Appendix
Introductory Computational Science Case Studies:

In this Appendix we display the implementation plans for two example problems to be covered in the Institute.

Predator-Prey

An example of one problem to be covered is the predator-prey-environment problem which is simple in its formulation, but which can quickly generate chaotic solutions (while the initial formulation is based on a biological example, the model is actually applicable to the study of a variety of analogous problems) or misleading results depending on the integration method. The simplest formulation of such a problem is in a finite difference framework:

Consider the problem of ladybugs and aphids in a field. The number of ladybugs which feed upon the aphids is a sensitive function of the number of aphids in a particular part of the field and the presence or lack of sufficient ground cover on which the aphids are feeding. The aphid population tends to double every day if left unchecked, and the plant life could die if devastated by the aphids. Consequently, one has a set of three, coupled, ordinary (but not necessarily linear) differential equations.

Study of the problem could proceed as follows:

1. Write down "reasonable" expressions for the gain and loss terms for each subsystem (ladybugs, aphids, plants). The more information one has about the actual behavior of the subsystems and their interactions, the more closely one is able to construct a reasonable computational model. Look at using networked resources to find the missing data. For instance, knowledge about the birth and death rates of the aphids and ladybugs, the number of aphids that can be consumed by a ladybug per unit time, the amount of space that a single ladybug can traverse per unit time are all important inputs.
   
   
2. For a variety of initial conditions, look at the long-time behavior of the total ecosystem. Look at the behavior of the individual populations as a function of the coupling constants.
   
   
3. Study the onset of chaos by observing in detail the sensitivity of the coupled equations to changes in the initial conditions or changes in the coupling constants.
   
   
4. Visualize multiple simulations using a discrete color mapping allowing the simultaneous modeling of multiple parameter sets.
   
   
5. Explore the mapping of the solution of the coupled equations for various parameterizations on scalar, vector, and parallel machines.
   

Weather Modeling

An example of a coordinated, computational project appropriate for the introductory workshop is an examination of the variety of materials available to study weather related phenomena which could be used as examples in chemistry, physics, earth science, biology and mathematics courses at the undergraduate level. The experiences that the faculty will derive from this study are applicable in this variety of undergraduate classes to show a gradation of computational modeling opportunities for inquiry-based discovery and learning. An implementation plan for this example follows:

1. Use a browser and accessible search engines to locate and retrieve real-time or archived weather images and data.
   
   
2. Use the visualization and collaboration tool NCSA Collage to visualize images and to share observations about the images with other faculty and with weather experts on the network.
   
   
3. Use NIH Image to process such images and to produce animations of a variety of weather phenomena with QuickTime and HyperCard.
   
   
4. Use a range of interactive computer models, from the simplicity of SimEarth which runs in a stand-alone PC/Macintosh environment to the use of HPCC computing models such as NERSC's Climoman tools, for constructing numerical experiments to study climatic and weather changes and displaying the results of those models on a PC or Macintosh.