Database of Mathematical Models

Title: The Pendulum Problem
Math Content: ODE (single, linear vs. nonlinear), Physics, Calculus (Taylor series)
Subject Area: Physics, Basic Modeling
Reference: Abell & Braselton, Modern Differential Equations

Comments: This problem might be good for examining the modeling process in general, and illustrating the interplay of math and science in particular. A comparison between the solution of the linear pendulum equation and the nonlinear pendulum equation is considered.

Comments: Quantities such as the natural frequency, the maximum kinetic energy and the maximum potential energy of a rack and gear system are determined.

Comments: A model for a fox population with rabies is presented - the model consists of four coupled nonlinear differential equations. Numerical solutions are requested in an effort to determine the critical value for a single parameter which gives rise to a periodic function. It may be possible to perform a bifurcation analysis to determine the critical value analytically.???

Comments: A simple model for a kidney dialysis machine is presented. Students are requested to determine the solution of the linear system of equations, and then a variety of relevant quantities are determined.

Comments: A system of equations is developed to model the spread of an infectious disease. The system is then reduced to a single equation, which is a Bernoulli equation. A non-constant daily contact rate is considered - in the special form of periodic contact rates. Different approach to modeling the spread of a disease can be found in the material from Strogatz and the Wisconsin conference.

Comments: The motion of a downhill skier is modeled. Special transformations and substitutions are used to solve the resulting equation. Students are requested to analyze the effect of a variety of parameters on the skier's performance.

Comments: The model for the spread of an infectious disease presented earlier is reconsidered. An extra equation for people who have recovered from the disease and are now immune has been added. Two different scenarios, classified as "without vital dynamics" and "with vital dynamics" are presented.

Comments: This project starts simple and gradually increases in complexity by relaxing some of the less realistic assumptions. Questions of optimality are discussed. As with the other models taken from this source, several extension problems are suggested.

Comments: Once the model equations (one for sugar, the other for insulin) have been established, this is essentially a simulation problem. The response to different scenarios can readily be examined by changing inputs and sensitivity parameters. Several problems for further study and suggestions for improving the model are included. One extension of this model might be to incorporate some sort of optimization/control structure.

Title: Drilling holes with a laser
Math Content: PDE (heat equation, perturbation), Laplace transform
Subject Area: Lasers, manufacturing, heat transfer
Reference: Andrews and McLone, Mathematical Modeling

Comments: This problem combines thermodynamics, PDEs, asymptotics and perturbations. As a final section, the limitations of the model which arise because fluid flow has been ignored are discussed. There are several problems for further study given in the exercises.

Comment: This is perhaps the most interesting model from this source. Dealing with basic physics and basic ODEs, the maximum mass of an n-stage rocket, for several values of n, which can be used to orbit a 1 ton satellite is computed. Practical realization of the optimal rocket design is considered.

Comments: This project puts a twist onto the pendulum problem - it introduces notions of feedback, control and stabilization - with the objective being to stabilize an inverted pendulum.

Comments: This is a chapter which considers several different scenarios for population dynamics - logistic growth, discrete versus continuous, predator-prey, competition, combat, etc.

Comments: Investigation of the flow of traffic which has been lined up behind a red light after the light turns green. Several scenarios are considered.

Comments: This problem investigates the Van der Pol equation - it uses techniques similar to those described for the basic pendulum model noted above. The use of energy techniques is an interesting change from standard differential equation analysis.

Comments: A nonlinear system of differential equations for modeling the behavior of the Earth's magnetic field is developed. The equations are non-dimensionalized and then analyzed for the presence of erratic field reversals. Hysteresis is discussed.

Comments: I believe that this is a simplified version of the FitzHugh-Nagamo equations modeling the fast/slow dynamics of the beating of the human heart. Model equations are proposed and then analyzed for cusp catastrophe and a limit cycle representing the normal beating of the heart (systole and diastole). Different arrhythmias are discussed with respect to the models predictions.

Comments: A model for predicting the minimum size algae patch required for the algae population to not die out is developed. The model is investigated using both PDE and ODE techniques. This problem could be thought of as model for the spread of an infectious disease to outbreak into an epidemic.

Comments: This problem discusses the dynamics of the interaction between the spruce budworm and spruce trees. The analysis includes an interesting bifurcation diagram with bistable regions. This coverage briefly mentions including an equation modeling the foliage level on the trees. Strogatz also covers this problem in detail and I have several journal papers which discuss different aspects of the problem.

Comments: This model investigates the dynamics of the pendulum using potential and kinetic energy notions to derive the phase portrait. The use of energy principles is an interesting change of pace from the traditional direct analysis of the differential equation.

Comments: Investigating the existence of a limit cycle in a predator-prey population model. This model includes a term in the predator equation which models satiation of the predator during feeding; i.e., there is some maximum benefit to be gained for the predator.

Comments: This problem, in particular, presents a model for the evolution of a fish population subject to harvesting. A nonlinear system of equations, including an equation modeling the harvesting effort, is developed. The principle objective of this model is to investigate the existence of a limit cycle in the population-harvesting dynamics.

Comments: Morphogenesis - the process whereby form and pattern evolve in a biological system. A system of PDEs governing morphogenesis are developed and the steady state response of the system is investigated. The analysis is carried out on several levels and under several different sets of conditions.

Comments: The existence of a limit cycle in a model of a mass-spring system in which the mass lies on a frictional drive belt is investigated.

Comments: This problem involves a rotating pendulum and the bifurcation of a stable equilibrium into two stable equilibrium positions as the angular rotation speed is increased. There is a similar treatment of this problem in Strogatz' text.

Comments: This is a very interesting model, which would be excellent for use in class and for providing several mini-projects for homework. The model provides a straightforward extension to the "mixing" problems traditionally covered in a first ODE course. the exercises present several scenarios which would make excellent homework assignments. One of the UMAP modules addresses this same issue.

Comments: A system of differential equations which model the distance between cars in a line of traffic is developed and investigated. Circumstances which will result in collisions are considered. Several different stimulus response submodels are considered in the exercises. This might make a good homework project???

Comments: A linear system of differential equations (rate in minus rate out type equations) is developed to model the level of lead in the blood, bone and tissues. The use of antilead medications and sensitivity studies are considered. As with the other projects from Borelli and Coleman, the exercises list several variations on the theme. This could make an excellent homework problem???

Comments: This project involves investigating the periods of the planetary orbits. It is interesting in that data for the planets is supplied which allows for verification of the model's predictions. To perform the verification, some curve fitting is required.

Comments: A model for the performance of an artificial kidney machine is developed. The solution of the model equations is determined and an important design parameter, the "clearance" of the dialysis machine, is determined.

Comments: This project attempts to answer a very practical question: at night, or when you go on a trip, what should you do with your thermostat. A model is developed and two scenarios are considered. A variety of other scenarios are possible, including a determination of cut-off values for certain parameters: lower temperature setting, length of time at lower temperature, cost of heating fuel, insulation value of home, etc. In class problem???

Comments: This project considers the construction of a rocket for placing a satellite in orbit. This project includes some ODEs, some Calculus, some Physics - a little bit of everything. There is a similar model in one of the other books.

Comments: This project, though simple, has a lot of potential and possibilities. The project can be carried out from the experiment-then-data-fitting approach or the model-then-validate approach. It is also interesting for illustrating the multidisciplinary nature of most mathematical modeling. This could be an excellent in class activity.

Comments: This is another example of a cascade model of drug absorption into the bloodstream (see UMAP modules and Borelli and Coleman). Different dosing scenarios are considered.

Comments: A model for the pursuit curve followed by a missile is developed. A special substitution is used to solve the nonlinear equation and to then determine the capture time of the missile. This problem would also be possible in terms of numerical solutions. Variations on the theme are most likely bountiful.

Comments: This is (as are all of the models drawn from this paper) an open-ended problem. The basic question here is to determine the amount of lost wages for which a lawyer should sue. The dead math professor was killed by an 18 wheeler which ran a stop sign.

Comments: This is (as are all of the models drawn from this paper) an open-ended problem. The basic question here is to evaluate a lottery whose prize is $1000 per week for life. Several comparisons are mentioned, and many others are possible.

Comments: This is (as are all of the models drawn from this paper) an open-ended problem. The basic question here is to design a system for moving baggage from its arrival point at an airport to the correct gate or to the baggage claim area.

Comments: This is (as are all of the models drawn from this paper) an open-ended problem. The basic question here is to advise an airline on the best way to overbook flights to protect them from lost income due to customers reserving a seat, but then not showing up for the flight.

Comments: This is (as are all of the models drawn from this paper) an open-ended problem. The basic question here is to decide whether to buy a home or to rent an apartment. Students can do a lot of interesting legwork on this project to gather all of the relevant data for their area.

Comments: This is (as are all of the models drawn from this paper) an open-ended problem. The basic question here is to develop an estimate for the cost to a company for providing a pension to their employees for the lifetime of the employee. This is another project in which students can perform some exciting legwork.

Comments: This is (as are all of the models drawn from this paper) an open-ended problem. The basic question here is to design an optimal production schedule for a crayon manufacturer. Information regarding the products which the manufacturer produces and regarding the actual manufacture and packaging of the crayons is provided.

