Detection of mixed-culture growth in the total biomass data by wavelet transforms

We have shown elsewhere that the presence of mixed-culture growth of microbial species in fermentation processes can be detected with high accuracy by employing the wavelet transform. This is achieved because the crosses in the different growth processes contributing to the total biomass signal appear as singularities that are very well evidenced through their singularity cones in the wavelet transform. However, we used very simple two-species cases. In this work, we extend the wavelet method to a more complicated illustrative fermentation case of three microbial species for which we employ several wavelets of different number of vanishing moments in order to eliminate possible numerical artifacts. Working in this way allows to filter in a more precise way the numerical values of the H\"older exponents. Therefore, we were able to determine the characteristic H\"older exponents for the corresponding crossing singularities of the microbial growth processes and their stability logarithmic scale ranges up to the first decimal in the value of the characteristic exponents. Since calibrating the mixed microbial growth by means of their H\"older exponents could have potential industrial applications, the dependence of the H\"older exponents on the kinetic and physical parameters of the growth models remains as a future experimental task