Question

Combat Model.A simplified mathematical model for conventional versus guerrilla combat is given by the system$\mathbf{x}{\mathbf{\text{'}}}_{\mathbf{1}}\mathbf{=}\mathbf{-}\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{1}\mathbf{)}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}\mathbf{;} {\mathbf{x}}_{\mathbf{1}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{10}\mathbf{;} \mathbf{x}{\mathbf{\text{'}}}_{\mathbf{2}}\mathbf{=}\mathbf{-}{\mathbf{x}}_{\mathbf{1}}\mathbf{;}{\mathbf{x}}_{\mathbf{2}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{15}$ where${\mathbf{x}}_{\mathbf{1}}$ and${\mathbf{x}}_{\mathbf{2}}$ are the strengths of guerrilla and conventional troops, respectively, and 0.1 and 1 are the combat effectiveness coefficients.Who will win the conflict: the conventional troops or the guerrillas? [Hint:Use the vectorized Runge–Kutta algorithm for systems with h=0.1to approximate the solutions.]

Short answer

The solution is convention loop will win.

Step-by-step solution

Transform the equation

Write the equation as $x{\text{'}}_1=-\left(0.1\right)x_1{x}_2$and $x{\text{'}}_2=-x_1$

The initial conditions are:

localid="1664097278504" $$\begin{gathered} {\mathbf{x}}_{\mathbf{1}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{10} \\ {\mathbf{x}}_{\mathbf{2}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{15} \end{gathered}$$

Apply Runge –Kutta method

For the solution, apply the Runge-Kutta method in Matlab for h=0.1.

Now the algorithm is:

Function n[t,x] =Runge_Kutta(f,t0,t_end,init_cond,h)
%we begin at time t0 and end when we reach t_end
%init_cond(i) contains the initial value of x_i
%f contains functions such that x_i'=f_i(t,x1,x2,...)

t(:,1)=t0; % t0 is the initial value of t
x(:,1)=init_cond; % initial conditions are set

i=1;
whilet(:,i) < t_end

k1=f(t(i),x(:,i));
k2=f(t(i)+0.5*h,x(:,i)+0.5*h*k1);
k3=f(t(i)+0.5*h,x(:,i)+0.5*h*k2);
k4=f(t(i)+h,x(:,i)+h*k3);

x(:,i+1)=x(:,i)+(h/6)*(k1+2*k2+2*k3+k4);

t(:,i+1)=t(:,i)+ h;
i=i+1;

end

Now for the solution

clear all

init_cond=[10;15];
f=@(t,X) [-0.1*X(1)*X(2);-X(1)];
[t,x] = Runge_Kutta(f,0,10,init_cond,0.1);

plot(t,x(1,:),t,x(2,:));
legend('guerilla','conventional')

Therefore, the convention loop will win.