Physiologically structured population models:
Formulation, analysis and ecological insights
André M. De Roos
Institute for Biodiversity and Ecosystem Dynamics University of Amsterdam
in collaboration with
Lennart Persson
Ecology and Environmental Sciences
Umeå University, Sweden January 27, 2017
The size-structured population model
Size (s)
Den si ty
Juveniles
Adults
s b s m
Resource
µ J
µ A
ω A (R)s m ω J ( R)s
ν J ( R)s
ν A ( R)s m s b
The size-structured population model
Mass conservation:
Juvenile growth and adult reproduction proportional to body size:
g (s, R) = ⌫ J (R)s = ( ! J (R) T ) s b(s m , R) = ⌫ A (R)s m
s b = ( ! A (R) T ) s m s b
@ c(t, s)
@ t + ⌫ J (R) @(sc(t, s))
@ s = µ J c(t, s) for s b s < s m
⌫ J (R)s b c(t, s b ) = ⌫ A (R)s m
s b C A (t) dC A
dt = ⌫ J (R)s m c(t, s m ) µ A C A (t) dR
dt = ⇢(R max R) ! J (R)
Z s
ms
bsc(t, s)ds ! A (R)s m C A (t)
Deriving an approximate model
J =
Z s
ms
bs c(t, s) ds and A = s m C A
dJ
dt =
Z s
ms
bs @c(t, s)
@ t ds
=
Z s
ms
bs ⌫ J (R) @ sc(t, s)
@s ds
Z s
ms
bs µ J c(t, s) ds
= ⌫ J (R) s 2 c(t, s)
s
ms
b+ ⌫ J (R)
Z s
ms
bs c(t, s) ds µ J J
) dJ
dt = ⌫ A (R) A ⌫ J (R) s 2 m c(t, s m ) + ⌫ J (R) J µ J J
Deriving an approximate model
dA
dt = ⌫ J (R) s 2 m c(t, s m ) µ A A dJ
dt = ⌫ A (R) A ⌫ J (R) s 2 m c(t, s m ) + ⌫ J (R) J µ J J
dR
dt = ⇢(R max R) ! J (R) J ! A (R) A
How to approximate ? ⌫ J (R) s 2 m c(t, s m )
Deriving an approximate model
J ˜ =
Z s
ms
bs˜ c(s) ds = ˜ b
⌫ J (R)
Z s
ms
b✓ s s b
◆ ⌫ µ
JJ
(R)
ds
⌫ J (R) s 2 m c(s ˜ m ) = ˜ b s m
✓ s m s b
◆ ⌫ µ
JJ
(R)
) ⌫ J (R) s 2 m c(t, s m ) = (⌫ J (R), µ J ) J (⌫ J (R), µ J ) = ⌫ J (R) µ J
✓
1 z 1
µ
J⌫
J(R)
◆
dJ
dt = ⌫ A + (R)A + ⌫ J (R)J ⌫ J + (R), µ J J µ J J dA
dt = ⌫ A (R)A ⌫ A + (R)A + ⌫ J + (R), µ J J µ A A dR
dt = ⇢(R max R) (! J (R)J + ! A (R)A)
Starvation mortality
Reproduction Growth Maturation Mortality
Foraging Turn-over
§ Maturation function depends
on juvenile net biomass production and mortality
§ Mass-specific net biomass production is balance between assimilation and
maintenance
J A R
: Juvenile biomass : Adult biomass : Resource biomass
Stage-structured Yodzis-Innes model
dJ
dt = ⌫ A + (R)A + ⌫ J (R)J ⌫ J + (R), µ J J µ J J dA
dt = ⌫ A (R)A ⌫ A + (R)A + ⌫ J + (R), µ J J µ A A dR
dt = ⇢(R max R) (! J (R)J + ! A (R)A)
Starvation mortality
Reproduction Growth Maturation Mortality
Foraging Turn-over
§ Maturation function depends
on juvenile net biomass production and mortality
J A R
: Juvenile biomass : Adult biomass : Resource biomass
⌫ (R) = ( !(R) T ) (⌫ J + (R), µ J ) = ⌫ J + (R) µ J
✓
1 z 1
µ J
⌫ J + (R)
◆
Stage-structured Yodzis-Innes model
Juveniles
Adults
Fecundity
0.1 1 10 100
0 10 20 30 40 500 1000 1500 2000
Juvenile-driven cycles q = 0.4
Biomass (mg/L)
Time (days)
Adult-driven cycles q = 1.6
Ontogenetic asymmetry: Two types of cycles
Juveniles competitively superior (fast growth, high survival)
Adults dominate in biomass
Adults competitively superior (high fecundity, long lifespan) Juveniles dominate in biomass
Adult-driven cycles Juvenile-driven cycles
⌫ J ( ˜ R) > ⌫ A ( ˜ R) > 0 ⌫ A ( ˜ R) > ⌫ J ( ˜ R) > 0
All physiological rates depend on body size
Resource density
growth
starvation
Attack rate
Handling time
Maintenance costs increase faster with body mass than food intake rate
Maintenance
Smaller individuals are competitively superior to
larger ones
Size-dependent asymmetry in energetics
Resource density
growth
starvation Smaller individuals are competitively inferior to
larger ones
Attack rate
Handling time
Maintenance costs increase slower with body mass than food intake rate
Maintenance
Juvenile biomass overcompensation Adult biomass overcompensation
Adult-biased mortality
Adult-biased production
Juvenile-driven cycles Adult-driven cycles
Overcompensation and population cycles
• Stage-driven population cycles associated with stage-specific biomass overcompensation
• Overcompensation occurs under wider conditions than cycles
µ A = p 2 p µ J
! A (R) = q
2 q ! J (R)
8 >
> >
> >
> >
> >
<
> >
> >
> >
> >
> :
@ c(t, s)
@t + f (R) @c(t, s)
@s = ⌘
f (R) c(t, s) f (R) c(t, 0) = f (R)
Z 1
1
c(t, s) ds dR
dt = D f (R)
Z 1
0
c(t, s) ds
Life history driven population cycles
§ Necessary requirements for their occurrence:
1. Juvenile delay, possibly food dependent
2. Competitive difference between adults and juveniles
8 >
<
> :
Juveniles (0 s < 1): f (R) = R
Adults (s 1): f (R) = q R
Juvenile- and Adult-driven cycles
0 1 2 3 4 5
Time
0.1 1.0 10.0
Juveniles ⎯ Food
0.0001 0.001 0.01 0.1
Adults
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Size
0.0001 0.001 0.01 0.1 1
Consumer density
t = 2.5
t = 2.1
0 5 10 15 20
Time
0.1 1.0 10.0
Juveniles ⎯ Food
0.0001 0.001 0.01 0.1
Adults
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Size
0.0001 0.001 0.01 0.1 1
Consumer density
t = 6.3
t = 7.4
0 1 2 3 4 5
Time
0.0001 0.001 0.01 0.1
Adults
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Size
0.0001 0.001 0.01 0.1 1
Consumer density
t = 2.5
t = 2.1
1 2 3 4 5
Time
0.2 0.4 0.6 0.8 1.0 1.2
Size
t = 2.7
t = 2.1
5 10 15 20
Time
0.1 1.0 10.0
Juveniles ⎯ Food
0.2 0.4 0.6 0.8 1.0 1.2
Size
t = 6.3
t = 7.4
Juveniles more competitive (q = 0.125)
Juveniles more competitive (q = 0.25)
Adults more competitive (q = 4.0)
Resource Juveniles Adults
Juvenile- and Adult-driven cycles: mechanisms
2 -4 2 -3 2 -2 2 -1 1 2 1 2 2 2 3
Competitive coefficient q
0.1 1.0 10.0
Consumer density
Depend on indirect density-
dependence (via resource), disappear if:
dR
dt ! R ¯
PSS⇡ 1 J + qA
Depend on resource- dependent juvenile delay, disappear if:
⌧ (R) ! ⌧ ¯
De Roos & Persson (2003) Theor. Pop. Biol. 63: 1-16
How did I create this bifurcation diagram?
