 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 LotkaVolterra equations, the interaction between the two s...
 9.5.14: Applying a constanteffort model of harvesting to the Lotka Volte...
 9.5.15: If we modify the LotkaVolterra equations by including a selflimit...
 9.5.16: In this problem we apply a constantyield 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
ISBN: 9781119256007
Solutions for Chapter 9.5: Predator  Prey Equations
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 9.5: Predator  Prey Equations includes 16 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Differential Equations and Boundary Value Problems, edition: 11. Elementary Differential Equations and Boundary Value Problems was written by and is associated to the ISBN: 9781119256007. Since 16 problems in chapter 9.5: Predator  Prey Equations have been answered, more than 12232 students have viewed full stepbystep solutions from this chapter.

Back substitution.
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 Fn1c 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 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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.

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.

Lucas numbers
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.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Pivot.
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)jI and det V = product of (Xk  Xi) for k > i.

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).