(Solved): Suppose that there are two kinds of bacteria \( X \) and \( Y \). Their population sizes \( x=x(t) ...

Suppose that there are two kinds of bacteria \( X \) and \( Y \). Their population sizes \( x=x(t) \) and \( y=y(t) \) at time \( t \) (in days) satisfy the following differential equations \[ \left\{\begin{array}{l} \dot{x}=\frac{3}{10} x-\frac{1}{10} y \\ \dot{y}=-\frac{1}{5} x+\frac{1}{2} y . \end{array}\right. \] Initially, the number of \( X \) is \( x_{0} \) and the number of \( Y \) is \( 2 x_{0} \), where \( x_{0} \) is a positive real number. 1 (a) Find the solutions to \( x \) and \( y \) at time \( t \) in terms of \( x_{0} \) and \( t \). ( 6 marks) (b) Will the bacterium \( Y \) eventually die off? Please explain briefly. (2 marks) (c) Show that bacterium \( X \) will eventually die off. Find the corresponding time. (2 marks)