#Project: Hardy-Weinberg #Group Members: Nick DiMarzio, Anusha Nagar, Tim Nasralla, Deven Singh #Part 1: Equal Fitness: > NULL; > NULL; > read "C://Users/an646/Downloads/DMB.txt"; First Written: Nov. 2021 This is DMB.txt, A Maple package to explore Dynamical models in Biology (both discrete and continuous) accompanying the class Dynamical Models in Biology, Rutgers University. Taught by Dr. Z. (Doron Zeilbeger) The most current version is available on WWW at: http://sites.math.rutgers.edu/~zeilberg/tokhniot/DMB.txt . Please report all bugs to: DoronZeil at gmail dot com . For general help, and a list of the MAIN functions, type "Help();". For specific help type "Help(procedure_name);" ------------------------------ For a list of the supporting functions type: Help1(); For help with any of them type: Help(ProcedureName); ------------------------------ For a list of the functions that give examples of Discrete-time dynamical systems (some famous), type: HelpDDM(); For help with any of them type: Help(ProcedureName); ------------------------------ For a list of the functions continuous-time dynamical systems (some famous) type: HelpCDM(); For help with any of them type: Help(ProcedureName); ------------------------------ ; > Help(HWg); HWg(u,v,M): The Generalized Hardy-Weinberg unerlying transformation with (u,v), M is the survival matrix. Based on Ann Somalwar's HW3g(u,v,w) (only retain the first two components and replace w by 1-u-v) Try: HWg(u,v,[[1,2,1],[2,3,4],[1,3,2]]); > RNG := rand(0. .. 1.0); RNG := proc () options operator, arrow; RandomTools:-Generate(fl\ oat('range' = 0. .. 1.0, 'method' = 'uniform')) end proc ; > a11_1 := RNG(); a11_1 := 0.2342493224 ; > F11_1 := HWg(u, v, [[a11_1, 1, 1], [1, 1, 1], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.006266547873, 0.1461120777], [0.006262190332, 0.1460654202], [0.006257840217, 0.1460188232], [0.006253497507, 0.1459722865], [0.006249162181, 0.1459258100], [0.006244834215, 0.1458793936], [0.006240513602, 0.1458330370], [0.006236200301, 0.1457867403], [0.006231894314, 0.1457405032], [0.006227595610, 0.1456943256], [0.006223304170, 0.1456482075], [0.006219019987, 0.1456021486]] ; > a11_2 := RNG(); a11_2 := 0.06587642124 ; > F11_2 := HWg(u, v, [[a11_2, 1, 1], [1, 1, 1], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.005463511414, 0.1372276281], [0.005459718835, 0.1371835758], [0.005455932720, 0.1371395808], [0.005452153042, 0.1370956430], [0.005448379789, 0.1370517622], [0.005444612937, 0.1370079384], [0.005440852480, 0.1369641714], [0.005437098395, 0.1369204610], [0.005433350660, 0.1368768072], [0.005429609266, 0.1368332098], [0.005425874190, 0.1367896686], [0.005422145414, 0.1367461836]] ; > a11_3 := RNG(); a11_3 := 0.3157837057 ; > F11_3 := HWg(u, v, [[a11_3, 1, 1], [1, 1, 1], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 0, 5); [[0.3, 0.4], [0.2007847514, 0.5328101660], [0.1960914493, 0.5119690635], [0.1828746251, 0.5087925980], [0.1722653610, 0.5036549260], [0.1628570710, 0.4985999548], [0.1545306359, 0.4935232352]] ; > NULL; > HWg(u, v, [[a11_3, 1, 1], [1, 1, 1], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.006773867972, 0.1513805612], [0.006769152860, 0.1513323767], [0.006764445790, 0.1512842545], [0.006759746734, 0.1512361944], [0.006755055667, 0.1511881964], [0.006750372576, 0.1511402602], [0.006745697430, 0.1510923858], [0.006741030221, 0.1510445730], [0.006736370917, 0.1509968216], [0.006731719494, 0.1509491316], [0.006727075943, 0.1509015028], [0.006722440239, 0.1508539350]] ; > NULL; > a12_1 := RNG(); a12_1 := 0.3447191092 ; > a12_2 := RNG(); a12_2 := 0.9034558453 ; > a12_3 := RNG(); a12_3 := 0.8219614586 ; > F12_1 := HWg(u, v, [[1, a12_1, 1], [1, 1, 1], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.0005741665188, 0.04711809346], [0.0005735582856, 0.04709357353], [0.0005729513201, 0.04706909125], [0.0005723456182, 0.04704464653], [0.0005717411762, 0.04702023928], [0.0005711379903, 0.04699586939], [0.0005705360563, 0.04697153677], [0.0005699353704, 0.04694724132], [0.0005693359285, 0.04692298296], [0.0005687377273, 0.04689876159], [0.0005681407629, 0.04687457711], [0.0005675450309, 0.04685042944]] ; > F12_2 := HWg(u, v, [[1, a12_2, 1], [1, 1, 1], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.004776973309, 0.1290226334], [0.004771399148, 0.1289527671], [0.004765837418, 0.1288830094], [0.004760288070, 0.1288133601], [0.004754751066, 0.1287438190], [0.004749226374, 0.1286743857], [0.004743713945, 0.1286050601], [0.004738213747, 0.1285358417], [0.004732725730, 0.1284667304], [0.004727249864, 0.1283977259], [0.004721786109, 0.1283288280], [0.004716334428, 0.1282600362]] ; > ; > F12_3 := HWg(u, v, [[1, a12_3, 1], [1, 1, 1], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.002367662576, 0.09292737226], [0.002365007181, 0.09287774833], [0.002362357555, 0.09282820162], [0.002359713682, 0.09277873194], [0.002357075542, 0.09272933909], [0.002354443118, 0.09268002288], [0.002351816391, 0.09263078312], [0.002349195343, 0.09258161959], [0.002346579958, 0.09253253211], [0.002343970214, 0.09248352049], [0.002341366094, 0.09243458454], [0.002338767584, 0.09238572407] ] ; > Orb(HWg(u, v, [[1, 0.056, 1], [1, 1, 1], [1, 1, 1]]), [u, v], [0.3, 0.4], 1000, 1010); [[0.0003913316555, 0.03912433305], [0.0003909211413, 0.03910405076], [0.0003905114760, 0.03908379954], [0.0003901026573, 0.03906357930], [0.0003896946826, 0.03904338996], [0.0003892875490, 0.03902323146], [0.0003888812544, 0.03900310370], [0.0003884757955, 0.03898300662], [0.0003880711704, 0.03896294014], [0.0003876673765, 0.03894290418], [0.0003872644114, 0.03892289866], [0.0003868622719, 0.03890292350]] ; > ; > Orb(HWg(u, v, [[1, 0.658, 1], [1, 1, 1], [1, 1, 1]]), [u, v], [0.3, 0.4], 1000, 1010); [[0.001151653073, 0.06591265235], [0.001150403758, 0.06587797988], [0.001149157094, 0.06584336101], [0.001147913073, 0.06580879559], [0.001146671685, 0.06577428349], [0.001145432924, 0.06573982458], [0.001144196780, 0.06570541871], [0.001142963246, 0.06567106575], [0.001141732313, 0.06563676557], [0.001140503972, 0.06560251804], [0.001139278218, 0.06556832302], [0.001138055042, 0.06553418037] ] ; > ; > Orb(HWg(u, v, [[1, 0.85, 1], [1, 1, 1], [1, 1, 1]]), [u, v], [0.3, 0.4], 1000, 1010); [[0.002873715437, 0.1018123457], [0.002870458883, 0.1017577461], [0.002867209448, 0.1017032316], [0.002863967124, 0.1016488020], [0.002860731883, 0.1015944570], [0.002857503697, 0.1015401965], [0.002854282557, 0.1014860200], [0.002851068417, 0.1014319276], [0.002847861275, 0.1013779189], [0.002844661096, 0.1013239938], [0.002841467869, 0.1012701521], [0.002838281570, 0.1012163934]] ; > ; > Orb(HWg(u, v, [[1, 0.8504, 1], [1, 1, 1], [1, 1, 1]]), [u, v], [0.3, 0.4], 1000, 1010); [[0.002882450191, 0.1019576957], [0.002879183191, 0.1019030146], [0.002875923339, 0.1018484188], [0.002872670623, 0.1017939079], [0.002869425010, 0.1017394817], [0.002866186475, 0.1016851401], [0.002862955002, 0.1016308828], [0.002859730564, 0.1015767097], [0.002856513146, 0.1015226204], [0.002853302715, 0.1014686148], [0.002850099256, 0.1014146927], [0.002846902746, 0.1013608538]] ; > ; > NULL; > a13_1 := RNG(); a13_1 := 0.5091763968 ; > a13_2 := RNG(); a13_2 := 0.2999657008 ; > a13_3 := RNG(); a13_3 := 0.2195119739 ; > F13_1 := HWg(u, v, [[1, 1, a13_1], [1, 1, 1], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.2584758952, 0.4830482094], [0.2584758952, 0.4830482095], [0.2584758952, 0.4830482094], [0.2584758952, 0.4830482095], [0.2584758952, 0.4830482094], [0.2584758952, 0.4830482095], [0.2584758952, 0.4830482094], [0.2584758952, 0.4830482095], [0.2584758952, 0.4830482094], [0.2584758952, 0.4830482095], [0.2584758952, 0.4830482094], [0.2584758952, 0.4830482095]] ; > F13_2 := HWg(u, v, [[1, 1, a13_2], [1, 1, 1], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.2626896008, 0.4746207983], [0.2626896008, 0.4746207983], [0.2626896008, 0.4746207983], [0.2626896008, 0.4746207983], [0.2626896008, 0.4746207983], [0.2626896008, 0.4746207983], [0.2626896008, 0.4746207983], [0.2626896008, 0.4746207983], [0.2626896008, 0.4746207983], [0.2626896008, 0.4746207983], [0.2626896008, 0.4746207983], [0.2626896008, 0.4746207983]] ; > F13_3 := HWg(u, v, [[1, 1, a13_3], [1, 1, 1], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.2644312748, 0.4711374499], [0.2644312748, 0.4711374499], [0.2644312748, 0.4711374499], [0.2644312748, 0.4711374499], [0.2644312748, 0.4711374499], [0.2644312748, 0.4711374499], [0.2644312748, 0.4711374499], [0.2644312748, 0.4711374499], [0.2644312748, 0.4711374499], [0.2644312748, 0.4711374499], [0.2644312748, 0.4711374499], [0.2644312748, 0.4711374499]] ; > ; > a13_4 := RNG(); a13_4 := 0.9193194089 ; > a13_5 := RNG(); a13_5 := 0.6279008356 ; > F13_4 := HWg(u, v, [[1, 1, a13_4], [1, 1, 1], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.2512800990, 0.4974398032], [0.2512800990, 0.4974398032], [0.2512800990, 0.4974398032], [0.2512800990, 0.4974398032], [0.2512800990, 0.4974398032], [0.2512800990, 0.4974398032], [0.2512800990, 0.4974398032], [0.2512800990, 0.4974398032], [0.2512800990, 0.4974398032], [0.2512800990, 0.4974398032], [0.2512800990, 0.4974398032], [0.2512800990, 0.4974398032]] ; > F13_5 := HWg(u, v, [[1, 1, a13_5], [1, 1, 1], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.2562619730, 0.4874760542], [0.2562619730, 0.4874760542], [0.2562619730, 0.4874760542], [0.2562619730, 0.4874760542], [0.2562619730, 0.4874760542], [0.2562619730, 0.4874760542], [0.2562619730, 0.4874760542], [0.2562619730, 0.4874760542], [0.2562619730, 0.4874760542], [0.2562619730, 0.4874760542], [0.2562619730, 