Stage-structured Populationsweb.nmsu.edu/~brook/courses/conservation-biology/... · 2009-11-20 ·...
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Stage-structured Populations Brook Milligan Department of Biology New Mexico State University Las Cruces, New Mexico 88003 [email protected] Fall 2009 Brook Milligan Stage-structured Populations
Transcript of Stage-structured Populationsweb.nmsu.edu/~brook/courses/conservation-biology/... · 2009-11-20 ·...
Stage-structured Populations
Brook Milligan
Department of BiologyNew Mexico State University
Las Cruces, New Mexico [email protected]
Fall 2009
Brook Milligan Stage-structured Populations
Age-Structured Populations
All individuals are not equivalent to each other
Rates of survivorship and reproduction depend on age
No other structure within the population
Individuals of different sizes but of the same age are equivalentDifferent genotypes of the same age are equivalent
Closed population
Resources are unlimited
Brook Milligan Stage-structured Populations
Stage-Structured Populations
All individuals are not equivalent to each other
Rates of survivorship and reproduction depend on stage
No other structure within the population
Individuals of different sizes but of the same stage areequivalentDifferent genotypes of the same stage are equivalent
Closed population
Resources are unlimited
Stages not strictly ordered
Transitions to “previous” (e.g., smaller) stages are possibleFor example, plants categoried by size can become smalleroccasionally
Brook Milligan Stage-structured Populations
Stage-Structured Populations
All individuals are not equivalent to each other
Rates of survivorship and reproduction depend on stage
No other structure within the population
Individuals of different sizes but of the same stage areequivalentDifferent genotypes of the same stage are equivalent
Closed population
Resources are unlimited
Stages not strictly ordered
Transitions to “previous” (e.g., smaller) stages are possibleFor example, plants categoried by size can become smalleroccasionally
Brook Milligan Stage-structured Populations
Projecting Age-Structured Populations: Life Cycle Graph
P0,k−1
N0 N1 N2 Nk−1 NkP10 P21 Pk−1,2 Pk,k−1
P01
P02
P0k
P00
Brook Milligan Stage-structured Populations
Projecting Stage-Structured Populations: Life Cycle Graph
Pk−1,k
N0 N1 N2 Nk−1 NkP10 P21 Pk−1,2 Pk,k−1
P01
P02
P0k
P0,k−1
P20 Pk−1,1 Pk2
P12
Brook Milligan Stage-structured Populations
Projecting Age-Structured Populations
Projection equations
N0(t + 1) =k∑
j=0
bjNj(t) (1)
Ni+1(t + 1) = giNi (t) (2)
gx is the age-specific survivorship
bx is the age-specific reproduction
N0(t + 1) =k∑
j=0
P0jNj(t) (3)
Ni+1(t + 1) = Pi+1,iNi (t) (4)
Brook Milligan Stage-structured Populations
Projecting Age-Structured Populations
P =
P00 P01 P02 . . . P0k
P10 0 0 . . . 00 P21 0 . . . 0...
......
. . ....
0 0 0 Pk,k−1 0
N =
N0
N1
N2...
Nk
Brook Milligan Stage-structured Populations
Projecting Age-Structured Populations
P =
P00 P01 P02 . . . P0k
P10 0 0 . . . 00 P21 0 . . . 0...
......
. . ....
0 0 0 Pk,k−1 0
N(t) =
N0(t)N1(t)N2(t)
...Nk(t)
Brook Milligan Stage-structured Populations
Projecting Age-Structured Populations
P =
P00 P01 P02 . . . P0k
P10 0 0 . . . 00 P21 0 . . . 0...
......
. . ....
0 0 0 Pk,k−1 0
N(t) =
N0(t)N1(t)N2(t)
...Nk(t)
N0(t + 1) =k∑
j=0
P0jNj(t) (5)
Ni+1(t + 1) = Pi+1,iNi (t) (6)
Brook Milligan Stage-structured Populations
Matrices
A matrix is a rectangular array of numbers enclosed in brackets.
Example
The following are examples of matrices.
