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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 89 "nbdd.ms   Nicholson-Bailey density dependent model
.  \nExploration of stability regions.  " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 306 "Enter values of growth rate r and
 density dependence parameter q.  The program will generate plots of p
opulations vs time and phase plots.  It also will calculate the eigenv
alues of the linearized equations around the equilibrium values of hos
t and parasitoid for the particular values of r and q chosen.  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "res
tart;  with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "r:=1
.5;  q:=0.6; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " c:=1.0;  \+
 a:=0.2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "### WARNING: s
emantics of type `string` have changed\n### WARNING: semantics of type
 `string` have changed\ntxtr:=convert(evalf(r),string):   txtq:=conver
t(evalf(q),string):\nlabelr:=`r =  `:   labelq:=`,  q = `:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Phat:= r/a*(1-q);  Nhat:= Phat/(1-e
xp(-a*Phat));  K:= Nhat/q;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
60 "N[0]:= 11; P[0]:= 4;  # initial conditions (can be changed!)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "for i from 0 to 200 do\n   \+
N[i+1]:=N[i]* exp(r * (1 - N[i]/K) - a*P[i]);\n   P[i+1]:=c*N[i]*(1 - \+
exp(-a*P[i]));\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "full
title:= cat(labelr,txtr,labelq,txtq):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 95 "plot(\{ [ [t,P[t]] $ t = 0.. 200], [ [t,N[t]] $ t = 0
..200] \}, style = line, title=`fulltitle`);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 65 "plot([ [N[t],P[t]] $ t=0..200], style = line, ti
tle=`fulltitle`);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 242 "\nCompare t
he results of these simulations to Figure 2.7 in Hassell 1978.  Now go
 back and try this again with new values for r and q.  You should use \+
the Hassell reprint as a guide to pick values that yield qualitatively
 different behaviors. " }}{PARA 0 "" 0 "" {TEXT -1 465 "\nTo calculate
 the eigenvalues of the linearized system, we recast the equations in \+
more symbolic terms, converting N(t) to N and P(t) to P.  We then calc
ulate the partial derivatives of each equation with respect to both N \+
and P.   This can be done by hand, as done by calculating a11, a12, a2
1, a22.  Alternatively, the jacobian command will do the calculation d
irectly.  Once the matrix is assembled, the eigenvals command is used \+
to calclulate the eigenvalues.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 242 "Remember that since this is a difference
 equation model, stability occurs  (possibly after damped oscillation)
 when the eigenvalues are less than 1 in absolute value.  Oscillations
 occur when there are imaginary components to the eigenvalues." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "r:=1.5;  q:=0.6; " }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 39 "Host_rate:=N* exp(r * (1 - N/K) - a*P);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "a11:=diff(Host_rate, N); \+
  a12:=diff(Host_rate, P);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
85 "A11:= evalf(subs(\{N=Nhat, P=Phat\}, a11));   A12:= evalf(subs(\{N
=Nhat, P=Phat\}, a12));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "
Parasitoid_rate:=c*N*(1 - exp(-a*P));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "a21:= diff(Parasitoid_rate, N);   a22:= diff(Parasito
id_rate, P);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "A21:=evalf(
subs(\{N=Nhat, P=Phat\}, a21));  A22:= evalf(subs(\{N=Nhat, P=Phat\}, \+
a22));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with (linalg):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "A:= matrix( 2, 2, [ [A11, \+
A12], [A21, A22] ] );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "\nHere i
s the alternate method for computing the coefficient matrix using the \+
jacobian command." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "J:= jacobian( \+
[Host_rate, Parasitoid_rate], [N, P] );" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 92 "A:= evalf( subs( \{N=Nhat, P=Phat\}, evalm( J ) ) ); \+
 #  Note we get the same thing as before." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 13 "eigenvals(A);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 219 "How do you i
nterpret the behavior of the system if these are the eigenvalues?  Go \+
back to the beginning of this part of the worksheet, reset r and q, an
d see if the eigenvalue caculations lead to a different conclusion." }
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