Comments: The twist in these models is the consideration of harvesting and several different form of the population growth term. Three different types of curves are considered. Bifurcations and hysteresis, as well as conditions under which the drive to extinction is irreversible, are considered.

Comments: This project uses nonlinear dynamics to consider the management of a fishery. This particular situation involves limit cycles and the effect of harvesting levels. A model which also involves interspecies competition is included.

Comments: This project develops a model for determining the altitude of a parachutist during free fall, taking air resistance into account. Experimentally observed data is presented for validation. Taylor series are used to obtain an approximate formula for the altitude.

Comments: A population growth model, which includes constant harvesting, is considered. This model is interested in determining the time to extinction when the harvesting level is above a certain threshold, which is determined as part of the model. Results are compared with that of another model.

Comments: A model for a company whose objective is to maximize the profit, which is considered as a function of the capital investment of the company. There is more work in the analysis of the solution after the model equations have been solved than there is in the development of the model itself.

Comments: This particular project takes a different look at the rocket problem. Rather than trying to determine the construction of a rocket to go into orbit, this project develops a model to determine the maximum height of a rocket and the amount of time needed to achieve liftoff.

Comments: The problem of producing sheet steel is considered. Model equations are developed and a variety of analysis techniques are applied. This is a very interesting and practical question, but the content may be a bit too advanced for class.

Comments: The construction of a flagpole which would withstand the force of an earthquake is considered. Model equations are developed, requiring the use of information from strength of materials. The use of dimensional analysis, dimensionless groups and perturbation techniques are interesting. A variety of other interesting problems is presented in the exercises.

Comments: The "morning rush hour" considered here is not traffic, but rather on an elevator in a busy office building. Several performance criteria for a multiple elevator system are computed via Monte Carlo simulation. Pseudocode for an algorithm is provided. This might make an excellent example for the in-class discussion of Monte Carlo methods.

Comments: The problem of analyzing the utilization and performance of a harbor docking and unloading system is considered. A variety of performance criteria are established using Monte Carlo simulation. Different probability distributions are also considered. Other scenarios are proposed in the exercises. This might be an excellent problem for the in-class discussion of Monte Carlo methods.

Comments: This is (as are all of the models drawn from the 2nd edition of this textbook) an open-ended problem which was used in the Mathematics Contest in Modeling sponsored by COMAP. The basic question here is to model the storage of salt in a circular dome and recommend, based on safety considerations, a maximum storage height within the dome.

Comments: This is (as are all of the models drawn from the 2nd edition of this textbook) an open-ended problem which was used in the Mathematics Contest in Modeling sponsored by COMAP. The basic question here is to determine locations within a specific rectangle which should be avoided because the water is too shallow for your vessel.

Comments: This is (as are all of the models drawn from the 2nd edition of this textbook) an open-ended problem which was used in the Mathematics Contest in Modeling sponsored by COMAP. The basic question here is to determine a harvesting policy to optimize the value of the harvest of some population. Students are requested to select a population on which data is readily available.

Comments: This is (as are all of the models drawn from the 2nd edition of this textbook) an open-ended problem which was used in the Mathematics Contest in Modeling sponsored by COMAP. The basic question here is to determine a minimal cost tree for a network of nodes.

Comments: This is (as are all of the models drawn from the 2nd edition of this textbook) an open-ended problem which was used in the Mathematics Contest in Modeling sponsored by COMAP. The basic question here is to develop objective criteria and schedule the work needed to restore power after a storm.

Comments: This is (as are all of the models drawn from the 2nd edition of this textbook) an open-ended problem which was used in the Mathematics Contest in Modeling sponsored by COMAP. The basic question here is to determine locations for two emergency facilities which are going to be built. Information regarding the distribution of previous emergency calls is provided.

Comments: This is (as are all of the models drawn from the 2nd edition of this textbook) an open-ended problem which was used in the Mathematics Contest in Modeling sponsored by COMAP. The basic question here is to develop a search method for finding a drug runner whose radio transmission has been intercepted.

Comments: This is (as are all of the models drawn from the 2nd edition of this textbook) an open-ended problem which was used in the Mathematics Contest in Modeling sponsored by COMAP. The basic question here is to design the layout of parking spaces in a corner parking lot.

Comments: This is (as are all of the models drawn from the 2nd edition of this textbook) an open-ended problem which was used in the Mathematics Contest in Modeling sponsored by COMAP. The basic question here is to determine how to load two flatcars with seven different kinds of crates to minimize wasted floor space, subject to various constaints.

