Geoscience Reference
In-Depth Information
Therefore,
'
now becomes
8
<
0
for
q
0
e
q
q
0
=
p
s
2
ð
Þ
'
ðÞ¼
q
for
0
q
m
max
m
min
:
0
for
q
m
max
m
min
and the growth and mortality functions become
KAe
a
m
Ð
m
þ
m
min
0
e
q
gm
ð
;
t
;
x
Þ¼
'
ðÞ
q
nm
ð
q
;
t
;
x
Þ
dq
;
Ae
a
m
Ð
m
max
m
0
ð
m
m
1
Þ
m
m
ð
;
t
;
x
Þ¼
e
a
q
'
ðÞ
q
nm
ð
þ
q
;
t
;
x
Þ
dq
þ m
0
e
b
m
þ m
s
Þ
:
m
max
ð
m
The energy input to the modelled system is a primary production distribu-
tion associated with the mass range [e
m
min
,e
m
1
]. This is specified by the
function n
pp
(m, t,
x
).
We consider the model in two spatial dimensions so that
x
¼
(x, y) and
consider a rectangular region given by x
min
y
max
.
We impose Neumann boundary conditions at the edges of the region, ensur-
ing that there is no spatial flux of biomass across the boundary.
x
x
max
and y
min
y
E. Parameter Choices
For the numerical simulations, the growth and mortality parameter values
were based directly upon previously published size-spectrum models. The
spatial parameter value ranges were selected to reflect idealised fish move-
ment rates based upon an allometric volume search relationship (based on
Ware, 1978
,asin
Ben
ˆ
it & Rochet 2004
) given as 640e
0.82m
m
3
yr
1
. This
implies average annual swimming speeds of between 25e
0.41m
and
8.6e
0.27m
myr
1
, depending on how the volume is searched. The C
ps
and
C
pa
coefficients and the
g
ps
and
g
pa
exponents were chosen so that the
behavioural movement velocity, (c
ps
r
g
c
pa
rm
)n, lay within this range.
F. Simulations
We use an initial phytoplankton distribution given by
2
e
y
500
=
100
p
p
2
2
ð
Þ
ð
Þ
Þ
0
:
02e
m
e
x
500
=
100
for m
min
<
m
<
m
1
;
which represents a temporally invariant phytoplankton distribution, normally
distributed in space, centred at (500 m, 500 m) in our coordinate system, with a
standard deviation of 100 m. The number density decreases exponentially
with mass. This can be thought of as representing an idealised plankton
bloom. For simplicity, we assume that the phytoplankton distribution is not
n
pp
m
;
0
; x
ð
