Biofilms are spatially-structured communities of different microbes, which have a huge impact on both ecosystems and human life. Mathematical models are powerful tools for understanding the function and evolution of biofilms as diverse communities. In this article, we give a review of some recently-developed models focusing on the interactions of different species within a biofilm, the evolution of biofilm due to genetic and environmental causes and factors that affect the structure of a

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... . Processes 2017, 5, 5 2 of 17 tested by experiments. In this sense, progresses made in experimental and modeling research always promote the development of each other. Mathematical modeling of biofilm started in the 1970s with models studying substrate utilization and mass transport in a homogeneous slab of biofilm [4, 5] . In the 1980s, models including multispecies and the non-uniform distribution of different biomass types started to emerge [6,7], but they were still primarily for one space dimension and steady-state growth dynamics. Starting in the 1990s and up to today, aided by the fast advancement in computing power and better understanding of biofilms through experimental data, multidimensional, multispecies, multisubstrate models are being developed to incorporate realistic biofilm morphology, biofilm mechanics, interactions between biofilm and the environment and interactions between different species within a biofilm, as well as various time scales involved in biofilm-related processes [8] [9] [10] [11] [12] . There is a rich literature on the review of mathematical models for biofilms. A few representative ones are listed below. The IWAtask group gives an excellent review of mathematical modeling of biofilms [13] . The book explains the basic steps in creating a mathematical model, emphasizes that the "golden rule" of modeling is that "a model should be as simple as possible, and only as complex as needed" and presents the model derivation based on mass conservation in detail. Models are classified into analytical models (A), Pseudo-Analytic models (PA), Numerical one-dimensional (N1) and multi-dimensional Numerical models (N2 and N3). The features, definitions and equations, as well as the application of each type of model are discussed. The performances of all models for solving three characteristic benchmark problems are compared, which help identify the trade-offs inherent to using different types of models. The book also points out the significance of the definitions and units of model parameters. Klapper and Dockery [14] discuss how macroscale physical factors might influence the composition, structure and function of ecosystems within microbial communities from the modeling perspective and emphasizes that despite its difficulty and complexity, it is important to include the physical, chemical and biological processes at a variety of time and length scales in the model to fully understand the physiology and ecology of the microbial communities. Specific modeling aspects discussed include quorum sensing, growth, mechanics and antimicrobial tolerance mechanisms. Wang and Zhang [15] give a chronological review of some biofilm models developed from the 1980s to the early 2000s. Based on their dimensionality, the way in which diffusion is treated and the complexity in terms of the incorporation of the physics, chemistry and biological effects, models are classified into four main categories: one-dimensional continuum models, diffusion-limited aggregate models, continuum-discrete diffusion models and biofilm-fluid coupled models. In recent years, fast advancement in experimental technologies, such as microscopy and high-throughput sequencing, has provided an abundance of data at both the genetic and community level and helped researchers to understand biofilms much better. In particular, it is a common belief that biofilms should be viewed as spatially-structured communities of microbes, and the structure and function of the communities are determined by both the surrounding environment and the local interactions between different species within biofilms via complex metabolic networks [16] [17] [18] [19] [20] . To understand biofilms as a diverse community and the evolution of different species and its genetic causes, mathematical modeling again serves as a powerful tool. Experimental data with a high level of detail provide both opportunities and challenges for researchers working on biofilm modeling. On the one hand, there are more data available to improve conceptual understanding of biofilm and to compare with model predictions; on the other hand, more sophisticated models are demanded in order to accommodate the data. Song et al. [21] give a methodological review on mathematical modeling of microbial community dynamics. Widder et al. [22] address the challenges in building predictive models for understanding the function and dynamics of Microbial Communities (MCs). Several specific examples where model-experiment integration has already resulted in important insights into MC function and structure are discussed. These include inferring species interactions from proximal data, predicting species interactions using stoichiometric models and kinetic models for