> #OK to post #Timothy Nasralla, HW11, 10/11/21 [[0, 1], [0, 1]] > #Question 1: Use it to prove that k<3 has 1 stable fixed point. #SFPe returns > a sequence of sets for [fixed points, their stability]. The absolute value of > the stability being less than 1 represents the fixed point being stable. > SFPe(c*x*(1-x),x) [ [c - 1 ]] [[0, c], [-----, 2 - c]] [ [ c ]] > #If c is less than 1, then x = 0 will be a stable fixed point. When c is equal > to 1, then neither point is stable. When c is inbetween 1 and 3, 2-c is > between -1 to 1 which makes all c's between 1 and 3 stable fixed points. > #The only stable fixed point between c = 0 and c = 1 is x = 0 #The only stable > fixed point between c = 1 and c = 3 is x = (c-1)/c. #Question 2: Approximate > for which k the second bifurcation point occurs using Orb > Orb(3.1*x*(1-x),x,0.3,1000,1020) [0.7645665197, 0.5580141258, 0.7645665197, 0.5580141258, 0.7645665197, 0.5580141258, 0.7645665197, 0.5580141258, 0.7645665197, 0.5580141258, 0.7645665197, 0.5580141258, 0.7645665197, 0.5580141258, 0.7645665197, 0.5580141258, 0.7645665197, 0.5580141258, 0.7645665197, 0.5580141258, 0.7645665197, 0.5580141258] > Orb(3.2*x*(1-x),x,0.3,1000,1020) [0.7994554906, 0.5130445091, 0.7994554906, 0.5130445091, 0.7994554906, 0.5130445091, 0.7994554906, 0.5130445091, 0.7994554906, 0.5130445091, 0.7994554906, 0.5130445091, 0.7994554906, 0.5130445091, 0.7994554906, 0.5130445091, 0.7994554906, 0.5130445091, 0.7994554906, 0.5130445091, 0.7994554906, 0.5130445091] > Orb(3.3*x*(1-x),x,0.3,1000,1020) [0.8236032832, 0.4794270198, 0.8236032832, 0.4794270198, 0.8236032832, 0.4794270198, 0.8236032832, 0.4794270198, 0.8236032832, 0.4794270198, 0.8236032832, 0.4794270198, 0.8236032832, 0.4794270198, 0.8236032832, 0.4794270198, 0.8236032832, 0.4794270198, 0.8236032832, 0.4794270198, 0.8236032832, 0.4794270198] > Orb(3.4*x*(1-x),x,0.3,1000,1020) [0.4519632478, 0.8421543994, 0.4519632478, 0.8421543994, 0.4519632478, 0.8421543994, 0.4519632478, 0.8421543994, 0.4519632478, 0.8421543994, 0.4519632478, 0.8421543994, 0.4519632478, 0.8421543994, 0.4519632478, 0.8421543994, 0.4519632478, 0.8421543994, 0.4519632478, 0.8421543994, 0.4519632478, 0.8421543994] > Orb(3.5*x*(1-x),x,0.3,1000,1020) [0.3828196827, 0.8269407062, 0.5008842111, 0.8749972637, 0.3828196827, 0.8269407062, 0.5008842111, 0.8749972637, 0.3828196827, 0.8269407062, 0.5008842111, 0.8749972637, 0.3828196827, 0.8269407062, 0.5008842111, 0.8749972637, 0.3828196827, 0.8269407062, 0.5008842111, 0.8749972637, 0.3828196827, 0.8269407062] > Orb(3.45*x*(1-x),x,0.3,1000,1020) [0.4462302146, 0.8525253952, 0.4337541677, 0.8473596395, 0.4462274188, 0.8525243578, 0.4337566913, 0.8473607931, 0.4462246539, 0.8525233321, 0.4337591860, 0.8473619334, 0.4462219208, 0.8525223178, 0.4337616535, 0.8473630612, 0.4462192178, 