- 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: The systemx = y, y = y x(x 0.15)(x 2)results from an approximation ...
- 9.4.13: In each of 13 through 16:(a) Sketch the nullclines and describe how...
- 9.4.14: In each of 13 through 16:(a) Sketch the nullclines and describe how...
- 9.4.15: In each of 13 through 16:(a) Sketch the nullclines and describe how...
- 9.4.16: In each of 13 through 16:(a) Sketch the nullclines and describe how...
- 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 | 10th Edition
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
Remove row i and column j; multiply the determinant by (-I)i + j •
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
= Xl (column 1) + ... + xn(column n) = combination of columns.
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.