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,37

37% of the butterflies that were there at the beginning die. During each season,24

24new 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 formx

t+1=f(xt)xt+1=f(xt), wherext

xtis the number of butterflies per square kilometer at the beginning of seasont

t. Find the updating functionf

f, expressed as a function ofx

x.

Answer:f(x)=

f(x)=__________

b)If at the beginning of seasont=0

t=0we have100

100butterflies 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:x

1

=

x1=____________,

x2=

x2=____________

and

x3=

x3=____________.

c)Find the fixed pointx

xof 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 conditionx

0

=100

x0=100, expressed as a function oft