 9.4.1: Each of 1 through 6 can be interpreted as describing the interactio...
 9.4.2: Each of 1 through 6 can be interpreted as describing the interactio...
 9.4.3: Each of 1 through 6 can be interpreted as describing the interactio...
 9.4.4: Each of 1 through 6 can be interpreted as describing the interactio...
 9.4.5: Each of 1 through 6 can be interpreted as describing the interactio...
 9.4.6: Each of 1 through 6 can be interpreted as describing the interactio...
 9.4.7: Consider the eigenvalues given by Eq. (39) in the text. Show that (...
 9.4.8: Two species of fish that compete with each other for food, but do n...
 9.4.9: Consider the competition between bluegill and redear mentioned in 8...
 9.4.10: Consider the system (2) in the text, and assume that 12 12 = 0. (a)...
 9.4.11: Consider the system (3) in Example 1 of the text. Recall that this ...
 9.4.12: (a) To find the new critical point, we must solve the equations x(1...
 9.4.13: The system x = y, y = y x(x 0.15)(x 2) results from an approximatio...
 9.4.14: The system x = y, y = y x(x 0.15)(x 2) results from an approximatio...
 9.4.15: The system x = y, y = y x(x 0.15)(x 2) results from an approximatio...
 9.4.16: The system x = y, y = y x(x 0.15)(x 2) results from an approximatio...
 9.4.17: 17 through 19 deal with competitive systems much like those in Exam...
 9.4.18: 17 through 19 deal with competitive systems much like those in Exam...
 9.4.19: 17 through 19 deal with competitive systems much like those in Exam...
Solutions for Chapter 9.4: Competing Species
Full solutions for Elementary Differential Equations and Boundary Value Problems  9th Edition
ISBN: 9780470383346
Solutions for Chapter 9.4: Competing Species
Get Full SolutionsChapter 9.4: Competing Species includes 19 full stepbystep solutions. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9780470383346. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 9. Since 19 problems in chapter 9.4: Competing Species have been answered, more than 13258 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)ยท(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Outer product uv T
= column times row = rank one matrix.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.