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As obesity and its related health problems grow around the world, efforts to control and manage weight is increasing in importance. It is well known that altering and maintaining weight is problematic and this has led to specific studies trying to determine the cause of the difficulty. Recent research has identified that the body reacts to forced weight change by adapting individual total energy expenditure. Key factors are an adaptation of resting metabolic rate, non-exercise activity thermogenesis and dietary induced thermogenesis. We develop a differential equation model based on the first law of thermodynamics that incorporates all three adjustments along with natural age related reduction of the resting metabolic rate. Forward time simulations of the model compare well with mean data in both overfeeding and calorie restriction studies.

In the past few decades, efforts have been made to clean sites polluted by heavy metals such as chromium. One of the new innovative methods of eradicating metals from soil is phytoremediation. Phytoremediation uses plants to pull metals from the soil through the roots. This article develops a system of differential equations to model the plant metal interaction of phytoremediation. We prove there exists a threshold time, $t$*, where the amount of metals in the environment meet a prescribed EPA criteria. The cost of phytoremediating up to time $t$* is computed. The cost function can be used to estimate the feasibility of clearing a polluted site through phytoremediation as opposed to alternate techniques such as brown filling.