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In writing about undergraduate research in mathematical modeling, I draw on my 31 years as a mathematics professor at the University of Nebraska–Lincoln, where I mentored students in honors’ theses, REU groups, and research done in a classroom setting, as well as my prior experience. I share my views on the differences between research at the undergraduate and professional levels, offer advice for undergraduate mentoring, provide suggestions for a variety of ways that students can disseminate their research, offer some thoughts on mathematical modeling and how to explain it to undergraduates, and discuss the challenges involved in broadening research participation to include early career students and mid-tier students and how to deliver a research experience in a classroom setting. While different situations pose different challenges, different problems require different approaches, and different experiences lead to different conclusions, it is my hope that my experiences will be of broad value to a wide audience. [ABSTRACT FROM AUTHOR]

Syntrophic systems are common in nature and include forms of obligate mutualisms in which each participating organism or component of an organism obtains from the other an essential nutrient or metabolic product that it cannot provide for itself. Models of how these complementary resources are allocated between partners often assume optimal behavior, but whether mechanisms enabling global control exist in syntrophic systems, and what form they might take, is unknown. Recognizing that growth of plant organs that supply complementary resources, like roots and shoots, can occur autonomously, we present a theory of plant growth in which root-shoot allocation is determined by purely local rules. Each organ uses as much as it can of its locally produced or acquired resource (inorganic nitrogen or photosynthate) and shares only the surplus. Subject to stoichiometric conditions that likely hold for most plants, purely local rules produce the same optimal allocation as would global control across a wide range of environmental scenarios, with sharing the surplus being the specific mechanism stabilizing syntrophic dynamics. Our local control model contributes a novel approach to plant growth modeling because it assumes a simple mechanism of root:shoot allocation that can be considered a higher-level physiological rule, from which the optimal growth outcome emerges from the system's dynamics, rather than being built into the model. Moreover, our model is general, in that the mechanism of sharing the surplus can readily be adapted to many obligate syntrophic relationships. [ABSTRACT FROM AUTHOR]

Asymptotic methods can greatly simplify the analysis of all but the simplest mathematical models and should therefore be commonplace in such biological areas as ecology and epidemiology. One essential difficulty that limits their use is that they can only be applied to a suitably scaled dimensionless version of the original dimensional model. Many books discuss nondimensionalization, but with little attention given to the problem of choosing the right scales and dimensionless parameters. In this paper, we illustrate the value of using asymptotics on a properly scaled dimensionless model, develop a set of guidelines that can be used to make good scaling choices, and offer advice for teaching these topics in differential equations or mathematical biology courses. [ABSTRACT FROM AUTHOR]

Potential theory and Stokes' stream function techniques are used to investigate the flow structure around the recirculation system developed by Gvirtzman and Gorelick (1992, 1993), which consists of an extraction well and a gallery (trench) for the recharge of treated water to the aquifer. Analytical formulas are derived for the drawdown, velocity, and stream function for a model in which the extraction well is modeled as a uniformly distributed line sink and the gallery is modeled as a uniformly distributed ring source. Travel times are reported for water particles traveling along the streamlines containing 50 and 90% of the flow for various degrees of well penetration and various radii of the ring source. The travel times along the streamline resulting in the shortest travel time (not necessarily the shortest path) are also reported for various degrees of well penetration and various radii of the ring source. The method completely eliminates the use of numerical finite-difference or finite-element methods and can be used for optimization of technological parameters of this remediation system. [ABSTRACT FROM AUTHOR]

Resource allocation within trees is a zero-sum game. Unavoidable trade-offs dictate that allocation to growth-promoting functions curtails other functions, generating a gradient of investment in growth versus survival along which tree species align, known as the interspecific growth-mortality trade-off. This paradigm is widely accepted but not well established. Using demographic data for 1,111 tree species across ten tropical forests, we tested the generality of the growth-mortality trade-off and evaluated its underlying drivers using two species-specific parameters describing resource allocation strategies: tolerance of resource limitation and responsiveness of allocation to resource access. Globally, a canonical growth-mortality trade-off emerged, but the trade-off was strongly observed only in less disturbance-prone forests, which contained diverse resource allocation strategies. Only half of disturbance-prone forests, which lacked tolerant species, exhibited the trade-off. Supported by a theoretical model, our findings raise questions about whether the growth-mortality trade-off is a universally applicable organizing framework for understanding tropical forest community structure.