The logistic model of population growth uses the initial value problem:N(1-) . N(0) = NodNdtKwhere N(t) is the population at time t, K is the carrying capacity, and r is the growth rate.a) Determine the dimensions of the constant r, and show that by introducing appropriate dimen-sionless variables N and i we can rewrite this equation in the form dNb) Find the general solution to the non-dimensionalized equation.= Ñ(1 – Ñ).

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Answered: The logistic model of population growth…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Biologists stocked a lake with 300 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 5000. The number of fish tripled in the first year. (a) Assuming that the size of the fish population satisfies the logistic equation dP = kp (1-P). (1-²), kP dt determine the constant k, and then solve the equation to find an expression for the size of the population after t years. k= P(t)= (b) How long will it take for the population to increase to 2500 (half of the carrying capacity)? It will take years.
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An observation indicates that the initial population of grey seals P(t) nearby a harbor is 25 and satisfies the logistic equation P(t)' = 0.0225P(t) - 0.0003P(t)² (with t in months.) a) Apply Euler's method together with any computer program (compulsory) to approximate the solution for 10 years. Use the step size of h = 1 and then with h = 0.5. (LT-C3) b) Find out the percentage of the limiting population of 75 grey seals has been attained after 5 years and after 10 years. (LT-C2) c) Summarize your findings in (b). (LT-C4)

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