Stability Analysis of a Stochastic Model for Prey-Predator System with Disease in the Prey

2003 ◽  
Vol 8 (2) ◽  
pp. 83-92 ◽  
Author(s):  
D. Mukherjee
Keyword(s):  
Stability Analysis ◽  
Stochastic Model ◽  
Global Stability ◽  
Equilibrium Point ◽  
Stochastic Stability ◽  
Deterministic Model ◽  
Stochastic Perturbations ◽  
Stability Properties ◽  
Interior Equilibrium ◽  
Predator System

In this paper we consider a prey-predator system where the prey population is infected by a microparasite. Local as well as global stability properties of the interior equilibrium point are discussed. The stochastic stability properties of the model are investigated, suggesting that the deterministic model is robust with respect to stochastic perturbations.

scholarly journals Stochastic model of the transmission dynamics of COVID-19 pandemic

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Aychew Wondyfraw Tesfaye ◽  
Tesfaye Sama Satana
Keyword(s):  
Stochastic Model ◽  
Global Stability ◽  
Equilibrium Point ◽  
Basic Reproduction Number ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Index Formula ◽  
Sensitivity Index ◽  
Simulation Results ◽  
Disease Free

AbstractIn this paper, we formulate an SVITR deterministic model and extend it to a stochastic model by introducing intensity of stochastic factors and Brownian motion. Our basic qualitative analysis of both models includes the positivity of the solution, invariant region, disease-free equilibrium point, basic reproduction number, local and global stability of disease-free equilibrium point, endemic equilibrium point, and sensitivity. We obtain the stochastic reproduction number and local stability by using twice differentiable Itô’s formula. We prove the global stability of the disease-free equilibrium point by using a Lyapunov function. We determine the sensitivity of the effect of each parameter on basic reproduction number of the model by using a normalized sensitivity index formula. On the other hand, we demonstrate numerical simulation results of deterministic and stochastic models of COVID-19 by using Maple 18 and MATLAB software. Our simulation results indicate that reducing the contact between infected and susceptible individuals and improvement of treatment play a vital role in COVID-19 pandemic control.

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.

GLOBAL STABILITY ANALYSIS OF AN ECO-EPIDEMIOLOGICAL MODEL OF THE SALTON SEA

2006 ◽  
Vol 14 (03) ◽  
pp. 373-385 ◽  
Author(s):  
ZHEN JIN ◽  
MAINUL HAQUE
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Global Stability ◽  
Present Article ◽  
Geometric Approach ◽  
Epidemiological Model ◽  
Salton Sea ◽  
Interior Equilibrium ◽  
Global Stability Analysis ◽  
Bird Mortality

The Salton Sea, the largest lake of California, is now facing problems due to its massive fish and bird mortality. Haque and Chattopadhyay1 have proposed and analyzed an eco-epidemiological model on Salton Sea. The present article deals with the global stability analysis of the positive interior equilibrium of that eco-epidemiological model. We use the geometric approach to the global stability analysis for ordinary differential equations which is based on the use of higher-order generalization of Bendixson's criterion. Our conclusion is that for persistence of the pelican (main predator) populations in Salton Sea, harvesting of tilapia (specially infected) populations is an utmost important factor.

scholarly journals Analysis of the stochastic model for predicting the novel coronavirus disease

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Ndolane Sene
Keyword(s):  
Stochastic Model ◽  
Deterministic Model ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
The Novel ◽  
Stochastic Perturbations ◽  
Numerical Investigations ◽  
Rapid Spread ◽  
The World ◽  
Novel Coronavirus

Abstract In this paper, we propose a mathematical model to predict the novel coronavirus. Due to the rapid spread of the novel coronavirus disease in the world, we add to the deterministic model of the coronavirus the terms of the stochastic perturbations. In other words, we consider in this paper a stochastic model to predict the novel coronavirus. The equilibrium points of the deterministic model have been determined, and the reproduction number of our deterministic model has been implemented. The asymptotic behaviors of the solutions of the stochastic model around the equilibrium points have been studied. The numerical investigations and the graphical representations obtained with the novel stochastic model are made using the classical stochastic numerical scheme.

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.

Effect of multiple delays on the dynamics of cannibalistic prey–predator system with disease in both populations

2017 ◽  
Vol 10 (04) ◽  
pp. 1750049 ◽  
Author(s):  
Santosh Biswas ◽  
Sudip Samanta ◽  
Qamar J. A. Khan ◽  
Joydev Chattopadhyay
Keyword(s):  
Stability Analysis ◽  
Disease Transmission ◽  
Local Stability ◽  
Numerical Experiments ◽  
Sufficient Conditions ◽  
Multiple Delays ◽  
Predator Population ◽  
Interior Equilibrium ◽  
Local Stability Analysis ◽  
Predator System

In the present paper, we investigate a prey–predator system with disease in both prey and predator populations and the predator population is cannibalistic in nature. The model is extended by introducing incubation delays in disease transmission terms. Local stability analysis of the system around the biologically feasible equilibria is studied. The bifurcation analysis of the system around the interior equilibrium is also studied. The sufficient conditions for the permanence of the system are derived in the presence of delays. We observe that incubation delays have the ability to destabilize the cannibalistic prey–predator system. Finally, we perform numerical experiments to substantiate our analytical findings.

scholarly journals Adaptive Pinning Synchronization of Complex Networks with Stochastic Perturbations

2010 ◽  
Vol 2010 ◽  
pp. 1-21 ◽  
Author(s):  
Xinsong Yang ◽  
Jinde Cao
Keyword(s):  
Stability Analysis ◽  
Complex Networks ◽  
Stochastic Stability ◽  
Sufficient Conditions ◽  
Vector Form ◽  
Stochastic Perturbations ◽  
Analysis Theory ◽  
Stochastic Stability Analysis ◽  
Theoretical Results ◽  
Partial States

The adaptive pinning synchronization is investigated for complex networks with nondelayed and delayed couplings and vector-form stochastic perturbations. Two kinds of adaptive pinning controllers are designed. Based on an Lyapunov-Krasovskii functional and the stochastic stability analysis theory, several sufficient conditions are developed to guarantee the synchronization of the proposed complex networks even if partial states of the nodes are coupled. Furthermore, three examples with their numerical simulations are employed to show the effectiveness of the theoretical results.

scholarly journals Stability Analysis of a Novel Mathematical Model of Plasmodium Life Cycle in Mosquito Midgut

2019 ◽  
Vol 8 (9) ◽  
pp. 1811-1813
Keyword(s):  
Life Cycle ◽  
Stability Analysis ◽  
Equilibrium Point ◽  
Suitable Condition ◽  
Life Cycle Model ◽  
Cycle Model ◽  
Existence Of Equilibrium ◽  
Interior Equilibrium ◽  
Mosquito Midgut ◽  
Theoretical Results

Present article investigates the complexity and stability analysis of plasmodium life cycle model in mosquito midgut. The existence of equilibrium point of the system are presented. Analysis of global stability are investigated by constructing suitable condition, around the interior equilibrium point. Theoretical results are numerically supported and the diagrams are presented.

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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