MA1506 Tutorial 11 Solutions including question6

MA1506 Tutorial 11 Solutions• ‚1 31. Tr =¡1;det=¡5)SADDLE1 ¡2• ‚2 ¡2 2Tr =2, det=8, T ¡4det=¡28<0)SPIRAL SOURCEr4 0• ‚¡2 ¡4 2Tr =¡2, det=40, T ¡4det=¡36<0) SPIRAL SINKr10 0• ‚¡5 4 2Tr =¡4, det=3, T ¡4det=4>0) NODAL SINKr¡2 1• ‚5 ¡4 2Tr =4, det=3, T ¡4det=4>0) NODAL SOURCEr2 ¡1• ‚0 1 2Tr =0, det=10, T ¡4det=¡40<0) CENTREr¡10 02. Let E(t) be the number of elves, D(t) the number of Dwarves. Let B ;D be theE Ebirth and death rates per capita for Elves, similarly B ;D for Dwarves. We areD Dtold that B > B and D < D . So B ¡D > B ¡D > 0 (more birthsE D E D E E D Dthan deaths).dENow is controlled by (B ¡D )E in the usual Malthus model, but here we areE EdtdEtold that is reduced by the presence of Dwarves, so we proposedtdE=(B ¡D )E¡P DE E Edtwhere P is a constant that measures the prejudice of the elves. SimilarlyEdD=(B ¡D )D¡P ED D DdtWhere P represents the prejudice of the Dwarves. SoD0 1dE • ‚µ ¶dt (B ¡D ) ¡P EE E E@ A=¡P (B ¡D ) DD D DdDdtSo in a concrete case, the constants in the matrix should satisfy B ¡ D >E E• ‚5 ¡4B ¡D > 0, and we are told that P > P . Indeed satisfles all ofD D E D ¡1 22these. Tr = 7, det=6, T ¡4det = 49¡24 = 25 > 0 so we have a nodal source.rEigenvalues(5¡‚)(2¡‚)¡4=0)‚=1 or 6:1µ ¶ µ ¶1 1Eigenvectors are ; .1 ¡1=4So the phase plane diagram(we only care about theflrst quadrant, sinceD‚0;E‚0) isbisectedbythelineD =E. AllpointsABOVEthatlinewillmovealongtrajectoriesthat eventually hit the D axis (that is, E ...