{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Helvetica" 1 10 0 0 0 1 1 2 2 0 0 0 0 0 0 }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 10 128 0 128 1 2 1 2 0 0 0 0 0 0 }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 10 0 128 128 1 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "logist.mws" }}{PARA 0 "" 0 "" {TEXT -1 53 "Single species models (companion to May 1981, pp 6-7)" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "2.2.1 DENSITY INDE PENDENT GROWTH" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "The simplest model as written by May for density independ ent growth is" }}{PARA 0 "" 0 "" {TEXT -1 16 " dN/dt = rN" }} {PARA 0 "" 0 "" {TEXT -1 61 "In Maple, we write the derivative dN/dt \+ as \"diff(N(t),t)\" " }}{PARA 0 "" 0 "" {TEXT -1 83 " which means \"the derivative of N (which is a function of t), with respect to t\" " }}{PARA 0 "" 0 "" {TEXT -1 88 " we then equate this derivative to r*N because May was implying r*N when he wrote rN:" }}{PARA 0 "" 0 " " {TEXT -1 27 " diff(N(t),t)=r*N;" }}{PARA 0 "" 0 "" {TEXT -1 98 " Since we want to use this equation, we give it a name \"eq2.1 \" which is its name in May's article." }}{PARA 0 "" 0 "" {TEXT -1 55 " In Maple, to name an expression we use colon-equals so" }}{PARA 0 " " 0 "" {TEXT -1 36 " eq2.1 := diff(N(t),t) = r*N;" }}{PARA 0 " " 0 "" {TEXT -1 79 " means that we can use the name eq2.1 inter changeably with the equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eq2.1 := diff(N(t),t)=r*N(t);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 306 "Once we have the differential equ ation defined, we can solve it. The dsolve command allows you to solv e a differential equation. You have to define both the equation and w hat variable you want in the solution. Since we want to know what the population size is at time t, we ask for N(t) as the solution:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "dsolve(eq2.1,N(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 246 "Note that the solution has an arbitrary \+ constant \"_C1\" which appears in the solution. That constant is depe ndent upon the initial population size. We can see this by solving ag ain, but explicitly saying that the initial population size is No. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "The syn tax of dsolve requires a \"set\" of equations and initial conditions e nclosed in braces \{ \}." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "dsolve( \{eq2.1,N(0)=No\},N(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "Note \+ that this solution looks just like May's equation 2.2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 145 "We will now use Map le's plotting tools to plot graphs of this equation. First we must ca ll up the differential equations tool package (DEtools):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with (DEtools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "We use the program DEplot to graph the solutions of diffe rential equations. " }}{PARA 0 "" 0 "" {TEXT -1 52 "The syntax of DEp lot is similar to that of dsolve. " }}{PARA 0 "" 0 "" {TEXT -1 65 " \+ It requires a LIST of equations, enclosed in square brackets" }} {PARA 0 "" 0 "" {TEXT -1 68 " a LIST of variables, \+ enclosed in square brackets" }}{PARA 0 "" 0 "" {TEXT -1 83 " \+ a definition of the independent variable and its range (doma in)" }}{PARA 0 "" 0 "" {TEXT -1 66 " a SET of initi al conditions enclosed in braces" }}{PARA 0 "" 0 "" {TEXT -1 85 " \+ , each one being a LIST in the same order as the variables" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "equations:=subs(r=0.5,eq2.1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Here we have substituted the value 0.5 for \"r\" int o the equation" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "vars:=[N(t)];" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Here we have defined the variable s to be t (the independent variable), " }}{PARA 0 "" 0 "" {TEXT -1 83 " and N (the depend ent variable)." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "domain:=0..10;" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Here we have asked for 10 time u nits on the plot." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "init:=[N(0)=10 ];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "Here we have defined the i nitial conditions to be t=0 and N= 10, meaning that the population siz e at time zero was 10 individuals" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "DEplot([equations],vars,t=domain,\{init\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "What happened to th e population size? May claims that there is unbounded exponential gro wth when r>0, and exponential decrease if r<0. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Move your cur sor back to the line " }}{PARA 0 "" 0 "" {TEXT -1 75 " equations:=su bs(r=0.5,eq2.1); and change r to make it less than zero. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "Then step through the worksheet and rerun the DEplo t. What happened this time? Do you believe May's comment?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 337 " May also claims that the characteristic time scale of the growth proce ss is proportional to 1/r. Try several values of r and see how long i t takes to get to a population size of 1000. You may need to fiddle w ith the domain in order to get it to plot properly. Do you believe Ma y's claim about the time scale? What is your evidence? " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 177 "Use the cut and p aste functions to paste your graphs into your worksheet as you run the m, so you can save your results. Be sure to put labels in the workshe et above each graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "2.2.2 D ENSITY DEPENDENT GROWTH" }}{PARA 0 "" 0 "" {TEXT -1 152 "Since unbound ed growth is unlikely in nature, density dependent models have been d eveloped. The LOGISTIC equation is probably the simplest such model: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "May w rites equation 2.3 as dN/dt = rN(1-N/K)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 189 "You have to be a bit careful transl ating this into Maple, because there are implied multiplications in th is equation. May really means dN/dt = r * N * (1 - N/K), so that is \+ what we write:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "eq2.3:=diff(N(t),t) = r * N(t) * (1 - N(t)/ K);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 138 "Now that we have the equa tion, we can try to solve it as we did for the exponential growth equa tion. As before, we use the dsolve command" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "dsolve(eq2.3,N(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 196 "Note that this solution is much more complicated than the expo nential solution, but Maple solves it symbolically. To find out what \+ the constant C1 is, we tell Maple the initial population was No:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "dsolve(\{eq2.3,N(0)=No\},N(t));" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Now we will try for a graphical s olution, so we define the information we need for DEplot:" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 34 "equations:=subs(r=1,K=1000,eq2.3);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "vars:=[N(t)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "init1:=[0,10];init2:=[0,150];init3: =[0,1200];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Note that we have d efined 3 different initial population sizes (10,150, 1200)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "domain:=0..30;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 54 "DEplot([equations],vars,t=domain,\{init1,init2,init 3\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "What happened? " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 252 "May clai ms that the growth rate is positive if N< K, negative if N>K, so there is a globally stable equlibrium value at N* = K. He says that K can \+ be thought of as the carrying capacity of the environment, as determin ed by food, space, enemies, etc. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Do you buy this argument? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 327 "In o rder to see what the tendency of the population is over the whole rang e of times and population sizes, DEplot gives us a \"direction field \" which plots arrows corresponding to the directions the population w ill move from any arbitrary point on the graph. We will use this tech nique extensively in our analysis of models. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "What did you see on the plot?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 192 "May claims that there is a \"globally stable\" equilibrium at K, meaning that no matter where \+ you start, you will always end up at K. Based on the directions of th e arrows, do you believe this?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "May says that the \"characteristi c return time\" T of this model is related to 1/r. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "Go back to the \"equat ions:=subs(r=1,eq2.3);\" line and change r. Rerun the DEplots." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Paste you r new plots into the worksheet here, with labels, so you know which is which. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "What does May mean by return time? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Do you think that the return time \+ is indeed related to 1/r?" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{MARK "28" 0 }{VIEWOPTS 1 1 0 1 1 1803 }