{VERSION 5 0 "IBM INTEL NT" "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 "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 } {PSTYLE "List Item" 0 14 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 14 5 }{PSTYLE "" 0 256 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "List Subitem" 14 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 3 12 0 0 0 0 270 0 }} {SECT 0 {EXCHG {PARA 268 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 41 "ORDINARY DIFFERENTIAL EQUATIONS POWERTOOL" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {TEXT -1 44 "Unit 26 -- Applica tion: Predator-Prey Models" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {URLLINK 17 "Prof. Douglas B. Meade" 4 "http://www.math.sc .edu/~meade/" "" }}{PARA 257 "" 0 "" {URLLINK 17 "Industrial Mathemati cs Institute" 4 "http://www.math.sc.edu/~IMI/" "" }}{PARA 258 "" 0 "" {URLLINK 17 "Department of Mathematics" 4 "http://www.math.sc.edu/" " " }}{PARA 259 "" 0 "" {URLLINK 17 "University of South Carolina" 4 "ht tp://www.sc.edu/" "" }}{PARA 260 "" 0 "" {TEXT -1 19 "Columbia, SC 292 08\n" }}{PARA 262 "" 0 "" {TEXT -1 7 "URL: " }{URLLINK 17 "http://ww w.math.sc.edu/~meade/" 4 "http://www.math.sc.edu/~meade/" "" }}{PARA 263 "" 0 "" {TEXT -1 25 "E-mail: meade@math.sc.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 38 "Copyright \251 2001 b y Douglas B. Meade" }}{PARA 265 "" 0 "" {TEXT -1 19 "All rights reserv ed" }}{PARA 267 "" 0 "" {TEXT -1 0 "" }}{PARA 266 "" 0 "" {TEXT -1 67 "-------------------------------------------------------------------" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "Outline of Unit 26" } }{EXCHG {PARA 14 "" 0 "" {HYPERLNK 17 "26.A" 1 "" "26.A" }{TEXT -1 25 " The Lotka-Volterra Model" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "Initialization" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "with( DEtools ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( plots ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with( linalg \+ ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with( student ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "26.A" {TEXT -1 29 "26.A The Lotka-Volterra Model" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "R(t)" "6#-%\"RG6#%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "F(t)" "6#-%\"FG6#%\"tG" }{TEXT -1 48 " denote \+ the number of rabbits and foxes at time " }{XPPEDIT 18 0 "t" "6#%\"tG " }{TEXT -1 363 " in a closed ecosystem. The rabbits have an unlimited food supply but the foxes sole source of food is the rabbits. Assume \+ that in the absence of the foxes, the rabbits would grow exponentially and in the absence of rabbits, the fox population would decay exponen tially to zero. The fox population increases as a result of interactio ns between rabbits and foxes (" }{TEXT 256 4 "i.e." }{TEXT -1 230 ", w hen a fox eats a rabbit). Assume that the number of interactions betwe en rabbits and foxes occur at a rate proportional to the product of th e number of rabbits and foxes. These assumptions lead to the system of first-order ODEs" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 41 "f := (R,F) -> alpha * R - beta * F*R:" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "g := (R,F) -> -delta * F + epsilon * F*R:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "sys1 := diff( R(t), t ) \+ = f( R(t), F(t) )," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " diff( F(t), t ) = g( R(t), F(t) );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "with initial conditions " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ic1 := R(0) = R[0], F(0) = F[0];" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "wher e all four parameters (" }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "c" "6# %\"cG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "d" "6#%\"dG" }{TEXT -1 46 ") and the initial conditions are all positive." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "The equilibrium solutions to this model are" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "equil := solve( \{f(r0,f0)=0,g(r0,f0)=0\}, \+ \{r0,f0\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "The qualitative analysis of the equilibri um solutions can be performed as in " }{HYPERLNK 17 "Unit 23" 1 "unit2 3.mws" "" }{TEXT -1 169 ". Since no numeric values are assigned to the parameters, minor modifications are required to eliminate the plots a nd to improve the quality of the output. The result is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "U := [ R(t), F(t) ]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for X0 in [equil] do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " b := eval( [ f(r0, f0), g(r0,f0) ], X0 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 " A := ev al( evalm(jacobian( [f(r0,f0),g(r0,f0)], [r0,f0] )), X0 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " linsys1 := equate( diff( U, t ), evalm( A &* U + b ) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 " print( `------- ------------------------------------------------------` );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 " print( `Linearization at [r0,f0]`=eval([r 0,f0],X0) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " print( A = evalm( A ) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 " print( lambda = [evalf (eigenvals( A ))] );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 " print( `- ------------------------------------------------------------` );" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "end do:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 227 "Thus, the \+ trivial equilibrium at the origin is a saddle point. At the equilibriu m solution in the first quadrant, the linearized system is a center so it is not possible to classify this critical point for the nonlinear \+ system." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "To perform a graphical investigation of the predator-prey system, select numerical values for the parameters" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "param1 := \{ alph a=4.5, beta=0.9, delta=0.16, epsilon=0.08, R[0]=4, F[0]=3 \};" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "and obtain a numerical solution to the IVP" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "sol1 : = dsolve( eval( \{ sys1, ic1 \}, param1 ), U, type=numeric );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "A plot of the two components of this system is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "o deplot( sol1, [ [t,R(t)], [t,F(t)] ], 0..20, numpoints=250, legend=[\" Rabbits\", \"Foxes\"] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 167 "This suggests that solutions to the predator-prey system might be periodic. A second test of this \+ conjecture can be made with a plot of the solution in the phase plane " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "odeplot( sol1, [ R(t), F(t) ], 0..20, numpoints=250 ) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 161 "The direction field suggests that the periodic natu re of solutions is a general property of predator-prey systems that do es not depend on the initial conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "DEplot( \{eval( sys1 , param1 )\}, U, t=0..20, [ [eval(ic1, param1 )] ], stepsize=0.2 );" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 120 "Note that all solutions appear to be counter-clockwise closed orbits that enclose the nontrivial equilibrium solution ( " } {XPPEDIT 18 0 "R" "6#%\"RG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "F" "6#%\" FG" }{TEXT -1 14 " ) = ( 2, 5 )." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "eval( equil, param1 );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 217 "While this example is far from a proof, it is true that \+ all solutions to the predator-prey system are periodic. Most introduct ory texts in differential equations contain a detailed analysis of the predator-prey model." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 314 "There are many variations of the standard Lotka-Volte rra model for a predator-prey system. For example, the exponential gro wth/decay that occurs in the absence of the other species can be repla ced with a logistic or Gompertz model. The classic May model is anothe r example of a variation of a predator-prey system." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "[Back to " }{HYPERLNK 17 "ODE Power tool Table of Contents" 1 "unit00.mws" "" }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }