Lecture 6 review
• Direct estimation of total abundance is needed in
many contexts
• Preferred method is direct counting, followed by
density x area expansions, followed by direct U
estimation methods and mark-recapture
• Big assessment models require dangerous
assumptions about vulnerability and stationarity of
recruitment relationships, often drastically
overestimate abundance
Lecture 7: Foraging arena theory and
Stock-Recruitment relationships
• Theory (Foraging arena theory)
• Practice (Problems with data)
Tales from the foraging arena
Are we finally able to develop
useful predictive models for
ecosystem management?
Many patterns in population and community
dynamics can be explained by the proposition
that predation risk drives evolution of
foraging in spatially restricted “arenas”
• Food “limitation” in the presence of apparent plenty
• Beverton-Holt recruitment relationships (recruitment
independent of parental abundance)
• Weak dependence of natural mortality rates on predator
abundances, even when most mortality is due to predation
• Ratio- or predator-dependence in functional responses,
implying dynamic stability, positive correlations between
prey and predator abundances along productivity
gradients, and high biodiversity
Willie asked the right question...
• Why don’t the fish eat them all, dad?
Fine-scale arena dynamics: food
concentration seen by predators should be
highly sensitive to predator abundance
“Invulnerable”
prey (N-V)
k1
k2
“Vulnerable”
prey (V)
Predation
rate:
aVP
(mass action
encounters,
within arena)
This structure implies “ratio-dependent” predation rates:
V=k1N/(k1+k2+aP)
(rate per predator decreases with increasing
predator abundance P)
Food concentration in arenas should
be highly sensitive to density of
animals foraging there
dC/dt = (mixing in)-(mixing out)-(consumption)
=
kI
-mC
-aCN
Fast equilibration of concentration C implies
C = kI / ( m + aN )
Fast equilibration of concentration C
implies:
C = kI / ( m + aN )
Arena Food Density (C)
Effect of Local Competition on
Food Density
1.2
1
0.8
0.6
0.4
0.2
0
0
5
10
Competitor Density (N)
15
Strong effects at low densities:
600
Ungrazed, Lo Fry
Final Body Weight (g)
500
Ungrazed, Hi Fry
Grazed, Lo Fry
Grazed, Hi Fry
400
Power (Series5)
300
200
100
0
0
500
1000
1500
2000
Yearling Density (fish/ha)
2500
3000
Big effects from small changes in space/time scale
(size matters)
Reaction vat model
Prey
eaten
Foraging arena model
Prey
eaten
Predator handling
limits rate
Prey density
Prey behavior
limits rate
Prey density
Spatial organization in foraging
bouts by schooling fish
Properties of the water volumes actually searched:
1. Never cover the entire water column and prey population
2. Intense competition and localized prey depletion within
volumes actually searched as schools disperse
3. Larger, faster fish gain disproportinate access to “new”
prey as fish move upward in the water column
Do “detailed” IBMs account for these effects?
Moving Predictions to larger scales
Decade
Ideal Free Distn.,
simulations
Season/
Year
Population
Dynamics
Local
Recruitment
Day
Arena
Beverton-Holt
Dynamics equation
Hour
Meter
Patch
Reach
Landscape
Mesoscale overlap patterns may limit trophic
interactions; we rely mainly upon diet
composition data to “measure” such effects
Decade
Season/
Year
Day
Arena
Dynamics
Arenas
Hour
Meter
Patch
Reach
Landscape
Beverton-Holt shape and recruitment
“limits” far below trophic potential
(over 300 examples now):
Behavior implies Beverton-Holt recruitment model
(1) Foraging arena effect of density on food available:
Food
Strong empirical
density
support
Juvenile fish density
(2) implies linear effect on required activity and predation risk:
Activity,
mortality
Emerging empirical
support (Werner)
Juvenile fish density
(3) which in turn implies the Beverton-Holt form:
Net recruits
surviving
Massive empirical
support
Initial juvenile fish density
Arena competition + predation risk while
foraging -> Beverton-Holt
• Cases: vary foraging time to maintain constant
growth rate, or vary total time required to reach
defined recruitment size
• Same result both cases, but easiest to derive for
animals trying to maintain constant growth rate G
• Starts with growth rate G = e a C P
e=growth efficiency, aC=food intake/time
P=proportion of time spent in arena
Arena competition + predation risk while
foraging -> Beverton-Holt
• Maintaining growth rate G=eaCP constant means
varying feeding time P so that P=G/(eaC)
• But if C varies with N as C=kI/(m+aN), animals
must vary feeding time as P=(m+aN)(G/eakI)
• If predation rate is proportional to foraging time P,
ie loss=RP, should see Beverton-Holt linear
density dependence:
dN/dt = -RPN = -a1 N - a2 N2
where a1=Rm/eakI and a2=Ra/eakI
Thus the Beverton-Holt parameters
have a very special interpretation
• (Recruits)=K1(Eggs)/[1+K2(Eggs)] is the
integral of dN/dt = -a1 N - a2 N2
with N starting at Eggs
• the maximum survival rate K1 turns out to be
K1=exp(- a1T}=exp{-(Tm/eak)(R/I)}
where R/I is “risk ratio” of predation rate per time
feeding to overall food “supply” in the system
(see that m/g?)
One very scary prediction
Relative Recruitment
• Suppose maximum juvenile survival is
order 5% (eg, coho salmon)
• Then Recruitment should vary with risk
ratio as:
5.00
4.00
3.00
2.00
1.00
0.00
0.5
1
1.5
Relative Risk Ratio (R/I)
2
Have small changes in R/I been
responsible for coho declines?
Quinsam
Puntledge
Big Qualicum
Inch Ck.
Chehalis
Chilliwack
Black
Salmon
Mesachie
35
30
25
20
15
10
5
Brood Year
1993
1991
1989
1987
1985
1983
1981
1979
1977
1975
0
1973
Marine Survival Rate (%)
Trends in Marine Survival Rate for Georgia Strait
Coho Stocks
We confidently predicted that coho would double:
Coho Juvenile abundance
Smolts produced
(millions)
20
Hatchery
15
Wild
10
5
1998
1994
1990
1986
1982
1978
1974
1970
1966
1962
1958
1954
1950
0
Year
Here is what we finally got:
2500000
Spawners
2000000
Sport Catch
Comm. Catch
1500000
1000000
500000
1997
1993
1989
1985
1981
1977
1973
1969
1965
1961
1957
1953
0
Estimating recruitment relationships
• What exactly are we trying to estimate?
• What limits our ability to do estimation,
especially of compensation ratios?
– Lack of contrast
– Errors in variables effects
– Time series bias (mad scientist effects)
What is a “stock-recruit relationship” as used in
developing harvest management policies?
• It is NOT a deterministic prediction of the exact
recruitment to be seen at each adult stock size or egg
deposition.
• Rather, it is a collection of probability distributions
whose means vary in a relatively simple way with stock
size:
The “stock-recruit
function” is a
SURVIVING
model for
RECRUITS
predicting the
means of these
distributions
EGGS
The most obvious problem is lack of
statistical contrast (information)
High variance does NOT prevent us
from getting accurate estimates of mean
recruitment at high stock sizes
RECRUITS
?
EGGS
But this is a huge issue no matter how many observations we have
The “errors in variables” problem:
apparent contrast in stock size (X
values ) due to errors in measuring it
Actual
Measured
RECRUITS
EGGS
High recruits at apparently low Eggs make us think
compensation is strong, when in fact we actually do not know!
The time series (“mad scientist”) problem:
recruitment variation causes stock variation,
nonrepresentative sampling of R,E pairs
Too many high
points at low
stock size
RECRUITS
EGGS
Too many
low points at
high stock
size
High recruits at apparently low Eggs make us think
compensation is strong, when in fact we actually do not know!
Foraging arena theory predicts less variation in
natural mortality rate than would be expected
from changes in predator abundance
Predicted
predation
mortality
rate Mij
of type i
prey due
to type j Ecosim
predators
Traditional
(mass action)
Base estimate
from Ecopath
1
Predator abundance
EwE can incorporate many functional
groups (100+) and fisheries (20+)
Ecosim is being widely tested
against historical trend data
•
•
•
•
•
•
•
•
Georgia Strait
Northwest Hawaiian Shelf
North Sea
Gulf of Thailand
Great Lakes
Bering Sea
Gulf of Mexico
Chesapeake Bay
Time predictions from an ecosystem model of the
Georgia Strait, 1950-2000
With mass-action (Lotka-Volterra) interactions only:
With foraging arena interactions:
Time predictions from an ecosystem model of the
Georgia Strait, 1950-2000
Northwest Hawaiian Islands
(French Frigate Shoals)
Fishing effort:
Initial ecosim runs: fishing+
Trophic interactions only
did not explain monk seal
decline, predicted lobster
recovery
Satellite chlorophyll data
indicate persistent 40-50%
decline in primary production
around 1990; this “explains” both
continued monk seal decline and
persistent low lobster abundance
Lo chl
North Sea
time series
from
MSVPA
compared
to Ecosim
Gulf of
Thailand
survey data
compared to
Ecosim
Preliminary but promising results from
Lake Superior, showing that partial model “failures”
do not necessarily contaminate all predictions
Fish and fishing interactions in the Central North Pacific
Do measured changes in coastal
nutrient loading predict fish
abundance changes, Florida Bays?
Should we trust the policy predictions of these
models, about trophic interaction effects?
• Of course not, nor should we implement policies
that will only work if any particular model is
correct
• But they have a far better chance of directing us to
wise policy choices than some of the boneheaded
arguments that are now being used to justify
policies that ignore interaction effects:
– The mammals will simply find other food sources when
we appropriate the production of their favored prey
– Fishing at less than the natural mortality rate (F<M)
should not impair the ability of a species like menhaden
to support its valued predators like striped bass
Views of ecosystem dynamics
NATURE BENIGN
Stable production dynamics, unpredictable
change driven by environmental factors
PREDICTED BY ECOSIM TO BE COMMON
NATURE CHAOTIC
Unstable production dynamics, unpredictable
change driven by biological interactions
“FORBIDDEN” BY FORAGING ARENA
THEORY?
NATURE RESILIENT
Stable production dynamics within stability
domains, multiple stable states
PREDICTED BY FAT TO BE UNCOMMON
One of those really smart quotes:
“We believe the food web modelling approach
is hopeless as an aid to formulating management
advice; the number of parameters and assumptions
required are enormous.”
Hilborn and Walters (1992, p. 448)
Trophic interaction predictions are critical in
many applied settings, eg the Grand Canyon
Peregrine falcon
Water birds
Exotic fishes
Cowbird
Sparrows etc.
Native fishes
Aquatic insects
Terrestrial insects
Detritus
Benthic algae
Riparian vegetation
Flow
Turbidity Temperature
Water management regime
Why we started wondering...
• Density dependence where there shouldn’t
be:
Watching a fishery die…
99
19
96
93
19
19
90
19
87
84
Year
19
19
81
19
78
19
75
19
72
19
69
19
66
19
63
19
60
900
800
700
600
500
400
300
200
100
0
19
Effort (1000 boat days)
Sport Effort in the Georgia Strait, B.C.
Big effects from small changes in space/time scale
(size matters)
Reaction vat model
Prey
eaten
Foraging arena model
Prey
eaten
Predator handling
limits rate
Prey density
Prey behavior
limits rate
Prey density
Traditional Issues (Hairston, Smith and Slobodkin)
Predators
Who’s going to eat me?
Population
What’s for supper?
Resources
(Predation and resource availability
treated as separate issues)
Modern synthesis
Who’s going to eat me when I go out for supper?
Population
Arena
Predators
And the score at the
Coliseum after 11 innings
is Lions 11, Christians 0
Resources
(Predation and resource availability
inextricably linked)