(0 1 2
) (0 23 1
) π2
0.4
4 57 89 6
The numbers which compose a matrix are called its elements.Each horizontal string of elements is called a row and each verticalstring is a column. The rows of a matrix are assigned numbers(starting with one) from the top down and the columns areassigned numbers from left to right. Hence, each element of amatrix is specified by noting the row and column (in that order) towhich it belongs.
Brook Milligan Stage-structured Populations
Matrices
A general matrix can be represented as
A =
a11 a12 . . . a1n
a21 a22 . . . a2n...
.... . .
...am1 am2 . . . amn
(7)
and the i , jth element, aij , is the element in the ith row and jthcolumn.
The matrix A in (7) has m rows and n columns and is refered to asan “m by n” (written m× n) matrix; note that the number of rowsis always given first.
Brook Milligan Stage-structured Populations
Simple Matrix Operations
Equality Two matrices, say A = (aij) and B = (bij), are equal ifthey have the same dimensions (i.e., the same number of rows andcolumns) and if aij = bij for every i and j (i.e., elements in thecorresponding positions are equal).
Brook Milligan Stage-structured Populations
Simple Matrix Operations
Addition and Subtraction Only matrices of equal dimensions canbe added or subtracted.We define A + B = (aij + bij) and A− B = (aij − bij). That is,these operations are defined as addition (subtraction) of thecorresponding elements.
Example
(1 23 4
)+
(3 45 6
)=
(1 + 3 2 + 43 + 5 4 + 6
)=
(4 68 10
)(
1 23 4
)−
(3 45 6
)=
(−2 −2−2 −2
)Note: the order of addition makes no difference: A + B = B + A(i.e., addition is commutative).
Brook Milligan Stage-structured Populations
Simple Matrix Operations
Addition and Subtraction Only matrices of equal dimensions canbe added or subtracted.We define A + B = (aij + bij) and A− B = (aij − bij). That is,these operations are defined as addition (subtraction) of thecorresponding elements.
Example
(1 23 4
)+
(3 45 6
)=
(1 + 3 2 + 43 + 5 4 + 6
)=
(4 68 10
)(
1 23 4
)−
(3 45 6
)=
(−2 −2−2 −2
)Note: the order of addition makes no difference: A + B = B + A(i.e., addition is commutative).
Brook Milligan Stage-structured Populations
Simple Matrix Operations
Scalar multiplication If c is a number and A is a matrix, theproduct cA = Ac is defined by cA = (caij), i.e., multiply eachelement of A by c .
Example
12
(1 23 4
)=
(12 2436 48
)Note: scalar multiplication and addition (subtraction) aredistributive, so c(A + B) = cA + cB.
Brook Milligan Stage-structured Populations
Simple Matrix Operations
Scalar multiplication If c is a number and A is a matrix, theproduct cA = Ac is defined by cA = (caij), i.e., multiply eachelement of A by c .
Example
12
(1 23 4
)=
(12 2436 48
)Note: scalar multiplication and addition (subtraction) aredistributive, so c(A + B) = cA + cB.
Brook Milligan Stage-structured Populations
Simple Matrix Operations
Transpose The transpose of a matrix A is obtained byinterchanging its rows and columns and is denoted A>. Hence, thei , jth element of A> is the j , ith element of A. If A is an m × nmatrix, A> is n ×m.
Example
(1 23 4
)>=
(1 32 4
) 6 7
9 102 1
>
=
(6 9 27 10 1
)
Brook Milligan Stage-structured Populations
Simple Matrix Operations
Transpose The transpose of a matrix A is obtained byinterchanging its rows and columns and is denoted A>. Hence, thei , jth element of A> is the j , ith element of A. If A is an m × nmatrix, A> is n ×m.
Example
(1 23 4
)>=
(1 32 4
) 6 7
9 102 1
>
=
(6 9 27 10 1
)
Brook Milligan Stage-structured Populations
Matrix Multiplication
The basic operation in matrix multiplication is multiplying acolumn vector by a row vector, element by element, then summingthe products. This procedure is defined only when the columnvector and row vector have the same number of elements.
