The structure of mixed microbial cultures—such as the human gut microbiota—is influenced by a complex interplay of interactions among its community members. The objective of this study was to propose a strategy to characterize microbial interactions between particular members of the community occurring in a simulator of the human gastrointestinal tract used as the experimental system. Four runs were carried out separately in the simulator: two of them were fed with a normal diet (control system), and two more had the same diet supplemented with agave fructans (fructan-supplemented system). The growth kinetics of Lactobacillus spp., Bifidobacterium spp., Salmonella spp., and Clostridium spp. were assessed in the different colon sections of the simulator for a nine-day period. The time series of microbial concentrations were used to estimate specific growth rates and pair-wise interaction coefficients as considered by the generalized Lotka-Volterra (gLV) model. A differential neural network (DNN) composed of a time-adaptive set of differential equations was applied for the nonparametric identification of the mixed microbial culture, and an optimization technique was used to determine the interaction parameters, considering the DNN identification results and the structure of the gLV model. The assessment of the fructan-supplemented system showed that microbial interactions changed significantly after prebiotics administration, demonstrating their modulating effect on microbial interactions. The strategy proposed here was applied satisfactorily to gain quantitative and qualitative knowledge of a broad spectrum of microbial interactions in the gut community, as described by the gLV model. In the future, it may be utilized to study microbial interactions within mixed cultures using other experimental approaches and other mathematical models (e.g., metabolic models), which will yield crucial information for optimizing mixed microbial cultures to perform certain processes—such as environmental bioremediation or modulation of gut microbiota—and to predict their dynamics.