Problem 5: System of Differential Equations for Population Dynamics with Harvesting Consider the system of differential equations modeling the growth of two interacting species where species 1 is harvested at a rate proportional to its population size: dx1 =1121 dt (1-²) - h₁₁ ax1x2 Κι dx2 dt 22 = 12x2 1 K2 where x1 is the population of species 1, and 22 is the population of species 2. The parameters 11, 12 are the growth rates, K1, K2 are the carrying capacities, h₁ is the harvesting rate for species 1, and a, ẞ are interaction coefficients. The initial conditions and parameter values are given in the table below: Parameter/Condition T1 T2 Κι K2 h₁ a Value 0.5 0.4 100 80 0.1 0.02 β Initial 1(0) Initial 2(0) 0.03 40 30 1. Solve the system of equations numerically using the Runge-Kutta method. Compute x1(t) and 2(t) over the time interval t = [0,50]. 2. Plot the time evolution of both populations and discuss the long-term dynamics. Does one species dominate or do they coexist? 3. Create a table showing the population values at t = 10, 20, 30, 40, 50. Problem 6: Eigenvalue Problem in Vibrating String The transverse vibrations of a string of length I are governed by the partial differential equation: Ju მt2 მე2 with the boundary conditions: u(0,t) = 0, u(L,t) = 0, and the initial conditions: u(x, 0) = f(x), (x, 0) = g(x). Ət Consider the case where L = 5, c = 2, and the initial conditions are given in the table below: Initial Condition Condition 1 Condition 2 f(x) f(x)=sin() f(x)=0 g(x) 0 g(x) = cos 1. Solve the wave equation using the method of separation of variables for both sets of initial conditions. 2. Find the eigenvalues and eigenfunctions of the corresponding Sturm-Liouville problem. 3. Create a table of the first three eigenvalues and eigenfunctions.