In the 1970s the biologist Robert May wrote this 
<a href="https://people.ucsc.edu/~rmont/classes/chao/2013/Orig_Papers/May_bio.pdf"> paper </a> which changed the way many people  thought about population dynamics,  and dynamics  (how things change and move with time) more generally.    

You are responsible for reading and understanding May's paper.  Questions directly concerning the paper will occur on the final.  The last HW is on this paper.

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   May wrote the dependent variable,` X' and viewed it as a  population density.  It evolves in discrete time t= 0,1,2,3,4,..and its value at time t is denoted X_t - here perhaps  X(t).  X  evolves according to

<p>X(t+1)  = F(x(t)) </p>

For exponential growth   take F(X) = k X.   
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May concerned himself  mainly  with the logistic map,  which is a discrete analogue of the logistic differential equation: 
dN(t)/dt = N(t)(1 - (N(t))  which we briefly   discussed in class.

 By ``seeding'' the population model with an initial value X1  and   iterating the map, we get a sequence :     

<p> X1,  X2, X3, ... X(t),  X(t+1) , ... with X2= F(X1),  X3 = F(X2),  ....,  X(t+1) = F(X(t)), ...  </p>

This sequence is  called the ``orbit'' of X1   

In class, on Friday, Valentine's Day, 2020, we showed how to generate the orbit graphically using
cobweb plots.    <a href ="https://en.wikipedia.org/wiki/Cobweb_plot"> cobweb plots </a>.
Most of the class seemed to have gotten it.