Mathematical modeling of various diffusion mechanisms

Abstract

세포, 박테리아, 화학약품, 동물 등과 같은 입자들의 집합에서 각각의 입자들은 일반적으로 임의의 방향으로 움직인다. 입자들은 이 불규칙한 개별적인 입자의 움직임의 결과로 퍼져나간다. 이 미세한 불규칙적 움직임이 그 집합의 거시적 혹은 총체적인 규칙적 움직임으로 끝날때 우리는 이것을 확산과정이라고 생각할 수 있다. 하지만 개별적인 미세한 행동의 지식에서 거시적인 행동을 얻는것은 매우 힘들다. 그래서 우리는 입자의 밀도나 농도를 이용해서 전체적인 행동을 위한 연속체 모형 방정식을 유도한다. 그것은 확률적으로 우리가 초보적인 방법으로 시작하는데 도움을 주고, 그 후에 결정적인 모형을 유도한다.|In an assemblage of particles, for example, cells, bacteria, chemicals, animals and so on,each particle usually moves around in random way. The particles spread out as a result of this irregular individual particle's motion. when this microscopic irregular movement result in some macroscopic or gross regular motion of the group we can think of it as a diffusion process. Of course there may be interaction between particles, for example, or the environment mat give some bias in which case the gross movement is not simple diffusion. To get the macroscopic behaviour from a knowledge of the individual microscopic behaviour is much too hard so we drive a continuum model equation for the global behaviour in terms of a particle density or concentration. It is instructive to start with a random process which we look at probabilistically in an elementary way, and then drive a deterministic model. For simplicity we consider initially only one-dimensional motion and the simplest random walk process. the generalisation to higher dimensions is then intuitively clear from the one-dimensional equation. Diffusion models form a reasonable basis for studying insect and animal dispersal and invasion;this and other aspects of animal population models are discussed in detail, for example, by Okubo(1980, 1986), Shigesada(1980) and Lewis(1997). Dispersal of interacting species is discussed by Shigesada et al.(1979) and of competing species by Shigesada and Roughgarden(1982). Kareiva(1983) has shown that many species appear to disperse according to a reaction diffusion model with a constant diffusion coefficient. He gives actual values for the diffusion coefficient which he obtained from experiments on variety of insect species. Kot et al.(1996) studied dispersal of organisms in general and importantly incorporated real data(see also Kot 2001). A common feature of insect populations is their discrete time population growth. As would be expected intuitively this can have a major effect on their spatial dispersal. The model equations involve the coupling of discrete time with continuous space, a topic investigated by Kot(1992) and Neubert et al.(1995).