 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 13358 students have viewed full stepbystep solutions from this chapter. Differential Equations and Boundary Value Problems: Computing and Modeling was written by 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.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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].

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Kirchhoff's Laws.
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.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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

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).

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.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

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