Geoscience Reference
In-Depth Information
This means that the number of individuals in the mass range [M
1
,M
2
]
¼
[e
m
1
,e
m
2
], in a volume
, at a time t, is given by the formula:
ð
m
2
V
ð
nm
ð
;
t
;
x
Þ
dm d
x
:
V
m
1
All parameters in this and subsequent terms are summarised in
Table 1
.
We use a generalised McKendrick-von Foerster equation with an explicit
spatial flux term to express changes in the numbers of individuals through
time:
@
n
@
ðÞ
@
gn
t
¼m
n
m
r
J
;
@
where g(m, t,
x
) is the per capita growth rate,
m
(m, t,
x
) is the per capita
mortality rate and J(m, t,
x
) is the local population flux.
B. Growth and Mortality
We assume that predation processes are the dominant factors affecting
growth and mortality, and adapt the predation-based growth and mortality
terms from
Ben
ˆ
ıt and Rochet (2004)
to incorporate a spatial dimension. The
functions were constructed so that the individuals are able to feed on a range
of prey sizes (according to a preference function
) and so that the volume of
water searched by individuals increases allometrically with mass, reflecting
their increasing energy demands and capacity for movement. The two terms
are given by
'
KAe
a
m
Ð
1
1
e
q
gm
ð
;
t
;
x
Þ¼
'
ðÞ
q
nm
ð
q
;
t
;
x
Þ
dq
;
Ae
a
m
Ð
1
1
e
a
q
m
m
ð
;
t
;
x
Þ¼
'
ðÞ
q
nm
ð
þ
q
;
t
;
x
Þ
dq
;
where q represents the difference in mass between predator and prey species
(the derivations are described in
Ben
ˆ
ıt and Rochet, 2004
).
We take
'
(q), the predator-prey mass preference function to be given by
p
s
2
ð
Þ
e
q
q
0
=
'
ðÞ¼
q
;
2
, which
that is, an un-normalised Gaussian distribution, with variance
s
peaks at 1 for q
q
0
, the preferred predator-prey mass difference.
Non-predation mortality is accounted for by the extra mortality terms
m
0
(m, t,
x
) and
m
s
(m, t,
x
), where
¼
ð Þ m
0
e
b
m
represents juvenile mortality effects such as disease and so decreases
allometrically with mass and
m
0
m
;
t
;
x
