- 9.5.1: Each of 1 through 5 can be interpreted as describing the interactio...
- 9.5.2: Each of 1 through 5 can be interpreted as describing the interactio...
- 9.5.3: Each of 1 through 5 can be interpreted as describing the interactio...
- 9.5.4: Each of 1 through 5 can be interpreted as describing the interactio...
- 9.5.5: Each of 1 through 5 can be interpreted as describing the interactio...
- 9.5.6: In this problem we examine the phase difference between the cyclic ...
- 9.5.7: a. Find the ratio of the amplitudes of the oscillations of the prey...
- 9.5.8: a. Find the period of the oscillations of the prey and predator pop...
- 9.5.9: Consider the system dx dt = ax_1 y 2 _, dy dt = by_1 + x 3 _, where...
- 9.5.10: The average sizes of the prey and predator populations are defined ...
- 9.5.11: Consider the system x_ = x(1 x 0.5y), y_ = y(0.75 + 0.25x), where >...
- 9.5.12: Consider the system dx/dt = x(a x y), dy/dt = y(c + x), where a, , ...
- 9.5.13: In the Lotka--Volterra equations, the interaction between the two s...
- 9.5.14: Applying a constant-effort model of harvesting to the Lotka-- Volte...
- 9.5.15: If we modify the Lotka--Volterra equations by including a selflimit...
- 9.5.16: In this problem we apply a constant-yield model of harvesting to th...
Solutions for Chapter 9.5: Predator -- Prey Equations
Full solutions for Elementary Differential Equations and Boundary Value Problems | 11th Edition
Upper triangular systems are solved in reverse order Xn to Xl.
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.)
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 Fn-1c can be computed with ne/2 multiplications. Revolutionary.
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
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.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
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.
Every v in V is orthogonal to every w in W.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Solvable system Ax = b.
The right side b is in the column space of A.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).