Abstract
In this paper, a stochastic microbial flocculation model with regime switching is developed and analyzed. By proposing a suitable stochastic Lyapunov function, the existence and ergodicity of a stationary distribution for the system are proved. Then, the extinction of microorganisms is discussed under appropriate conditions and sufficient conditions for extinction are obtained. Finally, the results of the theoretical analysis are illustrated by numerical simulation.
Highlights
E above research is generally based on experiments to study the flocculation effect of different flocculants or develop new flocculants
Gupta and Ako explored the effect of guar gum flocculant in the treatment of drinking water or food processing water through experiments and found that guar gum flocculant can improve the quality of water in the process of drinking water clarification, and there is no residue of acrylamide in the water, which reduces the health risks of the population [5]
Wu et al used chitosan flocculant to treat the Mn(II) and suspended solids produced in the dual-alkali flue gas desulfurization regeneration process, which solved the problem that the traditional methods are difficult to remove heavy metal ions in the suspension [6]
Summary
For a right-continuous Markov chain r(t), the generator Υ (cij)n×n is determined by. ⎧⎨ cijΔ + o(Δ), if i ≠ j, Pr(t + Δt) j|r(t) i ⎩ 1 + cijΔ + o(Δ), if i j,. Where Δt > 0, cij ≥ 0 is the transition rate from i to j if i ≠ j while cii − i≠jcij. In order to ensure that r(t) is irreducible, here, we need to assume cij ≥ 0, for i ≠ j. Us, r(t) has a unique stationary distribution π π1, π2, . For a diffusion process (u(t), r(t)) expressed by stochastic differential equations, du(t) g(u(t), r(t))dt + h(u(t), r(t))dB(t),. For each k ∈ S, let V(·, k) be any twice continuously differentiable function and we can define the operator L as follows: LV(u, k) n.
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