Introduction to Biological Models

Abstract : The ocean is inhabited by innumerable individuals of many different genera and species, each having its own developmental/physiological state and each being immersed in its own environment. The organisms move, both because of water flow and because of their own swimming or buoyancy, and interact with their environment by gathering resources which they need and by excreting waste products. The assimilated material can be used for maintenance, growth, or reproduction. Finally, the organisms can die either from natural causes or because of attacks by another organism. Furthermore, the processes just described must generally be regarded as stochastic. For example, the probability of a predator capturing a prey item will depend on multiple factors, each with its own probability: finding a prey item in range, the choice to attack, success in the attack, competition against others. Such a description suggests an agent-based or individual-based model (IBM), with each agent carrying information about its position, its species, its physiological state, etc. Individuals can grow, reproduce, and die. Certainly, we can build small versions of such models, but the number of individuals is necessarily limited (compared to nearly 20,000 copepods per cubic meter or to phytoplankton densities on the order of 10 (exp 8) per cubic meter). However, such experiments may indeed give insight into the way in which the local, stochastic interactions translate into terms representing, for example, grazing rates in terms of average densities. Following an introduction, the report contains a section on basic biological models. This section discusses the exponential growth model, the resource limitation model, the predator-prey model, and the NPZD model. Section 3 discusses the stochastic dynamics of modeling an organism's population dynamics, with a focus on probability density functions and the Fokker-Planck Expansion.