Neco Model Questions Vol1 2017 Agricultural Science Question 34

A fish pond initially contains 10 000 fish. The population follows the discrete logistic model \[ N_{t+1}=N_t+rN_t\!\left(1-\frac{N_t}{K}\right)-H_t, \] with intrinsic growth rate \( r = 0.4\) per month and carrying capacity \( K = 50\,000\) fish.\n\nTwo harvesting strategies are considered:\n\nA) A constant harvest of \( H\) fish each month.\n\nB) A harvest of a fixed proportion \( p\) of the population each month, i.e. \( H_t = p\,N_t\).\n\n(i) Find the maximum constant harvest \( H_{\max}\) (fish per month) that allows the population to persist indefinitely.\n\n(ii) Choose the proportion \( p\) for strategy B that gives the same average annual harvest (over 12 months) as the constant strategy using \( H_{\max}\).\n\nGive \( H_{\max}\) (fish per month) and \( p\) (as a percentage).