essential resources consumption vectors are parallel (essential) r1r1 r2r2 c i1 c i2 c1c1
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Equilibrium: 1 sp. 2 resources
consumption vector equal &opposite supplyvector
R1
R2
Ci
Ci
Ci
U
S1,S2
U
U
Equilibrium
• Equilibrium (R1,R2) falls on isocline
• therefore, dN / N dt =0• U and C vectors equal in magnitude,
opposite direction• therefore dR1 / dt = 0 and dR2 / dt = 0
Competition for 2 resources
R1
R2
sp. 1
S1,S2
S1,S2
S1,S2
sp. 2
sp. 1 alwaysexcludes sp. 2
sp. 2 cannotsurvive
neither spp.can survive
Competition for 2 resources
R1
R2
S1,S2
S1,S2
S1,S2 neither spp.can survive
sp. 2 cannotsurvive
sp. 1 alwaysexcludes sp. 2
S1,S2
coexistence
sp. 1
sp. 2
sp. 2
sp. 1
Equilibrium
• sp. 1– needs less R1 (limited by R2)
– consumes more R2
• sp. 2– needs less R2 (limited by R1)
– consumes more R1
• consumes more of the resource limiting to itself
Competition for 2 resources
R1
R2
S1,S2
S1,S2
S1,S2 neither spp.
can survive sp. 2 cannotsurvive
sp. 1 alwaysexcludes sp. 2
S1,S2
one species eliminated
sp. 1
sp. 2sp. 2
sp. 1
Equilibrium
• sp. 1– needs less R1 (limited by R2)
– consumes more R1
• sp. 2– needs less R2 (limited by R1)
– consumes more R2
• consumes more of the resource limiting to its competitor
Substitutable resources (Tilman)
R1
R2
sp. 1
sp. 2
1 wins
R1
R2sp. 1
sp. 2
2 wins
R1
R2 sp. 1
sp. 2
stable
sp. 2
sp. 1
R1
R2 sp. 1
sp. 2
unstable
sp. 2
sp. 1
Displacement from
equilibrium
R1
R2 sp. 1
sp. 2
unstable
sp. 2
sp. 1
R1
R2 sp. 1
sp. 2
stable
sp. 2
sp. 1
Stable: each speciesconsumes more ofthe resource that most limits it
A digression: Conflicting diagrams
• Compare Fig. 27 C. of Tilman with Fig. 2.8 of Chase & Leibold
• Disagreement about what produces stable coexistence for substitutable resources
• Grover (1997) gives similar isoclines/consumption vectors to Tilman
Stable coexistence
R1
R2 sp. 1
sp. 2
Chase & Leibold
sp. 2
sp. 1
R1
R2 sp. 1
sp. 2
Tilman;Grover
sp. 2
sp. 1
Stable: each speciesconsumes more ofthe resource that most limits it
Chase & Leibold, p. 47Mathematical appendix to ch. 2
• For the equilibrium to be locally stable: “Verbally, the species with the shallowest slope to its ZNGI must have the steepest impact vector;…”
R1
R2 sp. 1
sp. 2
Chase & Leibold
sp. 2
sp. 1
The problem: what does it mean to be “most limited” by a resource?
R1
R2
sp. 1
sp. 2sp. 2
sp. 1
R1
R2 sp. 1
sp. 2sp. 2
sp. 1
Most limited at equilibrium
Most limited by a resource: For a unit increase of a resource, the most limiting resource produces the greatest increase in dN/dt.
most limited by R2 (R*2 < R*
1)
dN/dt=0dN/dt>0
R2
R1
I
dN1/dt=0
dN2/dt=0
R2
R1
I1
I2
Grover and Tilmanboth agree with the statement: “…the species with the shallowest slope to its ZNGI must have the steepest impact vector;…”
isoclines given by Grover
Stable coexistence
R1
R2 sp. 1
sp. 2
Tilman;Grover
sp. 2
sp. 1
Species 1 is most limited by R1 because a given increase in R1
yields a greater increase in dN1/dt compared to the same increase in R2; Species 2 is most limited by resource 2 by similar logic.
