Modelling size-structured populations

David Boukal (IMR, Bergen, Norway)

FishACE Methods Course, Mallorca, 2-3 May 2006

1. energy budget models

2. ecological dynamics of structured populations

3. evolution of age and size at maturation (tomorrow)

2

intricate reality

Individuals vs. ecosystems: What do we do?

3

1. Energy budget models

size-structured populations:

many individual properties depend on size

4

Size/physiological structure

individual life history driven by size: feeding capacity (gape size) digestive capacity (stomach size) fecundity (gonad size) predation risk (body size) maintenance (body size)

w = weight ... ‘typical’ exponents are

b=1 (processes ~ body volume: maintenance, fecundity) b=2/3 (processes ~ body surface, gape size: feeding)

‘general’ scaling laws (allometries)

F(w) ~ c wb

5

Slight diversion: von Bertalanffy growth

von Bertalanffy growth

simple derivation:

isomorphic (‘bubble’) growth ( W L3 )

ingestion ~ gape size ( ~ L2 ~ W2/3)

maintenance ~ body volume ( ~ L3 ~ W)

Age (d)

Leng

th (

mm

)

Daphnia(Kooijman, 1993)

6

How to model individual life histories?

Dynamic Energy Budget (DEB) models:

individuals = ‘engines’ converting food into body mass & offspring

individual state characterized by a few variables:structural biomass (soma), reserves, ...

energy allocation described by a set of (simple) rules

three main processes:growth, maintenance and reproduction

?

7

Energy allocation rules

net production models

net assimilation models

food

reproduction

structural biomass

(somatic) maintenance

maturity maintenance

storage

faeces

1-

-rule (Kooijman 1993)

food

reproduction

structural biomass

(somatic) maintenance

reserves

faeces

1-

8

-rule (Lika & Kooijman 2003)

time‘mat

urat

ion

inve

stm

ent’

Amat

starvation

max juv.

structural biomass

rese

rves

SBmat

max adult

Energy allocation rules II: maturation

net assimilation models (+ some net production models)

net production models (L. Persson & Co)

‘gonads’

maturation ~ threshold surplus energy

repro ~ dtto

gain ... from storage, as (1- ) proportionloss ... maintenance of matur. investment,

(starvation)

maturation ~ threshold surplus energy

repro ~ threshold relative energy

gain ... from net intake, as (1- ) proportionloss ... maintenance + starvation

starvation

gonads

9

Energy allocation III: -rule (Kooijman, 2000)

Age, d Age, d

Length, mm Length, mm

Cum

# of young

Length, m

mIngestion rate, 105

cells/h

O2 consum

ption,

g/h

Respiration Ingestion

Reproduction

VB growth

32 LkvL M 2fL

332 )/1( pMM LkfgLkvL

)( LLrLdt

dB

10

Energy allocation rules IV

net allocation models Kooijman (1993, 2000), Nisbet et al. (2000), Ledder et al. (JMB 2004), ... usually assume constant over lifetime (juveniles ‘prepare’ for repro)

difficulties when studying evolution of size at maturation starvation requires additional assumptions & equations

(in continuous time: repro/growth shuts down immediately / gradually)

net production models Lika & Nisbet (JMB 2000), Ledder et al. (JMB 2004), de Roos & Persson & co ... usually assume switch in easier to study evolution of size at maturation

... but Lika & Nisbet (2000): < 1 constant, repro starts at threshold reserves starvation also requires some thought

(my) conclusions: DEB models can be VERY complicateduse something sensible & not too complexavoid -rule ?

11

2. Dynamics of size-structured populations

size-structured populations:

individuals are usually not the same

12

How to model structured populations?

age-specific life histories (LHs) e.g. juveniles adults (time lags)

stage-specific LHs e.g. distinct larval stages (insects)

size/physiological structure (fish!) age and individual state decoupled

matrix models (Caswell 2001), delay differential equations,PDEs (Forster-McKendrick) and IBMs

PDEs (size rather than age),numerical approximations, IBMs, DEB models,physiologically structured population models (PSPMs)

13

PSPMs & role of ecological feedback

density dependence:

dense populations individuals grow slowly reproduce less (if fecundity linked to size) ... equilibration(?)

