5. Microscopic models (above) predict future distributions of particles by describing interactions or movement, while macroscopic models (below) consider average quantities such as density, and predict future density from actual density. When properly done, the macroscopic predictions can be retrieved by statistical averaging of the microscopic model predictions Soetaert, K. and P.M.J. Herman. 2009. A practical guide to ecological modelling using R as a simulation platform. Springer.
6. 3.2 Representing Space in Models Spatial dimension 実際の実空間は3次元。 工夫によって次元を下げることも可。 e.g., 変化の小さい方向は除外 対称性を活用 Discreet 空間を離散化して扱う Continuous 空間を離散化せずに扱う -> 物理法則になじむ
7. Examples of spatial configurations in models. A. In this landscape model, space is divided in discrete cells that have distinct properties. B. Patch model with 3 discrete patches. C. In some models, so-called Delaunay triangulation is used to discretise space, for instance to model the territory of birds. D. transition rules in a cellular automaton model. () occupied cells (v)= transition allowed; (x)=not allowed. Soetaert, K. and P.M.J. Herman. 2009. A practical guide to ecological modelling using R as a simulation platform. Springer.
8. 3.3 Transport in a Zero-Dimensional Model 濃度等が空間的に均一 輸送は、外部とのやりとりのみ
9. Schematic representation of 0-D models that include transport. A. a well-stirred tank. B. A lake, where a river brings in water on one side, and another carries the water out of the lake. C. A water mass in contact with the air. Soetaert, K. and P.M.J. Herman. 2009. A practical guide to ecological modelling using R as a simulation platform. Springer.
10. 3.3 Transport in a One-Dimensional Model 連続空間を対象とした1次元の輸送を表現する方程式を立てる。 次のステップで説明されている。 フラックス発散項 ＋移流・拡散項 ＋生成項 1次元化の例
13. Eq. 3.9 Deriving one-directional transport in a small box. x, x+∆x: position along the X-axis, A: surface, ∆V: volume of the box, J: flux. See text for details. Soetaert, K. and P.M.J. Herman. 2009. A practical guide to ecological modelling using R as a simulation platform. Springer.
16. Two types of advection: flow in a river or estuary(above) and sinking of particles out of a water column (below). B Three types of dispersion: molecular diffusion induced by random motion of particles (top left), eddy diffusion caused by turbulent mixing of particles (top right) and mechanical dispersion, induced by variations in flow velocities. C. Effect of advection and diffusion on a dye spill in a river. Soetaert, K. and P.M.J. Herman. 2009. A practical guide to ecological modelling using R as a simulation platform. Springer.
18. 適用例 入り江、河川、湖沼 -> どの方向に変化が大きいのかを見極める The transport in rivers, estuaries and lakes can often be represented by the 1-D advection-diffusion equation. For rivers and estuaries, the 1-D axis is the length axis, while for lakes it is the depth axis. Soetaert, K. and P.M.J. Herman. 2009. A practical guide to ecological modelling using R as a simulation platform. Springer.
19. Schematic representation of “one-dimensional” spatial volumes as used in models. Grey lines denote isosurfaces. A. One-dimensional shape with constant surface area. B. Cylindrical shape, with non-zero cylinder length. C. cylindrical shape with zero length of the cylinder. D. Spherical shape Soetaert, K. and P.M.J. Herman. 2009. A practical guide to ecological modelling using R as a simulation platform. Springer.
20. Schematic representation of sediments and overlying water with liquid and solid phase and bulk sediment. Porosity (Φ) is the volumetric proportion of liquid over bulk sediment. Sediment models are generally more complex than water column models, because the transport and reaction equations have to take into account the conversion between these phases. Soetaert, K. and P.M.J. Herman. 2009. A practical guide to ecological modelling using R as a simulation platform. Springer.
22. Boundaries in one-dimensional models of various shapes. A. One-dimensional shape with constant surface area. B. Cylindrical shape, with non-zero cylinder length. C. Cylindrical shape with zero length of cylinder. D. Spherical shape. Soetaert, K. and P.M.J. Herman. 2009. A practical guide to ecological modelling using R as a simulation platform. Springer.
23. Two ways of representing boundary conditions in a discrete 2-D model. A. organisms moving outside the model domain are removed. B. Organisms reaching the end bounce back. C. Organisms are displaced at the other side. This is equivalent to folding the surface such that the edges are removed, and a donut-shape is obtained. Soetaert, K. and P.M.J. Herman. 2009. A practical guide to ecological modelling using R as a simulation platform. Springer.
24. 追記：この図は正しくない。教科書の方が正しい。 Model description for 1-D sediment biogeochemical models, with typical boundary conditions. A. For a particulate substance such as organic matter, an upper flux boundary condition is often prescribed. B. For a dissolved substance, such as oxygen, the upper boundary is more often prescribed as a concentration. J denotes the flux, C the concentration. Boundary conditions are in bold, model equations are enclosed in a box. It is assumed that porosity is constant, thus it can be removed from the equation. Soetaert, K. and P.M.J. Herman. 2009. A practical guide to ecological modelling using R as a simulation platform. Springer.
