Abstract
This paper examines the dynamics of the exponential population growth system with mixed fractional Brownian motion. First, we establish some useful lemmas that provide powerful tools for studying the stochastic differential equations with mixed fractional Brownian motion. We offer some explicit expressions and numerical characteristics such as mathematical expectation and variance of the solutions of the exponential population growth system with mixed fractional Brownian motion. Second, we propose two sufficient and necessary conditions for the almost sure exponential stability and the k th moment exponential stability of the solution of the constant coefficient exponential population growth system with mixed fractional Brownian motion. Furthermore, we conduct some large deviation analysis of this mixed fractional population growth system. To the best of the authors’ knowledge, this is the first paper to investigate how the Hurst index affects the exponential stability and large deviations in the biological population system. It is interesting that the phenomenon of large deviations always occurs for addressed system when 1 / 2 < H < 1 . Moreover, several numerical simulations are reported to show the effectiveness of the proposed approach.
Highlights
Many scholars recently have paid considerable attention to stochastic differential equations (SDEs), as they can be applied in many fields such as mathematics, physics, mechanics, biology, economics, complex networks, control engineering, multiagent systems, and financial markets [1,2,3,4,5,6,7,8]
We consider the dynamics of the exponential population growth system with mixed fractional Brownian motion, which is a linear combination of independent Brownian motion and fBm. e mfBm is a non-Markovian, long-range dependent, mixed-self-similar, and correlated process [37, 42]
In view of the exponential stability of the system, we investigate the phenomenon of large deviations
Summary
Many scholars recently have paid considerable attention to stochastic differential equations (SDEs), as they can be applied in many fields such as mathematics, physics, mechanics, biology, economics, complex networks, control engineering, multiagent systems, and financial markets [1,2,3,4,5,6,7,8]. We consider the dynamics of the exponential population growth system with mixed fractional Brownian motion (mfBm), which is a linear combination of independent Brownian motion and fBm. e mfBm is a non-Markovian, long-range dependent, mixed-self-similar, and correlated process [37, 42].
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