1. The dynamics of a fish population is described by the Richard's growth law dN = rN1. dt (1) where N = N(t) is the population size depending on the continuous time variable 1, and r and K are positive constants. (i) Give a biological meaning of the terms rN and -rN VN/K in the right hand side of the equation () and of the parameters r and K. [4 marks] (ii) Sketch the graph of the function in the right hand side of equation (I) for N 2 0. Use your sketch to show that the system has a single non-zero equilibrium, determine its stability and clearly indicate the stability of the steady state on your graph using arrows. [4 marks] (iii) Use the substitution u N-/2 to show that the exact solution of the equation () satisfying the initial condition N(0) = Ng > 0 is given by NO) = [K-V2 + (N - K-2)*. %3D Determine the behaviour of the solution in the limit oft 0o, [7 marks] (iv) If the fish is subjected to constant effort harvesting E, the model (I) is modified as dN IN = rN dt (2) EN. K Sketch the graph of the function in the right hand side of the equation (2) for several values of E 2 0. Use your graph to explain how the population will respond to harvesting. [6 marks] (v) Find the maximum sustainable yield of the fish population described by the model given by the equation (2). [9 marks]