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 occasionally Brook Milligan Stage-structured Populations
Projecting Age-Structured Populations: Life Cycle Graph P0k P0,k−1 P02 P01 N0 N1 N2 Nk−1 Nk P10 P21 Pk−1,2 Pk,k−1 P00 Brook Milligan Stage-structured Populations
Projecting Stage-Structured Populations: Life Cycle Graph P0k P0,k−1 P02 P01 N0 N1 N2 Nk−1 Nk P10 P21 Pk−1,2 Pk,k−1 P12 Pk−1,k Pk2 P20 Pk−1,1 Brook Milligan Stage-structured Populations
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 0 j N j ( t ) (3) j =0 N i +1 ( t + 1) = P i +1 , i N i ( t ) (4) Brook Milligan Stage-structured Populations
Projecting Age-Structured Populations P 00 P 01 P 02 P 0 k N 0 . . . 0 0 0 P 10 N 1 . . . 0 0 0 P 21 N 2 P = . . . N = . . . . . ... . . . . . . . . . . 0 0 0 0 P k , k − 1 N k Brook Milligan Stage-structured Populations
Projecting Age-Structured Populations P 00 P 01 P 02 P 0 k N 0 ( t ) . . . 0 0 0 N 1 ( t ) P 10 . . . 0 0 0 N 2 ( t ) P 21 P = . . . N ( t ) = . . . . . ... . . . . . . . . . . 0 0 0 0 N k ( t ) P k , k − 1 Brook Milligan Stage-structured Populations
Projecting Age-Structured Populations P 00 P 01 P 02 P 0 k N 0 ( t ) . . . 0 0 0 N 1 ( t ) P 10 . . . 0 P 21 0 0 N 2 ( t ) P = N ( t ) = . . . . . . . . ... . . . . . . . . . . 0 0 0 0 N k ( t ) P k , k − 1 k � N 0 ( t + 1) = P 0 j N j ( t ) (5) j =0 N i +1 ( t + 1) = P i +1 , i N i ( 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 4 5 π � 2 � � 0 1 2 2 7 8 3 1 0 . 4 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 11 a 12 a 1 n . . . a 21 a 22 a 2 n . . . A = (7) . . . ... . . . . . . a m 1 a m 2 a mn . . . and the i , j th element, a ij , is the element in the i th row and j th 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 = ( 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). 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 = ( a ij + b ij ) and A − B = ( a ij − b ij ). 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 = ( a ij + b ij ) and A − B = ( a ij − b ij ). That is, these operations are defined as addition (subtraction) of the corresponding elements. Example � 1 � 3 � 1 + 3 � 4 � � � � 2 4 2 + 4 6 + = = 3 4 5 6 3 + 5 4 + 6 8 10 � 1 � 3 � − 2 � � � 2 4 − 2 = − 3 4 5 6 − 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 = ( ca ij ), 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 = ( ca ij ), i.e., multiply each element of A by c . Example � 1 � 12 � � 2 24 12 = 3 4 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 , j th element of A ⊤ is the j , i th 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 , j th element of A ⊤ is the j , i th element of A . If A is an m × n matrix, A ⊤ is n × m . Example � 1 � 1 � ⊤ � 2 3 = 3 4 2 4 ⊤ � 6 6 7 � 9 2 9 10 = 7 10 1 2 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 = ( a 1 , a 2 , . . . a n ) and b = ( b 1 , b 2 , . . . b n ) ⊤ , then b 1 n b 2 � ab = ( a 1 , a 2 , . . . a n ) = a 1 b 1 + a 2 b 2 + · · · + a n b n = a k b k . . . . k =1 b n Brook Milligan Stage-structured Populations
Matrix Multiplication Example 1 = 0 · 1 + 1 · 2 + 2 · 3 = 8 � � 0 1 2 2 3 and � � 1 � � 0 1 2 is not defined. 2 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 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 , j th element of the product AB is defined as the product of the i th row in A and the j th column in B . Let a 11 a 12 a 1 n a 1 . . . a 21 a 22 a 2 n a 2 . . . A = = . . . . ... . . . . . . . . a m 1 a m 2 a mn a m . . . and b 11 b 12 b 1 l . . . b 21 b 22 b 2 l . . . � � B = = b 1 b 2 b l . . . ... . . . . . . . . . . b n 1 b n 2 b nl . . . Brook Milligan Stage-structured Populations
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