0.8525213149, 0.4337640926, 0.8473641759, 0.4462165461, 0.8525203233] > Orb(3.44*x*(1-x),x,0.3,1000,1020) [0.4421929706, 0.8485047151, 0.4421929548, 0.8485047085, 0.4421929706, 0.8485047151, 0.4421929548, 0.8485047085, 0.4421929706, 0.8485047151, 0.4421929548, 0.8485047085, 0.4421929706, 0.8485047151, 0.4421929548, 0.8485047085, 0.4421929706, 0.8485047151, 0.4421929548, 0.8485047085, 0.4421929706, 0.8485047151] > #Seemingly, the second bifurcation point is k = 3.45 > #Question 3: Find the fixed points of each equation and then express the > condition for each fixed point. > #Equation 1: N(x+1) = lambda/alpha * N(x)^(1-beta) where lambda is the > reproductive rate > 1, and alpha and beta are both positive. #Equation 2: > N(x+1) = N(x)*e^(r*(1-N(x)/K)) where r and K are positive constants. #Equation > 3: N(x+1) = lambda*N(x)*(1+a*N(x))^-b where a, b, and lambda are positive > constants. > #Equation 1: SFPe(l*x^(-b)*x/a, x); simplify(-l*exp(-ln(a/l)/b)^(-b)*(b - > 1)/a) [ [ (-b) ]] [ [ / / /a\\\ ]] [ [ | | ln|-||| ]] [ [ / /a\\ | | \l/|| ]] [ [ | ln|-|| l |exp|- -----|| (b - 1)]] [ [ | \l/| \ \ b // ]] [[0, 0], [exp|- -----|, - ----------------------------]] [ [ \ b / a ]] (-b) / / 1\\ | |- -|| | \ b/| |/a\ | l ||-| | (b - 1) \\l/ / - ------------------------ a > #Fixed points are when: x = 0; and it is always a stable fixed point. #Or when > x = e^(ln(l/a)/b); and it is stable when a and b are both greater than 0 as > well as lambda being greater than 1. > #Equation 2: SFPe(x*e^(r*(1-x/k)),x); [[ r] ] [[0, e ], [k, 1 - r ln(e)]] > #Fixed points are when: x = 0; and it is only a fixed point when r is less > than 0. #and when x = k; and it is stable only when r is less than 1. > #Equation 3: > SFPe(l*x*(1+a*x)^(-b),x); simplify(-l*exp(ln(l)/b)^(-b)*(b*exp(ln(l)/b) - > exp(ln(l)/b) - b)/exp(ln(l)/b)) [ [ (-b) ]] [ [ /ln(l)\ / /ln(l)\\ / /ln(l)\ /ln(l)\ \]] [ [exp|-----| - 1 l |exp|-----|| |b exp|-----| - exp|-----| - b|]] [ [ \ b / \ \ b // \ \ b / \ b / /]] [[0, l], [--------------, - --------------------------------------------------]] [ [ a /ln(l)\ ]] [ [ exp|-----| ]] [ [ \ b / ]] (-b) / /1\\ / /b - 1\ \ | |-|| | |-----| | | \b/| | \ b / | \l / \l b - l b + l/ > #Fixed points are when: x = 0; and it is only stable when l is less than 1. > #Or when x = 1/a * (e^ln(l)/b - 1); and it is stable when l,b, and a are > positive. > #Question 4: Find the equilibrium points. The equilibrium point is seemingly > Orbk(2,x,(x[1]+x[2])/(x[1]+x[2]),[5.3,1.1],1000,1010) [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] > Orbk(2,x,(x[1]+x[2])/(2*x[1]+x[2]),[5.3,1.1],1000,1010) [0.6666666670, 0.6666666667, 0.6666666670, 0.6666666667, 0.6666666670, 0.6666666667, 0.6666666670, 0.6666666667, 0.6666666670, 0.6666666667, 0.6666666670] > Orbk(2,x,(x[1]+x[2])/(3*x[1]+x[2]),[5.3,1.1],1000,1010) [0.4999999998, 