{VERSION 1 0 "X11/Motif" "1.0"}{GLOBALS 3 1}{FONT 0 "-adobe-couri er-bold-r-normal--14-*" "courier" "Courier-Bold" 8 14 192 "Courie r-Bold" 12}{FONT 1 "-adobe-courier-bold-o-normal--14-*" "courier" "Courier-BoldOblique" 8 14 224 "Courier-BoldOb" 12}{FONT 2 "-ado be-courier-medium-r-normal--14-*" "courier" "Courier" 4 14 192 "C ourier" 12}{SCP_R 1 0 61{COM_R 2 0{TEXT 1 1304 " COMPETI NG SPECIES\015 A NONLINEAR AUTONOMOUS SYSTEM\015\015W. E. Boyc e\015Rensselaer Polytechnic Institute\01504/15/94\015\015PURPOSE. To show how a two-dimensional nonlinear autonomous system can b e investigated systematically.\015\015EXAMPLE. Consider the syst em of differential equations\015\015 dx/dt = x(3 - x - y), dy /dt = y(2 - y - x/2).\015\015We can interpret x(t) and y(t) i n this system as the populations of two species that compete with each other in a closed ecological system. We are interested pri marily in the long-term behavior of x(t) and y(t). In particu lar, at what levels, if any, can they coexist in a stable relatio nship?\015\015(a) Find all critical points (equilibrium solution s).\015\015(b) Plot a direction field on a rectangle in the firs t quadrant that includes all of the critical points.\015\015(c) Plot several trajectories of the system.\015\015(d) Plot the tim e response of x versus t.\015\015(e) Find the linear system t hat approximates the given system near the critical point in the positive first quadrant.\015\015(f) Find the eigenvalues and eig envectors of the coefficient matrix of this linear system, and de termine its stability characteristics.\015\015\015SOLUTION. Firs t we bring up the package DEtools; the colon after the command su ppresses the screen output, i.e., the list of commands in the pac kage.\015"}}{INP_R 3 0 "> "{TEXT 0 14 "with(DEtools):"}}{SEP_R 4 0}{COM_R 5 0{TEXT 1 158 "\015There are several ways to enter the system of equations. We will enter the right side of each equati on, calling these expressions f and g, respectively.\015"}} {INP_R 6 0 "> "{TEXT 0 34 "f := x*(3-x-y); g := y*(2-y-x/2);"}} {OUT_R 7 0 6{DAG :3n3\`f`*5n3\`x`j2x0001+7j2x0003p6p4i2x0001n3\`y `pDp6}}{OUT_R 8 0 6{DAG :3n3\`g`*5n3\`y`j2x0001+7j2x0002p6p4i2x00 01n3\`x`/3pDp9p6}}{SEP_R 9 0}{COM_R 10 0{TEXT 1 94 "\015(a) To f ind the critical points, we must solve the equations f=0 and g =0 simultaneously.\015"}}{INP_R 11 0 "> "{TEXT 0 27 "crit := solv e(\{f,g\},\{x,y\});"}}{OUT_R 12 0 11{DAG :3n4\`crit`,5%2,3=3n3\`x `j2x0000=3n3\`y`pA%2,3p7=3pDj2x0002%2,3pC=3p8j2x0003%2,3=3pDj2x00 01=3p8p15}}{SEP_R 13 0}{COM_R 14 0{TEXT 1 539 "\015We observe tha t there is one critical point for which both x and y are posi tive. This critical point corresponds to coexistence. For the o ther critical points one or both species become extinct.\015\015( b) Next we plot a direction field on a rectangle that includes a ll of the critical points. We choose the square x=0..4, y=0..4. The following command produces the direction field. Note that t he third argument in the command gives a t interval, and this a rgument is required, even though the direction field is independe nt of t.