By Isaac Todhunter

This Elibron Classics booklet is a facsimile reprint of a 1864 version by way of Macmillan and Co., Cambridge and London.

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**Example text**

1 Let X be a Hilbert space. Then a linear operator A : D(A) ⊂ X → X is dissipative if Re Ax, x ≤ 0 for all x ∈ D(A). 3 A linear operator A is dissipative if and only if |(λI − A)x| ≥ λ|x| for all x ∈ D(A) and λ > 0. This result yields that for dissipative operators we have λI − A is continuously invertible on R(λ − A) when λ > 0. Thus, λ > 0 implies λ ∈ ρ(A) whenever R(λ − A) is dense in X. 4 (Lumer-Phillips) Suppose A is a linear operator in a Hilbert space X. 1. If A is densely defined, A − ωI is dissipative for some real ω, and R(λ0 − A) = X for some λ0 with Re λ0 > ω, then A ∈ G(1, ω).

To show this, suppose there exist w1 and w2 such that Ax, y = x, w1 for all x ∈ D(A) Ax, y = x, w2 for all x ∈ D(A). x, w1 − w2 = 0 for all x ∈ D(A). and Then, If D(A) is dense in X, then w1 − w2 must equal zero. ) Therefore, w1 = w2 , and the adjoint of A is uniquely defined. 1 Computation of A∗ : An Example from the Heat Equation The computation of A∗ can be easy or impossible. Let X = L2 (0, l), then the operator A from the heat equation is defined by Aϕ = (Dϕ ) on D(A) = {ϕ ∈ H 2 (0, l) | (Dϕ ) ∈ L2 (0, l), ϕ(0) = ϕ (l) = 0}.

In the SS model µ(t, ξ) represents the mortality rate of mosquitofish, and the function Φ(ξ) represents the initial size density of the population, while K represents the fecundity kernel. The boundary condition at ξ = ξ0 is recruitment, or birth rate, while the boundary condition at ξ = ξ1 = ξmax ensures the maximum size of the mosquitofish is ξ1 . The SS model cannot be used as formulated above to model the mosquitofish population because it does not predict dispersion or bifurcation of the population in time under biologically reasonable assumptions [BBKW, BF, BFPZ].