Transcript Population dynamics

DEB theory for populations,
communities and ecosystems
(Background for chapter 9 of DEB3
….. and more)
Roger Nisbet
April 2015
Ecology as basic science
According to Google, ecology is:
•The study of how organisms interact with each other and
their physical environment.
• The study of the relationships between living things and their
environment.
•The study of the relationship between plants and animals
(including humans) and their environment.
•The science of the relationships between organisms and their
environments.
Ecological Application: Ecological Risk Assessment (ERA)
Definition1: the process for evaluating how likely it is that the environment
may be impacted as a result of exposure to one or more environmental
stressors.
ERA involves predicting effects of exposure on populations, communities and
ecosystems – including “ecosystem services” such as nutrient cycling.
Key approach uses process-based, dynamic models of exposure and response
to exposure to predict “step-by-step” up levels of organization.
•
•
•
•
1. http://www.epa.gov/risk_assessment/ecological-risk.htm
AOP: Adverse outcome pathway
TK-TD: Toxicokinetic-toxico-dynamic
DEB: Dynamic Energy Budget
IBM: Individual-based (population)
model
Stress at different levels of biological organization
few/year
100’s/year
1000’s/year 10,000’s/day 100,000’s/day
High Throughput Bacterial,
Cellular, Yeast, Embryo or
Molecular Screening
Expensive in vivo testing and
ecological experiments
Challenge for DEB theorists: to use information from organismal and
suborganismal studies to prioritize, guide design, and interpret ecological
studies Include those that inform applications such as ERA.
Environmental Challenges are Urgent
• Climate change effects already occur and will accelerate over decades
• Environmental Stress is rapid (e.g. nutrient enrichment, insecticides,
water supply, frequency of extreme events)
• Technology changes rapidly(e.g. engineered nanomaterials)
YET
• DEB is over 30 years old and had its origins in ecotoxicology, but only
a very few agencies or industries use it, in spite of focused
publications (e.g. OECD guidance document)
IMPLYING
• EITHER:
• OR:
DEB is “too complicated” for practical applications
We (DEB crowd) need to improve communication
Meeting the challenges
DEB is “too complicated” for practical applications
• Often true (unfortunately)
• “Keep it simple”, but NOT stupid
• Use both DEB-based and DEB-inspired models
Improving communication
• Know intellectual culture of users (e.g. ecology or
ecotoxicogy)
• Develop useful tools
DEB-BASED POPULATION MODELS
Two approaches to modeling population
dynamics
A population is a collection of individual organisms interacting
with a shared environment.
Individual-based models (IBMs). Simulate a large number of
individuals, each obeying the rules of a DEB model (i-state
dynamics).
• Structured population models. This involves modeling the
distribution of individuals among i-states. A large body of theory
has been developed1, and there is a powerful computational
approach – the “escalator boxcar train”2. .
1.
See for example many papers by J.A.J. Metz, O. Diekmann, A.M. de Roos
2.
See http://staff.science.uva.nl/~aroos/EBT/index.html
Feedbacks via environment
•
•
•
Environment: E-state variables
- resources, temperature, toxicants etc. experienced by all organisms.
- possible feedback from p-states
Individual Organism: i-state variables
- age, size, energy reserves, body burden of toxicant, etc.
Population dynamics: p-state variables
- population size, age structure, distribution of i-state variables
- derived from i-state and E-state dynamics (book-keeping)
Ind
Individuals
Ind
Environment
Ind
Population
Feedback
Simplest approach: use ordinary differential
equations or delay differential equations for p-state
dynamics
ODEs can be derived with “ontogenetic symmtery”1
1) All physiological rates proportional to biomass (in biomass budget
models) or to structural volume (in DEB models – V1 morphs)
2) All organisms experience the same per capita risk of mortality (hazard)
3) Include ODEs describing environment (E-state)
Resulting equations describe biomass dynamics
Delay differential equations (DDEs) follow if assumption 2 is relaxed to2,3:
2a) All organisms in a given life stage experience the same risk of mortality
1.
2.
3.
A.M. de Roos and L.Persson (2013). Population and Community Ecology of Ontogenetic
Development. Princeton University Press. See also lectures by de Roos:
http://www.science.uva.nl/~aroos/Research/Webinars
R.M. Nisbet. Delay differential equations for structured populations. Pages 89-118 in S.
Tuljapurkar, and H. Caswell, editors. Structured Population Models in Marine, Terrrestrial, and
Freshwater Systems. Chapman and Hall, New York.
Murdoch, W.W., Briggs, C.J. and Nisbet, R.M. 2003. Consumer-Resource Dynamics. Princeton
University Press.
Population dynamics and bioenergetics
– two bodies of coherent theory
Coming soon – de Roos keynote!
DEB
Biomass –based models
DEB-based IBMs*
*
B.T. Martin, E.I. Zimmer, V. Grimm and T. Jager (2012). Methods in Ecology and Evolution 3: 445-449
DEB-IBM
food
feces
b
assimilation
reserve
mobilisation
somatic maintenance
growth
structure

1-
maturity maintenance
maturation
maturity
p
reproduction
buffer
eggs
• Implemented in Netlogo (Free)
• Computes population dynamics in simple environments with minimal
programming
• User manual with examples
*
B.T. Martin, E.I. Zimmer, V.Grimm and T. Jager (2012). Methods in Ecology and Evolution 3: 445-449
Population model tests*
Low food (0.5mgC d-1)
* B.T. Martin, T. Jager, R.M. Nisbet, T.G. Preuss, V. Grimm(2013). Predicting population dynamics from the
properties of individuals: a cross-level test of Dynamic Energy Budget theory. American Naturalist, 181:506519.
Refining the model
• Martin et al. tested 3 size selective
food-dependent submodels
• Juveniles more sensitive
• Adults more sensitive
• Neutral sensitivity
• Fit submodels to low food level
compare GoF at all food levels
Theory
Data
Best model
Low food
Total
Abundance
400
High food
Neonates
400
300
300
200
200
100
100
0
Total
Neonates
Juveniles
Adults
0
Juveniles
300
Adults
400
300
200
200
100
100
0
0
0
10
20
30
40
Days
0
10
20
30
40
0
10
20
30
40
0
Days
10
20
30
40
Futher test: Daphnia populations in large lab
systems with dynamic food *
Large amplitude cycles
Small amplitude cycles
Maturity time
LA cycle
Cycle period
Maturity time
SA cycle
* McCauley, E., Nelson, W.A. and Nisbet, R.M. 2008. Small amplitude prey-predator cycles emerge from
stage structured interactions in Daphnia-algal systems. Nature, 455: 1240-1243.
DEB-IBM dynamics
Population density
5e-5
200
algae
daphnia
4e-5
150
3e-5
100
2e-5
50
1e-5
2D Graph 1
0
0
Maturation time
50
40
Col 9 vs Col 10
Col 9 vs Col 11
30
20
10
0
400
500
time (d)
600
700
1200
1300
1400
time (d)
1500
1600
Effects of a contaminant on Daphnia populations
5
Feeding
length
length(mm)
(mm)
44
33
2
2
1
1
0
Somatic
maintenance
Maturity
maintenance
x
Growth
3,4dichloranaline
Maturation
Reproduction
cumulative
reproduction
cumulative
reproduction
0
175
175
150
150
125
125
100
100
75
75
50
50
25
250
0
0
5
0
5
10
time10(d)
15
15
time (d)
Data from T.G. Preuss et al. J. Environmental Monitoring 12: 2070-2079 (2010)
Modeling from B.T. Martin et al. Ecotoxicology, DOI 10.1007/s10646-013-1049-x (2013)
20
20
Generalization: relating physiological mode of action of toxicants
to demography of populations near equilibrium1
1. Martin, B., Jager, T., Nisbet, R.M., Preuss, T.G., and Grimm, V. (2014). Ecological
Applications, 24:1972-1983.
Simplification – consider DEBkiss*?
Likelihood profiles
g
v
*Jager, T., B. T. Martin, and E. I. Zimmer. 2013. DEBkiss or the quest for the simplest
generic model of animal life history. Journal of Theoretical Biology 328:9-18.
DEB-INSPIRED MODEL
OF FEEDBACKS
INVOLVING METABOLIC
PRODUCTS
Bathch cultures of microalgae*
•
Citrate coated silver NPs were added to batch cultures of Chlamydomonas
reinhardtii after 1, 6 and 13 days of population growth.
