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The population of salmon in a lake can be modelled by the differential equation
where t is time in years since 2019, S is the number of salmon in the lake, and r,K are positive constants.
Show that dtdS is
positive when S<K,
negative when S>K.
Hence explain why K is called the carrying capacity of the lake.
Show that the maximum value of dtdS is 12rK.
In 2019, the population S is smaller than the carrying capacity K.
By solving the differential equation, show that S(t)=1+Ae−(rt)/3K, where A is some constant.
By observing the population of salmon when food is unlimited, scientists know that the growth rate r=ln4. The population of salmon is measured to be 1000 in 2019, and 1500 in 2022.
Find the values of A and K.
Determine the number of salmon that were in the lake in 2016.
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