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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 16
"Eat and Be Eaten" }}{PARA 257 "" 0 "" {TEXT -1 9 "Jim Herod" }}{PARA
258 "" 0 "" {TEXT -1 21 "School of Mathematics" }}{PARA 259 "" 0 ""
{TEXT -1 12 "Georgia Tech" }}{PARA 260 "" 0 "" {TEXT -1 21 "herod@math
.gatech.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 51 " In the Vol. 153, ( February 7, 1998) issue of " }{TEXT
258 12 "Science News" }{TEXT -1 38 ", page 86, there is an article tit
led " }{TEXT 267 38 "How low will we go fishing for dinner?" }{TEXT
-1 350 " The article explains that \"ecologists measure an organism's \+
niche in terms of its trophic level. In the sea, the base level contai
ns mainly seaweeds and phytoplankton. These serve as food for level tw
o organisms, whose predators, in turn, make up level three. And so it \+
goes up the marine food web to its apex, killler whales at trophic lev
el five.\"" }}{PARA 0 "" 0 "" {TEXT -1 57 " The following model is
suggested by this article in " }{TEXT 259 12 "Science News" }{TEXT
-1 38 ". We mimic the situation described in " }{TEXT 260 12 "Science \+
News" }{TEXT -1 6 " with " }{TEXT 262 15 "logistic growth" }{TEXT -1
47 " for the lowest level of fauna and with simple " }{TEXT 256 11 "ma
ss action" }{TEXT -1 48 " differential equations for interacting speci
es." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 " \+
Consider this lowest level of fauna which we take to be growing lo
gistically as a collection. We use the equation " }}{PARA 0 "" 0 ""
{TEXT -1 46 "(1) " }
{XPPEDIT 18 0 "diff(z,t)=a*z(t)-b*z(t)^2" "6#/-%%diffG6$%\"zG%\"tG,&*&
%\"aG\"\"\"-F'6#F(F,F,*&%\"bGF,*$-F'6#F(\"\"#F,!\"\"" }}{PARA 0 "" 0 "
" {TEXT -1 37 "for illustration, making choices for " }{TEXT 265 1 "a
" }{TEXT -1 5 " and " }{TEXT 266 1 "b" }{TEXT -1 80 ". In this equatio
n, z(t) represents the population of the lowest level of fauna." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 234 " The
solutions for this equation are used to produce graphs. These graphs \+
suggest how logistic growth might be understood. We draw three graphs
; the graphs have solutions with initial values z(0) = 16, z(0) = 12
, and z(0) = 10." }}{PARA 0 "" 0 "" {TEXT -1 46 " This equation ca
n be solved analytically." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71
"a:=28; b:=2;\nSol1:=dsolve(\{diff(z(t),t)=a*z(t)-b*z(t)^2,z(0)=zo\},z
(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "z1:=unapply(rhs(So
l1),(zo,t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "plot([z1(16
,t),z1(12,t),z1(10,t)],t=0..1/5,\n color=BLACK);" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 38 " An observation to make from this " }
{TEXT 257 3 "toy" }{TEXT -1 324 " model is that if there is room for g
rowth of this collection of fauna, the population size will rise to a \+
level which will not be exceeded. On the other hand, if the population
size is too large, this logistic model predicts a decrease in populat
ion size to a level sustainable by the environment. This level is call
ed the " }{TEXT 263 25 "steady state of the fauna" }{TEXT -1 8 ", or t
he" }{TEXT 264 36 " carrying capacity of the environmen" }{TEXT -1 2 "
t." }}{PARA 0 "" 0 "" {TEXT -1 184 " At this point, it is importan
t to identify the steady state level of the base fauna. This can be ob
served from the graph. The steady state can also be computed as the le
vel where " }{TEXT 261 9 "d z / d t" }{TEXT -1 253 " is zero. Thus, se
t the right side of the equation (1) equal to zero and solve. You get \+
two solutions. One of these solutions is the non-zero steady state lev
el of this model. You should find that this answer agrees with the obs
ervations from the graph." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21
"solve(a*z-b*z^2=0,z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 ""
}}}{PARA 0 "" 0 "" {TEXT -1 95 "Humans can solve this simple equation.
