If 0 < u < 1 and γ
inequality holds if and
0 <a< 1 b < 0
c>0 0<d<1
< 4(i+α) [(1 — α)(1 + g) — γ] then 70 holds.
only if γ < α(1-+5α+g). If 0 < u < 1 then
Last
J=
such
. In this case the two eigen values λi, λ2 are
that λ1 + λ2 = a + d > 0 and λ1 λ2 = ad — bc > 0. These two inequalities imply
λ1 > 0 and λ2 > 0. Now if λ1 + λ2 = a + d < 1 then we have that 0 < λ1 < 1
and 0 < λ2 < 1 which implies convergence to the steady state if both are real.
Using 69 we obtain:
λ1 + λ2 =
2Y + [(1 - α)(1+ g) — γ]U
γ + α[(1 - α)(1 + g )- γ] u '
(71)
Using 71 we have that λ1 + λ2 < 1 if and only if:
γ< ∩~^2[(1 — α)(1 + g) — γ]1. (72)
αu
If 0 < u < 1 and γ < (1 α) [(1 — α)(1 + g) — γ] then 72 holds. Last inequality
holds if and only if γ < (1 — α)2(1 + g). Finally we define
Y' = min( α(1-α5(α1+g) , (1 — a)2(1 + g)).
References
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