•
Response depended on culture history
•
Experiments showed that environment (not cells) changed between treatments
•
dynamic model included: algal growth, nanoparticle dissolution, bioaccumulation , DOC
production, DOC-mediated inactivation of nanoparticles and of ionic silver.
•
Model fits (red lines)
* L. M. Stevenson, H. Dickson, T. Klanjscek, A. A. Keller, E. McCauley & R. M. Nisbet (2013). Plos ONE DOI
10.1371/journal.pone.0074456
Batch cultures of microalgae*
88888
•
Citrate coated silver NPs were added to batch cultures of Chlamydomonas
reinhardtii after 1, 6 and 13 days of population growth.
•
Response depended on culture history
•
Experiments showed that environment (not cells) changed between treatments
•
dynamic model included: algal growth, nanoparticle dissolution, bioaccumulation , DOC
production, DOC-mediated inactivation of nanoparticles and of ionic silver.
•
Model fits (red lines)
* L. M. Stevenson, H. Dickson, T. Klanjscek, A. A. Keller, E. McCauley & R. M. Nisbet (2013). Plos ONE DOI
Dynamic Energy Budget (DEB) Perspective
DEB model equations characterize an organisms as a “reactor”
that converts resources into products
Resources
Algal mass
(M)
(CO2, light, nutrients)
Growth
Development
Division
Metabolic
Products
(DOC, N or P waste)
Rate of product (DOC) production  k DV M  hDV
dM
dt
So, what’s going on?
Fast
-2
-1
0
1
Slowing
-3
log10(Chlorophyll (ug/L))
2
Stationary
5 mg/L AgNP
Control
x x x Chl below detectable limit
5
10
Day
15
20
So, what’s going on?
Fast
-2
-1
0
1
Slowing
-3
log10(Chlorophyll (ug/L))
2
Stationary
5 mg/L AgNP
Control
x x x Chl below detectable limit
5
10
Day
15
20
Environmental Implication
Can algal-produced organic material protect
other aquatic species?
0.8
0.6
0.2
0.4
media with
organic material
from algae
freshwater
media
0.0
Daphnia 48-hr
survival
Proportion of Daphnia alive after 48 hours
1.0
Acute toxicity tests: Porportion of Daphnia alive after 48 hours
Control
Control
Control
1 ug/L
AgNP
1 ug/L
AgNP
1 μg/L
10 ug/L
AgNP
10 ug/L
AgNP
10 μg/L
100 ug/L
AgNP
100 ug/L
AgNP
100 μg/L
Red = standard medium; Blue = water from late algal
cultures
DEB theory for communities
Communities and Ecosystems
•
•
•
Community: collection of interacting species
Ecosystem: Focus on energy and material flows among
groups of species (e.g. trophic levels).
Overarching challenge – understanding biodiversity
• Community dynamics involves much more than
bioenergetic processes .
• No consensus on whether “biology matters” – neutral
theory
• Is DEB relevant?
RESOURCE COMPETITION
A little demography
Consider a population of females divided into discrete age classes
Let S a be the fraction of newborns that survive to age a
Let  a be the total number of offspring from individual aged a .
A little demography
Consider a population of females divided into discrete age classes
Let S a be the fraction of newborns that survive to age a
Let  a be the total number of offspring from individual aged a .
Then the average number of offspring expected in a lifetime is
R0 
S
a
a
all age
classes
This quantity is called net reproductive rate in many ecology texts
(N.B. not a rate)

In continuous time R0    (a ) S (a )da (changing summation  integral)
0
A little demography
Consider a population of females divided into discrete age classes
Let S a be the fraction of newborns that survive to age a
Let  a be the total number of offspring from individual aged a .
Then the average number of offspring expected in a lifetime is
R0 
S
a
a
all age
classes
This quantity is called net reproductive rate in many ecology texts
(N.B. not a rate)

In continuous time R0    (a ) S (a )da (changing summation  integral)
0
In standard DEB, we can compute  ( a ) and S ( a ) by solving a system of 6
differential equations (easy for mathematica or matlab – hard for humans). Then
we can compute R0 .
