Geoscience Reference
In-Depth Information
8
<
0
for m
<
m
1
ð
m
m
1
Þ
m
s
for m
1
m
m
max
m
s
m
ð
;
t
;
x
Þ¼
m
max
ð
m
Þ
:
1
for m
>
m
max
represents the effects of senescence and so increases sharply close to some
maximum attainable body mass, m
max
.
The final growth and mortality terms are therefore given by
KAe
a
m
Ð
1
1
gm
ð
;
t
;
x
Þ¼
e
q
'
ðÞ
q
nm
ð
q
;
t
;
x
Þ
dq
;
Ae
a
m
Ð
1
1
ð
m
m
1
Þ
e
a
q
þ m
0
e
b
m
m
m
ð
;
t
;
x
Þ¼
'
ðÞ
q
nm
ð
þ
q
;
t
;
x
Þ
dq
þ m
s
:
m
max
ð
m
Þ
C. Spatial Flux
The spatial flux term J is defined so that individuals move locally to maximise
their growth rates (and hence fitness), minimise their mortality and reduce
competition, consistent with optimal foraging theory. Thus, the flux term can
be written as:
n
J
¼
c
ps
r
g
c
pa
rm
ð
c
dd
r
n
Þ
n
þ
ðÞ
c
a
v
n
:
c
pa
rm
)n is a non-linear velocity term which drives indivi-
duals to move according to the payoff between maximising their growth and
minimising their mortality, determined by the prey-search cost function
c
ps
(m, t,
x
) and the predator-avoidance cost function c
pa
(m, t,
x
).
Density-dependent competition is governed by the (c
dd
r
Here, (c
ps
r
g
n)n term. This
drives competing individuals of the same size towards areas of lower popula-
tion density with the magnitude of the effect determined by the density-
dependence cost function c
dd
(m, t,
x
).
To include the effects of passive, abiotic, transport processes (i.e. small-
scale turbulence), we introduced a randomised velocity field
(t,
x
) to repre-
sent the local turbulent fluid movements. The extent to which individuals are
influenced by this velocity field and thus passively transported by the water is
given by the abiotic cost function c
a
(m, t,
x
).
The choice of cost functions reflects the behaviour of individuals and could
incorporate behaviours that change in space as well as time. However, in the
following work, we assume that cost functions are spatially and temporally
invariant and of the form
n
