Stochastic stability and bifurcation for the chronic state in Marchuk’s model with noise

• Published in: Applied mathematical modelling Vol. 35; no. 12; pp. 5842 - 5855
• Main Authors: Huang, Zaitang, Yang, Qigui, Cao, Junfei
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Inc 01.12.2011; Elsevier
• Subjects: Bifurcations; Biological and medical sciences; Boundaries; Density; Exact sciences and technology; Fundamental and applied biological sciences. Psychology; General aspects; Local/global stochastic stability; Marchuk’s model; Mathematical models; Mathematics; Mathematics in biology. Statistical analysis. Models. Metrology. Data processing in biology (general aspects); Modelling; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications); Stability; Stochastic pitchfork bifurcation; Stochasticity; Stochastic pitchfork bifurcation; Marchuk’s model; Local/global stochastic stability; Stochastic model; Singularity; Stability; Differential equation; Bifurcation; Modeling; Stationary condition; Global local method; Manifold; Invariant measure; Lyapunov exponent; Population dynamics; Probability measure; Stochastic equation; Marchuk's model
• ISSN: 0307-904X
• DOI: 10.1016/j.apm.2011.05.027

A stochastic differential equation modelling a Marchuk’s model is investigated. The stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modelling. Firstly, the stochastic Marchuk’s model has been simplified by applying stochastic center manifold and stochastic average theory. Secondly, by using Lyapunov exponent and singular boundary theory, we analyze the local stochastic stability and global stochastic stability for stochastic Marchuk’s model, respectively. Thirdly, we explore the stochastic bifurcation of the stochastic Marchuk’s model according to invariant measure and stationary probability density. Some new criteria ensuring stochastic pitchfork bifurcation and P-bifurcation for stochastic Marchuk’s model are obtained, respectively.