1. (12 marks) The rate of change of a population P of an environment is determined...

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1. (12 marks) The rate of change of a population P of an environment is determined by the logistic formula P 20000 dP dt = 0.04P where t is in years since the beginning of 2015. So P(1) is the population at the beginning of 2016. Suppose P(0) = 1000. (a) Calculate P' (0). Explain what this number means. (b) Use the number from the previous part to estimate the population in the middle of 2015. That is, estimate P(0.5). (c) What assumption is made in the computation in the previous part? Use the formula given for P' to see whether or not the assumption is true, to within 1%. (d) Confirm that P' is constant to within 1% over a time interval from t = 0 to t = 1/12, that is, over 1 month. (e) Using time increments of 1 month, use Euler's method to estimate the population at the beginning of 2019, that is P(4). [Use a spreadsheet or something similar.] (f) Using time increments of 1 month, use Euler's method to determine the population over a 150 year period. Use a computer to plot the population P against t in years over the 150 year period. 1. (12 marks) The rate of change of a population P of an environment is determined by the logistic formula P 20000 dP dt = 0.04P where t is in years since the beginning of 2015. So P(1) is the population at the beginning of 2016. Suppose P(0) = 1000. (a) Calculate P' (0). Explain what this number means. (b) Use the number from the previous part to estimate the population in the middle of 2015. That is, estimate P(0.5). (c) What assumption is made in the computation in the previous part? Use the formula given for P' to see whether or not the assumption is true, to within 1%. (d) Confirm that P' is constant to within 1% over a time interval from t = 0 to t = 1/12, that is, over 1 month. (e) Using time increments of 1 month, use Euler's method to estimate the population at the beginning of 2019, that is P(4). [Use a spreadsheet or something similar.] (f) Using time increments of 1 month, use Euler's method to determine the population over a 150 year period. Use a computer to plot the population P against t in years over the 150 year period.

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Solution Given logistic formula dPdt Pt 004Pt1 Pt20000 1 condition P0 1000 a Substituting t0 in eqn ... View the full answer