DYNAMICAL ANALYSIS OF A ALLELOPATHIC PHYTOPLANKTON MODEL

2006 ◽  
Vol 14 (02) ◽  
pp. 205-217 ◽  
Author(s):  
MALAY BANDYOPADHYAY
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Equilibrium Points ◽  
Toxic Substance ◽  
Model System ◽  
Dynamical Analysis ◽  
Local Asymptotic Stability ◽  
Stabilizing Effect

In this paper we have considered a two-species competitive phytoplankton system with one toxin producing phytoplankton. Local asymptotic stability of various equilibrium points are considered to understand the effect of toxic substance on the dynamics of the model system. By using a suitable Lyapunov function we have observed that the toxic substance has some stabilizing effect on the dynamics of model system.

scholarly journals Dynamical Analysis Within-Host and Between-Host for HIV\AIDS with the Application of Optimal Control Strategy

2020 ◽  
pp. 1173-1189
Author(s):  
Ahmed Ali Mohsen ◽  
Raid Kamel Naji
Keyword(s):  
Optimal Control ◽  
Asymptotic Stability ◽  
Control Strategy ◽  
Equilibrium Points ◽  
The Body ◽  
Dynamical Analysis ◽  
Optimal Control Strategy ◽  
Local Asymptotic Stability ◽  
External Sources ◽  
Two Stages

The aims of this paper is investigating the spread of AIDS both within-host, through the contact between healthy cells with free virus inside the body, and between-host, through sexual contact among individuals and external sources of infectious. The outbreak of AIDS is described by a mathematical model consisting of two stages. The first stage describes the within-host spread of AIDS and is represented by the first three equations. While the second stage describes the between-host spread of AIDS and represented by the last four equations. The existence, uniqueness and boundedness of the solution of the model are discussed and all possible equilibrium points are determined. The local asymptotic stability (LAS) of the model is studied, while suitable Lyapunov functions are used to investigate the global asymptotic stability (GAS) of the model. Optimal control strategy is used to control the outbreak of AIDS. Finally, a numerical simulation is carried out to confirm the analytical results and understand the effects of varying the parameters on the spread of disease.

scholarly journals DETERMINISTIC CHAOS VERSUS STOCHASTIC OSCILLATION IN A PREY-PREDATOR-TOP PREDATOR MODEL

2011 ◽  
Vol 16 (3) ◽  
pp. 343-364 ◽  
Author(s):  
Ranjit Kumar Upadhyay ◽  
Malay Banerjee ◽  
Rana Parshad ◽  
Sharada Nandan Raw
Keyword(s):  
Deterministic Model ◽  
Driving Forces ◽  
Equilibrium Points ◽  
Chaotic Oscillation ◽  
Three Dimensional ◽  
Deterministic Chaos ◽  
Model System ◽  
Dynamical Analysis ◽  
Interior Equilibrium ◽  
Predator Model

The main objective of the present paper is to consider the dynamical analysis of a three dimensional prey-predator model within deterministic environment and the influence of environmental driving forces on the dynamics of the model system. For the deterministic model we have obtained the local asymptotic stability criteria of various equilibrium points and derived the condition for the existence of small amplitude periodic solution bifurcating from interior equilibrium point through Hopf bifurcation. We have obtained the parametric domain within which the model system exhibit chaotic oscillation and determined the route to chaos. Finally, we have shown that chaotic oscillation disappears in presence of environmental driving forces which actually affect the deterministic growth rates. These driving forces are unable to drive the system from a regime of deterministic chaos towards a stochastically stable situation. The stochastic stability results are discussed in terms of the stability of first and second order moments. Exhaustive numerical simulations are carried out to validate the analytical findings.

Discontinuous Lyapunov functions for a class of piecewise affine systems

2018 ◽  
Vol 41 (3) ◽  
pp. 729-736 ◽  
Author(s):  
Farideh Cheraghi-Shami ◽  
Ali-Akbar Gharaveisi ◽  
Malihe M Farsangi ◽  
Mohsen Mohammadian
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Lyapunov Functions ◽  
Convex Polytopes ◽  
Sufficient Conditions ◽  
Monotonicity Condition ◽  
Piecewise Affine Systems ◽  
Affine Systems ◽  
Piecewise Affine ◽  
Local Asymptotic Stability

In this paper, a Lyapunov-based method is provided to study the local asymptotic stability of planar piecewise affine systems with continuous vector fields. For such systems, the state space is supposed to be partitioned into several bounded convex polytopes. A piecewise affine function, not necessarily continuous on the boundaries of the polytopic partitions, is proposed as a candidate Lyapunov function. Then, sufficient conditions for the local asymptotic stability of the system, including a monotonicity condition at switching instants, are formulated as a linear programming problem. In addition, when the problem does not have a feasible solution based on the original partitions of the system, a new partition refinement algorithm is presented. In this way, more flexibility can be provided in searching for the Lyapunov function. Owing to relaxation of the continuity condition imposed on the system boundaries, the proposed method reaches to less conservative results, compared with the previous methods based on continuous piecewise affine Lyapunov functions. Simulation results illustrate the effectiveness of the proposed method.

