 6.3.1: Starting with the Jacobian matrix of the system in (1), derive its ...
 6.3.2: Separate the variables in the quotient dy dx D 150y C 2xy 200x 4xy ...
 6.3.3: Let x.t / be a harmful insect population (aphids?) that under natur...
 6.3.4: Show that the coefficient matrix of the linearization x0 D 60x, y0 ...
 6.3.5: Show that the linearization of (2) at .0; 21/ is u0 D 3u, v0 D 63u ...
 6.3.6: Show that the linearization of (2) at .15; 0/ is u0 D 60u 45v, v0 D...
 6.3.7: Show that the linearization of (2) at .6; 12/ is u0 D 24u 18v, v0 D...
 6.3.8: Show that the linearization of (3) at .0; 14/ is u0 D 4u, v0 D 28u ...
 6.3.9: Show that the linearization of (3) at .20; 0/ is u0 D 60u 80v, v0 D...
 6.3.10: Show that the linearization of (3) at .12; 6/ is u0 D 36u 48v, v0 D...
 6.3.11: Show that the coefficient matrix of the linearization x0 D 5x, y0 D...
 6.3.12: Show that the linearization of (4) at .5; 0/ is u0 D 5u5v, v0 D 3v....
 6.3.13: Show that the linearization of (4) at .2; 3/ is u0 D 2u2v, v0 D 3u....
 6.3.14: Show that the coefficient matrix of the linearization x0 D 2x, y0 D...
 6.3.15: Show that the linearization of (5) at .0; 4/ is u0 D 6u, v0 D 4u C ...
 6.3.16: Show that the linearization of (5) at .2; 0/ is u0 D 2u 2v, v0 D 2v...
 6.3.17: Show that the linearization of (5) at .3; 1/ is u0 D 3u 3v, v0 D u ...
 6.3.18: Show that the coefficient matrix of the linearization x0 D 2x, y0 D...
 6.3.19: Show that the linearization of the system in (7) at .5; 2/ is u0 D ...
 6.3.20: Show that the coefficient matrix of the linearization x0 D 3x, y0 D...
 6.3.21: Show that the linearization of the system in (8) at .3; 0/ is u0 D ...
 6.3.22: Show that the linearization of (8) at .5; 2/ is u0 D 5u 5v, v0 D 2u...
 6.3.23: Show that the coefficient matrix of the linearization x0 D
 6.3.24: Show that the linearization of (9) at .7; 0/ is u0 D 7u7v, v0 D 2v....
 6.3.25: . Show that the linearization of (9) at .5; 2/ is u0 D 5u5v, v0 D 2...
 6.3.26: For each twopopulation system in 26 through 34, first describe the...
 6.3.27: For each twopopulation system in 26 through 34, first describe the...
 6.3.28: For each twopopulation system in 26 through 34, first describe the...
 6.3.29: For each twopopulation system in 26 through 34, first describe the...
 6.3.30: For each twopopulation system in 26 through 34, first describe the...
 6.3.31: For each twopopulation system in 26 through 34, first describe the...
 6.3.32: For each twopopulation system in 26 through 34, first describe the...
 6.3.33: For each twopopulation system in 26 through 34, first describe the...
 6.3.34: For each twopopulation system in 26 through 34, first describe the...
Solutions for Chapter 6.3: Ecological Models: Predators and Competitors
Full solutions for Differential Equations and Boundary Value Problems: Computing and Modeling  5th Edition
ISBN: 9780321796981
Solutions for Chapter 6.3: Ecological Models: Predators and Competitors
Get Full SolutionsChapter 6.3: Ecological Models: Predators and Competitors includes 34 full stepbystep solutions. Since 34 problems in chapter 6.3: Ecological Models: Predators and Competitors have been answered, more than 6322 students have viewed full stepbystep solutions from this chapter. Differential Equations and Boundary Value Problems: Computing and Modeling was written by Patricia and is associated to the ISBN: 9780321796981. This textbook survival guide was created for the textbook: Differential Equations and Boundary Value Problems: Computing and Modeling, edition: 5. This expansive textbook survival guide covers the following chapters and their solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Iterative method.
A sequence of steps intended to approach the desired solution.

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.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Solvable system Ax = b.
The right side b is in the column space of A.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).
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