Comments: This is (as are all of the models drawn from the 2nd edition of this textbook) an open-ended problem which was used in the Mathematics Contest in Modeling sponsored by COMAP. The basic question here is to develop a mathematical model for managing a stockpile of the strategic metal cobalt.

Comments: This is (as are all of the models drawn from the 2nd edition of this textbook) an open-ended problem which was used in the Mathematics Contest in Modeling sponsored by COMAP. The basic question here is to develop a model for scheduling aircraft use of runways for departure. The model should take into account both passenger and airline satisfaction.

Comments: This is (as are all of the models drawn from the 2nd edition of this textbook) an open-ended problem which was used in the Mathematics Contest in Modeling sponsored by COMAP. The basic question here is to determine a selection scheme by which a group of judges can decide the best papers submitted as part of a contest.

Comments: This is (as are all of the models drawn from the 2nd edition of this textbook) an open-ended problem which was used in the Mathematics Contest in Modeling sponsored by COMAP. The basic question here is to determine the power needed for a specific radar configuration to detect a standard passenger aircraft at a specified distance from the radar.

Comments: This is (as are all of the models drawn from the 2nd edition of this textbook) an open-ended problem which was used in the Mathematics Contest in Modeling sponsored by COMAP. The basic question here is to develop a method for detecting the presence of a moving submarine, its speed, its size and its direction using only information obtained by measuring changes to the ambient noise field.

Comments: This is (as are all of the models drawn from the 2nd edition of this textbook) an open-ended problem which was used in the Mathematics Contest in Modeling sponsored by COMAP. The basic question here is to design a compensation system for the college's faculty. Specifc design criteria and philosophies are given.

Comments: This is (as are all of the models drawn from the 2nd edition of this textbook) an open-ended problem which was used in the Mathematics Contest in Modeling sponsored by COMAP. The basic question here is to determine whether a concrete slab can be constructed which will maintain temperatures within a specified range. Radiation cooling and heating is considered - not convective heat transfer.

Comments: This is (as are all of the models drawn from the 2nd edition of this textbook) an open-ended problem which was used in the Mathematics Contest in Modeling sponsored by COMAP. The basic question here is to determine an optimal schedule for transfering files among computers connected on a specified network, where the transfer time for each file and the ability of each computer to transfer multiple files is given.

Comments: This is (as are all of the models drawn from the 2nd edition of this textbook) an open-ended problem which was used in the Mathematics Contest in Modeling sponsored by COMAP. The basic question here is to determine the mixture of inputs to a compost heap to optimize compost production.

Comments: This is (as are all of the models drawn from the 2nd edition of this textbook) an open-ended problem which was used in the Mathematics Contest in Modeling sponsored by COMAP. The basic question here is to estimate the distribution of medication within the brain based on measurements at certain locations.

Comments: This is (as are all of the models drawn from the 2nd edition of this textbook) an open-ended problem which was used in the Mathematics Contest in Modeling sponsored by COMAP. The basic question here is to estimate the flow rate from a tank, given depth measurements at a set of discrete times. The problem is complicated by the fact that when the level gets low, a pump turns on to refill the tank, so no depth measurements are given.

Comments: This is (as are all of the models drawn from the 2nd edition of this textbook) an open-ended problem which was used in the Mathematics Contest in Modeling sponsored by COMAP. The basic question here is to develop a scheme for classifying two types of insects based on measurements of two physical characteristics.

Comments: This is (as are all of the models drawn from the 2nd edition of this textbook) an open-ended problem which was used in the Mathematics Contest in Modeling sponsored by COMAP. The basic question here is to determine, in real time, all of the intersections between a helix and a plane in general positions in space.

Comments: This is (as are all of the models drawn from the 2nd edition of this textbook) an open-ended problem which was used in the Mathematics Contest in Modeling sponsored by COMAP. The basic question here is to determine the expected annual costs of the loading operation at a coal company.

Comments: This material presents an analysis of a model for "standing gradient" osmotically driven flow. Dimensional analysis for a basic functional relationship and nondimensionalization of a set of model equations (to determine fundamental dimensionless groups) are both considered.

Comments: The basics of dimensional analysis and nondimensionalization of equations is presented. Several examples are given to illustrate the process. Additional problems are posed in the exercises.

Comments: I really like this model for an in-class discussion - can possibly even have students produce data at home for comparison - early in the semester. It is a practical question which can be used to illustrate the modeling process. This problem can even be modified to include a determination of whether the counter measures from the lead reel, the take-up reel, or measures in real time.

Comments: A nice introductory problem to Monte Carlo simulation. More contrived, less practical, than the problems found in Giordano and Weir, but could be fun.

