Transcript New Directions in Ecological Complexity:

Modeling food-web dynamics
The time evolution of species’ biomasses in a food web:
• Basal species exhibit exponential growth bounded by a carrying
capacity
• All other species grow according to their feeding rates, feeding
preferences, and assimilation efficiencies
• All species lose energy according to their metabolic rates and the
rate at which they are consumed
• Functional responses determine how feeding rates vary with the
abundance of predator and/or prey species
(Based on a 3-species food-chain model proposed by Yodzis and Innes 1992 American Naturalist)
Nonlinear bioenergetic ecosystem model
The variation of Bi, the biomass of species i, is given by:
( x y
n
Bi’(t) = Gi (B) – xi Bi (t) +
i
ij
αij Fij (B) Bi (t) – xj yji αji Fji (B) Bj (t) / eji
j =1
Rate of change = Production rate – Loss of biomass +
in biomass
if species i is basal
to metabolism
Gain of biomass –
from resource spp.
Loss of biomass to
consumer spp.
What factors allow persistence of species in
dynamical models of complex food webs?
(the “devious strategies”)
)
Rate of change = Production rate – Loss of biomass +
in biomass
of basal spp.
to metabolism
Gain of biomass –
from resource spp.
Loss of biomass to
consumer spp.
( xi yij αij Fij (B) Bi (t) – xj yji αji Fji (B) Bj (t) / eji )

j =1
n
Bi’(t) = Gi (B) – xi Bi (t) +
3 species parameters:
Gi (B ) : production rate of basal species i
(Mass/Time)
For primary producers, Gi (B) = ri Bi (t) (1 – Bi (t) / K i ), where
ri : intrinsic growth rate of species i (1/Time)
Ki : carrying capacity of species i (Mass)
______________
xi : mass-specific metabolic rate of species i (Mass/Time * 1/Mass)
4 species interaction parameters:
eji : assimilation efficiency of species j consuming species i (fraction of biomass)
yij : rate of maximum biomass gain by species i consuming j normalized by
metabolic rate of species i (Mass/Time / Mass/Time)
αij : relative preference of species i for species j (fraction of diet)
(αij = 0 for producers and sums to 1 for consumers)
Fij (B) : non-dimensional functional response (based on parameters q or c)
(relative consumption rate of predator species i consuming prey species j
as a fraction of the maximum ingestion rate; function of species’ biomass)
Parameterized Functional responses
Type II (dominates nonlinear population dynamics modeling; q or c = 0)
- ƒ(prey density)
- function of predator search and prey handling times
Type III (q = 1)
- ƒ(prey density)
- predation on low-density prey relaxed; successful food searches increases
predator’s search effort
Predator Interference (c = 1)
- ƒ(prey & predator densities)
- increase in predators decreases predation due to interference among predators
- matches empirical data much better than Type II (Skalski & Gilliam 2001)
Gradation from Type II to Type III Functional Response
1 q
ij
0.8
Fij ( B ) 
q=1
0.7
0.6
Relative
consumption
rate of predator F
species
c=0
B j (t )
n
1 qij
ik Bk (t )
k 1
1 qij
 B0 ji
0.5
0.4
c=1
q=0
q = .1
0.3
q=1
0.2
q = .25
0.1
0
0
B0
B1
2B0
α: relative prey preference of predator
species
B: biomass
Bo: half saturation density of prey species
when consumed by predator species
q: controls form of functional response
q = 0 (Type II)
q = 1 (Type III)
Biomass of prey species
Addition of Predator Interference
to Type II Functional Response:
Fij (B ) 
Bj (t )
n

