Appendix S1
Detailed derivation of Logistic MPA equilibrium
This section provides a detailed derivation of equations (14) and (15) which
characterize the level of effort at which introduction of a reserve provides no
change to the fishing catch.
To determine whether the reserve introduces a positive change to the fishery it
is first necessary to consider the dynamics without a reserve. This establishes
what optimal management would have been without a reserve and what catch
this would have produced.
Population dynamics without a reserve
The population dynamics for the fished area without a reserve is a classic model
that has been broadly studied and is detailed in many introductory mathematical
ecology textbooks. The model is produced from our generalized single area
model (equation (4)) by substituting the equation for fishing dynamics (equation
(10)) and population dynamics (equation (3)):
é
æ N öù
N t+1 = ê1+ r ç 1- t ÷ ú N t - qEN t ,
K øû
è
ë
(1)
at equilibrium we have Nt+1 = Nt and this becomes:
é
æ
Nöù
N = ê1+ r ç 1- ÷ ú N - qEN.
Køû
è
ë
(2)
The non-zero solution of this is readily solved to give:
æ qE ö
N = K ç 1÷
è
r ø
(3)
The equilibrium catch is:
C = qEN
(4)
æ qE ö
C = qEK ç 1÷.
è
r ø
(5)
This is maximized when:
E = r / 2q,
(6)
which corresponds to the maximum sustainable yield (MSY).
Population dynamics with a reserve
The population dynamics of the two areas found by substituting the equation for
spillover (equation (8)), fishing dynamics (equation (10)) and population
dynamics (equation (3)) in the generalized two area model (equation (5)):
é
æ
R öù
æ
ö
a
Rt+1 = ê1+ r ç 1- t ÷ ú Rt - m ç Rt Mt ÷ 2
1- a
è
ø
è aKøû
ë
é
æ
Mt ö ù
qEM t
æ
ö
a
M t+1 = ê1+ r ç 1+ m ç Rt Mt ÷ .
ú Mt ÷
1- a
1- a
è
ø
è (1- a )K ø û
ë
(7)
(8)
To find the equilibrium solution we have M t+1 = M t and Rt+1 = Rt so the above
becomes:
é
æ
æ
ö
R öù
a
R = ê1+ r ç 1M÷
ú R - mç R ÷
1- a ø
è aK øû
è
ë
é
æ
æ
ö
M öù
qEM
a
M = ê1+ r ç 1+ mç R M÷ .
úM÷
1- a
1- a ø
è
è (1- a )K ø û
ë
(9)
(10)
For the introduction of this reserve to have no net impact on the catch, the catch
from the pre-reserve fishery must equal the catch from the post-reserve fishery:
qEM
1- a
M = N(1- a ) .
qEN =
(11)
(12)
Substitution in equation (10) yields:
é
æ N (1- a ) ö ù
æ
ö
qEN (1- a )
a
N (1- a ) = ê1+ r ç 1N
(1a
)
+
m
R
N
(1a
)
ú
çè
÷ø
1- a
1- a
è (1- a )K ÷ø û
ë
é
æ
Nöù
N (1- a ) = ê1+ r ç 1- ÷ ú N (1- a ) - qEN + m ( R - a N )
Køû
è
ë
(13)
Subtracting equation (13) from 1- a times equation (2) gives:
0 = 0 + qEN - (1- a )qEN - m ( R - a N )
a qEN = m ( R - a N )
(14)
(15)
As noted in equation (12) in the main manuscript, this implies that the spillover
must equal the surplus production of the original fishing grounds that has now
been encompassed in the reserve.
This can be re-arranged to give R as a function of N :
R=
a qEN
+aN
m
(16)
Substitution of equation (15) in equation (9) gives:
é
æ
R öù
R = ê1+ r ç 1ú R - a qEN
è a K ÷ø û
ë
(17)
æ
R ö
0 = r ç 1R - a qEN
è a K ÷ø
(18)
Substitution of equations (16) and (3) in equation (18) after simplification
yields:
æ 1 é mqE - mr + q 2 E 2 - qEr ù
é qE ùö
éë mqE - mr + q 2 E 2 - qEr ùû - qE ê10 = m 2 r ç ê1+
ú
mr
r úû÷ø (19)
èmë
ë
û
0 = m 2 (qE - r) - ( m + qE - r) ( mqE - mr + q 2 E 2 - qEr ) .
A cubic in qE which has a real, positive solution:
(
)
qE = r - 2m + r 2 + 4m 2 / 2
(20)
This is the level of effort at which a reserve does not alter the catch and is equal
to equation (14) in the main paper. The relative level of excess effort that this
corresponds to is found by dividing by the optimal effort, equation (6), and
subtracting 1, yielding:
Ê = 1+ (2m / r)2 - 2m / r,
(21)
equation (15) in the main paper.