Question 1: A population of butterflies lives on a meadow, surrounded by forest. We want to investigate the dynamics of the population. We denote the number of butterflies at the beginning of season t by xt . Over the course of a season, 30% of the butterflies that were there at the beginning die. During each season, 20 new butterflies arrive from other meadows. (a) Write the DTDS for the number of butterflies. What is the updating function? (b) Starting with 40 butterflies in season 0, calculate their number in seasons 1, 2, 3. (c) Calculate the fixed point of the DTDS.
Question 1: A population of butterflies lives on a meadow, surrounded by forest. We want to investigate the dynamics of the population. We denote the number of butterflies at the beginning of season t by xt . Over the course of a season, 30% of the butterflies that were there at the beginning die. During each season, 20 new butterflies arrive from other meadows.
(a) Write the DTDS for the number of butterflies. What is the updating function?
(b) Starting with 40 butterflies in season 0, calculate their number in seasons 1, 2, 3.
(c) Calculate the fixed point of the DTDS.
(d) Write the solution of the DTDS in terms of a general initial condition x0.
(e) Draw the cobweb for this DTDS, starting at x0 = 40. Also draw the solution as a function of time.
(f) Suppose that through some conservation measures, we can improve the quality of the pond and reduce the death rate of the butterflies. To which level do we have to reduce the death rate if we want the steady state butterfly population to be 100?
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