Conformable Fractional SEIR Epidemic Model With Treatment

2021 ◽  
Vol 14 ◽  
Author(s):  
Arti Malik ◽  
Nitendra Kumar ◽  
Khursheed Alam
Keyword(s):  
Differential Equation ◽  
Numerical Simulation ◽  
Asymptotic Stability ◽  
Epidemic Model ◽  
Analytical Study ◽  
Equilibrium Points ◽  
Treatment Modalities ◽  
Local Asymptotic Stability ◽  
Fractional Differential ◽  
Disease Free

Background: The present paper is based on models of conformable fractional differential equation to describe the dynamics of certain epidemics. Methods: In this paper we have divided the population in the susceptible, exposed, infectious, recovered and also describe the treatment modalities. Results: The analytical study of the model show two equilibrium points (disease free equilibrium and endemic equilibrium). Conclusion: For both cases local asymptotic stability has been proven. In the conclusion we have presented the numerical simulation.

scholarly journals Effect of time delay on a detritus-based ecosystem

2006 ◽  
Vol 2006 ◽  
pp. 1-28 ◽  
Author(s):  
Nurul Huda Gazi ◽  
Malay Bandyopadhyay
Keyword(s):  
Time Delay ◽  
Equilibrium Points ◽  
Equation Model ◽  
Model System ◽  
Trophic Levels ◽  
Dynamical Analysis ◽  
Discrete Time Delay ◽  
Growth Equations ◽  
The Stability ◽  
Delay Differential

Models of detritus-based ecosystems with delay have received a great deal of attention for the last few decades. This paper deals with the dynamical analysis of a nonlinear model of a detritus-based ecosystem involving detritivores and predator of detritivores. We have obtained the criteria for local stability of various equilibrium points and persistence of the model system. Next, we have introduced discrete time delay due to recycling of dead organic matters and gestation of nutrients to the growth equations of various trophic levels. With delay differential equation model system we have studied the effect of time delay on the stability behaviour. Next, we have obtained an estimate for the length of time delay to preserve the stability of the model system. Finally, the existence of Hopf-bifurcating small amplitude periodic solutions is derived by considering time delay as a bifurcation parameter.

Exploring Complex Dynamics of Spatial Predator–Prey System: Role of Predator Interference and Additional Food

2020 ◽  
Vol 30 (07) ◽  
pp. 2050102
Author(s):  
Vandana Tiwari ◽  
Jai Prakash Tripathi ◽  
Debaldev Jana ◽  
Satish Kumar Tiwari ◽  
Ranjit Kumar Upadhyay
Keyword(s):  
Asymptotic Stability ◽  
Equilibrium Solution ◽  
Model System ◽  
Positive Equilibrium ◽  
Additional Food ◽  
Predator Prey ◽  
Local Asymptotic Stability ◽  
Diffusive Model ◽  
Predator Interference

In this paper, an attempt has been made to understand the role of predator’s interference and additional food on the dynamics of a diffusive population model. We have studied a predator–prey interaction system with mutually interfering predator by considering additional food and Crowley–Martin functional response (CMFR) for both the reaction–diffusion model and associated spatially homogeneous system. The local stability analysis ensures that as the quantity of alternative food decreases, predator-free equilibrium stabilizes. Moreover, we have also obtained a condition providing a threshold value of additional food for the global asymptotic stability of coexisting steady state. The nonspatial model system changes stability via transcritical bifurcation and switches its stability through Hopf-bifurcation with respect to certain ranges of parameter determining the quantity of additional food. Conditions obtained for local asymptotic stability of interior equilibrium solution of temporal system determines the local asymptotic stability of associated diffusive model. The global stability of positive equilibrium solution of diffusive model system has been established by constructing a suitable Lyapunov function and using Green’s first identity. Using Harnack inequality and maximum modulus principle, we have established the nonexistence of nonconstant positive equilibrium solution of the diffusive model system. A chain of patterns on increasing the strength of additional food as spots[Formula: see text][Formula: see text][Formula: see text]stripes[Formula: see text][Formula: see text][Formula: see text]spots has been obtained. Various kind of spatial-patterns have also been demonstrated via numerical simulations and the roles of predator interference and additional food are established.

scholarly journals Dynamical Analysis of Two-Microorganism and Single Nutrient Stochastic Chemostat Model with Monod-Haldane Response Function

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Mengnan Chi ◽  
Wencai Zhao
Keyword(s):  
Lyapunov Function ◽  
Response Function ◽  
Existence And Uniqueness ◽  
Random Perturbation ◽  
Equilibrium Points ◽  
Dynamical Analysis ◽  
Deterministic System ◽  
Chemostat Model ◽  
Global Positive Solution ◽  
The Stability

In this paper, we formulate and investigate a two-microorganism and single nutrient chemostat model with Monod-Haldane response function and random perturbation. First, for the corresponding deterministic system, we introduce the conditions of the stability of the equilibrium points. Then, using Lyapunov function and Itô’s formula, we investigate the existence and uniqueness of the global positive solution of the stochastic chemostat model. Furthermore, we explore and obtain the criterions of the extinction and the permanence for the stochastic model. Finally, numerical simulations are carried out to illustrate our main results.

