Download N t+1 - Sara Parr Syswerda

FW364 Ecological Problem Solving
Class 13: Age Structure
Oct. 21, 2013
Outline for Today
Continue to make population growth models more realistic
by adding in age structure
Objectives for Today:
Continue with age structure
Matrix algebra primer
Introduce the Leslie Matrix
Objectives for Next Class:
Complete in class examples of age structured
population growth
Text (optional reading):
Chapter 4: Sections 4.1 – 4.5
Not covering life table information (sections 4.6+)
Recap from Last Class
Worked with helmeted honeyeater age-structured data
Table 4.1
General survival rate equation
(for age class, x):
xSt
= x+1Nt+1 / xNt
Given the assumption that
1F = 2F =… = 9F
General fecundity equation
(for all adult age classes):
adult xFt
= 0Nt+1 / adult xNt
Fecundity is the average # offspring per individual for each age class
Recap from Last Class
(xS)
Table 4.1
1995?
0N1995
1N1995
2N1995
3N1995
adult xF
= 0.48
Using the general survival and fecundity equations,
we developed a series of equations to predict population size for
each age class from one year to the next
Recap from Last Class
(xS)
Table 4.1
1995?
0N1995
1N1995
2N1995
3N1995
adult xF
= 0.48
How many age 0 individuals will there be in 1995?
General equation:
0Nt+1
= 0F * 0Nt + 1F * 1Nt + 2F * 2Nt + 3F * 3Nt
For honeyeaters, we are assuming: 0F = 0 and 1F = 2F = 3F = 4F =
0N1995
= adult xF * 1N1994 + adult xF * 2N1994 + adult xF * 3N1994
0N1995
= 0.48 * 20 + 0.48 * 14 + 0.48 * 10 = 21 age 0 honeyeaters
adult xF
Recap from Last Class
(xS)
Table 4.1
1995?
21
1N1995
2N1995
3N1995
adult xF
= 0.48
How many age 1 individuals will there be in 1995?
General equation:
1Nt+1
= 0S * 0Nt
For honeyeaters,
1N1995
= 0S * 0N1994
1N1995
= 0.703 * 29 = 20 age 1 honeyeaters
Used a similar process
to calculate N1995
for other age classes
Recap from Last Class
(xS)
Table 4.1
1995?
21
20
14
11
adult xF
= 0.48
Summary of Last Class:
We used four different equations to forecast age-structured population
size from one year to the next
Mentioned a method to manipulate and organize equations that is very
helpful in this situation…. MATRIX ALGEBRA!
Today!
Matrix Algebra
Matrix algebra is a very powerful technique to solve
related sets of equations
X
A B C
D E F
G H I
*
Y
Z
=
A*X + B*Y + C*Z
1 2 4
D*X + E*Y + F*Z
4 5 7
G*X + H*Y + I *Z
7 8 0
1
*
2
3
 Matrix algebra introduction / refresher
17
=
35
23
Matrix Algebra
Matrix Basics:
A matrix is a table of numbers arranged in rows and columns
(kind of like a spreadsheet)
Column 2
Column 1 Column 3
Row 1
Row 2
Row 3
1 2 4
Matrices have “dimensions”, expressed as:
# rows x # columns
4 5 7
This matrix is a 3 x 3 matrix
7 8 0
(“three-by-three matrix”)
Called a “square” matrix
because # rows = # columns
Each number in the matrix is called an element
Matrix Algebra
Matrix Basics:
A matrix is a table of numbers arranged in rows and columns
(kind of like a spreadsheet)
Column 2
Column 1 Column 3
Can have single column matrices:
17
Row 1
Row 2
Row 3
1 2 4
35
4 5 7
23
7 8 0
Column matrices are called column vectors
The above column vector has
dimensions of 3 x 1
Matrix Algebra
We can multiply matrices together
All of our ecological examples will involve
square matrices multiplied by column vectors
 Focus of my review
General Form:
Square matrix x Column vector = Column vector
1 2 4
4 5 7
7 8 0
1
*
2
3
17
=
35
23
 I’ll call the resulting column vector the “results vector”
Matrix Algebra
Order matters for matrix algebra!
1 2 4
4 5 7
7 8 0
1
*
2
3
1
≠
2
3
1 2 4
*
4 5 7
7 8 0
Our matrix multiplication
will use this order
Let’s do some matrix multiplication!
Matrix Algebra
General process:
A B
C D
*
X
Y
=
A*X + B*Y
C*X + D*Y
Step 1:
Multiply the first element in the first matrix row
by the top element in the column vector
Multiply the second element in the first matrix row
by the second element in the column vector
Add these products together to get top element in
results vector
Matrix Algebra
General process:
A B
C D
*
X
Y
=
A*X + B*Y
C*X + D*Y
Step 2:
Multiply the first element in the second matrix row
by the top element in the column vector
Multiply the second element in the second matrix row
by the second element in the column vector
Add these products together to get bottom element in
results vector
Matrix Algebra
Now with numbers:
1 2
3 4
*
100
10
=
1*100 + 2*10
3*100 + 4*10
=
120
340
Step 2:
Multiply the first element in the first matrix row
by the top element in the column vector
Multiply the second element in the first matrix row
by the second element in the column vector
Add these products together to get top element in
results vector
Matrix Algebra
Now with numbers:
1 2
3 4
*
100
10
=
1*100 + 2*10
3*100 + 4*10
=
120
340
Step 2:
Multiply the first element in the second matrix row
by the top element in the column vector
Multiply the second element in the second matrix row
by the second element in the column vector
Add these products together to get bottom element in
results vector
Matrix Algebra
The same process is used for larger matrices:
1 2 4
4 5 7
7 8 0
1
*
2
3
1*1 + 2*2 + 4*3
=
4*1 + 5*2 + 7*3
7*1 + 8*2 + 0*3
17
=
35
23
Note:
The results vector will always
have the same dimensions as the
column vector being multiplied
Matrix Algebra
The same process is used for larger matrices:
1 2 4
4 5 7
1
*
7 8 0
2
1*1 + 2*2 + 4*3
=
4*1 + 5*2 + 7*3
7*1 + 8*2 + 0*3
3
In-class exercise:
9 8 7
6 5 4
3 2 1
5
*
6
7
=
?
17
=
35
23
Matrix Algebra
The same process is used for larger matrices:
1 2 4
4 5 7
1
*
7 8 0
2
1*1 + 2*2 + 4*3
=
4*1 + 5*2 + 7*3
17
=
7*1 + 8*2 + 0*3
3
35
23
In-class exercise:
9 8 7
6 5 4
3 2 1
9*5 + 8*6 + 7*7
5
*
6
7
=
6*5 + 5*6 + 4*7
3*5 + 2*6 + 1*7
Let’s take this matrix algebra primer
and apply it to population growth
142
=
88
34
Leslie Matrix
We can take the equations we built last class:
Results vector of
age-structured
population size
at time t+1
0Nt+1
= 0F * 0Nt + 1F * 1Nt + 2F * 2Nt + 3F * 3Nt
1Nt+1
= 0S * 0Nt
2Nt+1
= 1S * 1Nt
3Nt+1
= 2S * 2Nt
Column vector of
age-structured
population size
at time t
and organize them using matrices:
0Nt+1
0F 1F 2F 3 F
0Nt
1Nt+1
0S
1Nt
2Nt+1
3Nt+1
=
0 0 0
0 1S 0 0
0
0 2S 0
*
2Nt
3Nt
Leslie Matrix
We can take the equations we built last class:
Total #
offspring
0Nt+1
= 0F * 0Nt + 1F * 1Nt + 2F * 2Nt + 3F * 3Nt
1Nt+1
= 0S * 0Nt
Survival from age 0 to age 1
2Nt+1
= 1S * 1Nt
Survival from age 1 to age 2
3Nt+1
= 2S * 2Nt
Survival from age 2 to age 3
and organize them using matrices:
0Nt+1
0F 1F 2F 3 F
0Nt
1Nt+1
0S
1Nt
2Nt+1
3Nt+1
=
0 0 0
0 1S 0 0
0
0 2S 0
*
2Nt
3Nt
Leslie Matrix
0Nt+1
0F 1F 2F 3 F
0Nt
1Nt+1
0S
1Nt
2Nt+1
3Nt+1
=
0 0 0
0 1S 0 0
0
*
3Nt
0 2S 0
Leslie Matrix (L):
Matrix of survival and fecundity rates
2Nt
A Leslie matrix is specific to
the population being studied
(note: matrix and vector abbreviations are denoted in bold)
The matrix multiplication above can be written as simply:
Nt+1 = L Nt
Look familiar?
Where L is the Leslie matrix,
and Nt and Nt+1 are vectors of age-structured population sizes
 Nt+1 = λ Nt
Leslie Matrix
0Nt+1
0F 1F 2F 3 F
0Nt
1Nt+1
0S
1Nt
2Nt+1
=
3Nt+1
0 0 0
*
0 1S 0 0
0
2Nt
3Nt
0 2S 0
Remember, ORDER MATTERS for matrix multiplication!
0Nt+1
0Nt
0F 1F 2F 3F
1Nt+1
1Nt
0S
2Nt+1
3Nt+1
=
2Nt
3Nt
*
0 0 0
0 1S 0 0
0
0 2S 0
Leslie Matrix
0Nt+1
0F 1F 2F 3 F
0Nt
1Nt+1
0S
1Nt
2Nt+1
3Nt+1
=
0 0 0
0 1S 0 0
0
0 2S 0
*
2Nt
3Nt
Fecundities are always in the first row of the Leslie matrix
When an age-class (e.g., age 0) has 0 fecundity,
the Leslie matrix is sometimes written as:
0 1F 2F 3F
0S
0 0 0
0 1S 0 0
0
0 2S 0
Leslie Matrix
0Nt+1
0F 1F 2F 3 F
0Nt
1Nt+1
0S
1Nt
2Nt+1
3Nt+1
=
0 0 0
0 1S 0 0
0
*
0 2S 0
2Nt
3Nt
Survival rates are on the sub-diagonal
When we talk about stage structure (next week),
we will have values on the diagonal
For age structure, the diagonal (aside from 0F) will always be zero
Leslie Matrix
A good way to remember where fecundities and survivals go
is to label the matrix positions (i.e., elements)
Each column represents an age class at time t
Each row represents an age class at time t+1
Time, t
0
1 2
3
0
0F 1F 2F 3F
1
0S
0 0 0
2
0 1S 0 0
3
0
0 2S 0
Time, t + 1
Time, t + 1
Age
Time, t
Age
0
1
2
3
0
0,0
1,0
2,0
3,0
1
0,1
1,1
2,1
3,1
2
0,2
1,2
2,2
3,2
3
0,3
1,3
2,3
3,3
Positions are labeled with the column first
(makes sense: time t first)
Leslie Matrix
Each labeled element represents a transition (from time t to t+1):
0, 0 : Transition from age 0 to age 0
0, 1 : Transition from age 0 to age 1
0, 2 : Transition from age 0 to age 2
Possible? Yes, if age 0 reproduce: 0F
Possible? Absolutely! 0S
Possible? No (can’t age 2 years in 1 year)
Time, t
0
1 2
3
0
0F 1F 2F 3F
1
0S
0 0 0
2
0 1S 0 0
3
0
0 2S 0
Time, t + 1
Time, t + 1
Age
Time, t
Age
0
1
2
3
0
0,0
1,0
2,0
3,0
1
0,1
1,1
2,1
3,1
2
0,2
1,2
2,2
3,2
3
0,3
1,3
2,3
3,3
Positions are labeled with the column first
(makes sense: time t first)
Leslie Matrix
A few more transitions:
1, 0 : Transition from age 1 to age 0
1, 1 : Transition from age 1 to age 1
Possible? Yes, if age 1 reproduce: 1F
Possible? Not for age structure
(possible for stage structure)
Time, t
0
1 2
3
0
0F 1F 2F 3F
1
0S
0 0 0
2
0 1S 0 0
3
0
0 2S 0
Time, t + 1
Time, t + 1
Age
Time, t
Age
0
1
2
3
0
0,0
1,0
2,0
3,0
1
0,1
1,1
2,1
3,1
2
0,2
1,2
2,2
3,2
3
0,3
1,3
2,3
3,3
Positions are labeled with the column first
(makes sense: time t first)
Leslie Matrix
One more transition:
3, 3 : Transition from age 3 to age 3
Not possible for age structure
Assume that the oldest age class (age 3 here) dies before next time step
We will relax this assumption for stage structure
Time, t
0
1 2
3
0
0F 1F 2F 3F
1
0S
0 0 0
2
0 1S 0 0
3
0
0 2S 0
Time, t + 1
Time, t + 1
Age
Time, t
Age
0
1
2
3
0
0,0
1,0
2,0
3,0
1
0,1
1,1
2,1
3,1
2
0,2
1,2
2,2
3,2
3
0,3
1,3
2,3
3,3
Positions are labeled with the column first
(makes sense: time t first)
Leslie Matrix
Leslie matrix includes all mathematically possible transitions
including individuals getting younger, e.g., 2,1 3,1 3,2
The impossible transitions have values of zero in Leslie matrix
Time, t
0
1 2
3
0
0F 1F 2F 3F
1
0S
0 0 0
2
0 1S 0 0
3
0
0 2S 0
Time, t + 1
Time, t + 1
Age
Time, t
Age
0
1
2
3
0
0,0
1,0
2,0
3,0
1
0,1
1,1
2,1
3,1
2
0,2
1,2
2,2
3,2
3
0,3
1,3
2,3
3,3
Positions are labeled with the column first
(makes sense: time t first)
Leslie Matrix
To summarize: There are two ways that biologically possible transitions
can occur in age-structured models
1) Cohorts age (the sub-diagonals)  proportion aging is the
survival rate (always < 1)
2) Cohorts reproduce (first row)
 per capita fecundity rates ( ≥ 0)
Time, t
0
1 2
3
0
0F 1F 2F 3F
1
0S
0 0 0
2
0 1S 0 0
3
0
0 2S 0
Time, t + 1
Time, t + 1
Age
Time, t
Age
0
1
2
3
0
0,0
1,0
2,0
3,0
1
0,1
1,1
2,1
3,1
2
0,2
1,2
2,2
3,2
3
0,3
1,3
2,3
3,3
Positions are labeled with the column first
(makes sense: time t first)
Honeyeater Leslie Matrix
Let’s build the Leslie matrix
for the honeyeaters!
Honeyeater Leslie Matrix
Honeyeater xS and xF
(xS)
0F = 0
1F = 2F = … = 9F = 0.48
General Honeyeater Leslie matrix
0F
1F
2F
3F
4F
5F
6F
7F
8F
9F
0S
0
0
0
0
0
0
0
0
0
0
1S
0
0
0
0
0
0
0
0
0
0
2S
0
0
0
0
0
0
0
0
0
0
3S
0
0
0
0
0
0
0
0
0
0
4S
0
0
0
0
0
0
0
0
0
0
5S
0
0
0
0
0
0
0
0
0
0
6S
0
0
0
0
0
0
0
0
0
0
7S
0
0
0
0
0
0
0
0
0
0
8S
0
Honeyeater Leslie Matrix
0
0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48 0.48
0.703
0
0
0
0
0
0
0
0
0
0
0.717
0
0
0
0
0
0
0
0
0
0
0.751
0
0
0
0
0
0
0
0
0
0
0.769
0
0
0
0
0
0
0
0
0
0
0.746
0
0
0
0
0
0
0
0
0
0
0.717
0
0
0
0
0
0
0
0
0
0
0.806
0
0
0
0
0
0
0
0
0
0
0.778
0
0
0
0
0
0
0
0
0
0
0.667
0
We can now forecast age-structured population growth
given the age distribution of the population (i.e., a column vector, Nt)!
We’ll do an example next class
Looking Ahead
Next Class:
Continue with Age Structure
Practice forecasting growth