## Mass Balance Models in Ecosystems

Ecological modeling has a long history. Various kinds of models have been proposed to describe the properties of ecosystems to date. Among them, the dynamic mathematical model that describes the mass balance in ecosystems is one of the most important ones.

The lake ecosystem is a good example to describe the mass balance model. Because of its closed characteristics and relative easiness to grasp the mass balance than other ecosystems, a large number of models have been proposed for lake ecosystems. Here, the mass balance model is described based on the lake model. The mass balance in a lake is shown in Figure 1. Of course, there are many other organisms and abiotic components in lake ecosystems, such as macrophytes and zoobenthos. However, only some important components are depicted in the figure for simplification.

The mass balance model describes the flow of materials in an ecosystem. In Figure 1, for example, the material flows between phytoplankton, zooplankton, fish, detritus, and dissolved matter (note that these components are called 'state variables' in the modeling practice), and resultant time variations of the biomass and concentrations of these organisms and abiotic compounds are calculated. The variations of these state variables are usually described by simultaneous differential equations (or in some cases it is described

Inflow

Atmospheric temperature, precipitation, etc.

Inflow

 Phytoplankton Zooplankton Fish \ 1 > / Dissolved matter Detritus Outflow Figure 1 An example of mass balance in a lake. by simultaneous difference equations) on the basis of the material flows (shown by arrows in Figure 1) among them. These material flows are described by functions consisting of state variables, kinetic parameters, and effects of environmental factors. The maximum growth rates of phytoplankton, the grazing rate of zooplankton on phy-toplankton, and the decomposition rate of detritus are examples of the parameters. Solar radiation, atmospheric temperature, inflow/outflow of dissolved matter, etc., are examples of environmental factors. These environmental factors are called forcing functions. The state variables are commonly defined by concentrations of the circulating materials such as carbon, phosphorus, and/or nitrogen. Carbon is the most important element to describe the actual biomass (of course, for carbon circulation, the flows about photosynthesis and respiration are also included, which are not shown in Figure 1 ), whereas phosphorus and nitrogen are the nutrients that control the primary production in an ecosystem. In some situations, another element such as silicon is included when it becomes a limiting factor of primary production (e.g., describing a kind of diatom growth in a lake). 