These are the correct isoclines for stable coexistence
Displacement from
equilibrium
R1
R2 sp. 1
sp. 2
unstable
sp. 2
sp. 1
R1
R2 sp. 1
sp. 2
stable
sp. 2
sp. 1
Stable: each speciesconsumes more ofthe resource that most limits it
Kinds of resources
• General predictions do not depend on kind of resource (mostly)
• Suggests competition between autotrophs or between heterotrophs should lead to similar community structure– actually may not be true
• Combinations of resources can yield multiple equilibria
Competition for 2 resources
sp. 1 excludes sp. 2
coexistence
sp. 2 excludes sp. 1
R1
R2
S1,S2
S1,S2
S1,S2
sp. 1
sp. 2
S1,S2
S1,S2
sp. 2
sp. 2
sp. 1
sp. 1
Some relevant references
• Grover, J.P. 1997. Resource competition. Chapman & Hall NY
• Leon, J. A. & Tumpson, D. B. 1975. Competition between two species for two complementary or substitutable resources. J. Theoretical Biology 50:185-201
Common pattern predicted
• Coexistence among competitors– requires specific intermediate ratio of two
resources– extreme ratios lead to elimination of one or
the other competitor– resource ratio hypothesis: competitive
coexistence or exclusion are products of specific environmental resource ratios
Assumptions
• Simplifying environmental– environment is homogeneous and constant
except for resources• Simplifying biological
– individuals identical, constant through time• Explanatory
– competition is expressed only through depression of resources
Experiments: Tilman (1982) • Diatoms Asterionella &
Cyclotella• Resources PO4 & SiO2
• Determine R*’s & C vectors for each alone
• Predicts stable coexistence possible R1
R2
sp. 1
sp. 2sp. 2
sp. 1
Experiments: Tilman (1982) • Results
– 5/5 supply points predict Asterionella correctly
– 4/4 supply points predict stable coexistence correctly
– 2/4 supply points predict Cyclotella correctly• 2/4 yield coexistence
See fig. 4.1 in Chase & Leibold
More experiments • Tilman (1982) summarizes many more
studies with phytoplankton• Grover (1997) summarizes recent work
with– phytoplankton– bacteria– terrestrial plants– zooplankton
• R* rule, resource ratio hypothesis, and specific predictions largely supported
Resource competition theory
• more precise statement of competitive exclusion principle
• R* rule• resource ratio hypothesis• ground work for models of multiple
interacting species
Testing the resource ratio hypothesis
• Competitive coexistence or exclusion are products of specific environmental resource ratios
• Miller et al. 2005– Predictions of the resource-ratio
hypothesis supported 75% of the time– Prediction that dominance changes with
resource ratio supported 13/16 tests– Many purported tests deemed inadequate
• Replication; Controls; Time scale
Competition in nature
• Miller et al.: Resource ratio hypothesis rarely tested in nature
• Is resource competition common?• Does R* rule predict outcome?• Does resource ratio affect coexistence?• What other mechanisms of coexistence
are observed?
Competition in ecological time
• Observe: coexistence in nature• Hypotheses:
– competition is not occurring– coexistence based on resource ratios or
limitation by different resources – heterogeneity of environments creates
refuges from competition
Demonstrating that competition occurs
• Observations– exclusive or abutting
distributions gradient– responses to unintentional
introductions, displacement of native species
Distributions of barnacles
• Rocky intertidal zone
• adult barnacles immobile on rocks
• larvae settle on rocks from plankton
• Joseph Connell (1961)Ecology 42:710-723
• see Fig. 8.7
Distributions of Balanus & Chthamalus
lowest low tide
highest high tide Balanus
Adults Larvae
Balanus
Chthamalus
Chthamalus
Adults LarvaeROCK
Chthamalus & Balanus
• Hypothesis: Balanus excludes Chthamalus in competition
• Hypothesis: Chthamalus cannot tolerate submergence in low intertidal
• Hypothesis: Balanus cannot tolerate desiccation in high intertidal
• Hypothesis: Different predators in high vs. low intertidal
Testing interspecific competition in nature
• Reynoldson & Bellamy 1971• 5 criteria
– Comparative distribution / abundance of species suggest competition
– Species share some resource (or interfere)
– Evidence for interspecific competition• performance of species + related to resources
• Observational criteria
Reynoldson & Bellamy 1971• 5 criteria (continued)
– Manipulation of the resource and each population yield effects consistent with intraspecific competition
resource
perf
orm
ance
sp. 1 density
sp. 1
per
f.
sp. 1 density
sp. 2
per
f.
– Manipulations of species abundances yield effects on the other species consistent with interspecific competition
Experimental criteria Controls, replication
Performance
• Surivival• Growth• Feeding success• Fecundity• Assumed to be correlates of population
rate of increase