2 -4 2 -3 2 -2 2 -1 1 2 1 2 2 2 3
Competitive coefficient q
0.1 1.0 10.0
Consumer density
Depend on indirect density-
dependence (via resource), disappear if:
dR
dt ! R ¯
PSS⇡ 1 J + qA
Depend on resource- dependent juvenile delay, disappear if:
⌧ (R) ! ⌧ ¯
De Roos & Persson (2003) Theor. Pop. Biol. 63: 1-16
What you have learned last week
Potential problem:
In case of alternative, dynamic attractors, some attractors might be
missed altogether, while for others this approach does not detect the
entire range of parameter values for which they occur
What you have learned last week
Better approach:
• Use the final state of a time simulation at a particular parameter value p as initial state for the time simulation at p+Δp.
• Carry out these time simulations both for increasing values of the
parameter p from its minimum to its maximum value, as well as for
decreasing p values from its maximum to minimum
0 1 2 3 4 5
Time
0.1 1.0 10.0
Juveniles ⎯ Food
0.0001 0.001 0.01 0.1
Adults
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Size
0.0001 0.001 0.01 0.1 1
Consumer density
t = 2.5
t = 2.1
0 5 10 15 20
Time
0.1 1.0 10.0
Juveniles ⎯ Food
0.0001 0.001 0.01 0.1
Adults
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Size
0.0001 0.001 0.01 0.1 1
Consumer density
t = 6.3
t = 7.4
0 1 2 3 4 5
Time
0.0001 0.001 0.01 0.1
Adults
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Size
0.0001 0.001 0.01 0.1 1
Consumer density
t = 2.5
t = 2.1
1 2 3 4 5
Time
0.2 0.4 0.6 0.8 1.0 1.2
Size
t = 2.7
t = 2.1
5 10 15 20
Time
0.1 1.0 10.0
Juveniles ⎯ Food
0.2 0.4 0.6 0.8 1.0 1.2
Size
t = 6.3
t = 7.4
Juveniles more competitive (q = 0.125)
Juveniles more competitive (q = 0.25)
Adults more competitive (q = 4.0)
Are these cycles ecologically relevant?
Resource Juveniles Adults
Daphnia dynamics with logistic algal growth
McCauley et al. (1999) Nature 402: 654-656
100 0 40 100
0 40 60 60
20 Days 80 20 Days 80
0 10 20 30 0 20 40 algal density Daphnia and Stage duration and cycle period
d
c e
b
5
1 × 10
–31 × 10
–11 × 10
–0.5 1.0 1.5 2.0 2.5 3.0
0.0 3.5
K (mg C l
–1)
a
Algal density (mg C l
–1)
Daphnia Algae
Predator-prey cycles and
small-amplitude generation
cycles occur under the same conditions
Figure 1 | Multiple limit-cycle attractors in the structured predator–prey model. a, Bifurcation diagram showing the transition from a stable steady state (solid black line) to a region of multiple coexisting limit cycles with increasing algal carrying capacity K (mg C L
-1). The range in algal density (mg C L
-1) over a cycle is shown. Stable small- amplitude cycles (blue) and large- amplitude cycles (red) are shown, separated by an unstable
cycle(dashed blue line). b, d, Large- and small-amplitude cycles of Daphnia (black) and algae (grey). c, e, Graphs showing a key diagnostic feature: the
relationship between cycle period
(dashed line) and the stage
duration of Daphnia (solid line)
during large- (c) and small-
amplitude cycles (e).
Predator-prey cycles: experimental evidence
Daphnia Algae
Predator-prey cycles and small- amplitude
generation cycles occur under the same conditions
McCauley et al. (1999) Nature 402: 654-656
Fe cu n d it y (E ggs /L it er) Fe cu n d it y (E ggs /L it er)
Period
Delay Delay
Predator-prey cycles
• Large fluctuations in Daphnia fecundity
• Fast juvenile growth
• Juvenile period shorter than cycle period
Generation cycles
• Small fluctuations in Daphnia fecundity
• Slow juvenile growth
• Juvenile period equal or longer than cycle period
McCauley et al. (2008) Nature 455: 1240-1243
Krill in the Antarctic food web
Is eaten by:
• 6 species of baleen whales
• 20 species of squid
• > 100 species of fish
• 35 species of birds
• 7 species of seals
Eats:
• Algae
• Protozoa
• Other small crustaceans
• Various larvae
• Cycle period 5-6 years, 2 years of good recruitment, 3-4 years without recruitment
• New strong cohort appears when old strong
cohort goes extinct Quetin & Ross (2003), Ross et al. (2014)
Abiotic drivers of krill abundance oscillations
Sea ice conditions
El Niño-Southern Oscillation (ENSO) cycle
Primary productivity
anomalies
Adults
Repr oduc tion Surviv al, gr owth,
matur ation
Larvae
Interannual climate variability
Food: Chl-a, ice algae
Krill population dynamics driven by food
Consumption
Consumption
Adults
Repr oduc tion Surviv al, gr owth,
matur ation
Larvae
Interannual climate variability
Food: Chl-a, ice algae
But the food availability can be affected either by external factors (climate variability) or by grazing (if the population is resource
limited), causing cycles in case of ontogenetic asymmetry
Biotic impacts on krill cycles
Significant negative effect of krill biomass on krill recruitment,
which implies resource limitation induced by the whole population
Model
• Model captures the effects of seasonality on reproduction and ontogenetic development
• Growth and fertility are proportional to difference between ingestion and maintenance rates
• In summer: all feeding stages compete for phytoplankton
• In winter: Adults can starve, larvae need to feed on ice algae, because larvae have high energy requirements
Krill cohorts increase in weight and age
Weight
Embr yos
Starvation mortality + background mortality
A bundanc e
Larvae Juveniles Adults
Reproduction
Reproductive weights
Hatchabilit y
Competition-induced starvation drives large- scale population cycles in Antarctic krill
A.B. Ryabov, A.M. de Roos, B. Meyer, S. Kawaguchi,
B. Blasius
Model predictions versus data
• In the model, population cycles can occur even in the absence of interannual
variability in phytoplankton productivity and captures cycle characteristics:
• Two successive years of successful recruitment followed by 3-4 years of unsuccessful recruitment
• New cohort appears when an old strong cohort dies
• Negative effect of krill
biomass on the juvenile
abundance one year later
The mechanism of the cycles
• Autumn phytoplankton
concentrations and duration of starvation period are strongly sensitive to total krill biomass
• Abundant krill population
(adults and/or larvae) depletes phytoplankton, leading to long starvation period of larvae
• Small krill population has smaller impact on
phytoplankton, which are
sufficient for larvae to survive
The mechanism of the cycles
• Cyclic changes in biomass lead to cyclic changes in (starvation) mortality
• Large biomass ->
high starvation mortality ->
low absolute recruitment ->
Decrease in biomass
• Small biomass ->
low starvation mortality ->
high absolute recruitment ->
Increase in biomass
Effects of climate variability
The six year oscillation cycle is retained in the model with among-year random variations in algal productivity
Power spectrum
Effects of climate variability
The correlation between summer chlorophyll
level and krill abundance next summer increases with increasing
perturbation level.
