# Applied maths

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❼ Include all the necessary mathematical details.

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8 1. The Krill which are tiny shrimplike creatures are the principal food source for the huge blue whale.

The intrinsic growth rate and the carrying capacity of the blue whale are estimated at 5% per year

and 150,000 respectively. The maximum sustainable population for krill is 600 tons/acre. In the

absence of predators, in uncrowded conditions, the krill population grows at a rate of 20% per year.

The pres-ence of 600 tons/acre of krill increases blue whale population growth rate by 2.5% per year,

and the presence of 150,000 blue whales decreases krill growth rate by 8% per year.

(a) Determine all the steady states of the system and analyze theri stability. Can the blue whales

and the krill coexist?

(b) Verify your conclusion of stability by plotting the solution and the vector fields for selected

values of initial populations.

(c) Describe what happens to the two populations over time. Assume that we start o with 10,000

blue whales and 500 tons/acre of krill.

(d) How sensitive are your conclusions in to the assumption of 20% growth rate per year for krill?i