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Solutions for Chapter 6.3: Ecological Models: Predators and Competitors

Differential Equations and Boundary Value Problems: Computing and Modeling | 5th Edition | ISBN: 9780321796981 | Authors: C. Henry Edwards, David E. Penney, David T. Calvis

Full solutions for Differential Equations and Boundary Value Problems: Computing and Modeling | 5th Edition

ISBN: 9780321796981

Differential Equations and Boundary Value Problems: Computing and Modeling | 5th Edition | ISBN: 9780321796981 | Authors: C. Henry Edwards, David E. Penney, David T. Calvis

Solutions for Chapter 6.3: Ecological Models: Predators and Competitors

Solutions for Chapter 6.3
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Textbook: Differential Equations and Boundary Value Problems: Computing and Modeling
Edition: 5
Author: C. Henry Edwards, David E. Penney, David T. Calvis
ISBN: 9780321796981

Chapter 6.3: Ecological Models: Predators and Competitors includes 34 full step-by-step solutions. Since 34 problems in chapter 6.3: Ecological Models: Predators and Competitors have been answered, more than 13358 students have viewed full step-by-step 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.

Key Math Terms and definitions covered in this textbook
  • 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. Block-diagonal: 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.

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