Essential resources consumption vectors are parallel (essential) R1R1 R2R2 C i1 C i2 C1C1

Substitutable resources consumption vectors are not parallel (substitutable) R1R1 R2R2 C i1 C i2 CiCi

Switching resources consumption vectors are perpendicular to isocline (switching) R1R1 R2R2 C1C1

Renewal for 2 resources supply vector: points at supply point S 1,S 2 R1R1 R2R2 S 1,S 2 U

Equilibrium: 1 sp. 2 resources consumption vector equal & opposite supply vector R1R1 R2R2 CiCi CiCi CiCi U S 1,S 2 U U

Equilibrium Equilibrium (R 1,R 2 ) falls on isocline therefore, dN / N dt =0 U and C vectors equal in magnitude, opposite direction therefore dR 1 / dt = 0 and dR 2 / dt = 0

Competition for 2 resources R1R1 R2R2 sp. 1 S 1,S 2     sp. 1 always excludes sp. 2  sp. 2 cannot survive  neither spp. can survive

Competition for 2 resources R1R1 R2R2 S 1,S 2     neither spp. can survive  sp. 2 cannot survive  sp. 1 always excludes sp. 2  S 1,S 2  coexistence sp. 1 sp. 2 sp. 1

Equilibrium sp. 1 – needs less R 1 (limited by R 2 ) –consumes more R 2 sp. 2 –needs less R 2 (limited by R 1 ) –consumes more R 1 consumes more of the resource limiting to itself

Equilibrium is stable R1R1 R2R2 sp. 1 sp. 2 sp. 1 Print starting here

Competition for 2 resources R1R1 R2R2 S 1,S 2     neither spp. can survive  sp. 2 cannot survive  sp. 1 always excludes sp. 2  S 1,S 2  one species eliminated sp. 1 sp. 2 sp. 1

Equilibrium sp. 1 – needs less R 1 (limited by R 2 ) –consumes more R 1 sp. 2 –needs less R 2 (limited by R 1 ) –consumes more R 2 consumes more of the resource limiting to its competitor

Equlibrium is unstable R1R1 R2R2 sp. 1 sp. 2 sp. 1

Substitutable resources (Tilman) R1R1 R2R2 sp. 1 sp. 2 1 wins R1R1 R2R2 sp. 1 sp. 2 2 wins R1R1 R2R2 sp. 1 sp. 2 stable sp. 2 sp. 1 R1R1 R2R2 sp. 2 unstable sp. 2 sp. 1

Displacement from equilibrium R1R1 R2R2 sp. 1 sp. 2 unstable sp. 2 sp. 1 R1R1 R2R2 sp. 2 stable sp. 2 sp. 1 Stable: each species consumes more of the 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 R1R1 R2R2 sp. 1 sp. 2 Chase & Leibold sp. 2 sp. 1 R1R1 R2R2 sp. 2 Tilman; Grover sp. 2 sp. 1 Stable: each species consumes more of the resource that most limits it

Chase & Leibold, p. 47 Mathematical 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;…” R1R1 R2R2 sp. 1 sp. 2 Chase & Leibold sp. 2 sp. 1

The problem: what does it mean to be “most limited” by a resource? R1R1 R2R2 sp. 1 sp. 2 sp. 1 R1R1 R2R2 sp. 2 sp. 1 Most limited at equilibrium

Stable coexistence R1R1 R2R2 sp. 1 sp. 2 Chase & Leibold sp. 2 sp. 1 Species 1 is most limited by resource 2 because it has a greater R * 2 than does species 2; Species 2 is most limited by resource 1 by similar logic. (this logic is wrong)

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 R 2 (R * 2 < R * 1 ) dN/dt=0 dN/dt>0 R2R2 R1R1 I

dN 1 /dt=0 dN 2 /dt=0 R2R2 R1R1 I1I1 I2I2 Grover and Tilman both 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 R1R1 R2R2 sp. 1 sp. 2 Tilman; Grover sp. 2 sp. 1 Species 1 is most limited by R 1 because a given increase in R 1 yields a greater increase in dN 1 /dt compared to the same increase in R 2 ; Species 2 is most limited by resource 2 by similar logic. These are the correct isoclines for stable coexistence

Displacement from equilibrium R1R1 R2R2 sp. 1 sp. 2 unstable sp. 2 sp. 1 R1R1 R2R2 sp. 2 stable sp. 2 sp. 1 Stable: each species consumes more of the 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 R1R1 R2R2 S 1,S 2    sp. 1 sp. 2  S 1,S 2  sp. 2 sp. 1

Some relevant references Grover, J.P Resource competition. Chapman & Hall NY Leon, J. A. & Tumpson, D. B Competition between two species for two complementary or substitutable resources. J. Theoretical Biology 50:

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

Laboratory environment: a chemostat nutrient input (S 1,S 2 )outflow (m)

Real Chemostat Reaction vessel Inflow Outflow

Experiments: Tilman (1982) Diatoms Asterionella & Cyclotella Resources PO 4 & SiO 2 Determine R * ’s & C vectors for each alone Predicts stable coexistence possible R1R1 R2R2 sp. 1 sp. 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 –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

Miller et al.

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

Any natural pattern could be explained in several ways

Distributions of barnacles Rocky intertidal zone adult barnacles immobile on rocks larvae settle on rocks from plankton Joseph Connell (1961)Ecology 42: see Fig. 8.7

Distributions of Balanus & Chthamalus lowest low tide highest high tide Balanus Adults Larvae Balanus Chthamalus Adults Larvae ROCK

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 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 criteria (continued) –  Manipulation of the resource and each population yield effects consistent with intraspecific competition resource performance sp. 1 density sp. 1 perf. sp. 1 density sp. 2 perf. –  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

Experimental studies Evidence is cumulative Density manipulations are now the standard Not always feasible –spatial scale –ethics Reviews of experiments –Connell 1983 –Schoener 1983 –Gurevitch et al. 1992