\015"}}{INP_R 15 0 "> "{TEXT 0 70 "DEplot([f,g],[x,y],t= 0..1,x=0..4,y=0..4,arrows=THIN,title=`Figure 1`);"}}{SEP_R 16 0} {COM_R 17 0{TEXT 1 1 "\015"}}{GRH_R 18 0{RASTER 399 335 1 50 0000 00000000000000000000000000000000000000F06700000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000 00000000306000000000800100000000000000000000000000000000000000000 0000000000000000000000000000000000000000000300000000000E001000000 00000000000000000000000000000000000000000000000000000000000000000 00000000000003060B88CD93C8001000000000000000000000000000000000000 0000000000000000000000000000000000000000000000003060EC8CF96680010 00000000000000000000000000000000000000000000000000000000000000000 000000000000000000F063C68C196680010000000000000000000000000000000 000000000000000000000000000000000000000000000000000003060C68C197E 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The following command is one way to do this.\0 15"}}{INP_R 20 0 "> "{TEXT 0 99 "init1:= seq([0,0.5,.4*i],i=0..6) ; init2:=seq([0,3,0.4*i],i=0..6); init3:=seq([0,0.4*i,0.5],i=0..6 );"}}{OUT_R 21 0 20{DAG :3n4\`init1`,8[2,4j2x0000e3j2x0005i2x0001 p7[2,4p7p9e3j2x0004pC[2,4p7p9e3j2x0008pC[2,4p7p9e3j2x0012pC[2,4p7 p9e3j2x0016pC[2,4p7p9e3j2x0020pC[2,4p7p9e3j2x0024pC}}{OUT_R 22 0 20{DAG :3n4\`init2`,8[2,4j2x0000j2x0003p7[2,4p7p9e3j2x0004i2x0001 [2,4p7p9e3j2x0008p13[2,4p7p9e3j2x0012p13[2,4p7p9e3j2x0016p13[2,4p 7p9e3j2x0020p13[2,4p7p9e3j2x0024p13}}{OUT_R 23 0 20{DAG :3n4\`ini t3`,8[2,4j2x0000p7e3j2x0005i2x0001[2,4p7e3j2x0004pDpA[2,4p7e3j2x0 008pDpA[2,4p7e3j2x0012pDpA[2,4p7e3j2x0016pDpA[2,4p7e3j2x0020pDpA[ 2,4p7e3j2x0024pDpA}}{SEP_R 24 0}{COM_R 25 0{TEXT 1 136 "\015The f ollowing command superimposes the trajectories passing through th e given sets of points on the direction field shown in Figure 1.\ 015"}}{INP_R 26 0 "> "{TEXT 0 76 "DEplot([f,g],[x,y],t=0..5,\{ini 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N ote the presence of the scene option. The initial conditions wer e chosen arbitrarily.\015"}}{INP_R 31 0 "> "{TEXT 0 78 "DEplot([f ,g],[x,y],t=0..5,\{[0,1/2,1/2],[0,3,2]\},scene=[t,x],title=`Figur e 3`);"}}{COM_R 32 0{TEXT 1 1 "\015"}}{GRH_R 33 0{RASTER 398 360 1 50 000000000000000000000000000000000000000000F83300000000000000 00000000000000000000000000000000000000000000000000000000000000000 00000000000000000183000000000F00100000000000000000000000000000000 00000000000000000000000000000000000000000000000000001800000000009 80300000000000000000000000000000000000000000000000000000000000000 000000000000000000000018305CC66C1E1803000000000000000000000000000 000000000000000000000000000000000000000000000000000000000183076C6 7C330003000000000000000000000000000000000000000000000000000000000 000000000000000000000000000F83163C60C33E0010000000000000000000000 00000000000000000000000000000000000000000000000000000000000000183 063C60C3F00030000000000000000000000000000000000000000000000000000 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00000000000000000000000000000000000000001000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000 000000000090000000000000000000000000000000000000000000000000}} {COM_R 34 0{TEXT 1 463 "Both solutions appear to be approaching t he value two, consistent with the phase portrait.\015\015In a sim ilar way one can also plot y versus t.\015\015(e) To find the linear system valid near the critical point (2,1) it is helpful to bring up the linalg package. Since this package contains a gr eat many commands, we save space by ending the next command with a colon. If you wish to see a list of the commands in the linalg package, replace the colon by a semicolon.\015"}}{INP_R 35 0 "> "{TEXT 0 13 "with(linalg):"}}{SEP_R 36 0}{COM_R 37 0{TEXT 1 255 " \015To find the required linear system we need to find the matrix of partial derivatives of f and g with respect to x and y , and then evaluate the elements of this matrix at the critical p oint (2,1). The following three commands perform these steps.