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Unformatted text preview: Math 17B
Kouba
Leslie Matrices ~ for Population Models with Discrete Breeding Seasons We now discuss populations with discrete breeding seasons, where reproduc
tion is limited to a particluar season of the year. For example, let’s consider
the number of female rubythroated hummingbirds in a population with an
annual breeding season of March to July. A female usually lays one clutch of
two eggs; sometimes two clutches are laid. We will assume that the average
life span of a female hummingbird is four years. We deﬁne the age of a bird
at the END of a breeding season as follows : age zero (0) : any bird that is born during the current breeding season age one (1) : any zero—year old bird which survives to the end of the next
breeding season age two (2) : any one—year old bird which survives to the end of the next
breeding season age three (3) : any two—year old bird which survives to the end of the
next breeding season Let Nx(t) represent the total number of female hummingbirds of age x at
the end of breeding season t for t 2: 0, 1, 2, 3, 4, 5,   . We make the following
assumptions about reproductive viability of female birds: age zero (0) : not yet reproductiver mature age one (1) : will produce an. average of 1.2 female offspring the next
breeding season which survive age two (2) : will produce an average of 1.5 female offspring the next
breeding season which survive age three (3) : will produce an average of 0.7 female offspring the next
breeding season which survive This can be summarized in the following equation : N0(t+1) = (1.2)N1(t)+(1.5)N2(t)+(0.7)N3(t) We make the following assumptions about the survival rates of female birds: 50% of age zero (0) females at time t survive to time t + 1 ' 7 35% of a e one 1 females at time t survive to time t+ 1 
g i 15% of age two (2) females at time t survive to time t + 1 ' 7 0% of age three (3) females at time t survive to time t+ 1 . This can be summarized in the following equations: N1(t + 1) = (0.5)N0(t) ,
N2(t + 1) = (O.35)Nl(t) ,
and N3(t + 1) = (0.15)Ng(t) . Let N (t) represent the total number of female hummingbirds at the end of
breeding season t for t = 0, 1, 2, 3, 4, 5, ~  . Using matrix notation we get the
following representations : N0“)
N10?)
N26)
N305)
by age group at the end of season if . N0(t + 1)
N1 (t + 1)
N2(t + 1)
N3(t + 1)
mingbirds by age group at the end of season 15+ 1 . 0 1.2 1.5 0.7
0 0
0 Let N (t) 2 represent the total number of female hummingbirds Let N (t + 1) 2 represent the total number of female hum .0
01
o . Combining these matrices gives 0
.0
w
01
o N0(t+ 1) 0 12 15 0 7 N0(t)
N1(t+ 1) _ 0 5 0 0 0 N1(t)
N2(t+ 1) ” 0 035 0 0 N2(t) ’le’
N3(t + 1) 0 0 015 0 N3(t) N(t + 1) = L1V(t) . The 4 by 4 matrix L is called a Leslie Matrix. ...
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This note was uploaded on 02/17/2012 for the course MATH 17B taught by Professor Kouba during the Winter '07 term at UC Davis.
 Winter '07
 Kouba

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