# Manual Mathematical Models in Environmental Problems

Mathematical models also cover different spatial and temporal scales: from the smallest tidepool ecosystem to the entire planet; from a single day to millions of years.

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Mathematical models and simulations are used scientifically as a tool for improving the understanding of the ecology of a region and managerially as a tool for making decisions regarding resource and environmental issues. Depending on the scope and sensitivity of the system, developing mathematical models of an environment can be an extremely interdisciplinary undertaking.

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It might require input from ecologists, biologists, chemists, physicists, engineers, and computer scientists, as well as social scientists, economists, and politicians. Results from mathematical models might be shared among scientists, policy makers, and various government officials, often from different localities and jurisdictions. Mathematical models usually have three basic parts.

## Mathematical Modeling in the Environment

These are the variables and their definitions, the equations into which the variables are incorporated, and starting values for the variables. When mathematical models are applied to ecological situations, more information is required. For example, an ecological model requires the user to assign a meaning to the variables, to know the units of the variables, and to bind the ranges of values over which the variables are realistic. It might also be useful to know the way that the values of the variables are measured, the ecological context of the variables, the reproducibility of the values of the variables, among other information.

There are two basic approaches to building a mathematical model of an ecosystem. The first is called the compartmental system approach or a conceptual model. This type of model is often depicted using box and flow illustrations.

The boxes represent a pool or a compartment of something found in an ecosystem. For example, a box might represent the total weight of carbon in the phytoplankton in a certain region of the ocean. The box is then impacted by several flows, visually represented by arrows into and out of the box.

## Mathematical models for environmental problems - Proceedings of an international conference

An arrow into the phytoplankton carbon box might represent the rate of conversion of atmospheric carbon dioxide into carbohydrate by photosynthesis. An arrow out of the box might represent the loss of carbon from the community of phytoplankton because of predation. Conceptual models most often put emphasis on the gross dynamics of a whole ecosystem. They tend to be rather general models that can be applied to many different systems.

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For example, the phytoplankton model discussed earlier may be altered for use in either the Pacific Ocean or the Atlantic Ocean. However, conceptual models tend to be imprecise. The rates of photosynthesis in the Atlantic and Pacific Oceans are likely very different. This is because the climate conditions, the nutrients that drive photosynthesis, and the species of phytoplankton all differ from place to place.

These factors are largely omitted from conceptual or compartmental system models. In contrast, the qualitative model, which is also referred to as the experimental components approach, control model, or stressor model, incorporates as many interactions and components of a system as possible.

Instead of using a single compartment to represent the phytoplankton carbon, a qualitative model would represent the various populations of phytoplankton individually. It would take experimental results of photosynthetic rates for the different species and use this information to come up with a rate of carbon dioxide assimilation for the entire community.

Qualitative models tend to be more precise than compartmental models, but they are also very specific. It would be difficult to use a qualitative model of phytoplankton growth designed to represent Pacific Ocean phytoplankton when modeling the phytoplankton in the Atlantic Ocean. Qualitative models can be extremely useful for understanding how ecosystems will respond to environmental stresses. Depending on the needs of the users and the ecological questions addressed, several types of models may be developed to represent a single ecosystem.

Ecological models can also be embedded within one another. One part of the system may be represented by a qualitative model, while another part is represented by a conceptual model. Many ecological models have a geographical component. Geographical information systems GIS and geospatial analyses are often used in conjunction with ecological simulations.

## mathematical modelling

Mathematical models and simulations are used to better understand a multitude of ecological issues. For example, simulations are commonly used to model biogeochemical cycling within aquatic systems. They are used to understand the way that toxic materials move through ecosystems. These toxic materials include pesticides, heavy metals, and radionuclides.

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Ecological models of ground water flow and plume diffusion in air are often incorporated into these ecotoxicology models. In terrestrial systems, scientists have developed agricultural models and forestry models. One of the most common types of mathematical models involves population dynamics. Until recently, critical questions about the mathematical theory for the existence of solutions for the equation were unresolved, and solution of this equation strained the resources of the most powerful completers. However, mathematical advances have now made its solution routine, allowing accurate predictions of wave evolution.

Early numerical techniques to solve the equation were slow and cumbersome. But now, several efficient techniques exist which can yield reliable results. Not only has the mathematical theory of water waves helped us to understand and protect our environment, but its insights have also had a significant impact on technological development. Although the solitary wave is now well understood, other water waves still have mysterious effects on our environment and remain objects of active mathematical research.

Basic mathematics - calculus, percents, ratios, graphs and charts, sequences, sampling, averages, a population growth model, variability and probability - all relate to current, critical issues such as pollution, the availability of resources, environmental clean-up, recycling, CFC's, and population growth. In January of this year the annual winter meeting of the national mathematics societies held theme sessions on Mathematics and the Environment.

Several presentations were made. Papers are available on request as described below. Mathematical methods first developed in the early stages of sequencing the DNA molecule have turned out to be useful in deciding when to give different streams of traffic a green light.

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Related mathematical methods are useful in deciding how to make streets one-way so as to move traffic more efficiently. Department of Energy sites, such as Hanford, WA. A project goal is to formulate and implement accurate and efficient algorithms for modeling biodegradation processes. Numerical simulation results that utilize realistic data and parallel computational complexity issues are discussed. Simon Levin - Section of Ecology and Systematics, Cornel The Problem of Scale in Ecology: Why this is Important in Resolving Global Problems Global environmental problems have local and regional causes and consequences, such as, linkages between photosynthetic dynamics at the leaf level, regional shifts in forest composition, and global changes in climate and the distribution of greenhouse gases.

The fundamental problem is relating processes that are operating on very different scales of space and time. Mathematical methods provide the only way such problems can be approached, and techniques of scaling, aggregation, and simplification are critical. Engineering Rutgers University P. The environmental emphasis is in recognition of the national and international increase in awareness of environmental issues and the key role mathematics plays in analyzing and interpreting environmental data.