 7.6.1: 16 In each phase plane the xnullclines are blue and the ynullclin...
 7.6.2: 16 In each phase plane the xnullclines are blue and the ynullclin...
 7.6.3: 16 In each phase plane the xnullclines are blue and the ynullclin...
 7.6.4: 16 In each phase plane the xnullclines are blue and the ynullclin...
 7.6.5: 16 In each phase plane the xnullclines are blue and the ynullclin...
 7.6.6: 16 In each phase plane the xnullclines are blue and the ynullclin...
 7.6.7: 715 A system of differential equations is given. (a) Construct the ...
 7.6.8: 715 A system of differential equations is given. (a) Construct the ...
 7.6.9: 715 A system of differential equations is given. (a) Construct the ...
 7.6.10: 715 A system of differential equations is given. (a) Construct the ...
 7.6.11: 715 A system of differential equations is given. (a) Construct the ...
 7.6.12: 715 A system of differential equations is given. (a) Construct the ...
 7.6.13: 715 A system of differential equations is given. (a) Construct the ...
 7.6.14: 715 A system of differential equations is given. (a) Construct the ...
 7.6.15: 715 A system of differential equations is given. (a) Construct the ...
 7.6.16: 1620 A system of differential equations is given. (a) Use a phase p...
 7.6.17: 1620 A system of differential equations is given. (a) Use a phase p...
 7.6.18: 1620 A system of differential equations is given. (a) Use a phase p...
 7.6.19: 1620 A system of differential equations is given. (a) Use a phase p...
 7.6.20: 1620 A system of differential equations is given. (a) Use a phase p...
 7.6.21: Hookes Law states that the force F exerted by a spring on a mass is...
 7.6.22: The van der Pol equation is a secondorder differential equation de...
 7.6.23: The KermackMcKendrick equations are firstorder differential equat...
 7.6.24: The KermackMcKendrick equations from Exercise 23 can be extended t...
 7.6.25: The MichaelisMenten equations describe a biochemical reaction in w...
 7.6.26: Metastasis of malignant tumors Metastasis is the process by which c...
 7.6.27: LotkaVolterra competition equations For each case, derive the equa...
 7.6.28: 2830 Consumerresource models often have the following general form...
 7.6.29: 2830 Consumerresource models often have the following general form...
 7.6.30: 2830 Consumerresource models often have the following general form...
 7.6.31: Hemodialysis is a process by which a machine is used to filter urea...
 7.6.32: FitzhughNagumo equations Consider the following alternative form o...
 7.6.33: The RosenzweigMacArthur model is a consumerresource model similar ...
Solutions for Chapter 7.6: Phase Plane Analysis
Full solutions for Biocalculus: Calculus for Life Sciences  1st Edition
ISBN: 9781133109631
Solutions for Chapter 7.6: Phase Plane Analysis
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 7.6: Phase Plane Analysis includes 33 full stepbystep solutions. This textbook survival guide was created for the textbook: Biocalculus: Calculus for Life Sciences , edition: 1. Biocalculus: Calculus for Life Sciences was written by and is associated to the ISBN: 9781133109631. Since 33 problems in chapter 7.6: Phase Plane Analysis have been answered, more than 27813 students have viewed full stepbystep solutions from this chapter.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

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.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(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 .

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).

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.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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.

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!

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

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