Section 1.7: Properties of the Leslie Matrix Definition: A matrix A whose entries are nonnegative (positive) is called a nonnegative (positive) matrix, denoted as A 0 (A > 0). Definition: A square m m matrix A = (a ij ) is said to be reducible if the index set {1, 2,..., m} can be divided into two nonempty disjoint sets S 1 = {i 1, i 2,..., i µ } and S 2 = {j 1, j 2,..., j ν } with µ + ν = m, such that a iαj β = 0 for α = 1, 2,..., µ and β = 1, 2,..., ν. A square matrix that is not reducible is said to be irreducible. Example: Consider the matrix Show that A is reducible. 1 0 0 A = 2 3 4 5 6 7 Definition: The digraph associated with a matrix A is called strongly connected if there exists a directed path from node i to node j for every node i and j in the digraph. Theorem 1.2: The digraph of matrix A is strongly connected iff matrix A is irreducible. Example: Consider the matrix Determine whether A is irreducible. 1 2 0 A = 3 0 0 0 4 5 Note: A Leslie matrix L is irreducible iff b m > 0. 1
Theorem 1.3: (Frobenius Theorem) An irreducible, nonnegative matrix A always has a positive eigenvalue λ that is a simple root (multiplicity one) of the characteristic equation. Moreover, λ is a dominant eigenvalue and the corresponding eigenvector has positive coordinates. Note: The Frobenius Theorem applies to a Leslie matrix with b m > 0. Theorem 1.4: (Perron Theorem) A positive matrix A always has a real, positive eigenvalue λ that is a simple root of the characteristic equation and is strictly dominant. The corresponding eigenvector has positive coordinates. Definition: If an irreducible, nonnegative matrix A has h eigenvalues λ 1,..., λ h of maximum modulus ( λ 1 = λ i, i = 1,..., h), then A is called primitive if h = 1 and imprimitive if h > 1. The value of h is called the index of imprimitivity. Theorem 1.5: A nonnegative matrix A is primitive iff some power of A is positive. That is, A p > 0 for some integer p 1. Example: Classify each Leslie matrix as reducible or irreducible and primitive or imprimitive. ( ) b1 b (a) L = 2 s 1 0 (b) L = ( 0 ) b2 s 1 0 2
Consider a Leslie matrix model, The Stable Age Distribution X(t + 1) = LX(t), where L is a nonnegative, irreducible, primitive matrix. Then there exists a strictly dominant eigenvalue λ 1. The corresponding eigenvector V 1 = (v 1, v 2,..., v m ) T has positive entries and is called the stable age distribution. Let P (t) = m i=1 denote the total population size at time t and ν 1 = m i=1 v i denote the sum of the entries of V 1. Then, it can be shown that X(t) lim t P (t) = V 1. ν 1 Moreover, if λ 1 < 1, then lim t P (t) = 0 and if λ 1 > 1, then lim t P (t) =. An explicit expression for the stable age distribution is given by 1 s 1 λ 1 s 1 s 2 V 1 = λ 2 1. s 1 s 2 s m 1 λ m 1 1 The first entry in this expression is normalized to one. Example: Consider the Leslie matrix ( ) 1 2 L = 3 0 8 Find the strictly dominant eigenvalue and the stable age distribution. 3
The characteristic equation associated with the Leslie matrix L has a nice form: b 1 λ b 2 b m 1 b m s 1 λ 0 0 det(l λi) = 0 s 2 0 0 = 0........ 0 0 s m 1 λ After expansion, we obtain p m (λ) = λ m b 1 λ m 1 b 2 s 1 λ m 2 b m s 1 s m 1 = 0. By Descartes s Rule of Signs, there is only one positive real root. This positive root is the dominant eigenvalue, λ 1. The limiting behavior of the population depends on whether λ 1 > 1 or λ 1 < 1. Now p m (λ) as λ, p m (0) < 0, and p m (λ) crosses the positive λ-axis at only one point, λ 1. So by checking the sign of p m (1), it can be seen that λ 1 > 1 iff p m (1) < 0 and λ 1 < 1 iff p m (1) > 0. Here, we have where p m (1) = 1 b 1 b 2 s 1 b m s 1 s m 1 = 1 R 0, R 0 = b 1 + b 2 s 1 + + b m s 1 s m 1. Definition: The quantity R 0 is called the inherent net reproductive number, the expected number of offspring per individual per lifetime. Theorem 1.6: Assume that L is an irreducible, primitive Leslie matrix. The characteristic polynomial is p m (λ) and L has a strictly dominant eigenvalue λ 1 > 0 satisfying λ 1 = 1 R 0 = 1 λ 1 > 1 R 0 > 1 λ 1 < 1 R 0 < 1 where R 0 is the inherent net reproductive number. 4
Theorem 1.7: An irreducible Leslie matrix L is primitive iff the birth rates satisfy g.c.d.{i b i > 0} = 1, where g.c.d. denotes the greatest common divisor. Note: This result applies only to Leslie matrices. Example: (a) Show that a Leslie matrix with b 1 > 0 and b m > 0 is primitive. (b) Show that a Leslie matrix with b i > 0, b i+1 > 0, and b m > 0 is primitive. 5
Example: Consider the Leslie matrix 0 1 2 L = 0.5 0 0. 0 0.25 0 (a) Show that L is irreducible and primitive. (b) Find the characteristic equation and inherent net reproductive number, R 0, of L. (c) Determine the long term behavior of the total population size. (d) Determine the minimum value of s 2 which guarantees that the total population size increases. 6