In general, here’s how it works.Let a = (a1, a2, . . . an) and b = (b1, b2, . . . bn)
>, then
ab = (a1, a2, . . . an)
b1
b2...
bn
= a1b1+a2b2+· · ·+anbn =n∑
k=1
akbk .
Brook Milligan Stage-structured Populations
Matrix Multiplication
The basic operation in matrix multiplication is multiplying acolumn vector by a row vector, element by element, then summingthe products. This procedure is defined only when the columnvector and row vector have the same number of elements.In general, here’s how it works.Let a = (a1, a2, . . . an) and b = (b1, b2, . . . bn)
>, then
ab = (a1, a2, . . . an)
b1
b2...
bn
= a1b1+a2b2+· · ·+anbn =n∑
k=1
akbk .
Brook Milligan Stage-structured Populations
Matrix Multiplication
Example
(0 1 2
) 123
= 0 · 1 + 1 · 2 + 2 · 3 = 8
and
(0 1 2
) (12
)is not defined.
Order is important in this procedure. As we’ll see ba is also definedbut the result is quite different.
Brook Milligan Stage-structured Populations
Matrix Multiplication
A general requirement in multiplying matrices is that the numberof columns of the matrix on the left equal the number of rows ofthe matrix on the right. When this condition holds, the i , jthelement of the product AB is defined as the product of the ith rowin A and the jth column in B. Let
A =
a11 a12 . . . a1n
a21 a22 . . . a2n...
.... . .
...am1 am2 . . . amn
=
a1
a2...
am
and
B =
b11 b12 . . . b1l
b21 b22 . . . b2l...
.... . .
...bn1 bn2 . . . bnl
=(
b1 b2 . . . bl
).
Brook Milligan Stage-structured Populations
Matrix Multiplication
Then
AB =
a1b1 a1b2 . . . a1bl
a2b1 a2b2 . . . a2bl...
.... . .
...amb1 amb2 . . . ambl
.
Thus the i , jth element of AB, denote it abij , is given by
abij = aibj =(
ai1 ai2 . . . ain
)
bj1
bj2...
bjn
=n∑
k=1
aikbkj .
Note that AB is an m× l matrix, i.e., (m× n)× (n× l) → (m× l).
Brook Milligan Stage-structured Populations
Matrix Multiplication
Example (1 23 4
) (14
)=
(1 · 1 + 2 · 43 · 1 + 4 · 4
)=
(919
)
Example (1 2 34 5 6
) 1 23 45 6
=
(22 2849 64
)
Brook Milligan Stage-structured Populations
Matrix Multiplication
Example (1 23 4
) (14
)=
(1 · 1 + 2 · 43 · 1 + 4 · 4
)=
(919
)Example (
1 2 34 5 6
) 1 23 45 6
=
(22 2849 64
)
Brook Milligan Stage-structured Populations
Matrix Multiplication
Example (1 23 4
) (14
)=
(1 · 1 + 2 · 43 · 1 + 4 · 4
)=
(919
)Example (
1 2 34 5 6
) 1 23 45 6
=
(22 2849 64
)
Brook Milligan Stage-structured Populations
Matrix Multiplication
Example (6 12 4
) (2 12 1
)=
(14 712 6
)
Example (2 12 1
) (6 12 4
)=
(14 614 6
)
These examples illustrate that matrix multiplication isnoncommutative; i.e., AB = BA is often false.
Brook Milligan Stage-structured Populations
Matrix Multiplication
Example (6 12 4
) (2 12 1
)=
(14 712 6
)
Example (2 12 1
) (6 12 4
)=
(14 614 6
)
These examples illustrate that matrix multiplication isnoncommutative; i.e., AB = BA is often false.
Brook Milligan Stage-structured Populations
Matrix Multiplication
Example (6 12 4
) (2 12 1
)=
(14 712 6
)
Example (2 12 1
) (6 12 4
)=
(14 614 6
)
These examples illustrate that matrix multiplication isnoncommutative; i.e., AB = BA is often false.
