The population of a rare insect species in a national park is modeled by the differential equation $\frac{dP}{dt} = kP(1 - \frac{P}{M}) - H$, where $P(t)$ is the population at time $t$ (in years), $k$ is the intrinsic growth rate, $M$ is the carrying capacity of the environment, and $H$ is a constant representing the harvesting rate of insects by researchers for study.
Given that $k = 0.5$, $M = 1000$, and $H = 100$, and that the initial population $P(0) = 200$, determine the population $P(t)$ as a function of time. Analyse the long-term behaviour of the population by evaluating $\lim_{t \to \infty} P(t)$, if it exists, and interpret the result in the context of the insect population.
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Create Free Account Log inThis is a free VCE Units 3 & 4 Mathematical Methods practice question worth 7 marks, testing your understanding of Application of intergration. It falls under Calculus in Unit 3: Mathematical Methods Unit 3. Submit your answer above to receive instant AI-powered marking and personalised feedback.
Extend introductory study of functions, algebra, calculus. Focus on functions, relations, graphs, algebra, and applications of derivatives.
Covers limits, continuity, differentiability, differentiation, and anti-differentiation.
application of integration to problems involving finding a function from a known rate of change given a boundary condition, calculation of the area of a region under a curve and simple cases of areas between curves, average value of a function and other situations.
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