Reaction–diffusion equations of two species competing for two complementary resources with internal storage

AbstractThis paper examines a system of reaction–diffusion equations arising from a mathematical model of two microbial species competing for two complementary resources with internal storage in an unstirred chemostat. The governing system can be reduced to a limiting system based on two uncoupled conservation principles. One of main technical difficulties in our analysis is the singularities in the reaction terms. Conditions for persistence of one population and coexistence of two competing populations are derived from eigenvalue problems, maximum principle and the theory of monotone dynamical systems

A graphical theory of competition on spatial resource gradients

Resource competition is a fundamental interaction in natural communities.However little is known about competition in spatial environments where organisms are able to regulate resource distributions. Here, we analyze the competition of two consumers for two resources in a one-dimensional habitat in which the resources are supplied from opposite sides. We show that the success of an invading species crucially depends on the slope of the resource gradients shaped by the resident. Our analysis reveals that parameter combinations which lead to coexistence in a uniform environment may favor alternative stable states in a spatial system, and vice versa. Furthermore, differences in growth rate, mortality or dispersal abilities allow a consumer to coexist stationarily with - or even outcompete - a competitor with lower resource requirements. Applying our theory to a phytoplankton model, we explain shifts in the community structure that are induced by environmental changes

Mathematical models of microbial growth and metabolism: A whole-organism perspective

This is the author accepted manuscript. The final version is available from Science Reviews 2000 via the DOI in this record.We review the principles underpinning the development of mathematical models of the metabolic activities of micro-organisms. Such models are important to understand and chart the substantial contributions made by micro-organisms to geochemical cycles, and also to optimise the performance of bioreactors that exploit the biochemical capabilities of these organisms. We advocate an approach based on the principle of dynamic allocation. We survey the biological background that motivates this approach, including nutrient assimilation, the regulation of gene expression, and the principles of microbial growth. In addition, we discuss the classic models of microbial growth as well as contemporary approaches. The dynamic allocation theory generalises these classic models in a natural manner and is readily amenable to the additional information provided by transcriptomics and proteomics approaches. Finally, we touch upon these organising principles in the context of the transition from the free-living unicellular mode of life to multicellularity.Olga Nev was funded through EU Research Framework programme 7 Marie Curie Actions, grant 316630 Centre for Analytical Science – Innovative Doctoral Programme (CAS-IDP)

Effects of Diabetes on Ovarian Cancer: Data Analysis and Modeling Study

Ovarian cancer has one of the highest mortality rates of all gynecological cancers [13]. Further knowledge of risk factors for the growth of ovarian tumors would be beneficial in both the treatment and prevention of this type of cancer. Previous research has shown a positive correlation between diabetes and prostate tumor growth [22], The first aim of this study was to determine the effect of diabetes of ovarian tumor growth. The second aim was to develop a model to predict ovarian tumor growth based on the microenvironment within a patient’s body. The hypothesis was that there would be a positive correlation between diabetes and ovarian tumor volume. Data from fifty patients was collected from charts at Grady Memorial Hospital in Atlanta, Georgia. Oxygen saturation, tumor volume, blood glucose level, and cancer stage were gathered for each patient. The results contradicted the hypothesis; there was a negative correlation found between blood glucose level and tumor volume. More data is needed to determine if increased blood glucose could be an effective treatment of ovarian cancer, particularly since there other health risks associated with elevated blood glucose levels. The proposed mathematical model also needs modification in order to effectively bridge the gap between the clinical and research aspects of the cancer field

Mathematical Modeling and Analysis of a Phytoplankton Competition Model Incorporating Preferential Nutrient Uptake

Phytoplankton live in a complex environment with two essential resources forming various gradients. Light supplied from above is never homogeneously distributed in a body of water due to refraction and absorption from biomass present in the ecosystem and from other sources. Nutrients in turn are typically supplied from below. In poorly mixed water columns, phytoplankton can be heterogeneously distributed forming various layering patterns. We present a new reaction-diffusion-taxis model describing the vertical distribution of two phytoplankton species competing for two nutrients, one of which is assumed to be preferred. The parameter space of the model is analyzed for parameter identifiability - the ability for a parameter\u27s true value to be recovered through optimization, and for global sensitivity - the influence a parameter has on model response. Using simulations, we exhibit evidence of thin layer formation for motile phytoplankton in poorly mixed environments. A game theoretic approximation is considered, where the depth of the phytoplankton layer is treated as the strategy the phytoplankton adopt. The evolutionary stable strategy (ESS) is the depth at which the phytoplankton are equally limited by both resources. We analytically derive the ESS of the proposed preferential uptake model along with a related two-species reaction-diffusion-taxis model which only considers one limiting nutrient