Brook Milligan Stage-structured Populations
Matrix Multiplication
Example (6 12 4
) (2 12 1
)=
(14 712 6
)
Example (2 12 1
) (6 12 4
)=
(14 614 6
)
These examples illustrate that matrix multiplication isnoncommutative; i.e., AB = BA is often false.
Brook Milligan Stage-structured Populations
Matrix Multiplication
Example (6 12 4
) (2 12 1
)=
(14 712 6
)
Example (2 12 1
) (6 12 4
)=
(14 614 6
)
These examples illustrate that matrix multiplication isnoncommutative; i.e., AB = BA is often false.
Brook Milligan Stage-structured Populations
Matrix Multiplication
Example 123
(4 5 6
)=
4 5 68 10 1212 15 18
Example
(4 5 6
) 123
= (32)
These examples vividly illustrate that matrix multiplication isnoncommutative.
Brook Milligan Stage-structured Populations
Matrix Multiplication
Example 123
(4 5 6
)=
4 5 68 10 1212 15 18
Example
(4 5 6
) 123
= (32)
These examples vividly illustrate that matrix multiplication isnoncommutative.
Brook Milligan Stage-structured Populations
Matrix Multiplication
Example 123
(4 5 6
)=
4 5 68 10 1212 15 18
Example
(4 5 6
) 123
=
(32)
These examples vividly illustrate that matrix multiplication isnoncommutative.
Brook Milligan Stage-structured Populations
Matrix Multiplication
Example 123
(4 5 6
)=
4 5 68 10 1212 15 18
Example
(4 5 6
) 123
= (32)
These examples vividly illustrate that matrix multiplication isnoncommutative.
Brook Milligan Stage-structured Populations
Matrix Multiplication
Example 123
(4 5 6
)=
4 5 68 10 1212 15 18
Example
(4 5 6
) 123
= (32)
These examples vividly illustrate that matrix multiplication isnoncommutative.
Brook Milligan Stage-structured Populations
Projecting Age-Structured Populations
P =
P00 P01 P02 . . . P0k
P10 0 0 . . . 00 P21 0 . . . 0...
......
. . ....
0 0 0 Pk,k−1 0
N(t) =
N0(t)N1(t)N2(t)
...Nk(t)
N0(t + 1) =k∑
j=0
P0jNj(t) (8)
Ni+1(t + 1) = Pi+1,iNi (t) (9)
=k∑
j=0
Pi+1,jNj(t) (10)
Brook Milligan Stage-structured Populations
Projecting Age-Structured Populations
P =
P00 P01 P02 . . . P0k
P10 0 0 . . . 00 P21 0 . . . 0...
......
. . ....
0 0 0 Pk,k−1 0
N(t) =
N0(t)N1(t)N2(t)
...Nk(t)
N0(t + 1) =k∑
j=0
P0jNj(t) (8)
Ni+1(t + 1) = Pi+1,iNi (t) (9)
=k∑
j=0
Pi+1,jNj(t) (10)
Brook Milligan Stage-structured Populations
Projecting Age-Structured Populations
P =
P00 P01 P02 . . . P0k
P10 0 0 . . . 00 P21 0 . . . 0...
......
. . ....
0 0 0 Pk,k−1 0
N(t) =
N0(t)N1(t)N2(t)
...Nk(t)
N(t + 1) = P · N(t) (11)
Brook Milligan Stage-structured Populations
Projecting Stage-Structured Populations: Life Cycle Graph
Pk−1,k
N0 N1 N2 Nk−1 NkP10 P21 Pk−1,2 Pk,k−1
P01
P02
P0k
P0,k−1
P20 Pk−1,1 Pk2
P12
Brook Milligan Stage-structured Populations
Projecting Stage-Structured Populations
N(t) =(
N0(t) N1(t) N2(t) . . . Nk−1(t) Nk(t))>
P =
0 P01 P02 . . . P0,k−1 P0k
P10 0 P12 . . . 0 0P20 P21 0 . . . 0 0...
......
. . ....
...0 Pk−1,1 Pk−1,2 . . . 0 Pk−1,k
0 0 Pk,2 . . . Pk,k−1 0
N(t + 1) = P · N(t) (12)
Brook Milligan Stage-structured Populations