SOME NEW STABILITY PROPERTIES OF DYNAMIC NEURAL NETWORKS WITH DIFFERENT TIME-SCALES

2006 ◽  
Vol 16 (03) ◽  
pp. 191-199 ◽  
Author(s):  
WEN YU ◽  
ALEJANDRO CRUZ SANDOVAL
Keyword(s):  
Neural Networks ◽  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Time Scales ◽  
Equilibrium Points ◽  
Singularly Perturbed ◽  
Dynamic Neural Networks ◽  
Numerical Examples ◽  
Theoretical Results ◽  
Different Time Scales

Dynamic neural networks with different time-scales include the aspects of fast and slow phenomenons. Some applications require that the equilibrium points of these networks to be stable. The main contribution of the paper is that Lyapunov function and singularly perturbed technique are combined to access several new stable properties of different time-scales neural networks. Exponential stability and asymptotic stability are obtained by sector and bound conditions. Compared to other papers, these conditions are simpler. Numerical examples are given to demonstrate the effectiveness of the theoretical results.

scholarly journals Global Stability Analysis of a Bioreactor Model for Phenol and Cresol Mixture Degradation

Processes ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 124
Author(s):  
Neli Dimitrova ◽  
Plamena Zlateva
Keyword(s):  
Specific Growth ◽  
Equilibrium Points ◽  
Nonlinear Ordinary Differential Equations ◽  
The Novel ◽  
Model Design ◽  
Stirred Bioreactor ◽  
Global Stability Analysis ◽  
Local Asymptotic Stability ◽  
Model Dynamics ◽  
Stability And Bifurcations

We propose a mathematical model for phenol and p-cresol mixture degradation in a continuously stirred bioreactor. The model is described by three nonlinear ordinary differential equations. The novel idea in the model design is the biomass specific growth rate, known as sum kinetics with interaction parameters (SKIP) and involving inhibition effects. We determine the equilibrium points of the model and study their local asymptotic stability and bifurcations with respect to a practically important parameter. Existence and uniqueness of positive solutions are proved. Global stabilizability of the model dynamics towards equilibrium points is established. The dynamic behavior of the solutions is demonstrated on some numerical examples.

scholarly journals Mathematical analysis of a measles transmission dynamics model in Bangladesh with double dose vaccination

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Md Abdul Kuddus ◽  
M. Mohiuddin ◽  
Azizur Rahman
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Transmission Rate ◽  
Equilibrium Points ◽  
Rank Correlation ◽  
Dynamical Analysis ◽  
Model Parameters ◽  
Progression Rate ◽  
Double Dose ◽  
The Basic Reproduction Number

AbstractAlthough the availability of the measles vaccine, it is still epidemic in many countries globally, including Bangladesh. Eradication of measles needs to keep the basic reproduction number less than one $$(\mathrm{i}.\mathrm{e}. \, \, {\mathrm{R}}_{0}<1)$$ ( i . e . R 0 < 1 ) . This paper investigates a modified (SVEIR) measles compartmental model with double dose vaccination in Bangladesh to simulate the measles prevalence. We perform a dynamical analysis of the resulting system and find that the model contains two equilibrium points: a disease-free equilibrium and an endemic equilibrium. The disease will be died out if the basic reproduction number is less than one $$(\mathrm{i}.\mathrm{e}. \, \, {\mathrm{ R}}_{0}<1)$$ ( i . e . R 0 < 1 ) , and if greater than one $$(\mathrm{i}.\mathrm{e}. \, \, {\mathrm{R}}_{0}>1)$$ ( i . e . R 0 > 1 ) epidemic occurs. While using the Routh-Hurwitz criteria, the equilibria are found to be locally asymptotically stable under the former condition on $${\mathrm{R}}_{0}$$ R 0 . The partial rank correlation coefficients (PRCCs), a global sensitivity analysis method is used to compute $${\mathrm{R}}_{0}$$ R 0 and measles prevalence $$\left({\mathrm{I}}^{*}\right)$$ I ∗ with respect to the estimated and fitted model parameters. We found that the transmission rate $$(\upbeta )$$ ( β ) had the most significant influence on measles prevalence. Numerical simulations were carried out to commissions our analytical outcomes. These findings show that how progression rate, transmission rate and double dose vaccination rate affect the dynamics of measles prevalence. The information that we generate from this study may help government and public health professionals in making strategies to deal with the omissions of a measles outbreak and thus control and prevent an epidemic in Bangladesh.

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