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,3636% of the butterflies that were there at the beginning die. During each season,2525new butterflies per square kilometer arrive from other meadows.

a)The number of butterflies per square kilometer can be described by a linear DTDS of the formxt+1=f(xt)xt+1=f(xt), wherextxtis the number of butterflies per square kilometer at the beginning of seasontt. Find the updating functionff, expressed as a function ofxx.

Answer:f(x)=f(x)=__________

b)If at the beginning of seasont=0t=0we have115115butterflies 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:x1=x1=____________,

x2=x2=____________

and

x3=x3=____________.

c)Find the fixed pointxxof this linear DTDS. Give the exact answer; do not give a rounded decimal answer.

Answer:x=x=__________

d)Write the general solution of the DTDS with the initial conditionx0=115x0=115, expressed as a function oftt.

Answer:xt=xt=__________