Comments: This problem revisits the docking training maneuver problem mentioned above - introducing uncertainty into the time taken to determine motion parameters, decide upon appropriate action and to make desired action.

Comments: Another Markov chain problem is considered - this time concerning the operation of a truck maintenance crew. The basic problem is interesting in its own right, but many variations on the theme are possible.

Comments: A discrete model for a docking training maneuver is developed. This problem is nice for developing a modeling methodology which is useful for a large class of problems. Several other scenarios are proposed in the exercises.

Comments: Like the other problems from this book , this one allows for a lot of variations on a theme. This might be a nice problem to illustrate the difference between a Monte Carlo simulation and an analytic simulation.

Comments: This problem poses a practical question regarding an inventory policy for a certain item sold at a pet store. The analysis involves Markov chains and the Poisson distribution. The sensitivity of the predictions to the key parameters is considered. A variety of different inventory policies could be considered - with the ultimate result being a decision on the best policy to adopt.

Comments: A model is developed for determining the optimal number of boats in a fishing fleet which should be operational. It is assumed that fish population is too low due to overharvesting, and the owner of the fishing boats wishes to determine the amount of time to keep his boats in port to allow recovery of the population, and then subsequently, how many boats should be used once the population level has risen to safe levels.

Comments: An improved predator-prey model, which includes a predator carrying capacity which depends on the prey population and a Michaelis-Menten law term for the effect of predation on the prey population, is presented. It is established that under certain conditions, a limit cycle in the two populations exists. Biological significance is attached to the determined conditions.

Comments: The Streeter-Phelps model for environmental purification is presented.

Comments: A differential equation model for the transport of glucose from the mother to the fetus across the placenta is developed and analyzed. The analysis considers phase portraits, quasi-equilibrium and dimensional analysis. The dimensional analysis is carried out to determine the conditions under which the quasi-equilibrium condition can be assumed to hold.

Comments: This particular example has a lot of practical import - given a fixed chunk of money, how should that money be distributed to individuals in the form of a raise. Students might find it interesting to know that at CNU, a process similar to the one developed in this problem is used to determine faculty merit pay raises.

Comments: A model is developed for determining the optimal amount of pesticide and the optimal time to spray for a grower who is trying to minimize economic loss.

Comments: Using probability considerations, this model attempts to determine conditions at a T-junction which would justify the addition of a left turn lane at the intersection. Students might find this model particularly practical.

Comments: Several variations of models for determining average line lengths at a checkout station are examined. In addition to line length, waiting time is also considered. Heavy duty probability.

Comments: Sections 3.3 and 3.4 of the text consider predator-prey models with several suggestions for realistic predation terms and forms for the predator equation. Analysis for a particular model is given in section 3.4. Additional problems are considered in the exercises at the end of the chapter.

Comments: The analysis of the cycle of a disease, with delay included in the model, is contained in section 1.5 of the text. Determination of equilibrium solutions and their stability is considered, as well as the existence of oscillatory solutions. Additional material is considered in the chapter exercises.

Comments: The analysis of a logistic growth model with delay is contained in section 1.4 of the text. Determination of equilibrium solutions and their stability is considered, as well as the existence of oscillatory solutions. Additional material is considered in the chapter exercises.

Comments: Chapter 5 of the textbook contains an "everything you ever wanted to know, but were afraid to ask" about reaction kinetics. A discussion of the process for converting reaction equations into differential equations is presented. Models include cooperative phenomena, autocatalysis, activation, inhibition, multiple steady states, mushrooms and isolas.

Comments: This set of problems is similar in vein to the tape deck counter problem. Using measurements which can be taken from the end of a roll of linoleum, carpeting, cardboard, etc., such as material thickness, number of turns of material, diameter of core, diameter of roll, etc., determine the length of material on the roll.

Comments: This model may be appropriate for early in the semester for illustrating the interconnections between the sciences for are needed to effectively develop mathematical models. The model and its analysis are fairly straightforward.

Title: Guerrilla combat model
Math Content: ODE (system, nonlinear, stability)
Subject Area: Warfare, Population
Reference: SIAM Review (March 1991)

Comments: A "Lanchesterian system" modeling the combat between unseen and unreinforced guerrilla forces and conventional forces is considered.

Title: Simulation of a Mass-Servicing System
Math Content: Probability, Monte Carlo simulation
Subject Area: Business, Communication
Reference: Sobol, A Primer for the Monte Carlo Method

Comments: The objective is simulate the operation of a mass-servicing system with a finite number of available lines. The basic premise is that if the first line is busy, the request is passed on to the second line, if the second line is busy, the request is passed on to the third line, etc. If all lines are busy, the request is rejected. More complex problems are suggested at the end of the section.

