From Discrete Time Model to Continuous Time Model

Abstract

AbstractThis chapter is about the idea to mathematically derive the continuous time model from the discrete time model in Chaps. 1 and 2. The derived model is written as an ordinary differential equation, since we focus on the model of single species population dynamics in this chapter. Logistic equation appears first in this chapter, although its modeling and nature will be described in the subsequent chapters again, Chaps. 4 and 5. Besides, as the basic idea to understand the meaning of time derivative, we describe the concept of momental velocity of population size change. It is the essential aspect for the continuous time population dynamics model, and is very likely to be forgotten in the biological translation of the mathematical results obtained for the continuous time model. In most case of biological phenomena, the continuous time modeling with the momental velocity of a biological quantity like population size could be regarded as a mathematical approximation or simplification.