{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 0 1 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R 3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 12 0 0 0 0 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 14 0 0 0 0 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 258 1 {CSTYLE " " -1 -1 "Courier" 1 12 0 0 0 0 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 13 "rKselect.mws " } {TEXT -1 69 " This worksheet is explores the paper Jonathan Roughga rden, 1971, " }{TEXT 257 35 "Density dependent natural selection" } {TEXT -1 11 " , Ecology " }{TEXT 258 2 "52" }{TEXT -1 153 ": 453-468, \+ in which Roughgarden merges population genetics with population ecolog y. He describes a 1-locus 2-allele system in which fitness is defined \+ as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 " \+ W= 1+ \{individual's contribution to population growth\}. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 495 "Assumi ng the population is growing logistically, Roughgarden defines the ind ividual's contribution to population growth as r*(1 - N/K), where th e growth rate r and the carrying capacity K depend on the individual's genotype, and N is the total population. Each genotype (AA, Aa, aa) \+ has a distinct fitness, dependent upon specific r and K:\n\n \+ W[AA] = 1 + r[AA] - (r[AA] / K [AA])*N\n W[Aa] = 1 + r[Aa] \+ - (r[Aa] / K[Aa]) *N\n W[aa] = 1 + r[aa] - (r[aa] / K[aa]) \+ *N\n" }}{PARA 0 "" 0 "" {TEXT -1 65 "Each group is represented in Hard y-Weinberg proportions. If the " }{TEXT 259 1 "A" }{TEXT -1 43 " alle le is present in frequency p, and the " }{TEXT 260 1 "a" }{TEXT -1 156 " allele is present at frequency q = 1 - p, then genotype frequenc ies are:\n\n AA p^2\n Aa 2pq\n \+ aa q^2" }}{PARA 0 "" 0 "" {TEXT -1 157 "\nThe avera ge fitness of the population is:\n\n Wbar = p^2 *W[AA] + 2*p*q* W[ Aa] + q^2* W[aa] \n = 1 + p^2* r[AA] + 2*p*q* r[AA] + q^ 2 *r[aa] " }}{PARA 0 "" 0 "" {TEXT -1 224 " - (p^ 2* (r[AA] / K[AA]) + 2*p*q* (r[AA] / K[Aa]) + q^2 *(r[aa] / K[aa])) * \+ N\n\nwhich implies that the average growth rate is rbar = p^2* r[AA] \+ + 2*p*q* r[AA] + q^2 *r[aa] when the population N is small." }}{PARA 0 "" 0 "" {TEXT -1 179 "\nTo calculate the changes in population size \+ and allele frequency over time, we use two difference equations. It i s tempting to use expressions like W[AA] * N[AA] to predict the " } {TEXT 269 2 "AA" }{TEXT -1 33 " population at time t+1 from the " } {TEXT 270 2 "AA" }{TEXT -1 671 " population at time t. But these indi viduals do not clone themselves. Rather the growth they contribute to the population is due to gametes they produce, which are then randoml y paired with the gametes produced by all the subgroups to produce the next generation. Also we don't need to make up special notation for \+ the subpopulations, because we are assuming that they are in Hardy-Wei nberg proportions. In othe words, N[AA] = p^2 * N, N[Aa] = 2*p*q * N, \+ and N[aa] = q^2 * N. To produce W[AA] * N[AA] offspring there must ha ve been twice as many gametes, and the same for the other groups. The total number of gametes (or equivalently, alleles) in the gamete pool is" }}{PARA 0 "" 0 "" {TEXT -1 71 "\n 2* W[AA] *p^2 * N + 2* W[A a] * 2*p*q * N + 2 *W[aa] * q^2 * N " }}{PARA 0 "" 0 "" {TEXT -1 79 " = (W[AA] *p^2 + W[Aa] *2*p*q + W[aa]* q ^2) * 2*N " }}{PARA 0 "" 0 "" {TEXT -1 44 " \+ = Wbar * 2* N.\n" }}{PARA 0 "" 0 "" {TEXT -1 180 "Note that every ga mete produced by an AA individual is A, but only half the gametes prod uced by Aa individuals are A, and of course none come from aa individu als. The frequency of " }{TEXT 261 1 "A" }{TEXT -1 160 " alleles in th e population at time t+1 is related to the frequency at time t. If t he current population N is in Hardy-Weinberg proportion, then the numb er of " }{TEXT 262 1 "A" }{TEXT -1 71 " alleles in the gamete pool ava ilable to form the next generation is " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 " 2 *W[AA] * p^2* N + 0.5 * (2 \+ *W[Aa] *2*p*q * N) " }}{PARA 0 "" 0 "" {TEXT -1 71 " \+ = (W[AA] * p^2 + W[Aa] *p*q) * 2 * N . