(Solved): 4. The spruce budworm larva shown here is a serious pest of coniferous trees. The population of thi ...

4. The spruce budworm larva shown here is a serious pest of coniferous trees. The population of this bug can be modeled with a logistic equation: \[ S^{\prime}(t)=\frac{1}{2} S\left(1-\frac{S}{10}\right)-\frac{S^{2}}{1+S^{2}} \] Notice that this differential equation is autonomous which means we do not see the indepentent variable \( t \) in the right-hand-side. Our methods take in anonymous functions where the right-hand-side depends on \( t \) first and \( y \) second. For this problem, you should not change any of your algorithms. MATLAB doesn't mind at all if we define a function like \( f(y)=y^{2} \) as follows: \( f=e(t, y) y^{-2} \); \& this fsn't the model, it's just, an example We can do this for autonomous differential equations and not need to modify our RK4 code to deal with only one variable instead of two. (a) (2 points) Use a root finding algorithm or fzero to find the 3 positive equilibrium solutions of the differential equation (values of \( S \) for which \( S^{+}=0 \) ).