Title: Calculating the Quality and Reliability of Devices
Math Content: Probability, Monte Carlo simulation
Subject Area: Business, Manufacturing
Reference: Sobol, A Primer for the Monte Carlo Method

Comments: The objective is assess the reliability of devices which are composed of several composed whose probabilistic characteristics are known. The specific example of estimating the length of time a device will function without breakdown, given the failure characteristics of the components, is considered.

Title: Computation of Neutron Transmission through a Plate
Math Content: Probability, Monte Carlo simulation
Subject Area: Physics
Reference: Sobol, A Primer for the Monte Carlo Method

Comments: This problem requires the estimation of the probability that a neutron incident upon a plate will be transmitted through the plate, that it will be absorbed by the plate and that it will be reflected from the plate. Two simulation schemes are considered - a direct approach and an approach which uses weights to replace the absorption of neutrons by the plate.

Comments: This is another interesting project with practical considerations. Like the cost of guarantees problem, this is a project which could be redone for another situation - data could be gathered for a local restaurant and the optimal service rate could be determined. This project involves a good combination of probability, limits, basic optimization and numerical computation.

Comments: A large portion of this project is devoted to understanding the basic probability which is needed to determine average tire life and failure rate. This is, however, a fairly interesting problem with practical implications. Students might be able to contact companies, gather data, and determine the cost of other guarantees which are built into products which they use.

Comments: This problem would make an excellent student project. The author has left the problem deliberately open-ended. The dynamics of a two-dimensional nonlinear system are to be investigated. The system has three parameters and might have a stable limit cycle under to be determined conditions on those parameters.

Comments: A predator-prey model is presented and an analysis of the fixed points is requested. In particular, the conditions for a Hopf bifurcation to occur are to be determined, and through the use of a computer, the bifurcation is to be classified as either subcritical or supercritical.

Comments: The laser model from the exercises in Chapter 3 is revisited. There, the system was reduced to a single equation; here, the full system is considered. The objectives are to classify the fixed points, determine the number of qualitatively different phase portraits, plot the stability diagram and characterize the bifurcations which occur.

Comments: This is a continuation of the insect outbreak problem from Chapter 3. In this exercise from Chapter 8, the dynamics of the forest are considered. A two-dimensional system of nonlinear ODEs is presented and an analysis of the fixed points is requested.

Comments: A model for oscillation in a particular chemical reaction is presented. A four-variable model is reduced to a two-variable model, which is then analyzed. The presence of a limit cycle is established and a stability diagram as a function of the two system parameters is drawn. Numerical integration is used to establish the type of Hopf bifurcation which occurs.

Comments: A two-dimensional system of nonlinear ODEs is presented as a model for a genetic control system. The stability of fixed points is investigated as a function of the two system parameters.

Comments: A model for glycolysis (biochemical process by which living cells obtain energy by breaking down sugar) is presented. The objective of the analysis is to establish the existence of oscillations in the glycolysis process. A trapping region is established and the Poincare-Bendixson theorem is used to guarantee the existence of a limit cycle for certain values of the system parameters.

Comments: This project, which is presented as an exercise in Chapter 6, is an extension of the previous projects on lasers. Here, a two-mode laser, in which two different kinds of photons are produced, is considered. The objective is to perform a stability analysis on the fixed points and characterize all qualitatively different phase portraits.

Comments: A variety of populations models for two competing species is presented in the exercises of Chapter 6, section 4. Several different scenarios are considered.

Comments: A model for investigating the synchronization of flashing fireflies is presented. A complete analysis of the fixed points as a function of the frequency difference between the coupled oscillators is presented.

Comments: A simple model for the evolution of an epidemic is presented in the exercises for Chapter 3. A system of three coupled nonlinear equations is given and then reduced to a single nonlinear ODE. Several references for further discussion, background material and/or discussion of specific epidemics are given.

Comments: A model for a biochemical switch, in which a gene is activated by a biochemical signal substance, is presented. The model consists of a single nonlinear ODE. The equation is nondimensionalized and then the stability of the fixed points is analyzed in the space of two dimensionless parameters. Several references for further discussion and background material are given.

Comments: This is a more advanced version of the fishery model found in the exercises of Chapter 3 - in fact, this is just the next exercise. The refinement to the model consists of a more realistic harvesting term. Bifurcation analysis is performed, a stability diagram is constructed and the possibility of hysteresis is investigated.

Comments: This problem is posed in the exercises of Chapter 3, and is similar to a problem given in Clark. A single nonlinear equation is postulated to model the dynamics of a fishery. The equation is nondimensionalized and a bifurcation analysis is performed.

