Advanced level maths

Aleaeia is an ecologist working for Mediterranean fishing authorities. She is interested in whether the mackerel population density ls likely to fall below 5000 mackerel per loll". as this Is the minimum value required for sustainable fishing. She believes that the primary factor affecting the mackerel population Is the interaction of mackerel with sharks. their main predator. The population densities of mackerel (M thousands per km') and sharks (5 per km') in the Mediterranean Sea are modelled by the coupled differential equations: where r is measured in years. and a. . 3r and 6 are parameters. This model assumes that no other factors affect the mackerel or shark population densities. The term all! models the population growth rate of the mackerel in the absence of sharks. The term MS models the death rate of the mackerel due to being eaten by sharks. (a) Suggest similar interpretations tor the lollowing terms. 0) J'MS (ii) 63 dd! (b) An equilibrium point is a set of values of M and S. such that W = 0 and '15 a = - Given that both species are present at the equilibrium point. (i) show that. at the equilibrium point. the value of the mackerel population density is g: [3] (ii) nd the value of the shark population density at the equilibrium point. (e) The equilibrium point found in part (b) gives the average values of M and 3 over time. Use the model to predict how the following events would affect the average value of M. Justify your answers. (i) Toxic sewage is added to the Mediterranean Sea. Nessie claims this reduces the shark population growth rate and hence the value of y is halved. No other parameter changes. (ii) Global warming Increases the temperature of the Mediterranean Sea. Nessie claims that this promotes the mackerel population growth rate and hence the value of a is doubled. No other parameter changes. [1] [1] [2] [2] [2]