Geoscience Reference
In-Depth Information
C
ps
e
g
ps
m
c
ps
m
ð
;
t
;
x
Þ¼
C
pa
e
g
pa
m
c
pa
m
ð
;
t
;
x
Þ¼
c
dd
m
ð
;
t
;
x
Þ¼
C
dd
e
g
dd
m
0
0
1
1
1
m
m
t
@
@
A
A
c
a
m
ð
;
t
;
x
Þ¼
1
þ
exp
:
C
a
The first three increase allometrically with body size. For the c
ps
and c
pa
cost functions, this represents an increased ability (e.g. visual acuity, swim-
ming speed) to respond to prey and predators with size (
Blaxter, 1986
). For
the c
dd
cost function, we assume that the cost function is inversely propor-
tional to the local carrying capacity (
MacCall, 1990
):
1
K
c
m
c
dd
m
ð
;
t
;
x
Þ¼
Þ
;
ð
;
t
;
x
where K
c
is a measure of carrying capacity at that point in space of indivi-
duals of mass e
m
. We assume that carrying capacity decreases allometrically
with mass (
Brown et al., 2004
), which naturally leads to the expression for the
density-dependent cost function c
dd
. For the aspatial cost function c
a
,we
chose a smoothed step-like function that simulates a 'switching' behaviour at
a transition mass e
m
1
. The movement of individuals smaller than e
m
1
is
dominated by the abiotic velocity field, whereas individuals larger than e
m
1
are hardly affected by the abiotic processes. The coefficient C
a
determines the
sharpness of this transition with larger values leading to more sudden
transitions.
Owing to the way that growth and mortality are calculated in this model,
prey-seeking behaviour is equivalent to an individual moving so as to locally
maximise its growth. Similarly, predator-avoiding behaviour is equivalent to
an individual locally minimising its mortality.
D. Numerical Solution
In practice, we must simulate the model on some bounded intervals, so we
assume a mass interval [e
m
min
,e
m
max
] that defines the minimum and maximum
sizes of the individuals considered. We also choose to prevent individuals
consuming prey that are larger than themselves. To achieve this, we truncate
the predator-prey mass ratio distribution function,
'
(q), and impose variable
limits on the convolution integrals.
