Assignment 1

Determine r and K to fit human population data (since ~1950) to a continuous logistic model for the United States and for the total world population-and perhaps a few other countries. (The Web is one good source for census data -- e.g., www.popin.org and www.census.gov/statab/www/pop.html.) Show your data and graphics on an Excel spreadsheet.

Carry out a simulation of a population that grows according to the discrete-time logistic model. Use K = 100, and explore the time-dependent behavior over a range of values of r. (Use an Excel spreadsheet for this simulation.)

Assignment 2

Calculate and graph, on an Excel spreadsheet, curves showing the feasible steady states of n1 and n2 for the model used in class for the case of two competing species. Use the parameter values shown in slide 3 of the PowerPoint file 2c_two_competitors. Show n1 and n2 versus a12 (for a12 running from 0 to 7) on two separate graphs. (Such curves are called "bifurcation curves", and the parameter on the abscissa-; a12 in this case-; is called the "bifurcation parameter". See assignment #4.)

Assignment 3

Use Mathcad to repeat the simulation (i.e., to obtain curves of n1 and n2 versus time) shown by the smooth curves on slide 3 of the PowerPoint file 2c_two_competitors) for the case involving two competing bacteria. Vary the parameters so as to test the nature of the response in each of the four regions of the a12,a21 plane shown in slide 5 of that PowerPoint file. Briefly summarize your observations for the four regions.

Assignment 4

(not assigned; worked out in class) Obtain expressions for the bifurcation points on the curves of assignment #2. Prove that there are no Hopf bifurcations for this model and hence no oscillatory states.

Assignment 5

Review the notes accompanying the short PowerPoint file introduction_to_mutualism in the assignments folder. That file gives background information about a type of species population interaction known as mutualism. Then

(a) Obtain explicit expressions for the steady-state solutions, n1s and n2s, in terms of the model parameters.

(b) Show the feasible region(s) for the coexistent state in the a12,a21 plane.

(c) Use linearized time-dependent equations to draw conclusions about the stability of the steady states.

(d) Use Mathcad to simulate the unsteady state for this model and to explore the global time-dependent behavior

You should report all of this on a Mathcad sheet. If necessary, you may use Excel spreadsheets also, but refer to them on your Mathcad sheet.

Assignment 6

In class we examined a "resource-based" model for the dynamics of two competing species (producers) with populations n1 and n2. We reached the conclusion that the model would not allow for the coexistence of both species. Reconsider the problem using a similar model, but suppose that there are two nutrient sources, with amounts s1 and s2, generated from the detritus reservoir. Would it now be possible for the species to coexist? What general conclusion can you reach about the relationship between m, the number of producer (basal) species, and s the number of different nutrients?

Assignment 7

In this assignment you will apply the methods of analysis that we have covered to a chemical reactor problem.

An irreversible exothermic liquid-phase reaction between reactants A and B is governed by the following stoichiometric equation.

A + 2B --> products + heat

The reaction is second-order -- i.e., first-order with respect to each reactant. The reaction takes place in a continuous-flow stirred reactor, a CSTR, for which the parameter values listed below apply. Heat is removed from the stirred mixture though an immersed cooling coil. All fluid properties may be assumed constant.

Reactor volume, V = 1010 ml

Mixture heat capacity, Cp = 1.0 cal/g K

Mixture density, r = 1.0 g/ml

Feed concentration of A, Caf = 0.90 mol/liter (in aqueous solution)

Feed concentration of B, Cbf = 1.35 mol/liter (in aqueous solution)

Feed temperature, Tf = 273 K

Coolant temperature, Tc = 260 K

Overall heat transfer coefficient, U = 0.062 cal/(s.cm2.K)

Heat of reaction, H = -146,400 cal/mol of A

Reaction rate expression, R(Ca,Cb,T) = (1.6 x10^10) Ca Cb exp(-8140/T ) mol of A/liter.sec.

(a) Consider a specific case for which the feed flow rate is 18.2 ml/sec, and the desired steady-state temperature is 303 K. Use steady-state material balances on A and B to calculate their effluent concentrations, Ca and Cb, and use a steady-state energy balance to calculate the heat-transfer surface area, A, required to achieve the desired steady-state temperature.

(b) Use a mathematical model consisting of three ordinary differential equations -- two component material balances and an energy balance -- to test the stability of the operating state in (a) and to carry out a simulation of the unsteady state. Show the eigenvalues of the Jacobian matrix for that particular steady state. Show also the curves of T, Ca, and Cb versus time, t, and show a trajectory on the T,Ca phase plane. (In your choice of initial conditions, always take the initial concentration of A to be in 44% excess, as it is in the feed solution.)

Assignment 8

Consider a food chain with a single species population at each of three trophic levels (a "tritrophic" chain). The species at the lowest level is a producer (basal species); the species at the second level consumes that at the first, and it in turn is prey for the top predator at the highest trophic level.

(a) Present the differential equations describing the dynamics of this system. In those equations, assign a carrying capacity, K1, only to the basal species, and use the Holling II response function in the interaction terms. Use the symbols shown and definitions given in slides 3 and 4 of the handout "Biota Dynamics, Some Generalizations." Thus

Fij = ninj/(ni + Ai) for i < j

Fij = Fji for i > j

(b) Obtain explicit expressions for all of the steady-state solutions. That is, obtain expressions for n1s, n2s, and n3s in terms of the parameters.

(c) Use the following set of parameter values and calculate the feasible steady states for an arbitrary value (your choice) of K1 on the range 0 < K1 < 2. (We'll follow this up in class to see how "wise" your choice turns out to be. You might think of K1 as being a measure of the size of the habitat for this ecosystem.)

r1 = 0.535

r2 = -0.40

r3 = -0.01

a12 = -a21 = -1.67

a23 = -a32 = -0.05

A1 = 0.333

A2 = 0.50

All of your work should be reported on a Mathcad sheet.

Any additonal assigned work will be given in class.