A Size-Structured Competition Model

Data from [WiJ - particularly figures 3, 6, 8, 10, 18, 19, and 21 - leave no doubt that, at least for certain populations of algae, individual cell volume varies significantly during the course of experiments in the chemo-stat. These data also suggest that steady-state size distributions are reached which have remarkably stable shapes with respect to changes in the control parameters for the chemostat (flow rates, temperature, CO2). 

The purpose of this chapter is to present a model of competition in the chemostat for a single resource that takes these factors into account, and to determine the extent to which these factors influence the outcome of competition. The model presented here is a special case of a more general class of models formulated in [DMKH] and treated in [MD, sec. 1.3]. This simplified model is one of several considered in an elegant paper by Cushing [Cu2]. Most of the results of this chapter are taken from [Cu2]. 

A model that accounts for individual variation in one or more characteristics - such as age, size, or class - is often called a structured population model, and the particular characteristics allowed to vary are called the structure variables. In this chapter, a size-structured population model is presented. There is a large and rapidly developing literature on structured 

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In Section 4, competition between two populations is analyzed. Again, the equations can be reduced to a system that can be directly compared to the systems derived in Chapters I and 2. Section 5 explores the evolution in time of the population average length, surface area, and volume in Section 6 we formulate the conservation principle, which played such a crucial role in earlier chapters. The steady-state size distribution of a population is determined in Section 7. Our findings are summarized in a discussion section, where a comparison is made between the conclusions derived from the size-structured model and the unstructured models considered in Chapters 1 and 2. 

Cu2] J, M, Cushing (1989), A competition model for size-structured species, SIAM Journal on Applied Mathematics 49 838-58,... 

The importance of including soil-based parameters in rhizosphere simulations has been emphasized (56). Scott et al. u.sed a time-dependent exudation boundary condition and a layer model to predict how introduced bacteria would colonize the root environment from a seed-based inoculum. They explicitly included pore size distribution and matric potential as determinants of microbial growth rate and diffusion potential. Their simulations showed that the total number of bacteria in the rhizosphere and their vertical colonization were sensitive to the matric potential of the soil. Soil structure and pore size distribution was also predicted to be a key determinant of the competitive success of a genetically modified microorganism introduced into soil (57). The Scott (56) model also demonstrated that the diffusive movement of root exudates was an important factor in determining microbial abundance. Results from models that ignore the spatial nature of the rhizosphere and treat exudate concentration as a spatially averaged parameter (14) should therefore be treated with some caution. 

BFD from Pseudomonas putida has been characterized in detail with respect to its biochemical properties [4, 5] and 3D structure [6, 7]. Like other enzymes of this class, BFD is a homotetramer with a subunit size of about 56 kDa. The four active sites are formed at the interfaces of two subunits. The structure was published in 2003 [7] and contains the competitive inhibitor (R)-mandelate bound to the active sites, allowing model-based predictions about the interactions between active site residues and the substrate. 

The form of the isotherms of the mixtures is largely independent of the cation distribution within the cage, i.e., whether the cation-poor or cation-rich model is used. This result is somewhat surprising, especially in view of the different adsorbate structures predicted by single-component isotherms (118-120). Only nonpolar adsorbates were considered in this study and the insensitivity to cation arrangement may well change if one component possesses a permanent dipole. These simulations were based on simple spherical molecules, but the competition for pore space as it depends on size, shape, and polarizability may be extended to other adsorbates. Indeed, Santilli et al. (129) observed experimentally that a branched hydrocarbon adsorbs in preference to a linear one at low loading. 

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