Try it. See that you get that one steady state will be " }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "L1:=a/b;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 349 " We inject a second level of fauna, y(t), which is
predatory toward the first level of fauna. We suppose that this secon
d level of fauna cannot survive in the absence of the first; its survi
vability depends on and is enhanced by the presence of the first level
of fauna. We model such interaction in a standard way using the follo
wing equations:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 38 " " }{XPPEDIT 18 0 "d
iff(z(t),t)=a*z(t)-b*z(t)^2-c*z(t)*y(t)" "6#/-%%diffG6$-%\"zG6#%\"tGF*
,(*&%\"aG\"\"\"-F(6#F*F.F.*&%\"bGF.*$-F(6#F*\"\"#F.!\"\"*(%\"cGF.-F(6#
F*F.-%\"yG6#F*F.F7" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 3 "(2)
" }}{PARA 0 "" 0 "" {TEXT -1 38 " \+
" }{XPPEDIT 18 0 "diff(y(t),t)=-d*y(t)+e*z(t)*y(t)" "6#/-%%diffG6$-%
\"yG6#%\"tGF*,&*&%\"dG\"\"\"-F(6#F*F.!\"\"*(%\"eGF.-%\"zG6#F*F.-F(6#F*
F.F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 47 "What does your intuition tell you might happen:" }}
{PARA 0 "" 0 "" {TEXT -1 114 " (a) the lower level fauna might bec
ome more plentiful because the predator removes competitors for resour
ces?" }}{PARA 0 "" 0 "" {TEXT -1 121 " (b) the lower level fauna m
ight stay at the same level because it is not competing with its preda
tors for resources?" }}{PARA 0 "" 0 "" {TEXT -1 105 " (c) the lowe
r level fauna might decrease to a lower level population level because
of the predation?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 128 "A mathematical way to predict the long-range forecast fo
r the equilibrium level for these two populations would be to ask wher
e " }{XPPEDIT 18 0 "diff(z(t),t)=0" "6#/-%%diffG6$-%\"zG6#%\"tGF*\"\"!
" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "diff(y(t),t)=0" "6#/-%%diffG6$-%
\"yG6#%\"tGF*\"\"!" }{TEXT -1 71 ". This happens when the left side o
f the above two equations are zero." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39
"a:=28;b:=2; c:=2;d:=1; e:=1/3; L1:=a/b;" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 46 "solve(\{a*z-b*z^2-c*z*y=0,-d*y+e*z*y=0\},\{y,z\});" }
}}{PARA 0 "" 0 "" {TEXT -1 234 "The solutions for these two equations \+
indicate there are two non-trivial solutions. We wonder if some initia
l distributions lead to one and other initial solutions lead to the ot
her. We record one of these levels for a future problem." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Ly2:=(a*e-b*d)/(c*e);\nLz2:=d/e; "
}}}{PARA 0 "" 0 "" {TEXT -1 206 " Another feature of this more com
plicated problem is that it is unlikely we can solve these differentia
l equations in closed form -- in terms of elementary functions. We sol
ve the equations numerically." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 167 "Sol2:=dsolve(\{diff(z(t),t)=a*z(t)-b*z(t)^2-c*z(t)*y(t),\n \+
diff(y(t),t)=-d*y(t)+e*z(t)*y(t),y(0)=1/4,z(0)=L1\},\{y(t),z(t)\},\n \+
type=numeric,output=listprocedure);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 41 "z2:=subs(Sol2,z(t)); y2:=subs(Sol2,y(t));" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot(['y2(t)','z2(t)'],'t'=0..20,co
lor=[RED,GREEN]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 109 " You
should observe that the populations are approaching a steady state fo
r the two interacting species. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 464 " We could now add three more trophic