Comments: This is an even more sophisticated model of a laser than the one presented in "An Improved Model of a Laser." A system of three nonlinear ODEs is considered. The analysis is simplified by considering the simplest case: two of the variables relax more rapidly than the third, reducing the system to a single nonlinear equation.

Comments: This is an exercise from Chapter 3. The laser threshold problem from chapter 2 is revisited - this time with a system of nonlinear equations being considered. If the basic laser problem is used in class, this would be an excellent follow-up problem for illustrating the evolution of a model.

Comments: A model is developed for the outbreak of an insect population. A complete analysis is given, including: dimensionless formulation, analysis of fixed points, calculation of bifurcation curves, comparison with observations. I have several papers which investigate other aspects of this model.

Comments: This is a model discussed in Chapter 3. A single nonlinear differential equation is developed for the number of photons in the laser field. A transcritical bifurcation of the system as the pumping strength is increased is identified. The analysis in this problem is straightforward - might be an excellent in-class example for an introductory discussion of nonlinear equations and bifurcations.

Comments: This is taken from an exercise in Chapter 2. This considers a different model for effective growth than found in the logistic model. Students are requested to compare the results of this model with that of the logistic equation on a qualitative basis. A reference for further discussion is given.

Comments: A model for the growth of cancerous tumors is presented in the exercises of Chapter 2. A standard analysis based on fixed points, direction fields and stability is requested. References for further discussion and investigation are provided.

Comments: This actually isn't much of a model. The objective is to derive the exact solution of the logistic equation in two different manners. Taken in combination with some of the other entries which use the logistic equation, a decent project could be developed.

Comments: The objective here is to determine whether a letter carrier should deliver the mail by going from side to side as he/she moves along the street or by going up one side of the side and return along the other side. A simple inequality can be derived for the relationship between the width of the street and the length of the street upon which the decision can be made.

Comments: The objective is to determine the optimal number of customers that a lawncare company should strive for, given the current number of customers, the current price for services charged to those customers and the companies discount plan if the number of customers rises above a certain threshold. Here, the problem is worked out by building a table of revenue values for a wide range of customer bases. The problem can also be done using parabolas and completing the square - the revenue function will be a downward opening parabola, hence the maximum occurs at the vertex.

Comments: Stockpiles of gasoline need to be established along a 1000 mile route through the desert. The objective is to determine where along the route the stockpiles should be placed and how much gasoline must be stockpiled at each location so that a jeep can travel the full 1000 miles. A minimum number of gallons of gasoline should be used and the fuel economy and fuel carrying capacity of the jeep are given.

Comments: Information about the survival and birth rates of a deer population are given. The population is divided into males and females, and each group is further subdivided into newborn fawns, one year old fawns and adults. Given an initial population census, it is desired to know how the population will evolve over a ten year period. Additional scenarios which include harvesting of the deer are suggested. A difference equation approach is developed - a differential equation model should also be possible.

Comments: This problem involves setting up a linear programming problem for determining the amount of money that should be spent on radio and TV advertising for a concert.

Comments: A manufacturer needs to determine the optimal distance between irrigation heads on a moving irrigation system to provide as uniform as possible coverage of a field. This problem can be done using a graphing calculator or computer graphing program or by resorting to differential calculus.

Comments: The planning commission must determine the best way to orient parking spaces relative to the sidewalk in order to allow room for on-street parking and two-way traffic in the downtown area.

Title: A Mathematical Model of a Universal Joint
Math Content: Trigonometry, Calculus (differentiation, integration, max-min, improper integral)
Subject Area: Engineering
Reference: UMAP module # 566

Comments: A model of a universal joint is presented. The abstract indicates that the purpose of the model is to study the output pulsations, as well as the angle of coupling which will alleviate the pulsations on a second shaft. An excellent example for improper integrals arises when trying to determine the average angular velocity of the output.

Comments: This project requires only first year calculus. The flow through a pipe is considered and Poiseuille's law is used. A discussion of Poiseuille's law (the assumptions for which the result is valid) are presented. This might be a good problem for a calculus class just learning about integration and Riemann sums.

Comments: This might make an interesting interdisciplinary project for a calculus class. It combines biology, physics and mathematics to determine the contracted radius of the trachea which will maximize airflow during a cough.

Comments: Methods for solving both first-order and second-order linear difference equations, both homogeneous and nonhomogeneous, are presented. Applications of the theory include: compound interest, population growth, annuities, the Tower of Hanoi problem, the cobweb theorem of economics and the Fibonacci numbers.

Comments: Two models (a simplified one and a more complicated one) are developed relating atmospheric pressure to altitude and temperature. An application to meteorology is given.

