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Aquaponics are complex technical and biological systems. For example, possible explanations for fish not growing properly can be small food rations, adverse water quality, technical problems causing stress, etc. Due to the inherently slow biology, scientific investigations of the validity of these explanations would be tedious and require several experimental trials to get all important factors and their interactions, demanding a lot of facilities, expertise, research time and financial assets. Therefore, the issue of modelling aquaponic systems was addressed in this chapter. In aquaponics, modelling is required for different objectives: (i) insight/understanding, (ii) analysis, (iii) estimation and (iv) management and control. For all these objectives, appropriate models are required. For example, to achieve objectives (ii) and (iii), an empirical approach can be utilized which uses statistical models to analyse data from previous experimental trials with the objective of extracting as much information as possible without conducting new experiments. Statistical models can reveal the most important factors affecting fish and crop production in the aquaponic systems. Future experiments could concentrate on these factors, thus making the utilization of costly research assets more effective.
The complexity of aquaponic systems, due to their feedback character and the interactions between RAS and hydroponic system, water treatment and fish growth, implies that in order to fulfil objectives (i) and (iv), i.e. to understand or optimize a plant (configuration, size, fish, feed, flows, etc.) with respect to cost, stability, robustness and water quality, non-trivial theoretical models of most of the system components described in this chapter are required. The advantage of these theoretical models presented over statistical models is their stronger ability to analyse the process underlying the aquaponics and the possibility to model the time aspect (dynamics). Statistical models just confirm or refute a hypothesis and to what extent variables covary but give no evidence of the underlying processes. On the other hand, theoretical models allow us to simulate the processes according to a hypothesis, compare simulated with observed data, evaluate both the hypothesis and the model and make adaptions. The validity of statistical models may not be beyond the operational range they were trained for, whereas theoretical models can be defined and used for a wide range of environments, provided that the models are validated for these ranges before application. For example, the multiple regression model used to assess relationships between fish growth with Oreochromis niloticus as fish species and environmental variables in an aquaponics facility in Germany cannot be easily applied to Spain with Cyprinus carpio, whereas a theoretical model describing the underlying processes (e.g. fish behaviour, aquaculture, freshwater ecology) as mathematical equations can be adjusted relatively easily because the fish and ecological process underlying that model are basically the same for the two sites.
Nevertheless, theoretical models also require some parameters such as reaction constants and substance settling velocity in settling tank to be determined. This is achieved commonly based on empirical study of one facility or very few facilities or in most cases from previously published studies (secondary sources). Studies based on secondary sources have limitations imposed by the given structure and amount of the available data, which is not existent when the data come from an experimental setup designed ad hoc for the study. However, estimating model parameters using experimental data from one aquaponics facility only can have problems regarding generalizability and replication of the results due to particular conditions present in the study. The data scarcity sometimes imposes strong restrictions to models that limit their practicality. The development of studies for parameter estimation with primary data that use a larger number of aquaponics facilities than earlier studies does help to overcome the present limitations and provide better and reliable results. This, however, is not an easy challenge for aquaponics researchers.
Simulation of aquaponics with the mathematical models under a wide range of management conditions will improve the understanding of aquaponics, verify different aquaponics configurations and point the way to the most promising strategies for improving aquaponics facilities. Again, this can lead to a more efficient way of conducting experiments.
Some modelling tools were also presented in this chapter. Traditionally, stock and flow diagrams (SFD) have been used for understanding processes as support tools for quantitative analysis. They are used to comprehend the flow and fluxes of quantities but lack the ability to illustrate the information associated to the flow and fluxes. Causal loop diagram (CLD) can be used to transfer complex SFD system into understandable simplified feedback structures. Together, the SFDs and the CLDs fully define the differential equation system. If only a simple qualitative understanding of the system is required, then CLD and SFD may be enough, but if the answer requires a numerical accuracy, then the problem can be investigated further with system dynamic tool diagrams (SDTD) and subsequently be modelled in a software tool for numerical simulation.