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Therefore the frequenc y of " }{TEXT 263 1 "A" }{TEXT -1 40 " alleles in the gamete pool, nu mber of " }{TEXT 264 1 "A" }{TEXT -1 29 " alleles / total alleles, is \n" }}{PARA 0 "" 0 "" {TEXT -1 67 "(1) p[t+1] = (W[AA] *p[t] ^2 + W[Aa]* p[t]*q[t]) / Wbar." }}{PARA 0 "" 0 "" {TEXT -1 90 "\nUsi ng the average fitness (growth rate), we get that the population size \+ at time t+1 is " }}{PARA 0 "" 0 "" {TEXT -1 33 "\n(2) N[t+1] = Wbar * N[t]" }}{PARA 0 "" 0 "" {TEXT -1 100 "\nWe iterate the two di fference equations to see the changes in population size and allele fr equency." }}{PARA 0 "" 0 "" {TEXT -1 506 "\nIf we consider that both h igh growth rate and high carrying capacity are unlikely to co-occur wi thin one genotype, we can hypothesize that the AA genotype has high r \+ and low K and the aa genotype has low r and high K. Let us assume in \+ the first instance that the heterozygote has the same expression as th e homozygote AA.\n\n r[AA] = r[AA] = 0.8 K[A A] = 8000\n r[Aa] = r[AA] = 0.8 K[Aa] = 8000 \n r[aa] = r[aa] = 0.6 K[aa] = 12000" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "res tart: with (plots): W:='W';" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 302 "\n(The curious business about W is that Maple reserves this letter fo r the Lambert W function. We can insist, however, that W be just the v ariable W ; this is also useful for cases in a long program where you want to re-use a letter (like x or y) that has already taken on a spe cific numerical value.)\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "r[AA]: = 0.8; r[Aa]:= 0.8; r[aa]:=0.6;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "K[AA]:=8000; K[Aa]:=8000; K[aa]:=12000;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 550 "\nIt is convenient for future use to make a procedure of the plotting routine . The number defined above (rates and carrying capacities) are treate d as global variables; they retain their values throughout the procedu re (better programming would pass them in as parameters, but I'm lazy) . All the other variables in the procedure are local, and are forgott en once the procedure has been executed, with the exception of the las t assignment of a value to plotlist--this is the result that is return ed by the procedure. It happens to be a list of plots." }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 "makeplot:= proc()" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "local N, wbar, pp, p, P, i, Wbar, W;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 220 "N[0]:=100; # Note the initial population is s mall in comparison to the carrying capacity.\nplotlist:= NULL:\nwbar:= 1+ p^2*r[AA] + 2*p*q*r[Aa] + q^2*r[aa] - (p^2*(r[AA]/K[AA]) + 2*p*q*(r [Aa]/K[Aa]) + q^2*(r[aa]/K[aa]))*N;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "for pp from 0 to 1 by 0.1 do # compute population vs allele A f requency for different initial p\nP[0]:= pp;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 350 " for i from 0 to 50 do # loop to compute traject ory given initial A frequency pp\n Wbar:= subs( \{ p = P[i], q = 1 \+ - P[i], N = N[i] \}, wbar):\n W[AA]:=1 + r[AA] - (r[AA]/K[AA])*N[i]: \n W[Aa]:=1 + r[Aa] - (r[Aa]/K[Aa])*N[i]:\n P[i+1]:= (W[AA] * P[i] + W[Aa] * (1-P[i])) * P[i] / Wbar:\n N[i+1]:=Wbar * N[i]:\n od: # loop terminator for i" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 252 "plot list:=plotlist, plot([[P[t], N[t]] $t = 0 .. 50], style=line): # plot \+ trajectory for init freq pp and add to the list\nod: # loop terminat or for pp\nplotlist:=[ plotlist ]; # make it a real list, and return this as the value of the procedure\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "\nWe overlay the plots of population vs " }{TEXT 265 1 "A " }{TEXT -1 141 " allele frequency for the various initial values of p . The vertical axis is population size and the horizontal axis is the frequency of the " }{TEXT 266 1 "A" }{TEXT -1 50 " allele in the popu lation (ranging from 0 to 1.0)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 " display(makeplot(), title=`Stable environment: homozygote aa highest K `);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 177 "\nWhat has happened in this case? Is there fixat ion of one allele? If so, which one? Does it have the higher r or the \+ higher K? Does the outcome depend upon initial conditions?" }}{PARA 0 "" 0 "" {TEXT -1 108 "\nConsider the case where the heterozygote has the highest carrying capacity and an intermediate growth rate." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "r[AA] := 0.8; r[Aa]:= 0.7; r[aa] :=0.6;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "K[AA]:=8000; K[Aa]:=1500 0; K[aa]:=12000;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "displa y( makeplot(), title=`Stable environment: heterozygote highest K`);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 219 "\nWhat has happened in this case where the heterozygot e has the highest K? Is there fixation of one allele? If so, which o ne? Does it have the higher r or the higher K? Does the outcome depen d upon initial conditions?" }}{PARA 0 "" 0 "" {TEXT -1 73 "\nNow let u s consider the case in which the heterozygote has the lowest K:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "r[AA] := 0.8; r[Aa]:= 0.7; r[aa]: =0.6;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "K[AA]:=8000; K[Aa]:=5000; \+ K[aa]:=12000;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "display( m akeplot(), title=`Stable environment: heterozygote lowest K`);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 222 "\nWhat has happened in this case of the heterozygote ha ving the lowest K? Is there fixation of one allele? If so, which one ? Does it have the higher r or the higher K? Does the outcome depend upon initial conditions? " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 375 " \nWe will now consider a seasonal environment. The population grows f or 2 time steps and then is killed back by seasonal factors to a popul ation size of N_min. In harsh environments, N_min is small (very few individuals survive), and in benign environments N_min is large (many individuals survive). Roughgarden derives an analytical form of the \+ equilibrium proportion of " }{TEXT 267 1 "A" }{TEXT -1 186 " alleles a s a function of environmental harshness measured in terms of N_min. W e will first simulate such an environment and plot the relation betwee n p_hat (equilibrium proportion of " }{TEXT 268 2 "A " }{TEXT -1 232 " alleles) to N_min.\n\nWe first examine the case in which the high r- low K condition is dominant. We run each simulation for 200 time step s to let it get close to equilibrium. We will vary N_min from 1500 on up in increments of 400." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "restar t: with(plots): W:= 'W':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "r[AA] := 0.8; r[Aa]:= 0.8; r[aa]:=0.6;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "K[AA]:=8000; K[Aa]:=8000; K[aa]:=12000;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "p_final:=NULL: plotlist:= N ULL:\nSample:= [ 1500 + k * 350 $ k = 0 .. 12]; # sample values for N _min\nnumberofsteps:= 200;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "for N_min in Sample do # environmental harshness loop\nP[0 ]:=0.5; N[0]:= N_min; # initial p = 0.5" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 " for i from 0 to numberofsteps do \n \+ p:=P[i]; q:=1.0 - p;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 560 " Wbar := 1 + p^2 * r[AA] + 2*p*q*r[Aa] + q^2*r[aa] - (p^2*(r[AA]/K[AA]) + 2* p*q*(r[Aa]/K[Aa]) + q^2*(r[aa]/K[aa]))*N[i]; \n W[AA]:=1 + r[AA ] - (r[AA]/K[AA])*N[i]; # fitness of homozygote\n W[Aa]:=1 + r [Aa] - (r[Aa]/K[Aa])*N[i]; # fitness of heterozygote\n P[i+1]: = (p*W[AA] + q*W[Aa])*p / Wbar; # A allele freq at t+1\n N[i+ 1]:= Wbar * N[i]; # pop size at t+1\n if i mo d 2 = 0 then N[i+1]:= N_min; fi; # set pop to N_min every 2 timest eps\n od: # end time loop" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 356 "plotlist:= plotlist, plot([ [P[j ],N[j] ] $j = 0 .. numberofsteps], style=line, title=cat(`Environmenta l index: N_min = `, convert(N_min, string))):\np_last:= p: \+ # record final stable proportion of A allele\np_final:=p_final , [N_min, p_last]: # add equilibrium data to plot \nod: \+ # end env harshness loop" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 "display([plotlist], insequence = true );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "plot([p_final], style=line, \+ title=`Freq of high \"r\" alleles vs N_min`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 295 "\nWhat \+ is the relation between environmental harshness and the equilibrium pr oportion of high \"r\" alleles in the population? On the horizontal \+ axis of the graph, in which direction does environmental harshness INC REASE? What happens if the heterozygote has an intermediate growth ra te instead?