0.4874760542], [0.2562619730, 0.4874760542]] ; > ; > NULL; > a21_1 := RNG(); a21_1 := 0.4128601644 ; > a21_2 := RNG(); a21_2 := 0.4705674223 ; > a21_3 := RNG(); a21_3 := 0.9333260133 ; > F21_1 := HWg(u, v, [[1, 1, 1], [a21_1, 1, 1], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.0006449562773, 0.04984505426], [0.0006442707179, 0.04981907710], [0.0006435865911, 0.04979313986], [0.0006429038922, 0.04976724245], [0.0006422226171, 0.04974138477], [0.0006415427615, 0.04971556672], [0.0006408643205, 0.04968978818], [0.0006401872897, 0.04966404907], [0.0006395116655, 0.04963834928], [0.0006388374426, 0.04961268872], [0.0006381646176, 0.04958706729], [0.0006374931856, 0.04956148489]] ; > NULL; > F21_2 := HWg(u, v, [[1, 1, 1], [a21_2, 1, 1], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.0007199407857, 0.05256666026], [0.0007191728929, 0.05253922308], [0.0007184066085, 0.05251182812], [0.0007176419279, 0.05248447526], [0.0007168788462, 0.05245716440], [0.0007161173582, 0.05242989543], [0.0007153574593, 0.05240266826], [0.0007145991444, 0.05237548276], [0.0007138424083, 0.05234833884], [0.0007130872467, 0.05232123639], [0.0007123336542, 0.05229417530], [0.0007115816261, 0.05226715546]] ; > F21_3 := HWg(u, v, [[1, 1, 1], [a21_3, 1, 1], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.007411281003, 0.1576974830], [0.007402406021, 0.1576117816], [0.007393551094, 0.1575262121], [0.007384716156, 0.1574407742], [0.007375901143, 0.1573554675], [0.007367105984, 0.1572702917], [0.007358330615, 0.1571852465], [0.007349574977, 0.1571003315], [0.007340838996, 0.1570155466], [0.007332122627, 0.1569308912], [0.007323425784, 0.1568463652], [0.007314748424, 0.1567619681]] ; > ; > HWg(u, v, [[1, 1, 1], [0.1, 1, 1], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.0004113391731, 0.04008309451], [0.0004109071747, 0.04006230672], [0.0004104760703, 0.04004155078], [0.0004100458575, 0.04002082661], [0.0004096165339, 0.04000013412], [0.0004091880962, 0.03997947325], [0.0004087605419, 0.03995884390], [0.0004083338686, 0.03993824599], [0.0004079080729, 0.03991767945], [0.0004074831527, 0.03989714420], [0.0004070591051, 0.03987664016], [0.0004066359273, 0.03985616724]] ; > NULL; > HWg(u, v, [[1, 1, 1], [0.0001, 1, 1], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.0003685425649, 0.03800065074], [0.0003681564722, 0.03798095992], [0.0003677711774, 0.03796129925], [0.0003673866784, 0.03794166864], [0.0003670029718, 0.03792206803], [0.0003666200558, 0.03790249734], [0.0003662379278, 0.03788295649], [0.0003658565852, 0.03786344541], [0.0003654760259, 0.03784396402], [0.0003650962475, 0.03782451226], [0.0003647172476, 0.03780509004], [0.0003643390236, 0.03778569729]] ; > ; > HWg(u, v, [[1, 1, 1], [0.66, 1, 1], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.001159009611, 0.06611438410], [0.001157751994, 0.06607960137], [0.001156497047, 0.06604487241], [0.001155244760, 0.06601019708], [0.001153995126, 0.06597557523], [0.001152748135, 0.06594100675], [0.001151503780, 0.06590649148], [0.001150262052, 0.06587202929], [0.001149022944, 0.06583762006], [0.001147786447, 0.06580326364], [0.001146552551, 0.06576895991], [0.001145321251, 0.06573470873] ] ; > ; > NULL; > ; > NULL; > a22_1 := RNG(); a22_1 := 0.2557169579 ; > a22_2 := RNG(); a22_2 := 0.2753988326 ; > a22_3 := RNG(); a22_3 := 0.4750188771 ; > F22_1 := HWg(u, v, [[1, 1, 1], [1, a22_1, 1], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.2500000000, 0.4999999999], [0.2500000000, 0.4999999999], [0.2500000000, 0.4999999999], [0.2500000000, 0.4999999999], [0.2500000000, 0.4999999999], [0.2500000000, 0.4999999999], [0.2500000000, 0.4999999999], [0.2500000000, 0.4999999999], [0.2500000000, 0.4999999999], [0.2500000000, 0.4999999999], [0.2500000000, 0.4999999999], [0.2500000000, 0.4999999999]] ; > F22_2 := HWg(u, v, [[1, 1, 1], [1, a22_2, 1], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[ -7 ] [[1.652327186 10 , 0.001546225989], [ -7 ] [1.648627018 10 , 0.001544497018], [ -7 ] [1.644939262 10 , 0.001542771908], [ -7 ] [1.641263864 10 , 0.001541050646], [ -7 ] [1.637600766 10 , 0.001539333218], [ -7 ] [1.633949912 10 , 0.001537619612], [ -7 ] [1.630311251 10 , 0.001535909816], [ -7 ] [1.626684728 10 , 0.001534203816], [ -7 ] [1.623070288 10 , 0.001532501601], [ -7 ] [1.619467879 10 , 0.001530803157], [ -7 ] [1.615877447 10 , 0.001529108471], [ -7 ]] [1.612298936 10 , 0.001527417533]] ; > F22_3 := HWg(u, v, [[1, 1, 1], [1, a22_3, 1], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.2500000000, 0.5000000001], [0.2500000000, 0.5000000001], [0.2500000000, 0.5000000001], [0.2500000000, 0.5000000001], [0.2500000000, 0.5000000001], [0.2500000000, 0.5000000001], [0.2500000000, 0.5000000001], [0.2500000000, 0.5000000001], [0.2500000000, 0.5000000001], [0.2500000000, 0.5000000001], [0.2500000000, 0.5000000001], [0.2500000000, 0.5000000001]] ; > ; > NULL; > a23_1 := RNG(); a23_1 := 0.1399264738 ; > a23_2 := RNG(); a23_2 := 0.09952040545 ; > a23_3 := RNG(); a23_3 := 0.9791486628 ; > F23_1 := HWg(u, v, [[1, 1, 1], [1, 1, a23_1], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.9585512150, 0.04101744677], [0.9585729495, 0.04099616625], [0.9585946508, 0.04097491794], [0.9586163190, 0.04095370174], [0.9586379540, 0.04093251660], [0.9586595550, 0.04091136494], [0.9586811225, 0.04089024710], [0.9587026572, 0.04086916028], [0.9587241590, 0.04084810492], [0.9587456275, 0.04082708323], [0.9587670638, 0.04080609250], [0.9587884675, 0.04078513217]] ; > NULL; > F23_2 := HWg(u, v, [[1, 1, 1], [1, 1, a23_2], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.9595166615, 0.04007222846], [0.9595378745, 0.04005144747], [0.9595590550, 0.04003069752], [0.9595802028, 0.04000998047], [0.9596013190, 0.03998929335], [0.9596224030, 0.03996863751], [0.9596434540, 0.03994801272], [0.9596644720, 0.03992742182], [0.9596854582, 0.03990686139], [0.9597064125, 0.03988633155], [0.9597273345, 0.03986583437], [0.9597482255, 0.03984536598]] ; > F23_3 := HWg(u, v, [[1, 1, 1], [1, 1, a23_3], [1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.6781794250, 0.2907352233], [0.6783575555, 0.2905951689], [0.6785354388, 0.2904552800], [0.6787130752, 0.2903155580], [0.6788904655, 0.2901760009], [0.6790676098, 0.2900366093], [0.6792445092, 0.2898973829], [0.6794211640, 0.2897583204], [0.6795975740, 0.2896194239], [0.6797737412, 0.2894806910], [0.6799496655, 0.2893421201], [0.6801253462, 0.2892037152]] ; > NULL; > NULL; > NULL; > a31_1 := RNG(); a31_1 := 0.2396346609 ; > a31_2 := RNG(); a31_2 := 0.8658728984 ; > a31_3 := RNG(); a31_3 := 0.02195544416 ; > F31_1 := HWg(u, v, [[1, 1, 1], [1, 1, 1], [a31_1, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.2639887370, 0.4720225260], [0.2639887370, 0.4720225258], [0.2639887370, 0.4720225260], [0.2639887370, 0.4720225258], [0.2639887370, 0.4720225260], [0.2639887370, 0.4720225258], [0.2639887370, 0.4720225260], [0.2639887370, 0.4720225258], [0.2639887370, 0.4720225260], [0.2639887370, 0.4720225258], [0.2639887370, 0.4720225260], [0.2639887370, 0.4720225258]] ; > NULL; > F31_2 := HWg(u, v, [[1, 1, 1], [1, 1, 1], [a31_2, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.2521502795, 0.4956994410], [0.2521502795, 0.4956994410], [0.2521502795, 0.4956994410], [0.2521502795, 0.4956994410], [0.2521502795, 0.4956994410], [0.2521502795, 0.4956994410], [0.2521502795, 0.4956994410], [0.2521502795, 0.4956994410], [0.2521502795, 0.4956994410], [0.2521502795, 0.4956994410], [0.2521502795, 0.4956994410], [0.2521502795, 0.4956994410]] ; > F31_3 := HWg(u, v, [[1, 1, 1], [1, 1, 1], [a31_3, 1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.2690479205, 0.4619041602], [0.2690479205, 0.4619041602], [0.2690479205, 0.4619041602], [0.2690479205, 0.4619041602], [0.2690479205, 0.4619041602], [0.2690479205, 0.4619041602], [0.2690479205, 0.4619041602], [0.2690479205, 0.4619041602], [0.2690479205, 0.4619041602], [0.2690479205, 0.4619041602], [0.2690479205, 0.4619041602], [0.2690479205, 0.4619041602]] ; > NULL; > NULL; > NULL; > a32_1 := RNG(); a32_1 := 0.6218815182 ; > a32_2 := RNG(); a32_2 := 0.2991735440 ; > a32_3 := RNG(); a32_3 := 0.5740904174 ; > F32_1 := HWg(u, v, [[1, 1, 1], [1, 1, 1], [1, a32_1, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.9364085228, 0.06255847300], [0.9364424818, 0.06252563030], [0.9364763880, 0.06249283794], [0.9365102418, 0.06246009378], [0.9365440410, 0.06242740418], [0.9365777885, 0.06239476306], [0.9366114835, 0.06236217170], [0.9366451255, 0.06232963213], [0.9366787155, 0.06229714089], [0.9367122530, 0.06226470133], [0.9367457395, 0.06223230903], [0.9367791732, 0.06219996857]] ; > NULL; > F32_2 := HWg(u, v, [[1, 1, 1], [1, 1, 1], [1, a32_2, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.9539381408, 0.04552700416], [0.9539623790, 0.04550333148], [0.9539865795, 0.04547969475], [0.9540107425, 0.04545609471], [0.9540348680, 0.04543253085], [0.9540589558, 0.04540900383], [0.9540830065, 0.04538551409], [0.9541070215, 0.04536205689], [0.9541309988, 0.04533863724], [0.9541549398, 0.04531525229], [0.9541788435, 0.04529190427], [0.9542027115, 0.04526858957]] ; > F32_3 := HWg(u, v, [[1, 1, 1], [1, 1, 1], [1, a32_3, 1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.9402773220, 0.05881402188], [0.9403091082, 0.05878321212], [0.9403408445, 0.05875245062], [0.9403725310, 0.05872173594], [0.9404041675, 0.05869107099], [0.9404357558, 0.05866045096], [0.9404672948, 0.05862987926], [0.9404987852, 0.05859935343], [0.9405302268, 0.05856887364], [0.9405616190, 0.05853844300], [0.9405929632, 0.05850805687], [0.9406242590, 0.05847771776]] ; > NULL; > NULL; > a33_1 := RNG(); a33_1 := 0.9749715494 ; > a33_2 := RNG(); a33_2 := 0.3055679837 ; > a33_3 := RNG(); a33_3 := 0.4850963949 ; > F33_1 := HWg(u, v, [[1, 1, 1], [1, 1, 1], [1, 1, a33_1]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.5496701432, 0.3835374612], [0.5497929922, 0.3834573567], [0.5499157115, 0.3833773180], [0.5500383012, 0.3832973452], [0.5501607615, 0.3832174390], [0.5502830930, 0.3831375980], [0.5504052952, 0.3830578234], [0.5505273688, 0.3829781148], [0.5506493135, 0.3828984724], [0.5507711305, 0.3828188948], [0.5508928195, 0.3827393832], [0.5510143808, 0.3826599367]] ; > NULL; > F33_2 := HWg(u, v, [[1, 1, 1], [1, 1, 1], [1, 1, a33_2]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.8426177978, 0.1506774081], [0.8426704450, 0.1506294273], [0.8427230220, 0.1505815084], [0.8427755290, 0.1505336520], [0.8428279668, 0.1504858572], [0.8428803352, 0.1504381234], [0.8429326342, 0.1503904506], [0.8429848635, 0.1503428404], [0.8430370242, 0.1502952914], [0.8430891168, 0.1502478020], [0.8431411402, 0.1502003736], [0.8431930948, 0.1501530080]] ; > F33_3 := HWg(u, v, [[1, 1, 1], [1, 1, 1], [1, 1, a33_3]]); > Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.8262737258, 0.1654764466], [0.8263316818, 0.1654242482], [0.8263895608, 0.1653721170], [0.8264473630, 0.1653200528], [0.8265050888, 0.1652680553], [0.8265627382, 0.1652161245], [0.8266203118, 0.1651642599], [0.8266778095, 0.1651124610], [0.8267352308, 0.1650607294], [0.8267925768, 0.1650090635], [0.8268498475, 0.1649574629], [0.8269070430, 0.1649059282]] ; > NULL; > NULL; > NULL; > NULL; > Help(); DMB.txt: A Maple package for exploring Dynamical models in Biology The MAIN procedures are ComK, Dis, EquP, FP, Orb, OrbF, Orbk, OrbkF, PhaseDiag, SEquP, SFP, TimeSeries ; > Help(TimeSeries); TimeSeries(F,x,pt,h,A,i): Inputs a transformation F in the list of variables x The time-series of x[i] vs. time of the Dynamical system approximating the the autonomous continuous dynamical process dx/dt=F(x(t)) by a discrete time dynamical system with step-size h from t=0 to t=A Try: TimeSeries([x*(1-y),y*(1-x)],[x,y],[0.5,0.5], 0.01, 10,1); > ; > Help(FP); FP(F,x): Given a transformation F in the list of variables finds all the fixed point of the transformation x->F(x), i.e. the set of solutions of the system {x[1]=F[1], ..., x[k]=F[k]}. Try: FP([5/2*x*(1-x)],[x]); evalf(FP([(1+x+y)/(2+3*x+y), (3+x+2*y)/(5+x+3*y)],[x,y])); > NULL; > FP(F11_1, [u, v]); / { [-1.142763354, 4.285526707], [0., 0.], \ [ -9] [0.9999999972, 4.459891018 10 ], [1.142763356, -0.2855267131] \ } / ; > Orb(F11_1, [u, v], [-1.1, 4.3], 2000, 2001); [[0.003888065596, 0.1170957740], [0.003886717274, 0.1170767641], [0.003885370104, 0.1170577667]] ; > Orb(F11_1, [u, v], [-1.2, 4.1], 2000, 2001); [[0.003888886340, 0.1171073438], [0.003887537317, 0.1170883263], [0.003886189449, 0.1170693213]] ; > NULL; > Orb(F11_1, [u, v], [-0.1, -0.1], 2000, 2001); [[0.004051740841, 0.1193762189], [0.004050248406, 0.1193556640], [0.004048757326, 0.1193351234]] ; > Orb(F11_1, [u, v], [0.1, 0.1], 2000, 2001); [[0.003646268630, 0.1136249775], [0.003645117589, 0.1136081488], [0.003643967448, 0.1135913300]] ; > NULL; > Orb(F11_1, [u, v], [1.14, -0.28], 2000, 2001); [[0.003902987634, 0.1173059096], [0.003901626528, 0.1172867612], [0.003900266597, 0.1172676254]] ; > NULL; > Orb(F11_1, [u, v], [1.1, 0.001], 2000, 2001); [[0.003889084114, 0.1171101316], [0.003887734926, 0.1170911122], [0.003886386885, 0.1170721054]] ; > NULL; > NULL; > FP(F11_2, [u, v]); / { [-1.034660412, 4.069320824], [0., 0.], \ [ -9] [0.9999999958, 7.800220058 10 ], [1.034660416, -0.06932083309] \ } / ; > Orb(F11_2, [u, v], [-1.1, 4.3], 2000, 2001); [[0.003387649440, 0.1097950737], [0.003386479085, 0.1097772236], [0.003385309729, 0.1097593852]] ; > Orb(F11_2, [u, v], [-1.3, 4.4], 2000, 2001); [[0.003390336944, 0.1098360502], [0.003389164298, 0.1098181730], [0.003387992648, 0.1098003076]] ; > Orb(F11_2, [u, v], [-0.1, -0.1], 2000, 2001); [[0.003499228061, 0.1114812188], [0.003497960387, 0.1114622332], [0.003496693846, 0.1114432606]] ; > Orb(F11_2, [u, v], [0.1, 0.1], 2000, 2001); [[0.003210761071, 0.1070568394], [0.003209735675, 0.1070407230], [0.003208711088, 0.1070246164]] ; > Orb(F11_2, [u, v], [0.9, -0.01], 2000, 2001); [[0.003386551711, 0.1097783313], [0.003385382288, 0.1097604922], [0.003384213867, 0.1097426648]] ; > Orb(F11_2, [u, v], [0.99, -0.0001], 2000, 2001); [[0.003390036034, 0.1098314631], [0.003388863644, 0.1098135890], [0.003387692254, 0.1097957266]] ; > Orb(F11_2, [u, v], [1.1, -0.3], 2000, 2001); [[0.003390961472, 0.1098455698], [0.003389788293, 0.1098276864], [0.003388616116, 0.1098098148]] ; > Orb(F11_2, [u, v], [1.3, -0.4], 2000, 2001); [[0.003389787744, 0.1098276780], [0.003388615564, 0.1098098064], [0.003387444389, 0.1097919465]] ; > NULL; > NULL; > FP(F11_3, [u, v]); / { [-1.208935954, 4.417871909], [0., 0.], \ [ -9] [1.000000002, -2.776950411 10 ], [1.208935953, -0.4178719050] \ } / ; > Orb(F11_3, [u, v], [-1.1, 4.0], 2000, 2001); [[0.004183151236, 0.1211501972], [0.004181708056, 0.1211306881], [0.004180266100, 0.1211111918]] ; > Orb(F11_3, [u, v], [-1.0, 4.1], 2000, 2001); [[0.004230923577, 0.1217938344], [0.004229439486, 0.1217739018], [0.004227956676, 0.1217539824]] ; > Orb(F11_3, [u, v], [-0.1, -0.1], 2000, 2001); [[0.004287649343, 0.1225527374], [0.004286115785, 0.1225322964], [0.004284583576, 0.1225118693]] ; > Orb(F11_3, [u, v], [0.1, 0.1], 2000, 2001); [[0.003915962200, 0.1174706972], [0.003914735527, 0.1174534804], [0.003913509798, 0.1174362739]] ; > Orb(F11_3, [u, v], [1.2, -0.5], 2000, 2001); [[0.004201228988, 0.1213942496], [0.004199770404, 0.1213745807], [0.004198313064, 0.1213549248]] ; > Orb(F11_3, [u, v], [1.1, -0.4], 2000, 2001); [[0.004199545287, 0.1213715447], [0.004198088140, 0.1213518908], [0.004196632244, 0.1213322497]] ; > NULL; > FP(F12_1, [u, v]); {[0., 0.], [0.5721188341, 0.3557623325], [1., 0.], [0.3389405831 - 0.7970710495 I, 0.8221188345 + 1.594142099 I], [0.3389405831 + 0.7970710495 I, 0.8221188345 - 1.594142099 I]} ; > NULL; > Orb(F12_1, [u, v], [-0.1, -0.1], 2000, 2001); [[0.0002337664279, -0.03088398758], [0.0002336545200, -0.03087652356], [0.0002335427200, -0.03086906502]] ; > Orb(F12_1, [u, v], [0.1, 0.1], 2000, 2001); [[0.0002712103735, 0.03256128077], [0.0002710719533, 0.03255306899], [0.0002709336731, 0.03254486335]] ; > Orb(F12_1, [u, v], [0.5, 0.3], 2000, 2001); [[0.0002782541668, 0.03297635549], [0.0002781085452, 0.03296782921], [0.0002779630745, 0.03295930946]] ; > Orb(F12_1, [u, v], [0.6, 0.4], 2000, 2001); [[ -9] [[0.9999999988, 1.192250650 10 ], [ -9] [0.9999999992, 1.192250650 10 ], [ -10]] [1.000000000, -8.077493516 10 ]] ; > NULL; > Orb(F12_1, [u, v], [0.9, -0.1], 2000, 2001); [[ -11] [[1.000000000, -5.624521223 10 ], [ -11] [1.000000000, -5.624521223 10 ], [ -11]] [1.000000000, -5.624521223 10 ]] ; > Orb(F12_1, [u, v], [1.1, 0.1], 2000, 2001); [[ -9] [[1.000000004, -3.273288542 10 ], [ -9] [1.000000004, -3.273288540 10 ], [ -9]] [1.000000004, -3.273288538 10 ]] ; > NULL; > FP(F12_2, [u, v]); {[0., 0.], [0.5637931436, 0.3724137180], [1., 0.], [0.3431034295 - 2.247082275 I, 0.8137931462 + 4.494164549 I], [0.3431034295 + 2.247082275 I, 0.8137931462 - 4.494164549 I]} ; > Orb(F12_2, [u, v], [-0.1, -0.1], 2000, 2001); [[0.001411399378, -0.07780235658], [0.001410769332, -0.07778439943], [0.001410139862, -0.07776645498]] ; > Orb(F12_2, [u, v], [0.1, 0.1], 2000, 2001); [[0.001868160426, 0.08285917050], [0.001867250447, 0.08283986755], [0.001866341330, 0.08282057777] ] ; > Orb(F12_2, [u, v], [0.5, 0.3], 2000, 2001); [[0.002235328645, 0.09026551112], [0.002234038914, 0.09024071549], [0.002232750628, 0.09021593975] ] ; > Orb(F12_2, [u, v], [0.6, 0.4], 2000, 2001); [[ -9] [[0.9999999980, 1.895979256 10 ], [ -9] [0.9999999980, 1.895979254 10 ], [ -9]] [0.9999999980, 1.895979252 10 ]] ; > Orb(F12_2, [u, v], [0.9, -0.1], 2000, 2001); [[ -8] [[0.9999999802, 1.888173297 10 ], [ -8] [0.9999999802, 1.888173282 10 ], [ -8]] [0.9999999802, 1.888173267 10 ]] ; > Orb(F12_2, [u, v], [1.1, 0.1], 2000, 2001); [[ -8] [[1.000000012, -1.185630959 10 ], [ -8] [1.000000012, -1.185630965 10 ], [ -8]] [1.000000012, -1.185630971 10 ]] ; > NULL; > NULL; > FP(F12_3, [u, v]); {[0., 0.], [0.5649168660, 0.3701662652], [1., 0.], [0.3425415663 - 1.636774555 I, 0.8149168646 + 3.273549111 I], [0.3425415663 + 1.636774555 I, 0.8149168646 - 3.273549111 I]} ; > Orb(F12_3, [u, v], [-0.1, -0.1], 2000, 2001); [[0.0008071507130, -0.05827461542], [0.0008067781852, -0.05826083456], [0.0008064060072, -0.05824706362]] ; > Orb(F12_3, [u, v], [0.1, 0.1], 2000, 2001); [[0.001017714207, 0.06192710919], [0.001017202609, 0.06191201884], [0.001016691514, 0.06189693932] ] ; > Orb(F12_3, [u, v], [0.5, 0.3], 2000, 2001); [[0.001120941783, 0.06489333715], [0.001120323566, 0.06487601658], [0.001119706016, 0.06485870961] ] ; > Orb(F12_3, [u, v], [0.6, 0.4], 2000, 2001); [[ -9] [[0.9999999955, 4.195393264 10 ], [ -9] [0.9999999955, 4.195393258 10 ], [ -9]] [0.9999999955, 4.195393252 10 ]] ; > Orb(F12_3, [u, v], [1.1, 0.1], 2000, 2001); [[ -9] [[1.000000010, -8.981289540 10 ], [ -9] [1.000000010, -9.981289560 10 ], [ -9]] [1.000000009, -8.981289592 10 ]] ; > Orb(F12_3, [u, v], [0.9, -0.1], 2000, 2001); [[ -9] [[0.9999999991, 1.012725021 10 ], [ -9] [0.9999999993, 1.012725020 10 ], [ -9]] [0.9999999997, 1.012725019 10 ]] ; > NULL; > NULL; > ; > FP(F13_1, [u, v]); {[-1.538948730, 4.077897461], [0., 0.], [0.2584758953, 0.4830482094], [1., 0.], [1.280472835, -1.560945670]} ; > Orb(F13_1, [u, v], [-1.1, 3.3], 2000, 2001); [[0.9957672700, 0.004228236787], [0.9957694520, 0.004226060074], [0.9957716322, 0.004223884547]] ; > Orb(F13_1, [u, v], [-1.2, 3.2], 2000, 2001); [[0.000004390263202, 0.004179666937], [0.000004385782448, 0.004177539053], [0.000004381308540, 0.004175413326]] ; > Orb(F13_1, [u, v], [-0.1, -0.1], 2000, 2001); [[0.000004566515185, 0.004262518856], [0.000004561762640, 0.004260306100], [0.000004557017492, 0.004258095632]] ; > Orb(F13_1, [u, v], [0.1, 0.1], 2000, 2001); [[0.000004295910825, 0.004134626432], [0.000004291573388, 0.004132543996], [0.000004287242495, 0.004130463648]] ; > Orb(F13_1, [u, v], [0.2, 0.4], 2000, 2001); [[0.000004591917442, 0.004274326405], [0.000004587125290, 0.004272101419], [0.000004582340610, 0.004269878740]] ; > Orb(F13_1, [u, v], [0.3, 0.5], 2000, 2001); [[0.9956711585, 0.004324140321], [0.9956734398, 0.004321866155], [0.9956757198, 0.004319589800]] ; > Orb(F13_1, [u, v], [0.8, -0.7], 2000, 2001); [[0.000004630911128, 0.004292387858], [0.000004626057962, 0.004290144101], [0.000004621212405, 0.004287902680]] ; > Orb(F13_1, [u, v], [0.9, -0.8], 2000, 2001); [[0.2584758945, 0.4830482095], [0.2584758945, 0.4830482095], [0.2584758945, 0.4830482095]] ; > Orb(F13_1, [u, v], [1.1, 0.1], 2000, 2001); [[0.000005136628520, 0.004520046175], [0.000005130961415, 0.004517559103], [0.000005125303660, 0.004515074756]] ; > Orb(F13_1, [u, v], [0.9, -0.1], 2000, 2001); [[0.9958795955, 0.004116146008], [0.9958816632, 0.004114084242], [0.9958837298, 0.004112020945]] ; > NULL; > NULL; > FP(F13_2, [u, v]); {[-1.304693455, 3.609386909], [0., 0.], [0.2626896008, 0.4746207983], [1., 0.], [1.042003854, -1.084007708]} ; > Orb(F13_2, [u, v], [-1.2, 3.4], 2000, 2001); [[0.2626896005, 0.4746207985], [0.2626896005, 0.4746207985], [0.2626896005, 0.4746207985]] ; > Orb(F13_2, [u, v], [-1.3, 3.5], 2000, 2001); [[0.000002176654392, 0.002944825939], [0.000002174417862, 0.002943315647], [0.000002172184772, 0.002941806901]] ; > Orb(F13_2, [u, v], [-0.1, -0.1], 2000, 2001); [[0.000001278280621, 0.002257772383], [0.000001277272874, 0.002256883597], [0.000001276266318, 0.002255995510]] ; > Orb(F13_2, [u, v], [0.1, 0.1], 2000, 2001); [[0.000002085388731, 0.002882549244], [0.000002083291153, 0.002881102007], [0.000002081196735, 0.002879656219]] ; > Orb(F13_2, [u, v], [0.2, 0.4], 2000, 2001); [[0.000002183728680, 0.002949597967], [0.000002181481257, 0.002948082789], [0.000002179237297, 0.002946569163]] ; > Orb(F13_2, [u, v], [0.3, 0.5], 2000, 2001); [[0.9970251992, 0.002972582571], [0.9970267400, 0.002971044047], [0.9970282792, 0.002969507088]] ; > Orb(F13_2, [u, v], [0.24, 0.44], 2000, 2001); [[0.000002227892095, 0.002979214982], [0.000002225576271, 0.002977669300], [0.000002223264052, 0.002976125217]] ; > NULL; > Orb(F13_2, [u, v], [0.26, 0.46], 2000, 2001); [[0.000002294712713, 0.003023471632], [0.000002292292113, 0.003021879803], [0.000002289875335, 0.003020289646]] ; > NULL; > FP(F13_3, [u, v]); {[-1.240731182, 3.481462364], [0., 0.], [0.2644312750, 0.4711374501], [0.9762999071, -0.9525998142], [1., 0.]} ; > Orb(F13_3, [u, v], [-1.6, 4.3], 2000, 2001); [[0.000001772191050, 0.002657549856], [0.000001770358078, 0.002656177695], [0.000001768527944, 0.002654806948]] ; > Orb(F13_3, [u, v], [-1.7, 4.4], 2000, 2001); [[0.2644312755, 0.4711374501], [0.2644312755, 0.4711374501], [0.2644312755, 0.4711374501]] ; > Orb(F13_3, [u, v], [-0.1, -0.1], 2000, 2001); [[0.000001833735626, 0.002703215520], [0.000001831806497, 0.002701795902], [0.000001829880406, 0.002700377772]] ; > Orb(F13_3, [u, v], [0.1, 0.1], 2000, 2001); [[0.000001671806813, 0.002581322679], [0.000001670127135, 0.002580027946], [0.000001668449984, 0.002578734509]] ; > Orb(F13_3, [u, v], [0.26, 0.49], 2000, 2001); [[0.9972876222, 0.002710533292], [0.9972890512, 0.002709106686], [0.9972904788, 0.002707680939]] ; > Orb(F13_3, [u, v], [0.25, 0.48], 2000, 2001); [[0.000001819802515, 0.002692945438], [0.000001817895298, 0.002691536564], [0.000001815991075, 0.002690129161]] ; > Orb(F13_3, [u, v], [1.1, 0.1], 2000, 2001); [[0.000001541727923, 0.002479043300], [0.000001540240157, 0.002477848938], [0.000001538754540, 0.002476655724]] ; > Orb(F13_3, [u, v], [0.9, -0.1], 2000, 2001); [[0.9974331565, 0.002565193278], [0.9974344368, 0.002563914464], [0.9974357160, 0.002562635924]] ; > Orb(F13_3, [u, v], [1.4, -1.8], 2000, 2001); [[0.2644312748, 0.4711374501], [0.2644312748, 0.4711374501], [0.2644312748, 0.4711374501]] ; > Orb(F13_3, [u, v], [1.5, -1.9], 2000, 2001); [[0.000001749999375, 0.002640888997], [0.000001748200672, 0.002639533951], [0.000001746404737, 0.002638180292]] ; > NULL; > NULL; > FP(F21_1, [u, v]); {[0., 0.], [0.5710236213, 0.3579527574], [1., 0.], [0.3394881894 - 0.8506598081 I, 0.8210236213 + 1.701319616 I], [0.3394881894 + 0.8506598081 I, 0.8210236213 - 1.701319616 I]} ; > Orb(F21_1, [u, v], [-0.1, -0.1], 2000, 2001); [[0.0002599798943, -0.03260537384], [0.0002598556717, -0.03259750019], [0.0002597315695, -0.03258963232]] ; > Orb(F21_1, [u, v], [0.1, 0.1], 2000, 2001); [[0.0003032569208, 0.03438844613], [0.0003031021578, 0.03437978537], [0.0003029475511, 0.03437113107]] ; > Orb(F21_1, [u, v], [0.5, 0.3], 2000, 2001); [[0.0003120778940, 0.03487838231], [0.0003119141047, 0.03486935040], [0.0003117504851, 0.03486032542]] ; > Orb(F21_1, [u, v], [0.6, 0.4], 2000, 2001); [[ -9] [[1.000000002, -1.560039608 10 ], [ -9] [1.000000002, -1.560039607 10 ], [ -9]] [1.000000002, -1.560039606 10 ]] ; > Orb(F21_1, [u, v], [1.1, 0.1], 2000, 2001); [[ -10] [[1.000000000, -3.878336825 10 ], [ -10] [0.9999999997, 6.121663174 10 ], [ -10]] [1.000000000, -3.878336828 10 ]] ; > Orb(F21_1, [u, v], [0.9, -0.1], 2000, 2001); [[ -10] [[0.9999999997, 3.734517310 10 ], [ -10] [0.9999999999, -6.265482691 10 ], [ -10]] [0.9999999994, 3.734517307 10 ]] ; > NULL; > NULL; > FP(F21_2, [u, v]); {[0., 0.], [0.5701138400, 0.3597723200], [1., 0.], [0.3399430800 - 0.9034564234 I, 0.8201138400 + 1.806912847 I], [0.3399430800 + 0.9034564234 I, 0.8201138400 - 1.806912847 I]} ; > Orb(F21_2, [u, v], [-0.1, -0.1], 2000, 2001); [[0.0002872876384, -0.03431136603], [0.0002871506338, -0.03430308788], [0.0002870137608, -0.03429481579]] ; > Orb(F21_2, [u, v], [0.1, 0.1], 2000, 2001); [[0.0003369198961, 0.03620296525], [0.0003367479855, 0.03619386052], [0.0003365762479, 0.03618476258]] ; > Orb(F21_2, [u, v], [0.5, 0.3], 2000, 2001); [[0.0003478262608, 0.03677577475], [0.0003476431800, 0.03676623619], [0.0003474602898, 0.03675670496]] ; > Orb(F21_2, [u, v], [0.6, 0.4], 2000, 2001); [[ -10] [[0.9999999997, 6.372380396 10 ], [ -10] [1.000000000, -3.627619605 10 ], [ -10]] [0.9999999997, 6.372380394 10 ]] ; > Orb(F21_2, [u, v], [1.1, 0.1], 2000, 2001); [[ -10] [[1.000000000, -2.873673894 10 ], [ -10] [0.9999999998, 7.126326106 10 ], [ -10]] [1.000000000, -2.873673895 10 ]] ; > Orb(F21_2, [u, v], [0.9, -0.1], 2000, 2001); [[ -10] [[0.9999999993, -1.733322383 10 ], [ -9] [0.9999999985, 1.826667762 10 ], [ -10]] [0.9999999993, 8.266677608 10 ]] ; > NULL; > FP(F21_3, [u, v]); {[0., 0.], [0.5633886833, 0.3732226410], [1., 0.], [0.3433056602 - 2.714680226 I, 0.8133886870 + 5.429360452 I], [0.3433056602 + 2.714680226 I, 0.8133886870 - 5.429360452 I]} ; > Orb(F21_3, [u, v], [-0.1, -0.1], 2000, 2001); [[0.001957401353, -0.09224501194], [0.001956552255, -0.09222419253], [0.001955703913, -0.09220338748]] ; > Orb(F21_3, [u, v], [0.1, 0.1], 2000, 2001); [[0.002664836812, 0.09805792276], [0.002663583942, 0.09803609564], [0.002662332212, 0.09801428267] ] ; > Orb(F21_3, [u, v], [0.5, 0.3], 2000, 2001); [[0.003455625668, 0.1108382596], [0.003453556439, 0.1108070984], [0.003451489596, 0.1107759626]] ; > Orb(F21_3, [u, v], [0.6, 0.4], 2000, 2001); [[ -8] [[0.9999999792, 2.014261888 10 ], [ -8] [0.9999999792, 2.014261871 10 ], [ -8]] [0.9999999792, 2.014261854 10 ]] ; > Orb(F21_3, [u, v], [1.1, 0.1], 2000, 2001); [[ -8] [[1.000000036, -3.544298026 10 ], [ -8] [1.000000036, -3.544298082 10 ], [ -8]] [1.000000036, -3.544298138 10 ]] ; > Orb(F21_3, [u, v], [0.9, -0.1], 2000, 2001); [[ -10] [[0.9999999983, 8.871617651 10 ], [ -9] [0.9999999976, 2.887161765 10 ], [ -9]] [0.9999999982, 1.887161761 10 ]] ; > NULL; > NULL; > FP(F22_1, [u, v]); {[-0.07956338770, 1.159126775], [0., 0.], [0.2500000000, 0.5000000000], [1., 0.], [1.079563388, -1.159126775]} ; > Orb(F22_1, [u, v], [-0.6, 2.4], 2000, 2001); [[ -8 ] [[2.937480180 10 , 0.0006772848736], [ -8 ] [2.934519429 10 , 0.0006769437503], [ -8 ]] [2.931563153 10 , 0.0006766029704]] ; > Orb(F22_1, [u, v], [-0.7, 2.3], 2000, 2001); [[0.9993217091, 0.0006782616249], [0.9993220515, 0.0006779186380], [0.9993223929, 0.0006775780565] ] ; > Orb(F22_1, [u, v], [-0.1, -0.1], 2000, 2001); [[ -8 ] [[2.916358286 10 , 0.0006748475291], [ -8 ] [2.913429400 10 , 0.0006745088556], [ -8 ]] [2.910504924 10 , 0.0006741705218]] ; > Orb(F22_1, [u, v], [0.1, 0.1], 2000, 2001); [[ -8 ] [[2.894270018 10 , 0.0006722891947], [ -8 ] [2.891374333 10 , 0.0006719530830], [ -8 ]] [2.888482989 10 , 0.0006716173072]] ; > Orb(F22_1, [u, v], [0.2, 0.4], 2000, 2001); [[ -8 ] [[2.918222388 10 , 0.0006750629907], [ -8 ] [2.915290695 10 , 0.0006747241010], [ -8 ]] [2.912363418 10 , 0.0006743855513]] ; > Orb(F22_1, [u, v], [0.3, 0.6], 2000, 2001); [[0.9993252659, 0.0006747049661], [0.9993256045, 0.0006743671724], [0.9993259436, 0.0006740267759] ] ; > Orb(F22_1, [u, v], [0.25, 0.49], 2000, 2001); [[ -8 ] [[2.960912949 10 , 0.0006799786316], [ -8 ] [2.957916714 10 , 0.0006796347906], [ -8 ]] [2.954925025 10 , 0.0006792912971]] ; > Orb(F22_1, [u, v], [1.1, 0.1], 2000, 2001); [[0.9993251281, 0.0006748425131], [0.9993254667, 0.0006745045577], [0.9993258055, 0.0006741649996] ] ; > Orb(F22_1, [u, v], [0.9, -0.1], 2000, 2001); [[0.9993276610, 0.0006723099315], [0.9993279970, 0.0006719739491], [0.9993283328, 0.0006716383602] ] ; > Orb(F22_1, [u, v], [1.6, -2.3], 2000, 2001); [[0.9993217091, 0.0006782616249], [0.9993220515, 0.0006779186380], [0.9993223929, 0.0006775780565] ] ; > Orb(F22_1, [u, v], [1.7, -2.4], 2000, 2001); [[0.9992903718, 0.0007095951086], [0.9992907455, 0.0007092232292], [0.9992911205, 0.0007088478095] ] ; > NULL; > NULL; > FP(F22_2, [u, v]); {[-0.08738180520, 1.174763611], [0., 0.], [0.2500000003, 0.4999999999], [1., 0.], [1.087381806, -1.174763611]} ; > Orb(F22_2, [u, v], [-0.2, 1.4], 2000, 2001); [[0.9992719662, 0.0007279961795], [0.9992723488, 0.0007276144582], [0.9992727319, 0.0007272322156] ] ; > Orb(F22_2, [u, v], [-0.3, 1.5], 2000, 2001); [[0.9993057222, 0.0006942439892], [0.9993060706, 0.0006938956244], [0.9993064187, 0.0006935486763] ] ; > Orb(F22_2, [u, v], [-0.1, -0.1], 2000, 2001); [[ -8 ] [[3.316935429 10 , 0.0006935029166], [ -8 ] [3.313602776 10 , 0.0006931547306], [ -8 ]] [3.310275143 10 , 0.0006928068940]] ; > Orb(F22_2, [u, v], [0.1, 0.1], 2000, 2001); [[ -8 ] [[3.289829288 10 , 0.0006906658395], [ -8 ] [3.286537389 10 , 0.0006903204952], [ -8 ]] [3.283250426 10 , 0.0006899754960]] ; > Orb(F22_2, [u, v], [0.2, 0.4], 2000, 2001); [[ -8 ] [[3.318080120 10 , 0.0006936224702], [ -8 ] [3.314745743 10 , 0.0006932741642], [ -8 ]] [3.311416387 10 , 0.0006929262077]] ; > Orb(F22_2, [u, v], [0.3, 0.5], 2000, 2001); [[0.9993052083, 0.0006947580578], [0.9993055574, 0.0006944090779], [0.9993059064, 0.0006940605156] ] ; > Orb(F22_2, [u, v], [1.1, 0.1], 2000, 2001); [[0.9993064627, 0.0006935042157], [0.9993068110, 0.0006931557355], [0.9993071587, 0.0006928086716] ] ; > Orb(F22_2, [u, v], [0.9, -0.1], 2000, 2001); [[0.9993092835, 0.0006906845804], [0.9993096299, 0.0006903374627], [0.9993099752, 0.0006899917580] ] ; > Orb(F22_2, [u, v], [1.2, -1.4], 2000, 2001); [[0.9992719662, 0.0007279961795], [0.9992723488, 0.0007276144582], [0.9992727319, 0.0007272322156] ] ; > Orb(F22_2, [u, v], [1.3, -1.5], 2000, 2001); [[ -8 ] [[3.324041505 10 , 0.0006942447544], [ -8 ] [3.320698141 10 , 0.0006938958234], [ -8 ]] [3.317359817 10 , 0.0006935472429]] ; > NULL; > NULL; > FP(F22_3, [u, v]); {[-0.1900779659, 1.380155931], [0., 0.], [0.2499999998, 0.5000000000], [1., 0.], [1.190077966, -1.380155932]} ; > Orb(F22_3, [u, v], [-0.1, 1.1], 2000, 2001); [[ -7 ] [[1.098172262 10 , 0.0009606866416], [ -7 ] [1.097065310 10 , 0.0009602028133], [ -7 ]] [1.095960031 10 , 0.0009597194719]] ; > Orb(F22_3, [u, v], [0., 1.2], 2000, 2001); [[0.9990522994, 0.0009475927929], [0.9990527693, 0.0009471235746], [0.9990532392, 0.0009466540335] ] ; > Orb(F22_3, [u, v], [-0.1, -0.1], 2000, 2001); [[ -7 ] [[1.092827373 10 , 0.0009583482240], [ -7 ] [1.091728489 10 , 0.0009578667467], [ -7 ]] [1.090631261 10 , 0.0009573857527]] ; > Orb(F22_3, [u, v], [0.1, 0.1], 2000, 2001); [[ -7 ] [[1.086114723 10 , 0.0009554032712], [ -7 ] [1.085025944 10 , 0.0009549247463], [ -7 ]] [1.083938799 10 , 0.0009544467003]] ; > Orb(F22_3, [u, v], [0.2, 0.4], 2000, 2001); [[ -7 ] [[1.100270298 10 , 0.0009616029853], [ -7 ] [1.099160174 10 , 0.0009611182342], [ -7 ]] [1.098051730 10 , 0.0009606339715]] ; > Orb(F22_3, [u, v], [0.3, 0.5], 2000, 2001); [[0.9990359764, 0.0009639130755], [0.9990364635, 0.0009634260939], [0.9990369502, 0.0009629388271] ] ; > Orb(F22_3, [u, v], [1.1, 0.1], 2000, 2001); [[0.9990415358, 0.0009583545718], [0.9990420171, 0.0009578737292], [0.9990424982, 0.0009573925885] ] ; > Orb(F22_3, [u, v], [0.9, -0.1], 2000, 2001); [[0.9990444829, 0.0009554091088], [0.9990449621, 0.0009549295600], [0.9990454404, 0.0009544507090] ] ; > Orb(F22_3, [u, v], [1.1, -1.1], 2000, 2001); [[0.9990391867, 0.0009607031945], [0.9990396703, 0.0009602199186], [0.9990401536, 0.0009597363498] ] ; > Orb(F22_3, [u, v], [1.0, -1.0], 2000, 2001); [[0.9989687427, 0.001031129776], [0.9989692992, 0.001030573649], [0.9989698552, 0.001030018395]] ; > NULL; > NULL; > FP(F23_1, [u, v]); {[0., 0.], [0.07554807178, 0.3489038564], [1., 0.], [-0.1627740359 - 0.6741252026 I, 0.8255480718 + 1.348250405 I], [-0.1627740359 + 0.6741252026 I, 0.8255480718 - 1.348250405 I]} ; > Orb(F23_1, [u, v], [-0.1, -0.1], 2000, 2001); [[ -979 -490] [[2.107962559 10 , -5.233693510 10 ], [ -980 -490] [6.847886940 10 , -2.983012894 10 ], [ -980 -490]] [2.224591482 10 , -1.700207685 10 ]] ; > Orb(F23_1, [u, v], [0.1, 0.1], 2000, 2001); [[ -976 -488] [[4.943157092 10 , 2.534422679 10 ], [ -976 -488] [1.605824579 10 , 1.444527754 10 ], [ -977 -489]] [5.216651080 10 , 8.233277146 10 ]] ; > Orb(F23_1, [u, v], [0.06, 0.3], 2000, 2001); [[ -974 -487] [[9.347915160 10 , 3.485252289 10 ], [ -974 -487] [3.036745880 10 , 1.986465676 10 ], [ -975 -487]] [9.865114705 10 , 1.132212407 10 ]] ; > Orb(F23_1, [u, v], [1.1, 0.1], 2000, 2001); [[1.026783339, -0.02696244712], [1.026776913, -0.02695593411], [1.026770493, -0.02694942847]] ; > Orb(F23_1, [u, v], [0.9, -0.1], 2000, 2001); [[0.9713168138, 0.02847695578], [0.9713241392, 0.02846973592], [0.9713314592, 0.02846252049]] ; > NULL; > NULL; > FP(F23_2, [u, v]); / { [0., 0.], [0.07624915953, 0.3475016809], [1., 0.], \ [-0.1631245800 - 0.6546191735 I, 0.8262491595 + 1.309238347 I], [-0.1631245800 + 0.6546191735 I, 0.8262491595 - 1.309238347 I], [ [0.9999999987 - 0.00003847976663 I, -9 ] [ 1.653794960 10 + 0.00003847976663 I], [ 0.9999999987 + 0.00003847976663 I, -9 ]\ 1.653794960 10 - 0.00003847976663 I] } / ; > Orb(F23_2, [u, v], [-0.1, -0.1], 2000, 2001); [[ -1042 -521] [[3.977374038 10 , -2.192812563 10 ], [ -1042 -521] [1.202106734 10 , -1.205521079 10 ], [ -1043 -522]] [3.633202680 10 , -6.627475125 10 ]] ; > Orb(F23_2, [u, v], [0.1, 0.1], 2000, 2001); [[ -1038 -519] [[1.271060369 10 , 1.239613743 10 ], [ -1039 -520] [3.841605580 10 , 6.814903024 10 ], [ -1039 -520]] [1.161072581 10 , 3.746562466 10 ]] ; > Orb(F23_2, [u, v], [1.1, 0.1], 2000, 2001); [[1.026182619, -0.02635387997], [1.026176336, -0.02634751511], [1.026170057, -0.02634115438]] ; > Orb(F23_2, [u, v], [0.91, -0.1], 2000, 2001); [[0.9719916642, 0.02781181691], [0.9719988075, 0.02780477256], [0.9720059440, 0.02779773722]] ; > Orb(F23_2, [u, v], [0.06, 0.3], 2000, 2001); [[ -1036 -518] [[2.533145202 10 , 1.749981010 10 ], [ -1037 -519] [7.656083838 10 , 9.620699144 10 ], [ -1037 -519]] [2.313946300 10 , 5.289077510 10 ]] ; > NULL; > NULL; > FP(F23_3, [u, v]); {[0., 0.], [0.06277583846, 0.3744483258], [0.9997471543, 0.0002528296800], [1., 0.], [1.000252900, -0.0002529162730], [-0.1563879041 - 4.883590409 I, 0.8127758320 + 9.767180816 I], [-0.1563879041 + 4.883590409 I, 0.8127758320 - 9.767180816 I]} ; > Orb(F23_3, [u, v], [-0.1, -0.1], 2000, 2001); [[ -21 -10] [[3.006036268 10 , -1.085114396 10 ], [ -21 -10] [2.943683130 10 , -1.073801353 10 ], [ -21 -10]] [2.882623365 10 , -1.062606256 10 ]] ; > Orb(F23_3, [u, v], [0.1, 0.1], 2000, 2001); [[ -19 -9] [[4.873710115 10 , 1.381682522 10 ], [ -19 -9] [4.772616488 10 , 1.367277558 10 ], [ -19 -9]] [4.673619810 10 , 1.353022775 10 ]] ; > Orb(F23_3, [u, v], [0.07, 0.3], 2000, 2001); [[ -16 -8] [[1.970895190 10 , 2.778498562 10 ], [ -16 -8] [1.930013621 10 , 2.749530861 10 ], [ -16 -8]] [1.889980043 10 , 2.720865168 10 ]] ; > Orb(F23_3, [u, v], [0.06, 0.3], 2000, 2001); [[ -17 -8] [[4.913241708 10 , 1.387274742 10 ], [ -17 -8] [4.811328095 10 , 1.372811475 10 ], [ -17 -8]] [4.711528432 10 , 1.358498998 10 ]] ; > Orb(F23_3, [u, v], [1.1, 0.1], 2000, 2001); [[1.146563130, -0.1515815557], [1.146535244, -0.1515518205], [1.146507376, -0.1515221063]] ; > Orb(F23_3, [u, v], [0.9, -0.1], 2000, 2001); [[0.8377989130, 0.1550389152], [0.8378271180, 0.1550133149], [0.8378553100, 0.1549877259]] ; > NULL; > NULL; > FP(F31_1, [u, v]); {[-1.255778033, 3.511556067], [0., 0.], [0.2639887370, 0.4720225259], [0.9917892964, -0.9835785928], [1., 0.]} ; > Orb(F31_1, [u, v], [-1.4, 3.9], 2000, 2001); [[0.000001844339823, 0.002711042683], [0.000001842444170, 0.002709651702], [0.000001840551434, 0.002708262145]] ; > Orb(F31_1, [u, v], [-1.5, 4.0], 2000, 2001); [[0.2639887370, 0.4720225260], [0.2639887370, 0.4720225258], [0.2639887370, 0.4720225260]] ; > Orb(F31_1, [u, v], [-0.1, -0.1], 2000, 2001); [[0.9972739565, 0.002724181174], [0.9972753628, 0.002722776437], [0.9972767678, 0.002721373521]] ; > Orb(F31_1, [u, v], [0.1, 0.1], 2000, 2001); [[0.000001762905422, 0.002650626697], [0.000001761133738, 0.002649296890], [0.000001759364720, 0.002647968415]] ; > Orb(F31_1, [u, v], [0.2, 0.4], 2000, 2001); [[0.000001839093365, 0.002707191209], [0.000001837205782, 0.002705804169], [0.000001835321102, 0.002704418546]] ; > Orb(F31_1, [u, v], [0.3, 0.5], 2000, 2001); [[0.9972717752, 0.002726359588], [0.9972731838, 0.002724952924], [0.9972745910, 0.002723547088]] ; > Orb(F31_1, [u, v], [1.1, 0.1], 2000, 2001); [[0.9973101890, 0.002687998354], [0.9973115582, 0.002686630379], [0.9973129260, 0.002685265076]] ; > Orb(F31_1, [u, v], [0.9, -0.1], 2000, 2001); [[0.9973637392, 0.002634519308], [0.9973650548, 0.002633205440], [0.9973663690, 0.002631893028]] ; > Orb(F31_1, [u, v], [1.2, -1.4], 2000, 2001); [[0.2639887370, 0.4720225258], [0.2639887370, 0.4720225260], [0.2639887370, 0.4720225258]] ; > Orb(F31_1, [u, v], [1.3, -1.5], 2000, 2001); [[0.9997633208, 0.0002366643405], [0.9997633310, 0.0002366543382], [0.9997633412, 0.0002366453382] ] ; > NULL; > NULL; > FP(F31_2, [u, v]); {[-2.847826401, 6.695652802], [0., 0.], [0.2521502794, 0.4956994411], [1., 0.], [2.595676121, -4.191352243]} ; > Orb(F31_2, [u, v], [-2.9, 6.8], 2000, 2001); [[0.01181523879, 0.1926310306], [0.01170701959, 0.1918581243], [0.01160003292, 0.1910895001]] ; > Orb(F31_2, [u, v], [-3.0, 6.9], 2000, 2001); [[0.9855884285, 0.01435921927], [0.9855951985, 0.01435250001], [0.9856019632, 0.01434578407]] ; > Orb(F31_2, [u, v], [-0.1, -0.1], 2000, 2001); [[0.9890990955, 0.01087101200], [0.9891029962, 0.01086713265], [0.9891068940, 0.01086325741]] ; > Orb(F31_2, [u, v], [0.1, 0.1], 2000, 2001); [[0.00006116546908, 0.01551131902], [0.00006110324622, 0.01550349347], [0.00006104111730, 0.01549567569]] ; > Orb(F31_2, [u, v], [0.25, 0.49], 2000, 2001); [[0.0001253191906, 0.02212229830], [0.0001251391306, 0.02210659111], [0.0001249594528, 0.02209090571]] ; > Orb(F31_2, [u, v], [1.1, 0.1], 2000, 2001); [[0.0001247213962, 0.02207010620], [0.0001245426040, 0.02205447138], [0.0001243641905, 0.02203885820]] ; > NULL; > NULL; > FP(F31_3, [u, v]); {[-1.118474273, 3.236948547], [0., 0.], [0.2690479199, 0.4619041602], [0.8494263534, -0.6988527068], [1., 0.]} ; > Orb(F31_3, [u, v], [-1.8, 4.6], 2000, 2001); [[0.2690479195, 0.4619041603], [0.2690479195, 0.4619041604], [0.2690479195, 0.4619041601]] ; > Orb(F31_3, [u, v], [-1.9, 4.7], 2000, 2001); [[0.9978858680, 0.002113012052], [0.9978869572, 0.002111923950], [0.9978880452, 0.002110836144]] ; > Orb(F31_3, [u, v], [-0.1, -0.1], 2000, 2001); [[0.000001071965437, 0.002067525109], [0.000001070883604, 0.002066483176], [0.000001069803407, 0.002065442291]] ; > Orb(F31_3, [u, v], [0.1, 0.1], 2000, 2001); [[0.000001057538919, 0.002053586979], [0.000001056478827, 0.002052559025], [0.000001055420327, 0.002051532099]] ; > Orb(F31_3, [u, v], [0.255, 0.488], 2000, 2001); [[0.000001167198913, 0.002157266132], [0.000001165969940, 0.002156131940], [0.000001164742906, 0.002154998938]] ; > Orb(F31_3, [u, v], [0.3, 0.49], 2000, 2001); [[0.9978989815, 0.002099912688], [0.9979000575, 0.002098838136], [0.9979011325, 0.002097763836]] ; > Orb(F31_3, [u, v], [1.1, 0.1], 2000, 2001); [[0.9977941498, 0.002204630431], [0.9977953355, 0.002203446856], [0.9977965200, 0.002202263884]] ; > Orb(F31_3, [u, v], [0.9, -0.1], 2000, 2001); [[0.9979632912, 0.002035668780], [0.9979643022, 0.002034658871], [0.9979653125, 0.002033651015]] ; > Orb(F31_3, [u, v], [1.5, -2.1], 2000, 2001); [[0.9978945052, 0.002104384812], [0.9978955855, 0.002103304873], [0.9978966650, 0.002102227204]] ; > Orb(F31_3, [u, v], [1.6, -2.0], 2000, 2001); [[0.000001097494764, 0.002091961573], [0.000001096374102, 0.002090894905], [0.000001095255153, 0.002089829322]] ; > NULL; > NULL; > FP(F32_1, [u, v]); {[0., 0.], [0.06780460951, 0.3643907810], [1., 0.], [-0.1589023048 - 1.092536278 I, 0.8178046095 + 2.185072556 I], [-0.1589023048 + 1.092536278 I, 0.8178046095 - 2.185072556 I]} ; > Orb(F32_1, [u, v], [-0.1, -0.1], 2000, 2001); [[ -367 -183] [[6.470984165 10 , -1.304680867 10 ], [ -367 -183] [4.255480412 10 , -1.058018893 10 ], [ -367 -184]] [2.798509945 10 , -8.579906441 10 ]] ; > Orb(F32_1, [u, v], [0.1, 0.1], 2000, 2001); [[ -364 -182] [[2.100729027 10 , 2.350737541 10 ], [ -364 -182] [1.381491747 10 , 1.906308886 10 ], [ -365 -182]] [9.085033922 10 , 1.545903575 10 ]] ; > Orb(F32_1, [u, v], [0.06, 0.3], 2000, 2001); [[ -362 -181] [[2.558399560 10 , 2.594199138 10 ], [ -362 -181] [1.682467292 10 , 2.103741818 10 ], [ -362 -181]] [1.106432409 10 , 1.706009987 10 ]] ; > Orb(F32_1, [u, v], [0.07, 0.4], 2000, 2001); [[0.9555325845, 0.04396575725], [0.9555444498, 0.04395416141], [0.9555563052, 0.04394257550]] ; > Orb(F32_1, [u, v], [1.1, 0.1], 2000, 2001); [[1.040039338, -0.04043544781], [1.040029826, -0.04042574897], [1.040020320, -0.04041605514]] ; > Orb(F32_1, [u, v], [0.9, -0.1], 2000, 2001); [[0.9566866242, 0.04283761034], [0.9566976035, 0.04282687306], [0.9567085735, 0.04281614643]] ; > NULL; > NULL; > FP(F32_2, [u, v]); {[0., 0.], [0.07286358693, 0.3542728261], [1., 0.], [-0.1614317935 - 0.7654463805 I, 0.8228635869 + 1.530892761 I], [-0.1614317935 + 0.7654463805 I, 0.8228635869 - 1.530892761 I]} ; > Orb(F32_2, [u, v], [-0.1, -0.1], 2000, 2001); [[ -752 -376] [[3.245039390 10 , -2.340330296 10 ], [ -752 -376] [1.369286474 10 , -1.520247602 10 ], [ -753 -377]] [5.777881928 10 , -9.875327324 10 ]] ; > Orb(F32_2, [u, v], [0.1, 0.1], 2000, 2001); [[ -749 -375] [[3.090368920 10 , 7.222246912 10 ], [ -749 -375] [1.304021262 10 , 4.691476058 10 ], [ -750 -375]] [5.502486900 10 , 3.047520788 10 ]] ; > Orb(F32_2, [u, v], [0.06, 0.3], 2000, 2001); [[ -747 -374] [[4.887689442 10 , 9.082783659 10 ], [ -747 -374] [2.062423975 10 , 5.900056118 10 ], [ -748 -374]] [8.702665550 10 , 3.832598408 10 ]] ; > Orb(F32_2, [u, v], [0.07, 0.4], 2000, 2001); [[0.9677869968, 0.03195214788], [0.9677954328, 0.03194384911], [0.9678038620, 0.03193555647]] ; > Orb(F32_2, [u, v], [1.1, 0.1], 2000, 2001); [[1.029626726, -0.02984532497], [1.029619629, -0.02983812332], [1.029612537, -0.02983092605]] ; > Orb(F32_2, [u, v], [0.9, -0.1], 2000, 2001); [[0.9682157912, 0.03153027214], [0.9682238982, 0.03152229468], [0.9682319985, 0.03151432470]] ; > NULL; > NULL; > FP(F32_3, [u, v]); / { [0., 0.], [0.06852211785, 0.3629557642], [1., 0.], \ [-0.1592610595 - 1.022457732 I, 0.8185221180 + 2.044915464 I], [-0.1592610595 + 1.022457732 I, 0.8185221180 - 2.044915464 I], [ [0.9999999980 - 0.00005595136184 I, -9 ] [ 2.753703263 10 + 0.00005595136183 I], [ 0.9999999980 + 0.00005595136184 I, -9 ]\ 2.753703263 10 - 0.00005595136183 I] } / ; > Orb(F32_3, [u, v], [-0.1, -0.1], 2000, 2001); [[ -419 -209] [[7.486264345 10 , -1.361953419 10 ], [ -419 -209] [4.637292790 10 , -1.071918913 10 ], [ -419 -210]] [2.872525390 10 , -8.436486444 10 ]] ; > Orb(F32_3, [u, v], [0.1, 0.1], 2000, 2001); [[ -416 -208] [[2.747907880 10 , 2.609340528 10 ], [ -416 -208] [1.702164498 10 , 2.053668960 10 ], [ -416 -208]] [1.054389049 10 , 1.616330315 10 ]] ; > Orb(F32_3, [u, v], [0.07, 0.3], 2000, 2001); [[ -413 -207] [[1.603567607 10 , 6.303377430 10 ], [ -414 -207] [9.933141755 10 , 4.961043004 10 ], [ -414 -207]] [6.152986922 10 , 3.904565125 10 ]] ; > Orb(F32_3, [u, v], [0.07, 0.4], 2000, 2001); [[0.9582397258, 0.04131867090], [0.9582508190, 0.04130781404], [0.9582619035, 0.04129696536]] ; > Orb(F32_3, [u, v], [1.1, 0.1], 2000, 2001); [[1.037793174, -0.03814664438], [1.037784176, -0.03813747922], [1.037775184, -0.03812831934]] ; > Orb(F32_3, [u, v], [0.9, -0.1], 2000, 2001); [[0.9591948515, 0.04038364632], [0.9592052122, 0.04037350055], [0.9592155650, 0.04036336326]] ; > NULL; > NULL; > FP(F33_1, [u, v]); {[0., 0.], [0.9824095538 - 0.01719134466 I, 0.01758827397 + 0.01703878229 I], [ 0.9824095538 + 0.01719134466 I, 0.01758827397 - 0.01703878229 I ], [1.017590007 - 0.01801711118 I, -0.01758763470 + 0.01817421122 I], [ 1.017590007 + 0.01801711118 I, -0.01758763470 - 0.01817421122 I ]} ; > Orb(F33_1, [u, v], [-0.1, -0.1], 2000, 2001); [[1.194933782, -0.2036049703], [1.194938282, -0.2036098532], [1.194942783, -0.2036147381]] ; > Orb(F33_1, [u, v], [0.1, 0.1], 6000, 6001); [[0.7365791842, 0.2433341302], [0.7365940630, 0.2433217074], [0.7366089388, 0.2433092864]] ; > NULL; > NULL; > FP(F33_2, [u, v]); {[0., 0.], [0.9888070067, 0.01116149603], [1.011347220, -0.01137922855], [0.9999229025 - 0.01126850027 I, 0.0001088432231 + 0.01126824474 I], [ 0.9999229025 + 0.01126850027 I, 0.0001088432231 - 0.01126824474 I]} ; > Orb(F33_2, [u, v], [-0.1, -0.1], 2000, 2001); [[0.8748329345, 0.1209955250], [0.8748540840, 0.1209758303], [0.8748752202, 0.1209561480]] ; > Orb(F33_2, [u, v], [0.1, 0.1], 2000, 2001); [[0.8750615608, 0.1207826092], [0.8750825588, 0.1207630520], [0.8751035418, 0.1207435086]] ; > Orb(F33_2, [u, v], [-0.1, -0.1], 6000, 6001); [[0.9135289720, 0.08451877665], [0.9135338075, 0.08451416430], [0.9135386415, 0.08450955340]] ; > Orb(F33_2, [u, v], [-0.1, -0.1], 10000, 10001); [[0.9271357305, 0.07148755860], [0.9271381698, 0.07148521310], [0.9271406090, 0.07148286710]] ; > NULL; > NULL; > FP(F33_3, [u, v]); {[0., 0.], [0.9947206128 - 0.005265451230 I, 0.005279387195 + 0.005251515266 I], [ 0.9947206128 + 0.005265451230 I, 0.005279387195 - 0.005251515266 I], [ 1.005279387 - 0.005293323160 I, -0.005279387195 + 0.005307259124 I], [ 1.005279387 + 0.005293323160 I, -0.005279387195 - 0.005307259124 I]} ; > Orb(F33_3, [u, v], [-0.1, -0.1], 2000, 2001); [[0.8619851635, 0.1329052846], [0.8620083472, 0.1328838794], [0.8620315152, 0.1328624887]] ; > Orb(F33_3, [u, v], [0.1, 0.1], 2000, 2001); [[0.8619532658, 0.1329347351], [0.8619764710, 0.1329133100], [0.8619996605, 0.1328919000]] ; > Orb(F33_3, [u, v], [-0.1, -0.1], 10000, 10001); [[0.9195342625, 0.07877986170], [0.9195369545, 0.07877728450], [0.9195396460, 0.07877470830]] ; > NULL; > NULL; > NULL; > NULL; #Part 2: Non-Random Mating with(LinearAlgebra) # ` This model for Hardy-Weinberg Equilibrium attempts to correct the assumption that mating is random.` A := Matrix([[u^2, u*v, u*w], [u*v, v^2, v*w], [u*w, v*w, w^2]]); N := Matrix([[p11, p12, p13], [p21, p22, p23], [p31, p32, p33]]); M := Matrix([[A[1][1]*p11, A[1][2]*p12, A[1][3]*p13], [A[2][1]*p21, A[2][2]*p22, A[2][3]*p23], [A[3][1]*p31, A[3][2]*p32, A[3][3]*p33]]); AA := M[1][1] + M[1][2]/2 + M[2][1]/2 + M[2][2]/4; Aa := M[1][2]/2 + M[1][3] + M[2][1]/2 + M[2][2]/2 + M[2][3]/2 + M[3][1] + M[3][2]/2; aa := M[2][2]/4 + M[2][3]/2 + M[3][2]/2 + M[3][3]; #` This is the Hardy-Weinberg Equilibrium function that accounts for nonrandom mating. The output is a discrete-time,first-order dynamical system with two quantities. The two quantities are u, the frequency of homozygous dominant genotypes, and v, the frequency of heterozygous genotypes. Each successive term in the system's trajectory represents each successive population after mating.