TOPICS IN MATHEMATICAL BIOLOGY

我们的研究工作将集中在使用数学技术来解决和分析在生物学、生态学和生理学的系统建模中出现的复杂的微分方程组。要研究的系统直接关系到生物科学中的重大问题和问题。基本目标是使用所得到的数学结果来更好地理解这些系统的动力学。感兴趣的特定系统包括(但不限于)：周期性疾病(离散模型)。肾脏浓缩机制。生化振荡器。反应-平流扩散过程。节食的模范。相互作用的种群动力学将得到精确的、近似的和数值的解，并与现有的数据/观测进行比较，以了解正在研究的特定系统并对其动态演化做出预测。所用的数学方法包括摄动(正则和奇异)和渐近级数、调和平衡法、相空间分析、混沌系统的“理论”和数值积分。这些数学工具中的许多都起源于PI以前的工作，特别是使用谐波平衡来确定振荡系统的周期解，以及使用非标准有限差分格式来计算微分方程组的数值解。第二，但也是重要的目标是让本科生和研究生接触他们通常不熟悉或不了解的生物科学研究领域，并在科学课程中引入数学生物科学入门课程。

Our research efforts will center on the use of mathematical techniques to solve and analyze the complex different and differential equations that arise in the modeling of systems in biology, ecology and physiology. The systems to be studied directly relate to significant problems and issues in the biosciences. The fundamental goal is to use the resulting mathematical results to provide a better understanding of the dynamics of these systems. Particular systems of interest include (but, are not limited to): . periodic diseases (discrete models) . the renal concentrating mechanism . biochemical oscillators . reaction-advection diffusion processes . modeling of dieting . interacting population dynamics Exact, approximate and numerical solutions will be obtained and compared with available data/observations to both understand the particular system being studied and to make predictions concerning its dynamical evolution. The mathematical methods to be used include perturbation (both regular and singular) and asymptotic series, harmonic balance procedures, phase-space analysis, the "theory" of chaotic systems, and numerical integration. Many of these mathematical tools have originated in previous work by the PI, in particular, the use of harmonic balancing for determining periodic solutions to oscillating systems and non-standard finite difference schemes for calculating numerical solutions to differential equations. Secondary, but also important objectives are to expose both undergraduate and graduate students to an area of research in the biosciences for which they are generally not familiar or knowledgeable and to introduce into the science curriculum an introductory course in mathematical biosciences.