0.5000000002, 0.4999999998, 0.5000000002, 0.4999999998, 0.5000000002, 0.4999999998, 0.5000000002, 0.4999999998, 0.5000000002, 0.4999999998] > Orbk(2,x,(x[1]+x[2])/(4*x[1]+x[2]),[5.3,1.1],1000,1010) [0.3999999996, 0.4000000004, 0.3999999996, 0.4000000004, 0.3999999996, 0.4000000004, 0.3999999996, 0.4000000004, 0.3999999996, 0.4000000004, 0.3999999996] > Orbk(2,x,(x[1]+2*x[2])/(x[1]+x[2]),[5.3,1.1],1000,1010) [1.500000000, 1.500000000, 1.500000000, 1.500000000, 1.500000000, 1.500000000, 1.500000000, 1.500000000, 1.500000000, 1.500000000, 1.500000000] > Orbk(2,x,(x[1]+2*x[2])/(2*x[1]+x[2]),[5.3,1.1],1000,1010) [1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000, 1.000000000] > Orbk(2,x,(x[1]+2*x[2])/(3*x[1]+x[2]),[5.3,1.1],1000,1010) [0.7499999997, 0.7500000002, 0.7499999994, 0.7500000002, 0.7499999997, 0.7500000002, 0.7499999994, 0.7500000002, 0.7499999997, 0.7500000002, 0.7499999994] > Orbk(2,x,(x[1]+2*x[2])/(4*x[1]+x[2]),[5.3,1.1],1000,1010) [0.5999999967, 0.6000000033, 0.5999999967, 0.6000000033, 0.5999999967, 0.6000000033, 0.5999999967, 0.6000000033, 0.5999999967, 0.6000000033, 0.5999999967] > Orbk(2,x,(x[1]+3*x[2])/(x[1]+x[2]),[5.3,1.1],1000,1010) [2.000000000, 2.000000000, 2.000000000, 2.000000000, 2.000000000, 2.000000000, 2.000000000, 2.000000000, 2.000000000, 2.000000000, 2.000000000] > Orbk(2,x,(x[1]+3*x[2])/(2*x[1]+x[2]),[5.3,1.1],1000,1010) [1.333333332, 1.333333335, 1.333333332, 1.333333335, 1.333333332, 1.333333335, 1.333333332, 1.333333335, 1.333333332, 1.333333335, 1.333333332] > Orbk(2,x,(x[1]+3*x[2])/(3*x[1]+x[2]),[5.3,1.1],1000,1010) [0.9470829022, 1.055845508, 0.9471335770, 1.055789100, 0.9471841051, 1.055732862, 0.9472344868, 1.055676792, 0.9472847232, 1.055620889, 0.9473348156] > Orbk(2,x,(x[1]+3*x[2])/(4*x[1]+x[2]),[5.3,1.1],1000,1010) [0.4226497313, 1.577350267, 0.4226497313, 1.577350267, 0.4226497313, 1.577350267, 0.4226497313, 1.577350267, 0.4226497313, 1.577350267, 0.4226497313] > Orbk(2,x,(x[1]+4*x[2])/(x[1]+x[2]),[5.3,1.1],1000,1010) [2.500000000, 2.500000000, 2.500000000, 2.500000000, 2.500000000, 2.500000000, 2.500000000, 2.500000000, 2.500000000, 2.500000000, 2.500000000] > Orbk(2,x,(x[1]+4*x[2])/(2*x[1]+x[2]),[5.3,1.1],1000,1010) [1.666666662, 1.666666671, 1.666666662, 1.666666671, 1.666666662, 1.666666671, 1.666666662, 1.666666671, 1.666666662, 1.666666671, 1.666666662] > Orbk(2,x,(x[1]+4*x[2])/(3*x[1]+x[2]),[5.3,1.1],1000,1010) [0.6339745969, 2.366025401, 0.6339745969, 2.366025401, 0.6339745969, 2.366025401, 0.6339745969, 2.366025401, 0.6339745969, 2.366025401, 0.6339745969] > Orbk(2,x,(x[1]+4*x[2])/(4*x[1]+x[2]),[5.3,1.1],1000,1010) [0.3819660117, 2.618033986, 0.3819660118, 2.618033985, 0.3819660117, 2.618033986, 0.3819660118, 2.618033985, 0.3819660117, 2.618033986, 0.3819660118]