Comments: This module presents a fairly complete predator-prey model. The model is first developed and then nondimensionalized. The inclusion of harvesting plays a central role. Equilibrium points are determined and stability is mentioned. The concept of maximum sustainable yield is introduced. This might make an excellent student project.

Comments: Various assumptions about the growth of a single population in isolation and the competition between two species are used to develop models of population growth under competition. A graphical analysis is performed to determine the stability of the equilibrium points. The significance of the various possible phase portraits is considered.

Comments: The process of dimensional analysis is explained in detail. The examples of an undamped and a damped pendulum are explored. Sections on testing the model and presenting results are given. Several references are listed which might contain other examples of dimensional analysis.

Comments: From the abstract: "This unit derives the [Ricker Salmon] model, shows how it can be modified and introduces the concept of maximum sustainable yield. It also shows how difference equations may lead to periodic and chaotic behavior, and a computer program enables one to explore the periods and chaos. The technique of dynamic programming is introduced to show how to maximize income from fishing over a finite period."

Comments: Models for the relationship between excitation intensity and stimulus intensity are developed. The cases of weak stimulus, moderate stimulus and strong stimulus are examined. Comparison of model predictions to experimental data drawn from the literature is presented.

Comments: From the abstract: "This unit introduces a differential equations model for the digestive processes of sheep. Students describe the digestive processes of ruminants, explain how the assumptions of the model are translated into equations, discuss what support there is for the validity of the model and discuss some possible conclusions to be drawn from the model."

Comments: From the abstract: "Cardiac output is defined. ... This module develops a formula for computing cardiac output from observations obtained by dye dilution. The development is an elementary illustration of Riemann sums, definite integration and approximation by the trapezoidal and Simpson's rule.

Comments: This module presents a fairly basic model for the level of drug in the bloodstream. The objectives of this module are: (1) determining the effect on blood concentration levels of repeated doses of a drug; and (2) scheduling drug administration to achieve safe but effective levels. This type of model has been mentioned in several other locations in this database. This might make an excellent in-class problem.

Comments: These modules develop methods and procedures for "solving" ordinary differential equations graphically and numerically. Applications from optics, engineering and ecology are considered.

Comments: This is a chapter from White's book - in particular, Chapter 5. He considers dimensional homogeneity, dimensional analysis, non-dimensionalizing equations and the use of scale models. Several illustrations and exercises, all drawn from fluid mechanics, of dimensional analysis are presented.

Comments: This project actually has students develop two different models - one involving flow of traffic at a traffic light and the other involving the flow of traffic at a stop sign. The objective is to compare the results and decide whether a traffic light is needed or whether it would be better to use a stop sign. The performance criteria which is considered is the minimization of time lost per car due to slowing down, stopping and accelerating.

Comments: This problem is similar to one given in Lomen and Lovelock. As mentioned with that model, I think this would be an excellent problem for in-class discussion and homework. The model can be developed from several possible viewpoints - curve fitting data, algebraic considerations, solving a differential equation.

Comments: This project was adapted from UMAP module #232. The work involved on this project seems to be in solving some simple separable first-order ODEs and then fitting data to determine the order of a reaction and the rate constant for the reaction.

Comments: This is a Monte Carlo simulation project. The objective is to decide among three possible checkout line configurations given information about speed of checkout personnel, the number of items a customer has upon arriving at the checkout line and the distribution of arrival times.

Comments: This project is based upon UMAP module 269 and contains three different simulation scenarios: random walk, grocery store checkout and a fictitious game called "Mathball." This might be good for in-class discussions of the topic of Monte Carlo simulation.

Comments: This project has students develop an algebraic model for determining the length of time a traffic light should remain yellow in order to allow drivers in the intersection and those too close to the intersection to stop to pass through the intersection before the light turns red.

Comments: This project is based on UMAP module #72 - look there for more details. This is also similar to the project from Borelli and Coleman on cascades. Basically poses the question: how often should medication be taken and at what dosage in order to keep the level of medication in the bloodstream within required limits?

Comments: This is an excellent multidisciplinary project. A system of two nonlinear ordinary differential equations are developed to model the action of a chemostat. The mathematical analysis of the problem involves approximation methods, stability analysis, nondimensionalizing an equation, parameter estimation, etc.

Comments: This is an interesting problem in dimensional analysis which was adapted from an exercise in Giordano and Weir. After dimensional analysis is performed, students must fit their function to data and make a prediction.

Comments: This paper presents several different models for the transmission of HIV and the outbreak of AIDS. The primary worth of this paper is in presenting different schemes for developing mathematical models. There is also an example of non-dimensionalizing a differential equation.