` NRM := proc(u, v, N) with(LinearAlgebra); [(u^2*N[1][1] + 1/2*v*u*N[1][2] + 1/2*v*u*N[2][1] + 1/4*v^2*N[2][2])/(u^2*N[1][1] + v*u*N[1][2] + v*u*N[2][1] + v^2*N[2][2] + u*(1 - u - v)*N[1][3] + v*(1 - u - v)*N[2][3] + u*(1 - u - v)*N[3][1] + v*(1 - u - v)*N[3][2] + (1 - u - v)^2*N[3][3]), (1/2*v*u*N[1][2] + u*(1 - u - v)*N[1][3] + 1/2*v*u*N[2][1] + 1/2*v^2*N[2][2] + 1/2*v*(1 - u - v)*N[2][3] + u*(1 - u - v)*N[3][1] + 1/2*v*(1 - u - v)*N[3][2])/(u^2*N[1][1] + v*u*N[1][2] + v*u*N[2][1] + v^2*N[2][2] + u*(1 - u - v)*N[1][3] + v*(1 - u - v)*N[2][3] + u*(1 - u - v)*N[3][1] + v*(1 - u - v)*N[3][2] + (1 - u - v)^2*N[3][3])]; end proc; #` A symbolic representation of the function is represented below.` NRM(u, v, N); [(u^2*p11 + 1/2*u*v*p12 + 1/2*u*v*p21 + 1/4*v^2*p22)/(u^2*p11 + u*v*p12 + u*v*p21 + v^2*p22 + u*(1 - u - v)*p13 + v*(1 - u - v)*p23 + u*(1 - u - v)*p31 + v*(1 - u - v)*p32 + (1 - u - v)^2*p33), (u*v*p12/2 + u*(1 - u - v)*p13 + u*v*p21/2 + v^2*p22/2 + v*(1 - u - v)*p23/2 + u*(1 - u - v)*p31 + v*(1 - u - v)*p32/2)/(u^2*p11 + u*v*p12 + u*v*p21 + v^2*p22 + u*(1 - u - v)*p13 + v*(1 - u - v)*p23 + u*(1 - u - v)*p31 + v*(1 - u - v)*p32 + (1 - u - v)^2*p33)] #` 1) We can imagine that parents with the dominant phenotype are more likely to mate than parents with the recessive phenotype. This can be modeled by reducing the probability of mating for each mating combination with a parent of a homozygous recessive phenotype.` S1 := NRM(u, v, [[1, 1, 0.5], [1, 1, 0.5], [0.5, 0.5, 0.5]]); [/ 2 1 2\// 2 2 S1 := [|u + u v + - v | \u + 2 u v + v + 1.0 u (1 - u - v) [\ 4 / 2\ / + 1.0 v (1 - u - v) + 0.5 (1 - u - v) /, |u v \ 1 2 \// 2 + 1.0 u (1 - u - v) + - v + 0.5000000000 v (1 - u - v)| \u 2 / 2 + 2 u v + v + 1.0 u (1 - u - v) + 1.0 v (1 - u - v) 2\] + 0.5 (1 - u - v) /] ] Orb(S1, [u, v], [0.3, 0.4], 1000, 1010); [[0.9959927574, 0.004003219860], [0.9959967638, 0.003999221632], [0.9960007621, 0.003995231254], [0.9960047527, 0.003991248928], [0.9960087351, 0.003987274254], [0.9960127097, 0.003983307734], [0.9960166763, 0.003979348966], [0.9960206351, 0.003975398150], [0.9960245859, 0.003971455089], [0.9960285290, 0.003967519979], [0.9960324643, 0.003963592523], [0.9960363918, 0.003959672822]] #` The homozygous dominant phenotype is heavily favored. After the 1000 generation, the frequency of the homozygous dominant genotype in the population is still increasing.` #` 2) We can also imagine that parents with different phenotypes are less likely to mate than parents with the same phenotype. This can be modeled by reducing the probability of mating for each mating combination with a parent of a homozygous recessive genotype and a parent with either a homozygous dominant or heterozygous genotype.` NRM(u, v, [[1, 1, 0.5], [1, 1, 0.5], [0.5, 0.5, 1]]); [/ 2 1 2\// 2 2 [|u + u v + - v | \u + 2 u v + v + 1.0 u (1 - u - v) [\ 4 / 2\ / + 1.0 v (1 - u - v) + (1 - u - v) /, |u v + 1.0 u (1 - u - v) \ 1 2 \// 2 2 + - v + 0.5000000000 v (1 - u - v)| \u + 2 u v + v 2 / 2\] + 1.0 u (1 - u - v) + 1.0 v (1 - u - v) + (1 - u - v) /] ] Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.9959852609, 0.004010701545], [0.9959892823, 0.004006688107], [0.9959932959, 0.004002682821], [0.9959973013, 0.003998685285], [0.9960012988, 0.003994695902], [0.9960052882, 0.003990714371], [0.9960092696, 0.003986740891], [0.9960132431, 0.003982775364], [0.9960172086, 0.003978817691], [0.9960211663, 0.003974867968], [0.9960251160, 0.003970925999], [0.9960290580, 0.003966991983]] #` The homozygous dominant phenotype is still heavily favored, but the frequency of this genotype in the population after the 1000 generation is slightly less than in the previous scenario. The behavior of the system is the same.` #` 3) We can also imagine the opposite, that parents with different phenotypes are more likely to mate than parents with the same phenotype.` NRM(u, v, [[0.5, 0.5, 1], [0.5, 0.5, 1], [1, 1, 0.5]]); [/ 2 2\// 2 [\0.5 u + 0.5000000000 u v + 0.1250000000 v / \0.5 u + 1.0 u v 2 + 0.5 v + 2 u (1 - u - v) + 2 v (1 - u - v) 2\ / + 0.5 (1 - u - v) /, \0.5000000000 u v + 2 u (1 - u - v) 2 \// 2 2 + 0.2500000000 v + v (1 - u - v)/ \0.5 u + 1.0 u v + 0.5 v 2\] + 2 u (1 - u - v) + 2 v (1 - u - v) + 0.5 (1 - u - v) /] Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.05051025725, 0.4494897429], [0.05051025725, 0.4494897429], [0.05051025725, 0.4494897429], [0.05051025725, 0.4494897429], [0.05051025725, 0.4494897429], [0.05051025725, 0.4494897429], [0.05051025725, 0.4494897429], [0.05051025725, 0.4494897429], [0.05051025725, 0.4494897429], [0.05051025725, 0.4494897429], [0.05051025725, 0.4494897429], [0.05051025725, 0.4494897429]] #` The population stabilizes in this scenario. The frequency of the homozygous recessive genotype remains the majority of the population. The frequency of the heterozygous genotype is ~45% of the population. The homozygous dominant genotype is almost eradicated from the population.` #` 4) We can imagine a population where the maternal parent with a recessive phenotype is more likely to mate.` NRM(u, v, [[0.5, 0.5, 0.5], [0.5, 0.5, 0.5], [1, 1, 1]]); [/ 2 2\// 2 [\0.5 u + 0.5000000000 u v + 0.1250000000 v / \0.5 u + 1.0 u v 2 + 0.5 v + 1.5 u (1 - u - v) + 1.5 v (1 - u - v) 2\ / + (1 - u - v) /, \0.5000000000 u v + 1.5 u (1 - u - v) 2 \// 2 + 0.2500000000 v + 0.7500000000 v (1 - u - v)/ \0.5 u 2 + 1.0 u v + 0.5 v + 1.5 u (1 - u - v) + 1.5 v (1 - u - v) 2\] + (1 - u - v) /] Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[ -250 -125] [[2.606427309 10 , 3.424751508 10 ], [ -250 -125] [1.466115361 10 , 2.568563631 10 ], [ -251 -125] [8.246898908 10 , 1.926422723 10 ], [ -251 -125] [4.638880635 10 , 1.444817042 10 ], [ -251 -125] [2.609370356 10 , 1.083612782 10 ], [ -251 -126] [1.467770826 10 , 8.127095865 10 ], [ -252 -126] [8.256210900 10 , 6.095321899 10 ], [ -252 -126] [4.644118631 10 , 4.571491424 10 ], [ -252 -126] [2.612316730 10 , 3.428618568 10 ], [ -252 -126] [1.469428160 10 , 2.571463926 10 ], [ -253 -126] [8.265533404 10 , 1.928597944 10 ], [ -253 -126]] [4.649362538 10 , 1.446448458 10 ]] #` The homozygous recessive genotype overtakes the entire population by the 1000th generation.` #` 5) Let's imagine a population where the paternal parent with a dominant phenotype is more likely to mate.` NRM(u, v, [[1, 1, 0.5], [1, 1, 0.5], [1, 1, 0.5]]); [/ 2 1 2\// 2 2 [|u + u v + - v | \u + 2 u v + v + 1.5 u (1 - u - v) [\ 4 / 2\ / + 1.5 v (1 - u - v) + 0.5 (1 - u - v) /, |u v \ 1 2 \// 2 + 1.5 u (1 - u - v) + - v + 0.7500000000 v (1 - u - v)| \u 2 / 2 + 2 u v + v + 1.5 u (1 - u - v) + 1.5 v (1 - u - v) 2\] + 0.5 (1 - u - v) /] ] Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.9919521638, 0.008031578870], [0.9919602271, 0.008023548059], [0.9919682743, 0.008015533474], [0.9919763054, 0.008007534730], [0.9919843206, 0.007999552022], [0.9919923198, 0.007991584954], [0.9920003032, 0.007983633874], [0.9920082706, 0.007975698307], [0.9920162220, 0.007967778791], [0.9920241578, 0.007959875061], [0.9920320780, 0.007951986647], [0.9920399823, 0.007944113790]] #` The homozygous dominant genotype overtakes the population by the 1000th generation.` #` 6) Further, we can imagine that a certain maternal-paternal mating combination is favored. Combining the previous two scenarios, we can imagine that a maternal parent with a recessive phenotype and a paternal parent with a dominant phenotype are most likely to mate. ` NRM(u, v, [[0.5, 0.5, 0.5], [0.5, 0.5, 0.5], [1, 1, 0.5]]); [/ 2 2\// 2 [\0.5 u + 0.5000000000 u v + 0.1250000000 v / \0.5 u + 1.0 u v 2 + 0.5 v + 1.5 u (1 - u - v) + 1.5 v (1 - u - v) 2\ / + 0.5 (1 - u - v) /, \0.5000000000 u v + 1.5 u (1 - u - v) 2 \// 2 + 0.2500000000 v + 0.7500000000 v (1 - u - v)/ \0.5 u 2 + 1.0 u v + 0.5 v + 1.5 u (1 - u - v) + 1.5 v (1 - u - v) 2\] + 0.5 (1 - u - v) /] Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.06350832699, 0.4364916733], [0.06350832699, 0.4364916733], [0.06350832699, 0.4364916733], [0.06350832699, 0.4364916733], [0.06350832699, 0.4364916733], [0.06350832699, 0.4364916733], [0.06350832699, 0.4364916733], [0.06350832699, 0.4364916733], [0.06350832699, 0.4364916733], [0.06350832699, 0.4364916733], [0.06350832699, 0.4364916733], [0.06350832699, 0.4364916733]] #` The population stabilizes in this scenario. The frequency of the homozygous recessive genotype remains the majority of the population. The frequency of the heterozygous genotype is ~44% of the population. The homozygous dominant genotype is almost eradicated from the population.` #` 7) We can also imagine the opposite of Scenario 6, where a maternal parent with a dominant genotype and a paternal parent with a recessive phenotype are most likely to mate.` NRM(u, v, [[0.5, 0.5, 1], [0.5, 0.5, 1], [0.5, 0.5, 0.5]]); [/ 2 2\// 2 [\0.5 u + 0.5000000000 u v + 0.1250000000 v / \0.5 u + 1.0 u v 2 + 0.5 v + 1.5 u (1 - u - v) + 1.5 v (1 - u - v) 2\ / + 0.5 (1 - u - v) /, \0.5000000000 u v + 1.5 u (1 - u - v) 2 \// 2 + 0.2500000000 v + 0.7500000000 v (1 - u - v)/ \0.5 u 2 + 1.0 u v + 0.5 v + 1.5 u (1 - u - v) + 1.5 v (1 - u - v) 2\] + 0.5 (1 - u - v) /] Orb(%, [u, v], [0.3, 0.4], 1000, 1010); [[0.06350832699, 0.4364916733], [0.06350832699, 0.4364916733], [0.06350832699, 0.4364916733], [0.06350832699, 0.4364916733], [0.06350832699, 0.4364916733], [0.06350832699, 0.4364916733], [0.06350832699, 0.4364916733], [0.06350832699, 0.4364916733], [0.06350832699, 0.4364916733], [0.06350832699, 0.4364916733], [0.06350832699, 0.4364916733], [0.06350832699, 0.4364916733]] #` The population stabilizes, and the frequencies of each genotype after the 1000th generation are exactly the same as those in Scenario 6.