Three of the four main stages of anaerobic digestion: acidogenesis, acetogenesis, and methanogenesis are described by a system of differential equations modelling the interaction of microbial populations in a chemostat. The microbes consume and/or produce simple substrates, alcohols and fatty acids, acetic acid, and hydrogen. Acetogenic bacteria and hydrogenotrophic methanogens interact through syntrophy. The model also includes the inhibition of acetoclastic and hydrogenotrophic methanogens due to sensitivity to varying pH-levels. To examine the effects of these interactions and inhibitions, we first study an inhibition-free model and obtain results for global stability using differential inequalities together with conservation laws. For the model with inhibition, we derive conditions for existence, local stability, and bistability of equilibria and present a global stability result. A case study illustrates the effects of inhibition on the regions of stability. Inhibition introduces regions of bistability and stabilizes some equilibria.

Objective Obesity prevalence in the United States (US) appears to be leveling, but the reasons behind the plateau remain unknown. Mechanistic insights can be provided from a mathematical model. The objective of this study is to model known multiple population parameters associated with changes in body mass index (BMI) classes and to establish conditions under which obesity prevalence will plateau. Design and Methods A differential equation system was developed that predicts population-wide obesity prevalence trends. The model considers both social and non-social influences on weight gain, incorporates other known parameters affecting obesity trends, and allows for country specific population growth. Results The dynamic model predicts that: obesity prevalence is a function of birth rate and the probability of being born in an obesogenic environment; obesity prevalence will plateau independent of current prevention strategies; and the US prevalence of obesity, overweight, and extreme obesity will plateau by about 2030 at 28%, 32%, and 9%, respectively. Conclusions The US prevalence of obesity is stabilizing and will plateau, independent of current preventative strategies. This trend has important implications in accurately evaluating the impact of various anti-obesity strategies aimed at reducing obesity prevalence.

We investigate a model for anaerobic digestion, a process used to produce biogas. The model, introduced in Weedermann et al. (J Biol Dyn 7:59-85, 2013), consists of differential equations describing the interactions of microbial populations involved in three main stages of anaerobic digestion: acidogenesis, acetogenesis, and methanogenesis. We show that this model predicts that an increased yield in biogas can be achieved in regions where operating parameters push the system into a bistable state. In some regions of bistability, biogas production occurs at only one of the steady states while in others both steady states result in biogas production with one state being more productive than the other. We demonstrate which operating parameters and state variables have the most significant impact on system performance. Surprisingly, the optimal biogas production does not always occur at a steady state where all classes of microorganisms coexist.

Hopf bifurcations in scalar neutral delay-differential equations ẋ(t) − a ẋ(t − τ ) = L(γ )x t + f (x t , γ ) are analysed for |a| < 1. We have derived explicit conditions for a Hopf bifurcation to occur. Using centre manifold reduction and normal form theory the stability coefficient of the periodic orbit on the centre manifold is determined explicitly in terms of the coefficients in the original equation. The results are illustrated in two standard examples, one with a Wright-type nonlinearity and one with a cubic nonlinearity.

No abstract