\01 5"}}{INP_R 38 0 "> "{TEXT 0 19 "A := vector([f,g]);"}}{OUT_R 39 0 38{DAG :3n3\`A`(3n4\`VECTOR`,2[2,3*5n3\`x`j2x0001+7j2x0003pDpBi2 x0001n3\`y`p14pD*5p16pD+7j2x0002pDp16p14pB/3p14p1EpD}}{SEP_R 40 0 }{COM_R 41 0{TEXT 1 137 "\015Note that there is only a space sepa rating the components of the vector A. Thus you need to be caref ul in reading output such as this.\015"}}{INP_R 42 0 "> "{TEXT 0 23 "B := jacobian(A,[x,y]);"}}{OUT_R 43 0 42{DAG :3n3\`B`(3n4\`MA TRIX`,2[2,3[2,3+7j2x0003j2x0001n3\`x`i2x0002n3\`y`i2x0001+3p11p17 [2,3+3p15/3p17j2x0002+7p22pFp15p13p11p20}}{SEP_R 44 0}{INP_R 45 0 "> "{TEXT 0 26 "C1 := subs(x=2,y=1,op(B));"}}{OUT_R 46 0 45{DAG :3n3\`C1`(3n4\`MATRIX`,2[2,3[2,3i2x0002pC[2,3/3i2x0001j2x0002p12} }{SEP_R 47 0}{COM_R 48 0{TEXT 1 349 "\015Note the presence of the `op' in the last command.\015\015(f) The analysis of this linea r system involves finding the eigenvalues and eigenvectors of the matrix C1. Both can be found at the same time by use of the com mand `eigenvects'. The option `radical' directs Maple to express (if possible) the eigenvalues and eigenvectors in terms of radic als.\015"}}{INP_R 49 0 "> "{TEXT 0 31 "q1 := eigenvects(C1,'radic al');"}}{OUT_R 50 0 49{DAG :3n3\`q1`,3[2,4+5/3i2x0003j2x0002j2x00 01*3j2x0005/3pCpAp11pC%2,2(3n4\`VECTOR`,2[2,3pC+5/3i2x0001j2x0004 pCpEp21[2,4+5p7pCpE/3p22pApC%2,2(3p19,2[2,3pC+5p21pCpE/3pCp24}} {SEP_R 51 0}{COM_R 52 0{TEXT 1 434 "\015To interpret Maple's last response observe that it consists of two lists. In each list th e first item is an eigenvalue, the second item is the multiplicit y of the eigenvalue, and the third item is the corresponding eige nvector. In reading the eigenvectors note that the first compone nt is one, with the following space separating the components.\01 5\015To obtain a numerical answer we can use `evalf', as shown in the following command.\015"}}{INP_R 53 0 "> "{TEXT 0 27 "evalf(q 1[1]); evalf(q1[2]);"}}{OUT_R 54 0 53{DAG [2,4e3i4x000381966011i2 x0009e3j2x0001j2x0000%2,2(3n4\`VECTOR`,2[2,3p9e3i4x008090169945i2 x0010}}{OUT_R 55 0 53{DAG [2,4e3i4x002618033989i2x0009e3j2x0001j2 x0000%2,2(3n4\`VECTOR`,2[2,3p9e3j4x003090169945i2x0010}}{SEP_R 56 0}{COM_R 57 0{TEXT 1 381 "\015Since the eigenvalues are both neg ative, the critical point is an asymptotically stable node of the linear system, and the same is true for the nonlinear system. F urther, trajectories approach the critical point tangent to the e igenvector associated with the larger eigenvalue (-0.38... in thi s case).\015\015To see tis graphically we can plot a direction fi eld for the linear system.\015"}}{INP_R 58 0 "> "{TEXT 0 79 "DEpl ot(C1,[x,y],t=0..1,\{[0,0,0]\},x=-1..1,y=-1..1,arrows=THIN,title= `Figure 4`);"}}{SEP_R 59 0}{COM_R 60 0{TEXT 1 1 "\015"}}{GRH_R 61 0{RASTER 398 323 1 50 000000000000000000000000000000000000000000 F8330000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000183000000000800300000000000000 00000000000000000000000000000000000000000000000000000000000000000 00000180000000000C00300000000000000000000000000000000000000000000 000000000000000000000000000000000000000018305CC66C1E6003000000000 00000000000000000000000000000000000000000000000000000000000000000 0000000000183076C67C333003000000000000000000000000000000000000000 000000000000000000000000000000000000000000000F83163C60C3318030000 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