` #Part 3: Mutations A := Matrix([[u^2 , u*v, u*w], [u*v, v^2 , v*w], [u*w, v*w, w^2]]); > #The matrix above represents the probability of of parents of genotypes AA, > Aa, and aa respectively mating. The probability of them mating is entirely > reliant on the frequency of each genotype, which works under the assumption of > random mating. > #The idea of mutations in genetics is pretty common and expected to occur in > just about every generation. The most common form of mutations would be those > that can affect allelic frequencies (relatively weakly) > #Starting with mutations that affect allelic frequency directly, assume a > mutation rate in the forward and backward direction (respectively, A-->a and a > --> A). #We could represent this change in allelic frequency with a discrete > time dynamical system. #Let h represent the forward mutation and g the > backward mutation #Then > #A(n) = A(n-1) + (g*a(n-1) - h*A(n-1)) #a(n) = a(n-1) + (h*A(n-1) - g*a(n-1)) > #Where a(n) represents the recessive allele and A(n) represents the dominant > allele and its changes over discrete time n representing generations. > #In order to apply this concept to the given Matrix, we create a function > based on p and q, which returns two matrices, the first being a Matrix that > has mutated and the 2nd being a Matrix that has not mutated. > NewAllelicFreq := proc(p,q,g,h) local pnew,qnew,u,v,w,NewNormal,NewMatrix; > with(LinearAlgebra); pnew := p + (g*q - h*p); qnew := q + (h*p - q*g); u := > pnew^2; v := 2*pnew*qnew; w := qnew^2; NewMatrix := Matrix([[u^2, > u*v,u*w],[u*v, v*v,v*w], [u*w, w*v, w^2]]); end: > NewAllelicFreq(p,q,0.01,0.01) > #It is worth noting that to return a Matrix with no mutations, allow g and h > to be 0 (that is, allow the mutation rates to be 0). > WhatKindaZygotes := proc(NewA) local HoDZyg, HoRZyg, HeZyg, sum; sum := > (NewA[1,1] + NewA[1,2] + NewA[1,3] + NewA[2,1] + NewA[2,2] + NewA[2,3] + > NewA[3,1] + NewA[3,2] + NewA[3,3])^(-1); HoDZyg := sum*(NewA[1,1] + > 0.5*NewA[1,2] + 0.5*NewA[2,1] + 0.25*NewA[2,2]); HeZyg := sum*(0.5*NewA[1,2] + > NewA[1,3] + NewA[3,1] + 0.5*NewA[2,1] + 0.5*NewA[2,2] + 0.5*NewA[3,2] + > 0.5*NewA[2,3]); HoRZyg := sum*(0.25*NewA[2,2] + 0.5*NewA[3,2] + 0.5*NewA[2,3] > + NewA[3,3]); [HoDZyg, HeZyg, HoRZyg]; end: > WhatKindaZygotes(NewAllelicFreq(p,q,0.01,0.01)) [/ 4 3 [\(0.99 p + 0.01 q) + 2.0 (0.99 p + 0.01 q) (0.99 q + 0.01 p) 2 2\// 4 + 1.00 (0.99 p + 0.01 q) (0.99 q + 0.01 p) / \(0.99 p + 0.01 q) 3 + 4 (0.99 p + 0.01 q) (0.99 q + 0.01 p) 2 2 + 6 (0.99 p + 0.01 q) (0.99 q + 0.01 p) 3 4\ / + 4 (0.99 q + 0.01 p) (0.99 p + 0.01 q) + (0.99 q + 0.01 p) /, \2.0 3 (0.99 p + 0.01 q) (0.99 q + 0.01 p) 2 2 + 4.0 (0.99 p + 0.01 q) (0.99 q + 0.01 p) 3 \// 4 + 2.0 (0.99 q + 0.01 p) (0.99 p + 0.01 q)/ \(0.99 p + 0.01 q) 3 + 4 (0.99 p + 0.01 q) (0.99 q + 0.01 p) 2 2 + 6 (0.99 p + 0.01 q) (0.99 q + 0.01 p) 3 4\ / + 4 (0.99 q + 0.01 p) (0.99 p + 0.01 q) + (0.99 q + 0.01 p) /, \1.00 2 2 (0.99 p + 0.01 q) (0.99 q + 0.01 p) 3 4\// + 2.0 (0.99 q + 0.01 p) (0.99 p + 0.01 q) + (0.99 q + 0.01 p) / \ 4 3 (0.99 p + 0.01 q) + 4 (0.99 p + 0.01 q) (0.99 q + 0.01 p) 2 2 + 6 (0.99 p + 0.01 q) (0.99 q + 0.01 p) 3 4\] + 4 (0.99 q + 0.01 p) (0.99 p + 0.01 q) + (0.99 q + 0.01 p) /] > #Now that we have two functions that when used together return the amount of > Homozygous dominant, heterozygous, and homozygous recessive people in a new > generation with mutations. > > #Example 1: Let p = 0.6, and q = 0.4 > WhatKindaZygotes(NewAllelicFreq(0.6,0.4,0,0)); > WhatKindaZygotes(NewAllelicFreq(0.6,0.4,0.01,0.01)) [0.3600000000, 0.4800000000, 0.1600000000] [0.3576040000, 0.4807920000, 0.1616040000] > NewAWD := proc(Set1) local p, q; p := sqrt(Set1[1]); q := sqrt(Set1[3]); [p, > q]; end > #So far, the NewAllelicFreq function creates a matrix designed to reflect on > how mutation will affect the progeny of parents of varying genotypes. Then the > function WhatKindaZygotes finds the values of u, v and w respectively to see > how much change there has been after one generation of mutations. Then NewAWD > converts this value into it's allelic state, p and q, to represent change. > Orb(NewAWD(WhatKindaZygotes(NewAllelicFreq(p,q,0.01,0.01))),[p,q],[0.6,0.4],10 > 00,1010) [[0.5000000050, 0.4999999949], [0.5000000048, 0.4999999949], [0.5000000048, 0.4999999950], [0.5000000049, 0.4999999951], [0.5000000050, 0.4999999950], [0.5000000050, 0.4999999949], [0.5000000048, 0.4999999949], [0.5000000048, 0.4999999950], [0.5000000049, 0.4999999951], [0.5000000050, 0.4999999950], [0.5000000050, 0.4999999949], [0.5000000048, 0.4999999949]] > #As seen in the above example from the frequency of p = 0.6 and q = 0.4, with > a mutation rate of 0.01 in both directions (which is rather high) leads to a > stable fixed point of 0.5 for both p and q, meaning that this community would > reach its Hardy Weinberg Equilibrium > SFP(NewAWD(WhatKindaZygotes(NewAllelicFreq(p,q,0.01,0.01))),[p,q]) {[0.5000000000, 0.5000000000]} > #This stable equilibrium is confirmed by the SFP function > > Orb(NewAWD(WhatKindaZygotes(NewAllelicFreq(p,q,0.00001,0.00001))),[p,q],[0.49, > 0.51],1000,1010) [[0.4901980146, 0.5098019852], [0.4901982108, 0.5098017894], [0.4901984068, 0.5098015932], [0.4901986027, 0.5098013972], [0.4901987986, 0.5098012012], [0.4901989946, 0.5098010054], [0.4901991906, 0.5098008094], [0.4901993866, 0.5098006135], [0.4901995827, 0.5098004176], [0.4901997787, 0.5098002214], [0.4901999748, 0.5098000255], [0.4902001707, 0.5097998294]] > > Orb(NewAWD(WhatKindaZygotes(NewAllelicFreq(p,q,0.00001,0.00001))),[p,q],[0.3,0 > .7],1000,1010) [[0.3039603047, 0.6960396951], [0.3039642256, 0.6960357744], [0.3039681464, 0.6960318536], [0.3039720670, 0.6960279330], [0.3039759876, 0.6960240124], [0.3039799080, 0.6960200920], [0.3039838284, 0.6960161716], [0.3039877488, 0.6960122512], [0.3039916690, 0.6960083311], [0.3039955892, 0.6960044110], [0.3039995091, 0.6960004908], [0.3040034291, 0.6959965708]] > > Orb(NewAWD(WhatKindaZygotes(NewAllelicFreq(p,q,0.00001,0.00001))),[p,q],[0.1,0 > .9],1000,1010) [[0.1079206086, 0.8920793915], [0.1079284502, 0.8920715499], [0.1079362916, 0.8920637085], [0.1079441328, 0.8920558674], [0.1079519740, 0.8920480259], [0.1079598150, 0.8920401848], [0.1079676559, 0.8920323443], [0.1079754965, 0.8920245034], [0.1079833369, 0.8920166629], [0.1079911773, 0.8920088228], [0.1079990175, 0.8920009827], [0.1080068575, 0.8919931424]] > #Unlike the first example, applying a more realistic factor of mutation, which > is approximately 10^-5, leads to a relatively unstable and changing amount of > genetic frequency. Although the change in allelic frequency is not drastic, it > does represent how mutations need to be set to 0 in order to have a stable > community that can reach a Hardy Weinberg Equilibrium. > > > Orb(NewAWD(WhatKindaZygotes(NewAllelicFreq(p,q,0.00001,0.000001))),[p,q],[0.3, > 0.7],1000,1010) [[0.3066633227, 0.6933366771], [0.3066699495, 0.6933300506], [0.3066765761, 0.6933234241], [0.3066832025, 0.6933167974], [0.3066898290, 0.6933101708], [0.3066964555, 0.6933035446], [0.3067030818, 0.6932969184], [0.3067097080, 0.6932902918], [0.3067163343, 0.6932836658], [0.3067229603, 0.6932770395], [0.3067295865, 0.6932704136], [0.3067362124, 0.6932637874]] > > Orb(NewAWD(WhatKindaZygotes(NewAllelicFreq(p,q,0.00001,0.000001))),[p,q],[0.5, > 0.5],1000,1010) [[0.5044753686, 0.4955246313], [0.5044798193, 0.4955201806], [0.5044842699, 0.4955157299], [0.5044887207, 0.4955112792], [0.5044931713, 0.4955068286], [0.5044976218, 0.4955023780], [0.5045020724, 0.4954979275], [0.5045065228, 0.4954934769], [0.5045109733, 0.4954890265], [0.5045154239, 0.4954845762], [0.5045198739, 0.4954801257], [0.5045243245, 0.4954756758]] > > Orb(NewAWD(WhatKindaZygotes(NewAllelicFreq(p,q,0.00001,0.000001))),[p,q],[0.1, > 0.9],1000,1010) [[0.1088512777, 0.8911487223], [0.1088600803, 0.8911399197], [0.1088688828, 0.8911311172], [0.1088776852, 0.8911223148], [0.1088864875, 0.8911135125], [0.1088952897, 0.8911047103], [0.1089040918, 0.8910959082], [0.1089128939, 0.8910871061], [0.1089216959, 0.8910783041], [0.1089304978, 0.8910695022], [0.1089392996, 0.8910607004], [0.1089481013, 0.8910518988]] > SFP(NewAWD(WhatKindaZygotes(NewAllelicFreq(p,q,0.00001,0.000001))),[p,q]) {[0.9090909091, 0.09090909091]} > > Orb(NewAWD(WhatKindaZygotes(NewAllelicFreq(p,q,0.00001,0.000001))),[p,q],[0.90 > 9,0.0909],1000,1010) [[0.9090909091, 0.09090909091], [0.9090909092, 0.09090909092], [0.9090909093, 0.09090909093], [0.9090909091, 0.09090909091], [0.9090909092, 0.09090909092], [0.9090909093, 0.09090909093], [0.9090909091, 0.09090909091], [0.9090909092, 0.09090909092], [0.9090909093, 0.09090909093], [0.9090909091, 0.09090909091], [0.9090909092, 0.09090909092], [0.9090909093, 0.09090909093]] > > Orb(NewAWD(WhatKindaZygotes(NewAllelicFreq(p,q,0.00001,0.000001))),[p,q],[0.91 > ,0.09],1000,1010) [[0.9099900545, 0.09000994545], [0.9099900446, 0.09000995534], [0.9099900346, 0.09000996523], [0.9099900250, 0.09000997515], [0.9099900148, 0.09000998501], [0.9099900051, 0.09000999493], [0.9099899952, 0.09001000483], [0.9099899853, 0.09001001472], [0.9099899754, 0.09001002461], [0.9099899655, 0.09001003450], [0.9099899556, 0.09001004438], [0.9099899457, 0.09001005427]] > #Changing the mutation rate to a much more realistic amount where the > mutations forward are not equal to the mutations reversed also yielded a > seemingly not stable equilibrium point (despite the SFP function yielding a > "stable" fixed point.