Individual state:

e.g. length and energy reserves

Processes: vital rates

(feeding, maintenance, energy allocation)

reproduction

mortality

environment

can be anything …

typically: resource density,

density of conspecifics …

evolution: AD easy to implement due to full feedback loop

density dependence via food supply, not restricted to juveniles

14

Simple model of (planktivorous) fish life cycle

size-structured fish population

individuals characterized by irreversible/reversible mass (incl. gonads)

individual growth is density- and size-dependent (no fixed age-size relationship)

pulsed reproduction & maturation at fixed size

population dynamics = sum of individual life histories

reproduction growth feeding

‘herring & copepods’

unstructured resource in deterministic, closed system

15

Size/physiological structure

individual life history driven by size:

net production model & indeterminate growth (i.e. also after maturation)

‘general’ scaling laws (allometries)

vital rates scaling with body size/weight w: F(w) ~ c wb

16

Size- & state-dependent life history

irreversible mass (‘bones and vital organs’)

growth

Persson et al. (1998)

reproduction

reve

rsib

le m

ass

(‘res

erve

s+go

nads

’)

starvation threshold

max. juvenile condition

max. adult condition

metabolic maintenance limit

starvation

maturation

general life history pattern (indeterminate growth & pulsed reproduction)

17

Mathematical formulation (Persson et al. 1998)

S t a n d a r d i z e d m a s s xqxxw j)( M a i n t e n a n c e

r e q u i r e m e n t s

21 )(),(

yxyxE m

B o d y l e n g t hex p)( )()(

lxwlxL c

E n e r g y b a l a n c e ),(),(),,( yxERxERyxE mag

A t t a c k r a t e )( )1exp()(0

)(

0

)(max w

xww

xwAxA F r a c t i o n o f n e t e n e r g y

E g a l l o c a t e d t o g r o w t h

i n i r r e v e r s i b l e m a s s

o therwise0

0and)( if

0and)( if

),,()1(

)1(

gma txaqaq

y

gma txjqjq

y

ELxL

ELxL

Ryx

H a n d l i n g t i m e 21 ))(()(

xwxH S t a r v a t i o n m o r t a l i t y

sx

y

sx

y

s

q

qyx y

x

sqs

if 0

if ),(

1

F o o d i n t a k e r a t e

RxHxA

RxARxI

)()(1

)(),(

T o t a l n a t u r a l m o r t a l i t y ),(),( 0 yxyx s

A s s i m i l a t e d e n e r g y ),(),( RxIkRxE ea F e c u n d i t y

( e g g s s p a w n e d )

o therwise0

and)( if/)(),(

xqyLxLwxqykyxF jma tbjr

Individual level processes (x = irreversible mass; y = reversible mass; R = resource density)

18

Mathematical formulation (Persson et al. 1998)

Individual level dynamics (x = irreversible mass; y = reversible mass; R = resource density)

Population level dynamics = sum of individual life histories

resource dR/dt = r(K-R) - i I(xi,R)

consumer dNi/dt = -(xi,yi) Ni

reproductive pulses with period T added newborn cohort (i=0):

N0= i F(xi,yi) Ni (+ resets of reversible mass)

dxi/dt = [(xi,yi,R) Eg(xi,yi,R)]+

dyi/dt = [(1-(xi,yi,R)) Eg(xi,yi,R)]+

consumer population composed of a number of discrete cohorts (iI) cohorts characterized by physiological state (xi, yi) and number of individuals Ni

typical approach: age-based cohorts (computational load)

19

Role of competition for a shared resource

size-dependent competition abilities min. requirements for the shared resource:

3 qualitative types of outcome

= scaling of size-dependent attack rate A(w)~b w

0.001 0.01 0.1 1 10 100 1000

0.00001

0.0001

0.001

0.01

standardized mass, w (g)

min

imum

res

ourc

e de

nsity

= 1.1

= 0.8

= 0.5

adults can outcompete juveniles

juveniles can outcompete adults

20

Population dynamics - role of exponent

low: single-cohort cycles

(recruit-driven cycles)

intermediate: equilibrium dynamics

high: non-recruit-driven cycles juveniles

adults

resource

6-year single-cohort cycle

juveniles

adults

resource

8-year cycle (not single-cohort)

juveniles

adults

resource

fixed-point dynamics (annually)

21

Population dynamics - role of environment

[once more] an example: recruit-driven, single-cohort cycles

cycle length can change only in discrete steps decreases with mortality and increases with maturation size

juveniles

adults

resource

6-year single-cohort cycle Cohort cycle length decreases with mortality

4yr3yr

2yr

1yr5yr

22

Conclusions

PSPMs cover phenotypic plasticity via environmental feedback loop (e.g. common food source)

based on individual-level, mechanistic rules

(sort of ...) easy to incorporate other ecological phenomena e.g. cannibalism, ontogenetic niche shifts, spatial variation in resources …

price to pay I: more complex extensions are computationally very intensive

price to pay II: the models require a number of parameter values (detailed life histories),key ones not a priori clear

setup amenable to study of evolutionary responses quantitative genetics, adaptive dynamics (examples tomorrow)

deterministic system: exhaustive check of parameter (sub)space doable (x IBMs, dyn.prog.)

price to pay III: usually heavy computational load (scan of parameter space)