Math  /  Calculus

QuestionUse the Beverton-Holt model R(Nt)=R01+aNtR(N_{t})=\frac{R_{0}}{1+a N_{t}} with a=0.05a=0.05, R0=5R_{0}=5. Find NtN_{t} for t=1,2,,5t=1,2,\ldots,5 and limtNt\lim_{t \to \infty} N_{t} for N0=2N_{0}=2.

Studdy Solution
Calculate the limit as tt approaches infinity.
t=±+0._{t} = \frac{- \pm \sqrt{ +}}{0.}t=±20._{t} = \frac{- \pm \sqrt{2}}{0.}We only consider the positive root since population cannot be negative.
t=+20.=10207.94_{t} = \frac{- + \sqrt{2}}{0.} =10 - \sqrt{20} \approx7.94The population for t=,2,3,4,5t=,2,3,4,5 are approximately 2.38,2.67,2.88,3.03,3.142.38,2.67,2.88,3.03,3.14 respectively. The limit of the population as tt approaches infinity is approximately 7.947.94.

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