{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 42 "Unit 27 -- Applica tion: Competition 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 Mathematics I nstitute" 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 27" } }{EXCHG {PARA 14 "" 0 "" {HYPERLNK 17 "27.A" 1 "" "27.A" }{TEXT -1 52 " Competition Model for Exponentially Growing Species" }}{PARA 14 "" 0 "" {HYPERLNK 17 "27.B" 1 "" "27.B" }{TEXT -1 51 " Competition Model \+ for Logistically Growing Species" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "Initializat ion" }}{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 "27.A" {TEXT -1 56 "27.A Competition Model for Exponentially Growi ng Species" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 203 "The classical compe tition model describes an ecosystem with two species that would grow e xponentially in isolation from the other species. But, when the two sp ecies interact, both populations suffer. If " }{XPPEDIT 18 0 "X(t)" "6 #-%\"XG6#%\"tG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Y(t)" "6#-%\"YG6#% \"tG" }{TEXT -1 44 " denote the size of the two species at time " } {XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 31 ", then the model takes the f orm" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "U := [ X(t), Y(t) ]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "f := (X,Y) -> alpha * X - beta * X*Y:" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 40 "g := (X,Y) -> delta * Y - epsilon * X*Y:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "sys1 := op(equate( diff( U, t ), [ \+ f( op(U) ), g( op(U) ) ] ));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ic1 := X(0) = X[0], Y(0) = Y[0];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 164 "Note that ex cept for a change of sign in both terms on the right-hand side of the \+ second ODE, this model has the same form as the Lotka-Volterra model d iscussed in " }{HYPERLNK 17 "Unit 26" 1 "unit26.mws" "" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "The e quilibrium solutions to this model are" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "equil := solve( \{f(x0, y0)=0,g(x0,y0)=0\}, \{x0,y0\} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Linearization about each equilibrium solution proceeds as usual:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for X0 in [equil] do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " b := eval( [ f(x0, y0), g(x0,y0) ], X0 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 " A := ev al( evalm(jacobian( [f(x0,y0),g(x0,y0)], [x0,y0] )), 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 [x0,y0]`=eval([x 0,y0],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 -1 179 "Thus, whil e the equilibrium in the first quadrant has the same classification as for the Lotka-Volterra model, the trivial equilibrium at the origin i s an unstable node (source). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 105 "To perform a graphical investigation of \+ the competition model, select numerical values for the parameters" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "param1 := \{ alpha=0.45, beta=0.09, delta=.16, epsilon=0.008, X[ 0]=4, Y[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, t ype=numeric ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "A plot of the two components of th is system is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "odeplot( sol1, [ [t,X(t)], [t,Y(t)] ], 0..20," } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " legend=[\"Species X\", \" Species Y\"] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "These solutions are, without a dou bt, not periodic. The corresponding plot in the phase plane is" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "odeplot( sol1, [ X(t), Y(t) ], 0..20, labels=[X,Y] );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 137 "This one example is insufficient to make any general con clusions. The direction field shows that solutions generally tend to s tates with " }{XPPEDIT 18 0 "X=0" "6#/%\"XG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Y" "6#%\"YG" }{TEXT -1 26 " increasing without bound. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "DEplot( \{eval( sys1, param1 )\}, U, t=0..20," }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 " [ [eval(ic1, param1 )] ], s tepsize=0.2 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "For a second example, consider the model in which the parameter " }{XPPEDIT 18 0 "delta" "6#%&deltaG" } {TEXT -1 29 " is reduced by a factor of 2:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "param2 := \{ alph a=0.45, beta=0.09, delta=.08," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 " \+ epsilon=0.008, X[0]=4, Y[0]=3 \};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "sol2 := dsolve( eval( \{ sys1, ic1 \}, param2 ), U , type=numeric ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "odeplo t( sol2, [ [t,X(t)], [t,Y(t)] ], 0..20," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " legend=[\"Species X\", \"Species Y\"] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "odeplot( sol2, [ X(t), Y(t) ], 0..2 0, labels=[X,Y] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "DEplo t( \{eval( sys1, param2 )\}, U, t=0..20," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 49 " [ [eval(ic1, param2 )] ], stepsize=0.2 );" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "In this example species " }{XPPEDIT 18 0 "Y" "6#%\"YG" } {TEXT -1 14 " dies out and " }{XPPEDIT 18 0 "X" "6#%\"XG" }{TEXT -1 28 " increases without bound as " }{XPPEDIT 18 0 "t" "6#%\"tG" }{TEXT -1 11 " increases." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 213 "Competitive exclusion is the special term used to descri be the general property that all but one species in a competition mode l (with exponential growth) tend to zero and the remaining species gro ws without bound." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "27.B" {TEXT -1 55 "27.B Competition Model for Logistically Growing Species" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 274 "The unbounded growth of the surviving sp ecies in the competition model with exponential growth raises the same concerns as the Malthusian model for a single population. One way to \+ attempt to address this issue is to use logistic models for the compet ition-free growth terms." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "f := (X,Y) -> alpha * ( 1 - X / M ) * X - beta * X*Y:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "g := (X,Y) -> delta * ( 1 - Y / N ) * Y - epsilon * X*Y:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "sys2 := op(equate( diff( U, t ), [ f( op(U) ), g( op( U) ) ] ));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ic2 := X(0) = X[0], Y(0) = Y[0];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {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(x0,y0)=0,g(x0,y0)=0\}, \{x0,y0 \} );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "There are now four equilibria to consider. Lin earization about each equilibrium solution proceeds as usual:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for X0 in [equil] do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 " b : = eval( [ f(x0,y0), g(x0,y0) ], X0 );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 " A := simplify( eval( evalm(jacobian( [f(x0,y0),g(x0,y0)], [x0, y0] )), X0 ) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 " linsys1 := equ ate( 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 [x0,y0]`=eval([x0,y0],X0) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " print( A = evalm( A ) );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 " print( lambda = [eval(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 -1 167 "The equilibrium at the origin is still a n unstable node (source). The two new equilibria consist of a finite p opulation of only one species. The additional assumption " }{XPPEDIT 18 0 "N" "6#%\"NG" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "alpha/beta" "6#*& %&alphaG\"\"\"%%betaG!\"\"" }{TEXT -1 35 " ensures that the equilibriu m ( 0, " }{XPPEDIT 18 0 "N" "6#%\"NG" }{TEXT -1 40 " ) is a stable nod e (sink); likewise, ( " }{XPPEDIT 18 0 "M" "6#%\"MG" }{TEXT -1 35 ", 0 ) is a stable node (sink) when " }{XPPEDIT 18 0 "M" "6#%\"MG" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "delta/epsilon" "6#*&%&deltaG\"\"\"%(epsilon G!\"\"" }{TEXT -1 36 ". These two assumptions ensure that " }{XPPEDIT 18 0 "alpha*delta - beta*epsilon*M*N" "6#,&*&%&alphaG\"\"\"%&deltaGF&F &**%%betaGF&%(epsilonGF&%\"MGF&%\"NGF&!\"\"" }{TEXT -1 73 " < 0 and he nce that the fourth equilibrium point is physically possibly (" } {TEXT 256 4 "i.e." }{TEXT -1 258 ", in the first quadrant). The eigenv alues for this case are rather complicated. Determination of the sign \+ of the eigenvalues is more easily accomplished by looking at the trace and determinant of the coefficient matrix for the linearization about this point:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "A = evalm( A );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "trA := simplify( trace( A ) );" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 17 "detA := det( A );" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 61 "assume(alpha-beta*N<0, delta-epsilon*M<0, alpha>0, \+ delta>0 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "is( trA < 0 ) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "is( detA < 0 );" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "unassign( 'alpha', 'beta', ' M', 'delta', 'epsilon', 'N' ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 170 "Because the determina nt is negative, the eigenvalues must have different signs and the \"co existence\" equilibrium is a saddle point for the linear, and nonlinea r, systems." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "To perform a graphical investigation of the competition model \+ with logistic growth, select numerical values for the parameters" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "param2 := \{ alpha=2, beta=0.5, M=5," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " delta=1, epsilon=0.5," }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 35 " N=10, X[0]=5, Y[0]=6 \};" }}}{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 "sol2 := dso lve( eval( \{ sys2, ic2 \}, param2 ), 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 45 "odeplot( sol2, [ [t,X(t)], [t,Y(t)] ], 0..20," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 " legend=[\"Species X\", \"Species Y\"] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "This solution exhibits competitive exclusion with the addition al property that the surviving species remains bounded." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 245 "The direction field shows the general behavior of solutions. The specific initial conditi ons have been chosen to illustrate how minor changes in the initial co nditions can lead to significant differences in the qualitative behavi or of solutions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "ic3 := [ [ X(0)=0.1, Y(0)=0.7 ], [ X(0)=0.1, \+ Y(0)=0.4 ]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 " [ X(0)=5, Y (0)=6.3 ], [ X(0)=5, Y(0)=6.6 ] ]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "DEplot( \{eval( sys2, param2 )\}, U, t=0..20, X=0..6, Y=0..12," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " ic3, stepsize= 0.1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 233 "Note that there is a curve along which the sol ution approaches the coexistence equilibrium. In practice, however, it is not possible to locate a point on this curve or to expect a numeri cal solver to be able to track these solutions." }}{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 3 0" 7 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }