AgeStructured Population Growth

1
Age-Structured Population Growth

• Projection Matrices

2
Age-Structured Matrices

• Some definitions
• A matrix is a shorthand way to write out an
  equation.
• It consists of n rows and n columns of numbers
• Simplifies the projection of age-structured
  populations

3
Vectors

• A vector is a matrix with just one row or column
• Vectors we are interested in are the vector (Sx)
  which describes the number of individuals in each
  age class at time x

4
Projection

• To project forward, multiply the vector of
  population size (Sx) times the age-projection
  matrix of survival and fertility probabilities
• Note-we use age classes and survival and
  fertility probabilities in matrices, ages and
  survivorship schedules and fecundity schedules in
  life-tables

5
Calculating Age Class Survival Probabilities

• Use probability of surviving from one age class
  to the next
• Similar to the gx calculations in Gotelli

6
Projecting forward
7
Calculating Age Class Fertility Probabilities

• Note that fertility probabilities include both
  the per capita reproduction and the probability
  that those individuals survive to the next age
  class

8
Projecting Forward-Number in first age class
9
The Age-Projection Matrix

• Has the fertility parameters (f1fn) in the top
  row
• The first column of row 2 has p21, the survival
  probability of going from age 1 to age 2.
• Survival probabilities of going from age class x
  to age class x1 are in a diagonal from p21 to
  the lower right.
• All other entries in the matrix are 0.

10
Example of an Age-Specific Matrix
11
Using Matrices to Project Populations
12
Matrix calculations-some rules

• In matrices, you do an element by element
  multiplication with the elements in the rows
  multiplied by the elements in the columns.
• For example, a 3 by 3 matrix has 3 rows and 3
  columns. It can be multiplied by a vector of 3
  rows and a resulting vector of 3 rows results.
• If the matrix is not square, ie rectangular, say
  4 by 7, the vector must be 4, not 7.
• n(t1)An(t) where A is a Leslie Matrix

13
Matrix projections

• Like life tables, matrix projections will show
  initial fluctuations in the rate of increase (r
  or ?) and the stable age structure.
• Like life tables, matrix projections will
  eventually show a stable age structure and ? (the
  dominant eigenvalue)
• You can also calculate the reproductive value
  using matrices (it is the left eigenvector of the
  matrix)

14
Matrix Modeling Software-Populus

• Populus actually uses the matrix format to model
  life-tables
• Can see the matrix equivalents of a life-table by
  viewing the Leslie matrix graph
• All the derived parameters (rate of increase,
  stable age structure, and reproductive values)
  are the same for life-tables and matrices

15
(No Transcript)
16
Matrix Modeling Software-PopTools add-in for Excel

• There is a free add-in for using Excel to do
  matrix calculations
• It is available at http//www.cse.csiro.au/poptool
  s
• Has a handy feature for making loop graphs

17
(No Transcript)
18
(No Transcript)
19
Life-Cycle Diagrams (made using PopTools)
Life-cycle diagrams show the links between nodes
(age classes) of survival (links from node x to
x 1 and fertility (from node xn to x0)
20
Matrix Modeling Software-Matlab

• Advanced programming language using matrices
• This is a course in itself but offers the most
  flexibility and customization in doing the
  modeling

21
(No Transcript)
22
PRIMER1 Master script for chapter 1 of
Jonathan Roughgarden, 1998, Primer of Ecological
Theory, Prentice Hall. Uses MATLAB and the
Symbolic Toolbox. (Does NOT use SIMULINK.) cd
c\primer diary on f 'q - h (b - a)' be
solve(f,'b') be expand(be) pretty(be)
latex(be) h 50 a 18 q 1000 eval(be) q
0 1000 1500 y eval(be)
23
Matrix Modeling Software-Ramas Ecolab
24
Select Age and Stage Structure
25
Enter Model Information

• Fill out the following (required)
• General information-Title, comments,
  replication, duration, stochasticity, constraints
• Stages-names, weights, and number of stages
• Stage Matrix-fertility and survival
  probabilities
• Initial Abundance-initial population vector
• Standard Deviation Matrix (Optional)
• Density Dependence (Optional)
• Management and Migration (Optional)

26
Run the model and examine results
27
(No Transcript)