PCLake

Key takeaways

PCLake belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect PCLake to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of PCLake from memory before moving on to harder problems.

Reference excerpt

PCLake is a dynamic, mathematical model used to study eutrophication effects in shallow lakes and ponds. PCLake models explicitly the most important biotic groups and their interrelations, within the general framework of nutrient cycles. PCLake is used both by scientist and water managers. PCLake is in 2019 extended to PCLake+, which can be applied to stratifying lakes.

Background Typically, shallow lakes are in one of two contrasting alternative stable states: a clear state with submerged macrophytes and piscivorous fish, or a turbid state dominated by phytoplankton and benthivorous fish. A switch from one state to the other is largely driven by the input of nutrients (phosphorus and nitrogen) to the ecosystem. If the nutrient loading exceeds a critical value, eutrophication causes a switch from the clear to the turbid state. As a result of urban water pollution and/or intensive agriculture in catchment areas, many of the world’s shallow lakes and ponds are in a eutrophic state with turbid waters and poor ecological quality. In this turbid state, the lake also becomes subject to algal blooms of toxic cyanobacteria (also called blue-green algae). Recovery of the clear state however is difficult as the critical nutrient loading for the switch back is often found to be lower than the critical loading towards the turbid state. Lowering the nutrient input thus does not automatically lead to a switch back to the clear water phase. Hence, the system shows hysteresis.

Application PCLake is designed to study the effects of eutrophication on shallow lakes and ponds. On one hand, the model is used by scientists to study the general behavior of these ecosystems. For example, PCLake is used to understand the phenomena of alternative stable states and hysteresis, and in that light, the relative importance of lake features such as water depth or fetch length. Also the potential effects of climate warming for shallow lakes have been studied. On the other hand, PCLake is applied by lake water resource managers that consider the turbid state as undesirable. They can use the model to define the critical loadings for their specific lakes and evaluate the effectiveness of restoration measures. For this purpose also a meta-model has been developed. The meta-model can be used by water managers to derive an estimate of the critical loading values for a certain lake based on only a few important parameters, without the need of running the full dynamical model.

Model content Mathematically, PCLake is composed of a set of coupled differential equations. With a large number of state variables (>100) and parameters (>300), the model may be characterized as relatively complex. The main biotic variables are phytoplankton and submerged aquatic vegetation, describing primary production. A simplified food web is made up of zooplankton, zoobenthos, young and adult whitefish and piscivorous fish. The main abiotic factors are transparency and the nutrients phosphorus (P), nitrogen (N) and silica (Si). At the base of the model are the water and nutrient budgets (in- and outflow). The model describes a completely mixed water body and comprises both the water column and the upper sediment layer. The overall nutrient cycles for N, P and Si are described as completely closed (except for in- and outflow and denitrification). Inputs to the model are: lake hydrology, nutrient loading, dimensions and sediment characteristics. The model calculates chlorophyll-a, transparency, cyanobacteria, vegetation cover and fish biomass, as well as the concentrations and fluxes of nutrients N, P and Si, and oxygen. Optionally, a wetland zone with marsh vegetation and water exchange with the lake can be included. PCLake is calibrated against nutrient, transparency, chlorophyll and vegetation data on more than 40 European (but mainly Dutch) lakes, and systematic sensitivity and uncertainty analysis have been performed. Although PCLake is primarily used for Dutch lakes, it is likely that the model is also applicable to comparable non-stratifying lakes in other regions, if parameters are adjusted or some small changes to the model are made.

Model development The first version of PCLake (by then called PCLoos) was built in the early 1990s at the Netherlands National Institute for Public Health and the Environment (RIVM), within the framework of a research and restoration project on Lake Loosdrecht. It has been extended and improved since then. Parallel to PCLake, PCDitch was created, which is an ecosystem model for ditches and other linear water bodies. The models were further developed by dr. Jan H. Janse and colleagues at the Netherlands Environmental Assessment Agency (PBL), formerly part of the RIVM. Since 2009, the model is jointly owned by PBL and the Netherlands Institute of Ecology, where further development and application of PCLake is taking place, related to aquatic-ecological research.

Worked examples

Example 1 — a first encounter with PCLake

Start with the simplest possible case. Write down what PCLake claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to PCLake before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about PCLake ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of PCLake

In research: PCLake appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses PCLake in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: PCLake is common in secondary-school and first-year university syllabi. It links to neighbouring topics Environmental chemistry, Mathematical modeling, so understanding it makes those chapters shorter.
In everyday life: Look for PCLake outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study PCLake in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what PCLake means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain PCLake out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is PCLake in simple terms?

PCLake is a dynamic, mathematical model used to study eutrophication effects in shallow lakes and ponds. PCLake models explicitly the most important biotic groups and their interrelations, within the general framework of nutrient cycles.

Why does PCLake matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study PCLake?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on PCLake.