Question

9. For the model 1 +32 (a) Find the formula(s) for the fixed point(s) if r > 2. (b) Why do we include the condition r > 2? What occurs to the fixed points if r < 2? You may want to use an online graphing tool to assist in your cobweb diagrams for some models! 45

Answer 1

- men 9. auto the for fined point uzunzitute Hence 2 2 Tute > (1422) apr2 = 0 » ntno niz zo > 2 (1+tz-rt) > 0 = n* (na pri +1) =0 nzo; e priti zo - nadrt 552_4 it A (2 | 2 P4 10 than na tht 64 . app na in imaginary - complen number Hence for real @fined point ano Bolore Cobweb nown for diagram in p22 and raa

clear all close all %1st example r=3; x0=0.2; figure(1) cob_web(r, x0) $1st example r=1.5; x0=0.5; figure(2) cob_web(r, x0) $Matlab function for Cobweb plot function cob_web (r, x0) alpha and x0 are the input for cobweb plot Function for which Cobweb plot have to draw f=@(x) r.*x.2./(1+x. *2); t=x0; plotting the function xx=linspace(0,1,100); yy=f(xx); Xplotting the lines for cobweb hold on plot (xx,yy, 'b', 'Linewidth',2) plot (xx, xx,'b', 'Linewidth',2) plot([x0 x0], [0 f(x0)], 'r') Xchanging points after each iterations for i=1:500 xl=x0; yl=f(xl); x2=f(x1); y2=f(xl); x3=f(x1); y3=f(f(xl)); plot([xl x2],[yl y2),'r') plot([x2 x3], [y2 y3], 'r') x0=x3; end $title and lebeling of curve title( sprintf('Cobweb plot for r=32.2f and x0=$2.2f',r,t)) xlabel('x') ylabel('y') box on fprintf('Stable equilibrium point for a=ff and x0=$f is $f \n',r,t,0) end 8888888888888888 End of Code $88888888888888 Stable equilibrium point for a=3.000000 and x0=0.200000 is 0.000000 Stable equilibrium point for a=1.500000 and x0=0.500000 is 0.000000

Cobweb plot for r=3.00 and x0=0.20 0 0.1 02 03 04 0.5 0.6 0.7 0.8 0. 9 1 Cobweb plot for r=1.50 and x0=0.50 y O 0.1 02 0.3 0.4 0.5 0.6 0.7 0.8 9 0. 1

clear all
close all
%1st example
r=3; x0=0.2;
figure(1)
cob_web(r,x0)
%1st example
r=1.5; x0=0.5;
figure(2)
cob_web(r,x0)
%Matlab function for Cobweb plot
function cob_web(r,x0)
%alpha and x0 are the input for cobweb plot

    %Function for which Cobweb plot have to draw
    f=@(x) r.*x.^2./(1+x.^2);
    t=x0;
    %plotting the function
    xx=linspace(0,1,100);
    yy=f(xx);
    %plotting the lines for cobweb
    hold on
    plot(xx,yy,'b','Linewidth',2)
    plot(xx,xx,'b','Linewidth',2)
    plot([x0 x0],[0 f(x0)],'r')
    %changing points after each iterations
    for i=1:500
        x1=x0;
        y1=f(x1);
        x2=f(x1);
        y2=f(x1);
        x3=f(x1);
        y3=f(f(x1));

        plot([x1 x2],[y1 y2],'r')
        plot([x2 x3],[y2 y3],'r')

        x0=x3;
    end
    %title and lebeling of curve
    title(sprintf('Cobweb plot for r=%2.2f and x0=%2.2f',r,t))
    xlabel('x')
    ylabel('y')
    box on
    fprintf('Stable equilibrium point for a=%f and x0=%f is %f\n',r,t,x0)
end
   %%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%