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 297 "Now let us consider the case where random catastrophes keep reducing the size of the population to a low value, say 0.1 of i ts level before the catastrophe. Let us imagine that there is a catas trophe in 15% of the years. We will use the same values for r and K t hat we used in our first example." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "restart; with (plots): W:= 'W': " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "_seed:= 3: # Suggestion: change the _seed value each time." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "r[AA] \+ := 0.8; r[Aa]:= 0.8; r[aa]:=0.6;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "K[AA]:=8000; K[Aa]:=8000; K[aa]:=12000;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "N[0]:=100; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "threshold:=0.85; # frequency of catastrophe avoi dance" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "numyears:= 100; \+ # number of years to run simulation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "frames:=NULL: plotlist:= NULL: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "for pp from 0 to 1 by 0.05 do # \+ repeat model for increasing initial p values\nP[0]:= pp; \+ # initial frequency of A allele" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 333 " for i from 0 to numyears do\n p:=P[i]; q:=1.0 - p;\n Wbar:=1+ p^2*r[AA] + 2*p*q*r[Aa] + q^2*r[aa] - (p^2*(r[AA]/K[AA ]) + 2*p*q*(r[Aa]/K[Aa]) + q^2*(r[aa]/K[aa]))*N[i];\n W[ AA]:= 1 + r[AA] - (r[AA]/K[AA])*N[i];\n W[Aa]:= 1 + r[Aa] - (r[Aa]/K[ Aa])*N[i];\n P[i+1]:= (p*W[AA] + q*W[Aa])*p / Wbar;\n N[i+1]:= Wbar \+ * N[i];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 187 " ping:=rand() / ( 10^ 12); # generate random number between 0 and 1\n if ( ping > t hreshold) then N[i+1]:=0.01*N[i]*Wbar; fi; # if catastrophe then kill \+ back to 10% of population" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 " od: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "frames:= frames, plot([ [P[j], N[j] ] $j = 0 .. numyears], style=line, axes=FRAMED):\nplotlist:= plo tlist, [pp, P[numyears]]:\nod: # terminat e loop on p" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "display([fr ames], insequence=true, title=`Catastrophic environment: N vs p`); # O h boy, is this neat!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "plo t( [plotlist], color=green, style=point, axes=framed, title=`final p v s. initial p`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 636 "\nWhat has happened in this case? Is there a polymorphism or has the system moved to fixation of one a llele? If there is fixation, which allele dominates? Does it have the higher r or the higher K? What effect does changing the threshold to 0.95 have? (This is a very interesting question!)\n\nWhat differences do you see between the cases where the environment is stable compared to those where the environment is subject to catastrophes? Is there \+ a difference in the outcome of selection? Which alleles tend to domin ate in each case? What are the r and K values of the winners?\n\nWhat is \"r-selection\" and what is \"K-selection\" ??? " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 897 "This is a dif ference equation model based on the logistic, so it should have simila r stability properties to those described by Robert May (see either ch aos.mws or the Release 3 worksheets bifurc.ms and chaos.ms). Have a l ook at the stability properties of this model under condtions like th ose examined by May. Remember that the logistic equation as used by R oughgarden is the same as Equation 2 in May, so the transitions from s tability to damped oscillations to stable cycles to chaos are in the E quation 2 column of Table 1 in May. Does this system show the same pa ttern of fixation of alleles when r is in the range where population d ynamics are chaotic? How about when r is in the range where dynamics \+ include multipoint cycles? Are there unrealistic or impossible condit ions that occur? How might you change Roughgarden's model to incorpor ate May's equation 1 for density dependence?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }