In this paper, taking into account the effect of a randomly fluctuating environment, we assume that fluctuations in the environment will manifest themselves mainly as fluctuations in the parameter β, as in [12], betarightarrowbeta+sigmadot{B}(t), where (B t)) is standard Brownian motions with (B 0) = 0), and with the intensity of the white noise (sigma^{2}>0).
In addition, for fluctuating environments, fluctuations of intermediate frequency appear most favorable for elevating mutation rate (Travis and Travis 2002; Palmer and Lipsitch 2006).
The fluctuation in environment helps to maintain the species diversity.
The results also supports the general discussions which propose that the temporal fluctuation of environment increase the species diversity of the ecological systems [6], [7].
Although the autonomous periodic and chaotic behaviors in population dynamics were reported to induce the increase of species diversity [13], [14], [15], there is no direct demonstration of the increase of the species diversity as a response to the temporal fluctuation in environment.
Next, we consider the effect of stochastic fluctuations of environment to the endemic equilibrium of the corresponding deterministic system.
Without considering the random fluctuations of environment, that is say, (sigma_{i} (i=1,2,3)=0), according to the proof of Theorem 7, it is easy to get the following conclusion.
The multiplicative noise c u ∘ d W ( t ) characterizes, to some extent, some of the minimal fluctuations among environment or a man-made complex system, which we should take into consideration in order to model perfectly the concrete problem.
It allows deployers to describe software using a hierarchical DSL, deploy it using an implementation of a multicloud distributed deployment framework, and provide autonomic rules to respond to fluctuations of environment, thereby ensuring availability and scalability.