levels, one at a time and observe the influence on the steady state a
t the lower levels. However, we add the remaining three. We suppose th
at each level profits at increasing level by encounter with species in
the lower level. The effect of other changes in the size of the coeff
icients would be an interesting exploration. Even more interesting wou
ld be to adjust the coefficients to fit the data provided in the origi
nal " }{TEXT 268 12 "Science News" }{TEXT -1 9 " article." }}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 30 "a:=28;b:=2; c:=2;d:=1; e:=1/3;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 31 "Ly2:=(a*e-b*d)/(c*e);\nLz2:=d/e;" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 379 "Sol3:=dsolve(\{diff(z(t),t)=a*z(t)
-b*z(t)^2-c*z(t)*y(t),\n diff(y(t),t)=-d*y(t)+e*z(t)*y(t)-c*x(t)*
y(t),\n diff(x(t),t)=-d*x(t)+2*e*y(t)*x(t)-c*w(t)*x(t),\n di
ff(w(t),t)=-d*w(t)+3*e*x(t)*w(t)-c*v(t)*w(t),\n diff(v(t),t)=-d*v
(t)+4*e*w(t)*v(t),\n v(0)=2,w(0)=2,x(0)=2,y(0)=Ly2,z(0)=Lz2\},\n \+
\{v(t),w(t),x(t),y(t),z(t)\},\n type=numeric,output=listproce
dure);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "z3:=subs(Sol3,z(
t));\ny3:=subs(Sol3,y(t));\nx3:=subs(Sol3,x(t));\nw3:=subs(Sol3,w(t));
\nv3:=subs(Sol3,v(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "
plot(['v3(t)','w3(t)','x3(t)','y3(t)','z3(t)'],'t'=0..40,\n co
lor=[BLACK,BLACK,BLUE,RED,GREEN]);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 98 "plot(['x3(t)','w3(t)','v3(t)'],'t'=0..40,\n v
iew=[0..20,0..2],color=[BROWN,MAGENTA,BLACK]);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 103 " This sugests that one should not expect many
killer whales in comparison to, say, tuna or salmon. " }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 25 "Exercise for
the student." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 334 "Repeat the above problem, only allow fishing from the th
ird level. What effect does this have on the fifth level? ... on the \+
lowest level? Simulate this problem twice. The first time, allow fishi
ng at a level depending on the amount in level 3. To simulate this, ta
ke away 1/10 of level three. This changes the equation for x(t) from"
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " \+
" }{XPPEDIT 18 0 "diff(x(t),t)=-d*x(t)+2*e*y(t)*x(t)-c*w(t)*x(t)" "
6#/-%%diffG6$-%\"xG6#%\"tGF*,(*&%\"dG\"\"\"-F(6#F*F.!\"\"**\"\"#F.%\"e
GF.-%\"yG6#F*F.-F(6#F*F.F.*(%\"cGF.-%\"wG6#F*F.-F(6#F*F.F1" }}{PARA 0
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "to" }}{PARA 0 ""
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT
18 0 "diff(x(t),t)=-d*x(t)+2*e*y(t)*x(t)-c*w(t)*x(t)-x(t)/10" "6#/-%%d
iffG6$-%\"xG6#%\"tGF*,**&%\"dG\"\"\"-F(6#F*F.!\"\"**\"\"#F.%\"eGF.-%\"
yG6#F*F.-F(6#F*F.F.*(%\"cGF.-%\"wG6#F*F.-F(6#F*F.F1*&-F(6#F*F.\"#5F1F1
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 171 "The second simulation is to allow a fixed amount to be t
aken from level 3. Change the equation to remove 1/10 of the original \+
value. Thus, the equation for x(t) changes to" }}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{XPPEDIT 18
0 "diff(x(t),t)=-d*x(t)+2*e*y(t)*x(t)-c*w(t)*x(t)-.2" "6#/-%%diffG6$-%
\"xG6#%\"tGF*,**&%\"dG\"\"\"-F(6#F*F.!\"\"**\"\"#F.%\"eGF.-%\"yG6#F*F.
-F(6#F*F.F.*(%\"cGF.-%\"wG6#F*F.-F(6#F*F.F1$F3F1F1" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Compare t
he two strategies." }}}}{MARK "1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1
1 }{PAGENUMBERS 0 1 2 33 1 1 }