k 1
ik
Bk (t )  (1  cij Bi (t ))B0 ji
3-species dynamics & functional response
Slight increases in Predator Interference or Type III stabilize dynamics
chaotic dynamics  period doubling reversals  stabilization of limit cycles  stable stationary solution
Type III
biomass local min/max
(top predator)
local min & max
Predator Interference
functional response parameters
When c or q = 0, the functional response is Type II
10-species dynamics & functional response
9
6
7
.5
.5
3
1
8
.8
.4
.4
4
.5
.5
1
5
.5
1
.5
1
2
min
.2
Bmax
.2
Strong Type II FR
may stress dynamics
by increasing feeding
on rarer species while
decreasing it on more
abundant species.
biomassBmin
, & max
10
At q = 0 (conventional strong Type II response),
only 4 taxa display persistent dynamics.
1
2
3
4
5
6
7
8
9
10
At q > 0.15 (very weak Type III response),
all 10 taxa are persistent.
At q > 0.3 (weak Type III response),
all 10 taxa are steady-state.
q
q
functional response
Stabilization of Dynamics of Ecological Networks
(S=30, C=0.15) with Functional Responses
 Effects of Structure on Dynamics 
Holling Type II / III
0.7
Robustness
0.6
Niche model
0.5
Cascade model
0.4
0.3
0.2
0.1
0
Random model
0
0.05
0.1
0.15
0.2
0.25
0.3
Beddington-DeAngelis Predator Interference
0.7
Niche model
0.6
0.5
Cascade model
0.4
0.3
0.2
0.1
0
Random model
0
0.4
0.8
c
q
 Effects of Dynamics on Structure 
1.2
1.6
2
0.8
Type III f unctional response, q = 0.2
BD functional response, c = 1
0.75
Linear vs. Hyperbolic
De-stabilization of
30-Species Dynamics
Due to increases in
Diversity and
Complexity
0.65
0.6
0.55
(a) C0 = 0.15
0.5
0.45
15
20
25
30
35
40
45
50
S
0
0.9
Type III f unctional response, q = 0.2
BD functional response, c = 1
0.8
Robustness
Back to May’s (1973)
Stability criteria:
i(SC)1/2<1
Robustness
0.7
0.7
0.6
0.5
(b) S0 = 30
0.4
0.3
0.05
0.1
0.15
0.2
C
0
0.25
0.3
3.8
B
0.4
3.6
0.35
3.4
3.2
B
Mean Trophic Level of Consumers
0.45
3
A
2.6
2.4
10
15
20
S
25
0.2
0.5
C
0.4
0.3
0.2
10
15
20
S
25
0.15
30
0.6
0.1
0.3
0.25
2.8
Omnivory as fraction of consumers
robust niche
webs have: (A)
consumers at
lower trophic
levels, (B)
more basal
species, and
(C) higher
fractions of
herbivores
Herbivory as fraction of consumers
q=0.2, c=0
30
10
15
20
S
25
30
10
15
20
S
25
30
0.9
0.8
0.7
0.6
0.5
0.4
0.6
SD of node connectivity
Effects
of
Dynamics
on
Structure
4
Dynamical model
Niche model
Random deletion of consumers
0.55
0.5
0.45
10
15
20
S
25
30
Effects of Omnivore Feeding Preference
among Trophic Levels
0.7
q = 0.2
Robustness
0.6
0.5
0.4
q=0
0.3
0.2
0.1
0
0.1
Low Trophic-level Prey
1
Skewness
10
High Trophic-level Prey
Factors increasing overall species persistence
•
Non-type II functional responses
- stabilizes chaotic & cyclic dynamics
- more ecologically plausible & empirically supported
•
Non-random network topology
- especially empirically well-corroborated niche model structure
•
Decreasing S & C
- supporting May’s early analyses
- but not fatal to persistence of diverse, complex networks
•
Consumption weighted to low trophic levels
- eat low on the food chain!
Current & future directions
•
Non-uniform distributions of functional responses, prey preferences,
etc.
•
Allometric scaling: distribution of metabolic parameters (Body Size!)
•
Add Nutrients etc. and conduct Invasion & extinction experiments
•
Data: Coupled human/natural systems (e.g., fisheries)
•
Ecoinformatics: Webs on the Web (WOW)
•
•
•
Large Diverse Complex Networks need Collaboration
Database: Who eats Whom, Functional Responses, Metabolic Parameters,
Analysis and Visualization
This work was supported by National Science Foundation grants:
 Scaling of Network Complexity with Diversity in Food Webs
 Effects of Biodiversity Loss on Complex Communities: A Web-Based Combinatorial Approach
 Webs on the Web: Internet Database, Analysis and Visualization of Ecological Networks
 Science on the Semantic Web: Prototypes in Bioinformatics
Willliams, R. J. and N. D. Martinez . 2000.
Simple rules yield complex food webs. Nature 404:180-183.
Willliams, R. J. and N. D. Martinez . 2001. Stabilization of
Chaotic and Non-permanent Food-web Dynamics. Santa Fe Inst. Working Paper 01-07-37.
Williams, R. J., E. L. Berlow, J. A. Dunne, A-L Barabási. and N. D. Martinez. 2002.
Two degrees of Separation in Complex Food Webs. PNAS 99:12917-12922
Dunne, J. A. R. J. Williams and N. D. Martinez. 2002.
Food-web structure and network theory: the role of size and connectance. PNAS 99:12917-12922
Brose, U., R.J. Williams, and N.D. Martinez. 2003.
The Niche model recovers the negative complexity-stability relationship effect in adaptive food webs.
Science 301:918b-919b
Williams, R.J., and N.D. Martinez.
Limits to trophic levels and omnivory in complex food webs: theory and data.
In press. American Naturalist.
Dunne, J.A., R.J. Williams, and N.D. Martinez
Network structure and robustness of marine food webs
In press Marine Ecology Progress Series