Emergent behaviour in a chlorophenol-mineralising three-tiered microbial `food web'

Anaerobic digestion enables the water industry to treat wastewater as a resource for generating energy and recovering valuable by-products. The complexity of the anaerobic digestion process has motivated the development of complex models. However, this complexity makes it intractable to pin-point stability and emergent behaviour. Here, the widely used Anaerobic Digestion Model No. 1 (ADM1) has been reduced to its very backbone, a syntrophic two-tiered microbial food chain and a slightly more complex three-tiered microbial food web, with their stability analysed as function of the inflowing substrate concentration and dilution rate. Parameterised for phenol and chlorophenol degradation, steady-states were always stable and non-oscillatory. Low input concentrations of chlorophenol were sufficient to maintain chlorophenol- and phenol-degrading populations but resulted in poor conversion and a hydrogen flux that was too low to sustain hydrogenotrophic methanogens. The addition of hydrogen and phenol boosted the populations of all three organisms, resulting in the counterintuitive phenomena that (i) the phenol degraders were stimulated by adding hydrogen, even though hydrogen inhibits phenol degradation, and (ii) the dechlorinators indirectly benefitted from measures that stimulated their hydrogenotrophic competitors; both phenomena hint at emergent behaviour.Comment: 19 pages, 8 figure

Further results on stabilization of periodic trajectories for a chemostat with two species,

Abstract We discuss an important class of problems involving the tracking of prescribed trajectories in the chemostat model. We provide new tracking results for chemostats with two species and one limiting substrate, based on Lyapunov function methods. In particular, we use a linear feedback control of the dilution rate and an appropriate time-varying substrate input concentration to produce a locally exponentially stable oscillatory behavior. This means that all trajectories for the nutrient and corresponding species concentrations in the closed loop chemostat that stay near the oscillatory reference trajectory are attracted to the reference trajectory exponentially fast. We also obtain a globally stable oscillatory reference trajectory for the species concentrations, using a nonlinear feedback control depending on the dilution rate and the substrate input concentration. This guarantees that all trajectories for the closed loop chemostat dynamics are attracted to the reference trajectory. Finally, we construct an explicit Lyapunov function for the corresponding global error dynamics. We demonstrate the efficacy of our method in a simulation

Mathematical Models of Microbial Evolution: Cooperative Systems

Microbes usually live in large communities, where they interact with other organisms and species. These interactions include cooperation, when individuals facilitate each others growth and reproduction. Such cooperation has been for instance observed within pathogens in the process of infection. Therefore, given the number and the frequency of infectious diseases, understanding the nature and the dynamics of microbial cooperation may be a crucial step in modern medicine. Microbes often secrete costly enzymes which extracellularly metabolise resources available in the environment. This external metabolism is a form of ’public good cooperation’, in which individuals invest their energy in producing ’public goods’, available to other organisms. To study this phenomenon we deploy mathematical models which are based on biologically relevant assumptions. Our models not only aim to capture the dynamics of studied microbial communities, but also to remove the natural complexity arising in the empirical studies and thus to provide a mechanistic understanding of their results. We first recover and explain the recent empirical finding, about mixed strain infections, showing that an addition of a low virulent strain which does not produce public goods (termed ’cheat’) may counter-intuitively enhance the total population virulence. What drives this result turns out to be an interaction of two different cooperative traits and the presence of spatial structure. Next we study the competition between the strains that do and do not produce public goods. Our results depend on environmental conditions, such as resource concentration and population density, but they are also determined by the degree of spatial structure - the ecological trait which so far has been treated only as a binary variable. Finally, we identify some environmental threats for the external metabolism feeding strategy, and we examine its competitiveness in comparison to ’internal metabolism’, in which the costly enzymes are private.EPSR

Holling Type I versus Holling Type II functional responses in Gram-negative bacteria

We consider how the double-membrane structure of the cell envelope of Gram-negative bacteria affects its functional response, which is the mathematical relationship that expresses how the nutrient uptake flux depends on environmental conditions. We show that, under suitable conditions, the Holling Type I functional response is a plausible model, as opposed to the Holling Type II (rectangular hyperbolic, ‘Michaelis–Menten’) response that is the default model in much of the literature. We investigate both diffusion-limited and capacity-limited regimes. Furthermore, we reconcile our findings with the preponderance in the established literature of hyperbolic models for the growth response, which are generally assumed to be valid, for both Gram-negative and Gram-positive bacteria. Finally, we consider the phenomenon of dynamic adjustment of investment of molecular building blocks in cellular components, and show how this will affect the functional response as observed by the experimenter