Stage-structured Populations Brook Milligan Department of Biology - - PowerPoint PPT Presentation

Stage-structured Populations

Brook Milligan

Department of Biology New Mexico State University Las Cruces, New Mexico 88003 brook@nmsu.edu

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 equivalent Different 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 are equivalent Different genotypes of the same stage 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 are equivalent Different genotypes of the same stage are equivalent

Closed population Resources are unlimited Stages not strictly ordered

Transitions to “previous” (e.g., smaller) stages are possible For example, plants categoried by size can become smaller

• ccasionally

Brook Milligan Stage-structured Populations

Projecting Age-Structured Populations: Life Cycle Graph

P0,k−1

N0 N1 N2 Nk−1 Nk

P10 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 Nk

P10 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 . . . P21 . . . . . . . . . . . . ... . . . Pk,k−1        N =        N0 N1 N2 . . . Nk       

Brook Milligan Stage-structured Populations

Projecting Age-Structured Populations

P =        P00 P01 P02 . . . P0k P10 . . . P21 . . . . . . . . . . . . ... . . . Pk,k−1        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 . . . P21 . . . . . . . . . . . . ... . . . Pk,k−1        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.

• 1

2

• 2

3 1

• 

 π 2 0.4     4 5 7 8 9 6   The numbers which compose a matrix are called its elements. Each horizontal string of elements is called a row and each vertical string is a column. The rows of a matrix are assigned numbers (starting with one) from the top down and the columns are assigned numbers from left to right. Hence, each element of a matrix is specified by noting the row and column (in that order) to which 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 jth column. The matrix A in (7) has m rows and n columns and is refered to as an “m by n” (written m × n) matrix; note that the number of rows is always given first.

Brook Milligan Stage-structured Populations

Simple Matrix Operations

Equality Two matrices, say A = (aij) and B = (bij), are equal if they have the same dimensions (i.e., the same number of rows and columns) and if aij = bij for every i and j (i.e., elements in the corresponding positions are equal).

Brook Milligan Stage-structured Populations

Simple Matrix Operations

Addition and Subtraction Only matrices of equal dimensions can be added or subtracted. We define A + B = (aij + bij) and A − B = (aij − bij). That is, these operations are defined as addition (subtraction) of the corresponding elements.

Brook Milligan Stage-structured Populations

Simple Matrix Operations

Addition and Subtraction Only matrices of equal dimensions can be added or subtracted. We define A + B = (aij + bij) and A − B = (aij − bij). That is, these operations are defined as addition (subtraction) of the corresponding elements. Example 1 2 3 4

• +

3 4 5 6

• =

1 + 3 2 + 4 3 + 5 4 + 6

• =

4 6 8 10

• 1

2 3 4

• −

3 4 5 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, the product cA = Ac is defined by cA = (caij), i.e., multiply each element of A by c.

Brook Milligan Stage-structured Populations

Simple Matrix Operations

Scalar multiplication If c is a number and A is a matrix, the product cA = Ac is defined by cA = (caij), i.e., multiply each element of A by c. Example 12 1 2 3 4

• =

12 24 36 48

• Note: scalar multiplication and addition (subtraction) are

distributive, so c(A + B) = cA + cB.

Brook Milligan Stage-structured Populations

Simple Matrix Operations

Transpose The transpose of a matrix A is obtained by interchanging its rows and columns and is denoted A⊤. Hence, the i, jth element of A⊤ is the j, ith element of A. If A is an m × n matrix, A⊤ is n × m.

Brook Milligan Stage-structured Populations

Simple Matrix Operations

Transpose The transpose of a matrix A is obtained by interchanging its rows and columns and is denoted A⊤. Hence, the i, jth element of A⊤ is the j, ith element of A. If A is an m × n matrix, A⊤ is n × m. Example 1 2 3 4 ⊤ = 1 3 2 4

• 

 6 7 9 10 2 1  

⊤

= 6 9 2 7 10 1

• Brook Milligan

Stage-structured Populations

Matrix Multiplication

The basic operation in matrix multiplication is multiplying a column vector by a row vector, element by element, then summing the products. This procedure is defined only when the column vector and row vector have the same number of elements.

Brook Milligan Stage-structured Populations

Matrix Multiplication

The basic operation in matrix multiplication is multiplying a column vector by a row vector, element by element, then summing the products. This procedure is defined only when the column vector 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

• 1

2

• 

 1 2 3   = 0 · 1 + 1 · 2 + 2 · 3 = 8 and

• 1

2 1 2

• is not defined.

Order is important in this procedure. As we’ll see ba is also defined but the result is quite different.

Brook Milligan Stage-structured Populations

Matrix Multiplication

A general requirement in multiplying matrices is that the number

• f columns of the matrix on the left equal the number of rows of

the matrix on the right. When this condition holds, the i, jth element of the product AB is defined as the product of the ith row in 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 2 3 4 1 4

• =

1 · 1 + 2 · 4 3 · 1 + 4 · 4

• =

9 19

• Brook Milligan

Stage-structured Populations

Matrix Multiplication

Example 1 2 3 4 1 4

• =

1 · 1 + 2 · 4 3 · 1 + 4 · 4

• =

9 19

• Example

1 2 3 4 5 6   1 2 3 4 5 6   =

Brook Milligan Stage-structured Populations

Matrix Multiplication

Example 1 2 3 4 1 4

• =

1 · 1 + 2 · 4 3 · 1 + 4 · 4

• =

9 19

• Example

1 2 3 4 5 6   1 2 3 4 5 6   = 22 28 49 64

• Brook Milligan

Stage-structured Populations

Matrix Multiplication

Example 6 1 2 4 2 1 2 1

• =

Brook Milligan Stage-structured Populations

Matrix Multiplication

Example 6 1 2 4 2 1 2 1

• =

14 7 12 6

• Brook Milligan

Stage-structured Populations

Matrix Multiplication

Example 6 1 2 4 2 1 2 1

• =

14 7 12 6

• Example

2 1 2 1 6 1 2 4

• =

Brook Milligan Stage-structured Populations

Matrix Multiplication

Example 6 1 2 4 2 1 2 1

• =

14 7 12 6

• Example

2 1 2 1 6 1 2 4

• =

14 6 14 6

• Brook Milligan

Stage-structured Populations

Matrix Multiplication

Example 6 1 2 4 2 1 2 1

• =

14 7 12 6

• Example

2 1 2 1 6 1 2 4

• =

14 6 14 6

• These examples illustrate that matrix multiplication is

noncommutative; i.e., AB = BA is often false.

Brook Milligan Stage-structured Populations

Matrix Multiplication

Example   1 2 3   4 5 6

• =

Brook Milligan Stage-structured Populations

Matrix Multiplication

Example   1 2 3   4 5 6

• =

  4 5 6 8 10 12 12 15 18  

Brook Milligan Stage-structured Populations

Matrix Multiplication

Example   1 2 3   4 5 6

• =

  4 5 6 8 10 12 12 15 18   Example

• 4

5 6

• 

 1 2 3   =

Brook Milligan Stage-structured Populations

Matrix Multiplication

Example   1 2 3   4 5 6

• =

  4 5 6 8 10 12 12 15 18   Example

• 4

5 6

• 

 1 2 3   = (32)

Brook Milligan Stage-structured Populations

Matrix Multiplication

Example   1 2 3   4 5 6

• =

  4 5 6 8 10 12 12 15 18   Example

• 4

5 6

• 

 1 2 3   = (32) These examples vividly illustrate that matrix multiplication is noncommutative.

Brook Milligan Stage-structured Populations

Projecting Age-Structured Populations

P =        P00 P01 P02 . . . P0k P10 . . . P21 . . . . . . . . . . . . ... . . . Pk,k−1        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)

Brook Milligan Stage-structured Populations

Projecting Age-Structured Populations

P =        P00 P01 P02 . . . P0k P10 . . . P21 . . . . . . . . . . . . ... . . . Pk,k−1        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 . . . P21 . . . . . . . . . . . . ... . . . Pk,k−1        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 Nk

P10 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 =          P01 P02 . . . P0,k−1 P0k P10 P12 . . . P20 P21 . . . . . . . . . . . . ... . . . . . . Pk−1,1 Pk−1,2 . . . Pk−1,k Pk,2 . . . Pk,k−1          N(t + 1) = P · N(t) (12)

Brook Milligan Stage-structured Populations