Department of Physics, Clark Atlanta University, Atlanta, GA 30314, USA.
J Biol Dyn. 2007 Oct;1(4):427-36. doi: 10.1080/17513750701605598.
Population models arising in ecology, epidemiology and mathematical biology may involve a conservation law, i.e. the total population is constant. In addition to these cases, other situations may occur for which the total population, asymptotically in time, approach a constant value. Since it is rarely the situation that the equations of motion can be analytically solved to obtain exact solutions, it follows that numerical techniques are needed to provide solutions. However, numerical procedures are only valid if they can reproduce fundamental properties of the differential equations modeling the phenomena of interest. We show that for population models, involving a dynamical conservation law the use of nonstandard finite difference (NSFD) methods allows the construction of discretization schemes such that they are dynamically consistent (DC) with the original differential equations. The paper will briefly discuss the NSFD methodology, the concept of DC, and illustrate their application to specific problems for population models.
在生态学、流行病学和生物数学中出现的群体模型可能涉及一个守恒定律，即总人口是恒定的。除了这些情况外，还可能出现其他情况，即总人口在时间上渐近趋近于一个常数。由于很少有情况可以通过解析求解运动方程来获得精确解，因此需要数值技术来提供解。然而，数值程序只有在能够再现建模感兴趣现象的微分方程的基本性质的情况下才是有效的。我们表明，对于涉及动态守恒定律的群体模型，使用非标准有限差分（NSFD）方法可以构建离散化方案，使得它们与原始微分方程在动力学上是一致的（DC）。本文将简要讨论 NSFD 方法、DC 的概念，并说明它们在特定群体模型问题中的应用。
