#Maple code for Lecture 1 of Dynamical Models in Biology (Rutgers university) , Math 336, taught by Dr. Z.
#ITALY, 12th CENTURY FIBONACCI
#A FEMALE RABBIT IS FERTILE IF SHE IS EITHER 1-year old or 2-year-old 
#
#R(n): The number of female rabbits born at year n
#
#INITIAL CODITIONS. R(0)=1, R(1)=1
#
#R(n)=R(n-1)+R(n-2)   SECOND-ORDER RECURRENCE  (aka DIFFERENCE) EQUATION
#R(0)=1, R(1)=1; R(2)=2; R(3)=3; R(4)=5; R(5)=8; R(6)=13; R(7)=21; R(8)=34;

R:=proc(n) option remember:
if n=0 or n=1 then
 1: #INITIAL CONDITIONS
else
R(n-1)+R(n-2): #REcurrence
fi:
end:
#A MORE REALISTIC MODEL
#
#ASSUMPTIONS: (i) ONLY 1-year old and two-year old females can have  children
#
#The prob. that a 1-year-old female has children is p1
#
#The prob. that a 2-year-old female has children is p2
#
#Rg(0)=c0; Rg(1)=c1; INITIAL CONDITIONS;
#Rg(n,p1,p2,c0,c1): THE EXPECTED NUMEBR OF FEMALES born at time n
#
Rg:=proc(n,p1,p2,c0,c1) option remember:
if n=0 then
 c0: 
elif n=1 then
  c1:
else
expand(p1*Rg(n-1,p1,p2,c0,c1)+p2*Rg(n-2,p1,p2,c0,c1)): #Recurrence
fi:
end: