Time-dynamic modeling, Ecosim

Biogeochemical processes and fish dynamics in food web models

International Centre for Theoretical Physics

Trieste, Nov 26-27, 2007

Villy Christensen & Carl Walters

UBC Fisheries Centre

Toward ecosystem management

• Consider feeding interactions: fish eat fish

• But living organisms are smart

– Behavior impacts interactions

– Human behavior impacts fishing

2

Main elements of Ecosim

• Includes biomass and size-structure dynamics

– mixed differential and difference equations

• Mass-balance (Ecopath) for initial state

• Variable speed splitting

– dynamics of both ‘fast’ (phytoplankton) and ‘slow’ groups (whales)

• Effects of micro-scale behaviors on macro-scale rates

Size-structured dynamics

• Multi-stanza size/age structure by monthly cohorts, density- and risk-dependent growth;

• Adult numbers, biomass, mean size accounting via delay-difference equations;

• Recruitment relationship as ‘emergent’ property of competition/predation interactions of juveniles.

Biomass dynamics in Ecosim

• Gross food conversion Efficiency, GE = Production / Consumption

• dB/dt = GE · Consumption - Predation - Fishery + Immigration - Emigration - Other Mort.

• Consumption = Σ micro-scale rates

• Predation = Σ micro-scale rates

6

Unavailable prey B-V

Available prey, V

Predator, P

Foraging arena

v = behavioral exchange rate (‘vulnerability’)

aVP

Walters, Christensen & Pauly 1997

7

Top-down vs. bottom-up control

Carrying capacity?

8

Predation mortality: effect of vulnerability

Predator abundance

Predicted predation mortality

Bottom-upTop-Down

High v Low v

Carrying capacity

0 Ecopath baseline

V = = 2

Foraging arena theory

• argues that the same fine-scale variation that drives us crazy when we try to survey abundances in the field is also critical to long-term, large-scale dynamics and stability

• Foraging arenas can now be shared between predators, (e.g., by different year-classes)

• EwE now allows bout feeding

Fine-scale arena dynamics: food concentration seen by predators should be highly sensitive to

predator abundance

‘Invulnerable’prey (N-V)

‘Vulnerable’prey (V)

Predationrate:

aVP

(mass actionencounters,within arena)

This structure implies ‘ratio-dependent’ predation rates:

V=v1N/(v1+v2+aP)

(rate per predator decreases with increasing predator abundance P)

v1

v2

Food concentration in arenas should be highly sensitive to density of animals

foraging there

dV/dt = (mixing in)-(mixing out)-(consumption)= vN - v’V - aVP

Fast equilibration of concentration implies

V = vN / ( v’ + aP )

Effect of local competition on food density

• Fast equilibration of concentration implies: V = vN / ( v’ + aP )

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15

Are

na

foo

d d

ensi

ty (V

)

Competitor density (P)

Predicts strong effects at low densities

0

100

200

300

400

500

600

0 500 1000 1500 2000 2500 3000

Fin

al b

od

y w

eig

ht (g

)

Yearling density (fish/ha)

Ungrazed, Lo Fry

Ungrazed, Hi Fry

Grazed, Lo Fry

Grazed, Hi Fry

Power (Series5)

Trout in BC lakes

Moving predictions to larger scales

Hour

Season/

Year

Day

Decade

Meter Patch Reach Landscape

Arena

Dynamics

Local

Recruitment

Population

Dynamics

Beverton-Holt

equation

Ideal Free Distn.,

simulations

Behavior implies Beverton-Holt recruitment model(1) Foraging arena effect of density on food available:

Food

density

Juvenile fish density(2) implies linear effect on required activity and predation risk:

(3) which in turn implies the Beverton-Holt form:

Net recruits

surviving

Initial juvenile fish density

Activity,

mortality

Juvenile fish density

Strong empirical

support

Emerging empirical

support (Werner)

Massive empirical

support

Beverton-Holt shape and recruitment “limits” far below trophic potential (over

600+ examples now):

Predicting consumption

Qij

=a

ij • v

ij • B

i• P

j• T

i• T

j• S

ij• M

ij/ D

j

vij

+ vij

• Ti

• Mij

+ aij

• Mij

• Pj • Sij

• Tj/ D

j

Q = consumptiona = effective search ratev = vulnerabilityB = biomass; P = predator biomass or number

Qij

=a

ij• v

ij• B

i• P

j

vij

+ vij

+ aij

• Pj

Basic consumption equation:

Adding additional realism to the consumption equation:

S = seasonality or long-term forcingM = mediationT = search timeD = f(handling time)

Mediation

• Non-trophic mediation of interaction between a consumer and a prey group

– Tuna – small pelagics – albatrosses

– Piscivores – macroalgae – juvenile fish

– Trawling – resuspension – primary production

– Primary production – shading – macroalgae

– …

18

“D = f(handling time)”Functional response?

