Stage-structured Populations
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1 Department of Biology New Mexico State University Las Cruces, New Mexico Fall 2009
2 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
3 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
4 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 occasionally
5 Projecting Age-Structured Populations: Life Cycle Graph P0k P02 P0,k 1 P01 N0 N1 N2 Nk 1 Nk P10 P21 Pk 1,2 Pk,k 1 P00
6 Projecting Stage-Structured Populations: Life Cycle Graph P01 P02 P0,k 1 P0k N0 N1 N2 Nk 1 Nk P10 P21 Pk 1,2 Pk,k 1 P12 Pk 1,k P20 Pk 1,1 Pk2
7 Projecting Age-Structured Populations Projection equations k N 0 (t + 1 = b j N j (t (1 j=0 N i+1 (t + 1 = g i N i (t (2 g x is the age-specific survivorship b x is the age-specific reproduction k N 0 (t + 1 = P 0j N j (t (3 j=0 N i+1 (t + 1 = P i+1,i N i (t (4
8 Projecting Age-Structured Populations P = P 00 P 01 P P 0k P P P k,k 1 0 N = N 0 N 1 N 2. N k
9 Projecting Age-Structured Populations P = P 00 P 01 P P 0k P P P k,k 1 0 N(t = N 0 (t N 1 (t N 2 (t. N k (t
10 Projecting Age-Structured Populations P = P 00 P 01 P P 0k P P P k,k 1 0 N(t = N 0 (t N 1 (t N 2 (t. N k (t k N 0 (t + 1 = P 0j N j (t (5 j=0 N i+1 (t + 1 = P i+1,i N i (t (6
11 Matrices A matrix is a rectangular array of numbers enclosed in brackets. The following are examples of matrices. ( ( π 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.
12 Matrices A general matrix can be represented as a 11 a a 1n a 21 a a 2n A = a m1 a m2... a mn (7 and the i, jth element, a ij, 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.
13 Simple Matrix Operations Equality Two matrices, say A = (a ij and B = (b ij, are equal if they have the same dimensions (i.e., the same number of rows and columns and if a ij = b ij for every i and j (i.e., elements in the corresponding positions are equal.
14 Simple Matrix Operations Addition and Subtraction Only matrices of equal dimensions can be added or subtracted. We define A + B = (a ij + b ij and A B = (a ij b ij. That is, these operations are defined as addition (subtraction of the corresponding elements.
15 Simple Matrix Operations Addition and Subtraction Only matrices of equal dimensions can be added or subtracted. We define A + B = (a ij + b ij and A B = (a ij b ij. That is, these operations are defined as addition (subtraction of the corresponding elements. ( ( ( ( = = ( ( = ( Note: the order of addition makes no difference: A + B = B + A (i.e., addition is commutative.
16 Simple Matrix Operations Scalar multiplication If c is a number and A is a matrix, the product ca = Ac is defined by ca = (ca ij, i.e., multiply each element of A by c.
17 Simple Matrix Operations Scalar multiplication If c is a number and A is a matrix, the product ca = Ac is defined by ca = (ca ij, i.e., multiply each element of A by c. ( = ( Note: scalar multiplication and addition (subtraction are distributive, so c(a + B = ca + cb.
18 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.
19 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. ( = = 2 1 ( (
20 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.
21 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 = (a 1, a 2,... a n and b = (b 1, b 2,... b n, then ab = (a 1, a 2,... a n b 1 b 2. b n = a 1b 1 +a 2 b 2 + +a n b n = n a k b k. k=1
22 Matrix Multiplication and ( ( ( 1 2 = = 8 is not defined. Order is important in this procedure. As we ll see ba is also defined but the result is quite different.
23 Matrix Multiplication A general requirement in multiplying matrices is that the number of 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 and a 11 a a 1n a 21 a a 2n A = = a m1 a m2... a mn a 1 a 2. a m b 11 b b 1l b 21 b b 2l B = = ( b 1 b 2... b l. b n1 b n2... b nl
24 Matrix Multiplication Then a 1 b 1 a 1 b 2... a 1 b l a 2 b 1 a 2 b 2... a 2 b l AB = a m b 1 a m b 2... a m b l Thus the i, jth element of AB, denote it ab ij, is given by ab ij = a i b j = ( a i1 a i2... a in b j1 b j2. b jn = n a ik b kj. k=1 Note that AB is an m l matrix, i.e., (m n (n l (m l.
25 Matrix Multiplication ( ( 1 4 = ( = ( 9 19
26 Matrix Multiplication ( ( 1 4 = ( = ( 9 19 ( =
27 Matrix Multiplication ( ( 1 4 = ( = ( 9 19 ( = (
28 Matrix Multiplication ( ( =
29 Matrix Multiplication ( ( = (
30 Matrix Multiplication ( ( = ( ( ( =
31 Matrix Multiplication ( ( = ( ( ( = (
32 Matrix Multiplication ( ( = ( ( ( = ( These examples illustrate that matrix multiplication is noncommutative; i.e., AB = BA is often false.
33 Matrix Multiplication ( =
34 Matrix Multiplication ( =
35 Matrix Multiplication ( = ( =
36 Matrix Multiplication ( = ( = (32
37 Matrix Multiplication ( = ( = (32 These examples vividly illustrate that matrix multiplication is noncommutative.
38 Projecting Age-Structured Populations P = P 00 P 01 P P 0k P P P k,k 1 0 N(t = N 0 (t N 1 (t N 2 (t. N k (t k N 0 (t + 1 = P 0j N j (t (8 j=0 N i+1 (t + 1 = P i+1,i N i (t (9
39 Projecting Age-Structured Populations P = P 00 P 01 P P 0k P P P k,k 1 0 N(t = N 0 (t N 1 (t N 2 (t. N k (t k N 0 (t + 1 = P 0j N j (t (8 j=0 N i+1 (t + 1 = P i+1,i N i (t (9 k = P i+1,j N j (t (10 j=0
40 Projecting Age-Structured Populations P = P 00 P 01 P P 0k P P P k,k 1 0 N(t = N 0 (t N 1 (t N 2 (t. N k (t N(t + 1 = P N(t (11
41 Projecting Stage-Structured Populations: Life Cycle Graph P01 P02 P0,k 1 P0k N0 N1 N2 Nk 1 Nk P10 P21 Pk 1,2 Pk,k 1 P12 Pk 1,k P20 Pk 1,1 Pk2
42 Projecting Stage-Structured Populations N(t = ( N 0 (t N 1 (t N 2 (t... N k 1 (t N k (t P = 0 P 01 P P 0,k 1 P 0k P 10 0 P P 20 P P k 1,1 P k 1, P k 1,k 0 0 P k,2... P k,k 1 0 N(t + 1 = P N(t (12
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