Fractional grey unequal-interval time-varying Lotka-Volterra model and its application for microbial communities in compost.

Aerobic compost is an effective method for the treatment of livestock manure, which is usually accompanied by complex interspecific competition. Describing these competitive relationships through mathematical models can help understand the interaction of microorganisms and analyze the effect of exogenous additive to regulate the composting process. The common model for analyzing competition problem is the Lotka-Volterra model. However, the fixed parameters of the Lotka-Volterra model are not suitable to reflect the dynamic variations of the competitive relationship when the environmental conditions change during composting process. Therefore, this paper establishes a novel fractional grey unequal-interval time-varying Lotka-Volterra model. Firstly, a fractional grey derivate operator is proposed on the basis of the unequal interval of composting data and historical dependence of microbial growth. Secondly, considering the influence of temperature, a time-varying parameter matrix is defined to reflect the variation of competitive relationship at different composting phases, and it is estimated by forgetting factor recursive least squares. Thirdly, the optimal coefficients are optimized by grey prediction evolution algorithm. Finally, the proposed model is employed to analyze the chicken manure composting experiment. The results show that the proposed model has lower error criteria and more accurate trend of fitting curve than the other five existing models. The parameter matrix describes the dynamical variation of microbial competitive relationship in two taxonomic levels and reveals that effect of the exogenous additive is principally reacted in the thermophilic phase and the competitive advantage is shifted from Bacteroidota to Firmicutes after treatment with the exogenous additive.