# ABACO: A New Model of Microalgae-Bacteria Consortia for Biological Treatment of Wastewaters
## 1 Introduction  
One of the most critical environmental challenges of the 21 st century envisaged by humanity is the expansion of the population, which will result in increased urban wastewater production [1] and large amounts of animal slurry caused by the rise in meat production [2, 3]. The world's growing population, along with (i) a rapid industrialization, (ii) intensive agriculture, (iii) the effluent discharged below an environmentally safe level, and (iv) the lack of technologies to reclaim used water could lead to a scarcity of clean water in many countries [4]. The current conventional wastewater treatment methods have become quickly outdated because they need a lot of land, intensive energy input, and a lot of money [5]. As an alternative strategy to beat these disadvantages, microalgae-based wastewater treatment is gaining an increased importance in the context of European bioe-conomy, because of its potential to treat wastewater, recover nutrients of wastewater, and produce a large variety of valuable compounds with applications in agriculture, aquaculture, and food production, among others [6, 7, 8]. The use of microalgae for wastewater treatment involves the emergence of complex microalgae-bacteria consortia which vary as functions of environmental and operational conditions [9].  
Microalgae are photosynthetic microorganisms that grow using inorganic carbon (CO\({}_{2}\)) as a carbon source, and light as an energy source. During this growth, microalgae release oxygen which can use by heterotrophic bacteria to oxidate the organic matter present in influent wastewater. At the same time, heterotrophic bacteria supply CO\({}_{2}\) for photosynthetic activity, completing the cycle. Besides, the oxygen produced by microalgae can be used by nitrifying bacteria to oxidize the ammonium to nitrate (nitrification process), consuming CO\({}_{2}\) as a carbon source too [10; 11; 12]. Since microalgae-bacteria consortia in wastewater treatment was described in 1953 by [13], multiple microalgae-bacteria models have been described and validated [14; 15; 16; 17]. These mathematical models offer great appeal to studying microalgae-bacteria interactions because they can provide useful tools for design and control purposes, in addition to model simulators, which can all lead to an increase the process efficiency [18].  
In most of the proposed mathematical models, the part related to the activity of the bacteria is widely obtained and validated through the Activated sludge models (ASM) [19]. However, information on microalgae parameters in wastewater treatment systems is scarce. Therefore, in this work, a new microalgae-bacteria mathematical model named ABACO is proposed; the characteristic parameters of microalgae in it were obtained experimentally in previous works [20; 21]. Thus, the main purpose of this study was to develop, calibrate, and validate the whole microalgae and bacteria model with experimental data from duplicate laboratory-scale photobioreactors using pig slurry as a nutrient source. The implementation of the microalgae-bacteria model has been performed in MATLAB software, and it allows one to simulate the dynamics of different components in the system and the relative proportions of microalgae and bacteria. Moreover, the model has a series of parameters whose exact values are unknown, being within a range. The calibration of these parameters has been carried out using genetic algorithms, which allow determining their values from minimizations of given cost functions. This calibration procedure provides a simple and fast adjustment method for the characterization of the model parameters, even allowing recalibration with different scenarios in a very easy way, such as for different strains and culture mediums. Moreover, notice that thanks to the proposed calibration process, it is possible to estimate the percentage of each species in the reactor, which is also a relevant contribution of the methodology proposed in this work.