A population of butterflies lives in a meadow, surrounded by forest. We want to investigate the dynamics of the population. Over the course of a season, 40% of the butterflies that were there at the beginning die. During each season, 22 new butterflies per square kilometer arrive from other meadows. a) The number of butterflies per square kilometer can be described by a linear DTDS ot the form mt+1 = f(zt), where (at is the number of butterflies per square kilometer at the beginning of season t. Find the updating function f, expressed as a function of :3. Answer: f(1:) = ab % J; M 1r sin(a) [Ill 0 B b) If at the beginning of season t = 0 we have 145 butterflies per square kilometer in the meadow, compute the number of butterflies per square kilometer at the beginning of seasons 1, 2, and 3. Your answers must be accurace to at least two decimal places. Answer: 11:1 = Number , m2 = Number and :33 = Number c) Find the fixed point a" of this linear DTDS. Give the exact answer; do not give a rounded decimal answer. Answer: *= Va a T sin (a) ? d) Write the general solution of the DTDS with the initial condition x0 = 145, expressed as a function of t. Answer: + = a Va a sin (a)