Christian Boire, Andrew Sisitzky, Genevieve Gibbons, Angel Chavez

   Growth Models

   Your project paper on the basic ideas of "the systems perspective" is 
   reasonably good, but I think your oral presentation was stronger. You 
   have an OK conclusion as well, rather than just sort of "petering out" 
   without tying together the points you have made.  I know it can be hard 
   to manage things like this when you are "writing by committee." Making 
   sure you are communicating and coordinating your efforts is an important 
   part of working in a team and that aspect of the paper was pretty 
   successful.  
 
   However, the major criticism I have is that you have not done a good
   job of identifying the sources of the individual pieces of information
   you present.  You have the required list of references at the end, but
   you didn't cite those sources in foot- or endnotes to document where
   you were taking the information from.  That is also required for this
   sort of writing and it's the major thing I am marking you down for.   

   Specific comments:

   (1)  Page 1:  "Malthusian" population models are synonymous with 
                 exponential models.  It's not clear that's true from 
                 what you are saying.  

   (2)  Page 2:  Don't understand what you meant by "simplifies the 
                 predicted rate of growth or decay."

   (3)  Page 3:  Exponential models are "very inaccurate" if they 
                 are used indiscriminately.  However, they can also 
                 be fine over short time spans, in situations where
                 the basic assumptions are met (in particular where there
                 are sufficient resources to allow growth at the start).
                 So your comments are essentially overstated here.  

   (4)  Pages 4-6:  There is a technical point about the way you are
                    comparing the logistic and Gompertz models that
                    requires more explanation.  Namely, the version of the
                    logistic model that you are discussing is a discrete-time
                    model described by a difference equation.  The Gompertz
                    model presented on the next page is a continuous-time model
                    (i.e. a function of a real variable t) whose value represents 
                    the population size at time t.  The difference is the same
                    as the difference between giving the exponential difference 
                    equation Q(n+1) = aQ(n) and the exponential function 
                    Q(t) = Q(0) a^t.  This is not explained very well, and
                    the discussion of the Gompertz and related models is pretty
                    skimpy.  Some graphs (for particular values of A, k_G, T_i)
                    would have been a good addition.  It's not immediately clear
                    from the formulas that these models give populations that
                    evolve similarly to the ones from logistic models.  You
                    also don't really explain why biologists look at different
                    models beyond the logistic models.  If you look at Exercise
                    14 on page 136 in the course textbook, you'll see one
                    particular possible criticism of the logistic models and
                    that is one reason to look beyond to something different.

   (5)  Page 6:  The "matrix models" you are mentioning are discussed (but not
                 using that name or the matrix infrastructure) in Chapter 8 of
                 the course textbook.  I called them population models with
                 stratification by age group.  See in particular Example 8.2
                 on pages 142-143 of the textbook.  

   (6)  Page 7:  Do you know what "dampen the value for this group in the matrix"
                 means?  If not, it's best to avoid getting that technical.

   (7)  Pages 8 and 9:  Organization in this section is not optimal.   I think
                        the discussion of the "Malthusian model" should probably
                        come before the discussion of its shortcomings.  

   (8) Page 10:  The formula for the exponential model for the bee colonies
                 is not correct.  See the comment on the hardcopy for the correct
                 formula.  Remember that the a in the exponential function is
                 given by a = (1 + r/100) if r is the percent growth/decay rate
                 per unit time.  


   Grade:  88 (B+)