Effects of climate variability
Synchronization of two uncoupled population by climate (Moran effect)
The correlation between two separate populations increases with the level of
environmental disturbances, leading ultimately to a complete synchronization of two
uncoupled populations
0 1 2 3 4 5
Time
0.1 1.0 10.0
Juveniles ⎯ Food
0.0001 0.001 0.01 0.1
Adults
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Size
0.0001 0.001 0.01 0.1 1
Consumer density
t = 2.5
t = 2.1
0 5 10 15 20
Time
0.1 1.0 10.0
Juveniles ⎯ Food
0.0001 0.001 0.01 0.1
Adults
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Size
0.0001 0.001 0.01 0.1 1
Consumer density
t = 6.3
t = 7.4
0 1 2 3 4 5
Time
0.0001 0.001 0.01 0.1
Adults
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Size
0.0001 0.001 0.01 0.1 1
Consumer density
t = 2.5
t = 2.1
1 2 3 4 5
Time
0.2 0.4 0.6 0.8 1.0 1.2
Size
t = 2.7
t = 2.1
5 10 15 20
Time
0.1 1.0 10.0
Juveniles ⎯ Food
0.2 0.4 0.6 0.8 1.0 1.2
Size
t = 6.3
t = 7.4
Juveniles more competitive (q = 0.125)
Juveniles more competitive (q = 0.25)
Adults more competitive (q = 4.0)
Juvenile- and Adult-driven cycles
Resource Juveniles Adults
Consumers with pulsed reproduction
Feeding (continuous)
Growth & mortality (continuous)
Reproduction
(once a year)
Competitiveness explains dynamics
0 2 4 6 8 10
Years
10-610-5
Resource
104 106 108
Consumers
< 1 year Juveniles > 1 year Adults
growth
starvation
0 2 4 6 8 10
Years
10-510-4
Resource
102 104 106 108
Consumers
< 1 year Juveniles > 1 year Adults
growth
starvation
Juvenile- driven
cycles
Adult-
driven
cycles
Yellow perch in Chrystal Lake, Wisconsin Juvenile-driven cycles
Sanderson et al. (1999) CJFAS 56: 1534-1542
1985 1986 1987
Per cen tage
1988 1989
40 80 120 160
0 1990 10 20 30 40
0 10 20 30
0 10 20 30
1991
40 80 120 160 40 80 120 160
Length (mm)
1992
Yellow perch in Chrystal Lake, Wisconsin Dominant cohorts
Sanderson et al. (1999) CJFAS 56: 1534-1542
1980 1982 1984 1986 1988 1990 1992 0
40 80 120 160
Length (mm)
Year
Thanks to…
§ Lennart Persson, University of Umeå, Sweden
§ Numerous collaborators, PhD and MSc students:
Hans Metz, Odo Diekmann, Alex Ryabov, Mats Gyllenberg, David Claessen, Tobias van Kooten, Karen van de Wolfshaar, Tim Schellekens, Reinier HillerisLambers, Anieke van
Leeuwen, Arne Schröder, Magnus Huss, Birte Reichstein, Jens Anderson, Nika Galic, and many others
Population and Community Ecology of Ontogenetic Development
André M. de Roos & Lennart Persson Princeton Monographs 51
http://staff.fnwi.uva.nl/a.m.deroos/Research/Monograph
Thank you!
ERC Executive Agency Grant Preparation Unit
Place Rogier 16, BE-1049 Brussels, Belgium I [email protected] I http://erc.europa.eu
Brussels, 09/11/2012
Ares(2012) 1322359
Prof. ANDRE MARC D E ROOS Universiteit van Amsterdam F NWI-IB E D
Science Park 904 NL-1098 X H Amsterdam
Sent by electronic mail only Programme - Call identifier: E R C-2012-AdG
Proposal No 322814 - E C O E V O D E V O Dear Prof. DE ROOS,
I am pleased to inform you that the European Research Council Executive Agency (ERCEA) is now in a position to initiate the preparation of the grant agreement for your proposal entitled: "Eco- evolutionary dynamics of community self-organization through ontogenetic asymmetry".
The ERCEA intends to follow the Evaluation Report advice which has already been transmitted to you. Consequently, it is estimated that the maximum financial contribution of the Union to your project could be up to 1.779.635,00 Euro for a period of up to 60 months.
During the granting process, we will request additional information and documents as detailed in the appendix attached to be provided before the 3rd December 2012 at the latest.
Failure to respect this deadline without justification may be considered by the ERCEA as an indication that you do not wish to enter into the preparation for a grant agreement and therefore withdraw your proposal. In such a case, the ERCEA will initiate 8 weeks after the procedures to reject your proposal.
We expect the granting process to be completed as soon as possible, and by end of March 2013 at the latest.
This letter should by no circumstances be regarded as a formal commitment by the ERCEA to provide financial support, as this depends on the satisfactory conclusion of the preparation of the grant agreement and the finalisation of the ethical evaluation (where applicable).
