Question

Using MatLab, complete the following:

A model for extermination of a roach population in an area p(t) is based on the assumption that their growth rate decreases over time at a rate of 3/50 per day, as the result of a toxic substance introduced in their living area. The population is also reduced at a rate of 1 roach per day, due to predation. Then, the roach population over time p(t) is modeled by the ODE 1 3t 60 50 where t is time in da ys In your prob2): (a) Apply the midpoint method with h = 0.01, T = 20 to solve the ODE numerically, assuming that p(0) = 9. (b) Apply the midpoint method with h = 0.01, T = 20 to solve the ODE numerically, assuming that p(0) = 50 (c) Propose a change in the extermination strategy (write it in a comment in your code), so that the 50 roaches are exterminated in fewer days. Support your suggestion by deriving the corresponding IVP and solving it numerically with the midpoint method with h = 0.01, T = 20· (There is more than one correct strategy!) Output: Suppress all output and comment all fprintf messages from intermediate steps and function calls an (a) Print a message in the standard output stating the time it takes to exterminate a population of 9 roaches b) Print a message in the standard output stating the time it takes to exterminate a population of 50 roaches (c) Print a message in the standard output stating the time it takes to exterminate a population of 50 roaches with the improved strategy you proposed. All values are in the metric system

Answer 1

Matlab code for Midpoint method clear all close all %function for solving ODE using midpoint method fun- (t,p1/60)-((3*t)./50)).*p-1; initial conditions %initial t t in-t(1); t_max 20; Midpoint h-0.01; %Final t iterations n- (t_max-t in) /h; for i-l:n kl-h*fun(t(i),p(i)) k2-h fun (t (i)+(1/2) h, p(i)+(1/2) *kl); t (i+1)-t inti*h; p(i+1)-double (p(i)+k2) hold on lot (t,p,'linewidth',2) fprintf For step size h-82.2f and p (0)-9\n ",h) fprint f ("\n\tAt t-%f value of pe&E.\n\n',t(end ),p(end)) clear t; clear p; t(1)-0;p(1)-50; t_in-t (1) %initial conditions %Initial t Final t t max-20 %Midpoint iterations h-0.01; in)/h; n= ( t for i-1: n max-t k1-h * fun ( t (i),p(i)); p ( i+1 )-double ( p ( İ )-k2 ) ; end fprint f ( 'For step size h-82.2f and p(0)-20\n ',h) fprintf( "\n\tAt t= % f value of p= % f . \n\n",t(end ),p(end)) plot (t,p, 'linewidth',2) box on xlabel( time t' ylabel p(t) ) title( p(t) vs. t plot using midpoint method) legend( 'p(0)-9', p(0)-50) 88888888%88888888 End of Code 888888888888888888

For step size h-0.01 and p(0)-9 At t-20.000000 value of p-0.887411 For step size h-0.01 and p(o)-20 At t-20.0000 value of p-0.887059 p(t) vs. t plot using midpoint method 50 40 30 20 10 -10 0 24681014 16 18 20 ime t Published with MATLAB® R2018a

%%Matlab code for Midpoint method
clear all
close all
%function for solving ODE using midpoint method
fun=@(t,p) ((1/60)-((3*t)./50)).*p-1;
    t(1)=0;p(1)=9;    %initial conditions
    t_in=t(1);        %Initial t
    t_max=20;         %Final t
    %Midpoint iterations
    h=0.01;
    n=(t_max-t_in)/h;
    for i=1:n
     
        k1=h*fun(t(i),p(i));
        k2=h*fun(t(i)+(1/2)*h,p(i)+(1/2)*k1);
      
        t(i+1)=t_in+i*h;
        p(i+1)=double(p(i)+k2);
       
    end
    hold on
    plot(t,p,'linewidth',2)
    fprintf('For step size h=%2.2f and p(0)=9\n ',h)
    fprintf('\n\tAt t=%f value of p=%f.\n\n',t(end),p(end))
    clear t; clear p;
  
    t(1)=0;p(1)=50;    %initial conditions
    t_in=t(1);        %Initial t
    t_max=20;         %Final t
    %Midpoint iterations
    h=0.01;
    n=(t_max-t_in)/h;
    for i=1:n
     
        k1=h*fun(t(i),p(i));
        k2=h*fun(t(i)+(1/2)*h,p(i)+(1/2)*k1);
      
        t(i+1)=t_in+i*h;
        p(i+1)=double(p(i)+k2);
       
    end

    fprintf('For step size h=%2.2f and p(0)=20\n ',h)
    fprintf('\n\tAt t=%f value of p=%f.\n\n',t(end),p(end))
    plot(t,p,'linewidth',2)
    box on
    xlabel('time t')
    ylabel('p(t)')
    title('p(t) vs. t plot using midpoint method')
    legend('p(0)=9','p(0)=50')
  
   %%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%%