findings. The biological implications of our analytical and numerical findings are discussed.

scholarly journals Dynamical Analysis of a Parasite-Host Model within Fluctuating Environment

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Kun Xu ◽  
Ziang Zhou ◽  
Huitao Zhao
Keyword(s):  
Stochastic Model ◽  
Global Stability ◽  
Numerical Simulations ◽  
Positive Solution ◽  
Deterministic Model ◽  
Dynamical Analysis ◽  
Boundedness Of Solutions ◽  
Fluctuating Environment ◽  
Global Positive Solution ◽  
Positivity And Boundedness

A parasite-host model within fluctuating environment is proposed. Firstly, the positivity and boundedness of solutions of the model within deterministic environment are discussed, and, then, the asymptotical stability and global stability of equilibria of deterministic model are investigated. Secondly, we show that the stochastic model has a unique global positive solution; furthermore, we show that the stochastic model has a stationary distribution under certain conditions. Finally, we give some numerical simulations to illustrate our analytical results.

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