A little population ecology
• Ultimate fate of a closed population that does not influence its environment is
unbounded growth or extinction.
• Without feedback, the long-term average pattern of growth or decline of
populations is exponential – even in fluctuating environments
• The long term rate of exponential growth, r, is obtained as the solution of the
equation

1    (a ) S (a )e  ra da
0
(Note similarity to equation for R0 )
A little population ecology
• Ultimate fate of a closed population that does not influence its environment is
unbounded growth or extinction.
• Without feedback, the long-term average pattern of growth or decline of
populations is exponential – even in fluctuating environments
• The long term rate of exponential growth, r, is obtained as the solution of the
“Euler-Lotka” equation1

1    (a ) S (a )e  ra da
0
(Note similarity to equation for R0 )
• Feedback from organisms in focal population to the environment may lead to
an equilibrium population (R0 = 1) or to more exotic population dynamics such as
cycles.
1. A.M. de Roos (Ecology Letters 11: 1-15, 2009) contains a computational approach (with sample code)
for solving this equation when (a) and S(a) come from a DEB model.
Resource competition
Consider two species competing for a single food
resource, X.
For each species, R0 is a function of X., and at
equilibrium, R0=1.
Thus equilibrium coexistence is unlikely.
Idea behind competitive exclusion principle
Resource competition
Consider two species competing for a single food
resource, X.
For each species, R0 is a function of X., and at
equilibrium, R0=1.
Thus equilibrium coexistence is unlikely.
Idea behind competitive exclusion principle (CEP)
Coexistence at equilibrium of N species requires N
resources
Theory behind CEP is sound
David Tilman (1977) Resource Competition between Plankton Algae: An
Experimental and Theoretical Approach. Ecology, 58, 338-348.
• 2 algal species, 2 substrates (P and Si);
• Described by Droop model (evolutionary ancestor of DEB)
• Chemostat dynamics + labe experiments
• Field data from Lake Michigan
LAB
LAKE
Possible mechanisms for species coexistence
DEB3 page 337
Bas’s List in bigger print
(1) mutual syntrophy, where the fate of one species is
directly linked to that of another
(2) nutritional `details': The number of substrates is
actually large, even if the number of species is small
(3) social interaction, which means that feeding rate is
no longer a function of food availability only
(4) spatial structure: extinction is typically local only and
followed by immigration from neighbouring patches;
(5) temporal structure
SYNTROPHIC SYMBIOSIS
MUTUAL EXCHANGE OF PRODUCTS
CORALS
FREE LIVING
INTEGRATION
FULLY MERGED
FREE LIVING HOST
FREE LIVING SYMBIONT
SHARING THE SURPLUS
ENDOSYMBIOSIS
• HOST RECEIVES PHOTOSYNTHATE SYMBIONT CANNOT USE
• SYMBIONT RECEIVES NITROGEN HOST CANNOT USE
Model predictions
E.B. Muller et al. (2009)JTB , 259: 44–57. ; P. Edmunds et al. Oecologia, in
review; Y. Eynaud et al. (2011) Ecological Modelling, 222: 1315-1322.
•Stable host;symbiont ratio at level consistent with data synthesis from 126
papers describing 37 genera, and at least 73 species
•Dark respiration rates also consistenT with data
Bas’s List in bigger print
(1) mutual syntrophy, where the fate of one species is
directly linked to that of another
(2) nutritional `details': The number of substrates is
actually large, even if the number of species is small
(3) social interaction, which means that feeding rate is
no longer a function of food availability only
(4) spatial structure: extinction is typically local only and
followed by immigration from neighbouring patches;
(5) temporal structure
Example of (6) Temporal structure
Daphnia galeata competing with Bosmina longirostris
Experiments by Goulden et al. (1982).
•Low-food, 2-day transfers: Bosmina dominated
•High-food, 4-day transfers: Daphnia dominated
Note: experiments only ran for ~70 days,
so long-term coexistence not known
BUT CEP--> outcome of competition independent of
enrichment.
EXPLANATION: Temporal variability due to experimenter!
Competition between Daphnia and Bosmina Fine line, Daphnia; bold line, Bosmina
NOTE SMALL COEXISTENCE REGION – CONSISTENT WITH ASSERTION IN
DEB3
IS THIS GENERAL?