• Serves to limit consumption rates when moving away from initial condition

• Foraging arena provides a flexible representation of functional response

• Foraging arena: main difference from Holling’s functional response formulation– Incorporates prey behavior

19

Ecosim seeks to predict changes in mortality rates, Z

• Zi = Fi + sum of Mij

(predation components of M)

where Mij is Qij/Bi

(instantaneous risk of being eaten)

– Mij varies with

– Changes in abundance of type j predators

– Changes in relative feeding time by prey i and predator j

Time predictions from an ecosystem model of the Georgia Strait, 1950-2000

With mass-action (Lotka-Volterra) interactions only:

With foraging arena interactions:

How can we ‘test’ complex ecosystem models?

• No model fully represents natural dynamics, and hence

every model will fail if we ask the right questions;

• A ‘good’ model is one that correctly orders a set of

policy choices, i.e. makes correct predictions about the

relative values of variables that matter to policy choice;

• No model can predict the response of every variable to

every possible policy choice, unless that model is the

system being managed (experimental management

approach).

So how can we decide if a given model is likely to correctly order a set of

specific policy choices?

• Can it reproduce the way the system has responded to similar choices/changes in the past (temporal challenges)?

• Can it reproduce spatial patterns over locations where there have been differences similar to those that policies will cause (spatial challenges)?

• Does it make credible extrapolations to entirely novel circumstances, (e.g., cultivation/depensation effects)?

26

Fitting to time series: learning from ecosystem history

• Numerous ecosystem (EwE) models have in recent years produced credible fit to historical data, and made plausible policy predictions

–have clearly shown that this requires inclusion of environmental as well as fisheries impact

Formal estimation

Ecosystem model (predation,

competition, mediation,

age structured)

Climate Nutrientloading

Fishing

Predicted C, B, Z, W, diets

ObservedC,B,Z,W, diets

Log Likelihood

( BCC/B0)

(Diet0)

(Z0)

Habitat area

Errorpattern

recognition

Choice of parametersto include in final

estimation (e.g., climate anomalies)

Judgmental evaluation

Search for

minimi-zation

Christensen & Walters 2005

Modeling process: fitting & drivers

Confounding of fishery, environment, and trophic effects: monk seals in NWHI

• Initial Ecosim runs: fishing & trophic interactions together could not explain monk seal decline. Predicted lobster recovery

Low Chl

Fishing effort:

1970 2000

• Satellite chlorophyll data indicated persistent ~40% decline in primary production around 1990. ‘Explains’ both continued monk seal decline and persistent low lobster abundance

Are seals causing fish declines in the Georgia Strait?

Is it fishing?

Is it environ-mental change?

Or, is it all three?

1950 19502000 2000‘50 ‘50 ‘90‘90

PP anomaly and climate indicators

• Strait of Georgia– Race Rocks Salinity, Feb-May

0.7

0.9

1.1

1.3

1950 1960 1970 1980 1990 2000

pro

duct

ion a

nom

aly

-20

-10

0

10

upw

ellin

g (m

3/1

00m

/s)

PPA 10

MJJ 10

0.7

0.9

1.1

1.3

1950 1960 1970 1980 1990 2000

pro

duct

ion a

nom

aly

-1.3

-0.9

-0.5

-0.1

0.3

0.7

1.1

PD

O in

dex

PPA 30

AMJJ 30

• BC shelf– Upwelling, May-July, 54°N

• NE Pacific– PDO, April-July

Preikshot (UBC FCRR) 200730

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

1950 1960 1970 1980 1990 2000

Nu

trie

nt

loa

din

g (

rela

tiv

e)

Environmental forcing • Force nutrient supply

to environment

• Force primary production – or secondary, or any other

• Force temperature, salinity, or any other environmental parameter through mediation

• Force (residual) currents

• Long time series should be used for contrast – Often, detailed hydrographic models only cover

short time period; not desirable31

Nutrient loading, Chesapeake Bay

Year