A forward–backward closure identity for nonlocally coupled size-structured optimal harvesting
Jiguang Yu, Louis Shuo Wang, Ye Liang, Jilin ZhangAbstract
We study optimal harvesting of a size-structured population whose growth and mortality respond to the population itself only through a single scalar environmental variable
E
(
t
) =
∫χ
(
l
)
x
(
t
,
l
) d
l
– a weighted biomass, stand density or basal area. Because the same scalar closes the state equation, its linearisation and its adjoint, the control problem is governed throughout by one dimensionless number: the loop gain Φ′(
E
*) of the stationary environmental feedback. Our main result is the forward–backward closure identity
B
(0) = Φ′(
E
*), which states that the sensitivity of the stationary closure map and the zero-discount gain of the nonlocal adjoint feedback are the same quantity. Consequently the return difference 1 − Φ′(
E
*) simultaneously governs (i) uniqueness of the stationary environment, (ii) solvability of the linearised state system with its nonlocal feedback term, and (iii) solvability of the stationary adjoint, which we show is a rank-one – hence explicitly invertible – Fredholm problem with denominator 1 −
B
(
r
). The adjoint decomposes as
λ
=
λ
red
+ Γ
ψ
r
with Γ =
A
(
r
)/(1 −
B
(
r
)), so the harvesting switching function is a rank-one correction
S
=
S
red
− Γ
ψ
r
of its frozen counterpart; small corrections preserve a single size threshold, while corrections producing alternating signs destroy it. We prove local well-posedness and Gâteaux differentiability of the control-to-state map on
W
1,1
, identify the corner compatibility condition