Other patterns in communities

• Macroecology: relationships of – geographic distribution and body size– species number and body size

• Latitudinal gradients: changes in S with latitude

• Species-Area relations: Island biogeography and related questions

Species-area relationships• Islands, either oceanic or habitat• Selected areas within continents• How is number of species related to area?

S

A

Mathematics

S = c Az

–S is number of species–A is area sampled–c is a constant depending on the taxa

& units of area–z is a dimensionless constant

• often 0.05 to 0.37

Often linearized• ln (S ) and ln (A )• ln (S ) = ln (c ) + z ln (A )

– z is now the slope– ln (c ) is now the intercept

ln (S )

ln (A )

Theory & Hypotheses• Area per se hypothesis

– why S goes up with A– why S = c A z

– why z takes on certain values• Habitat heterogeneity hypothesis

– why S goes up with A• Passive sampling hypothesis

– why S goes up with A

Area per se• large heterogeneous assemblage log

normal distribution of species abundances • assume log normal ("canonical log normal")

– Abundance class for most abundant species = abundance class with most individuals

– constrains variance (s2) of the distribution• assume that N increases linearly with A• Yield: unique relationship: S = c Az

• for "canonical" with S > 20: S = c A0.25

ni

Sn

Area per se• z varies systematically

– larger for real islands vs. pieces of contiguous area

• z does not take on any conceivable value– if log normal had s2 = 0.25 (very low)– then z 0.9 … which is virtually unknown in

nature– implies constraints on log normal distributions

Dynamics of the area per se hypothesis

• open island of a given area• rate of immigration

(sp. / time) = I initially high• once a species is added, I

declines• nonlinear:

– 1st immigrants best dispersers– last are poorest dispersers

S

I

ST

Dynamics of the area per se hypothesis

• rate of extinction (sp. / time) = E initially 0

• as species are added, E increases

• nonlinear: – lower n as S increases– more competition as S

increases

S

E

ST

Dynamic equilibrium

• equilibrium when E = I

• determines S*

• how are rates related to area?

S

RATE

S*

E

I

Effect of area on S*

• 2 islands equally far from mainland

• large & small• extinction rate

greater on small– smaller n’s– greater competition

• under this hypothesis I is not related to area

S

RATE

S*large

Elarge

I

Esmall

S*small

Area per se• Neutral hypotheses vs. Niche hypotheses• Neutral hypotheses – presume that biological

and ecological differences between species, though present, are not critical determinants of diversity

• Area per se is a neutral hypothesis– S depends only on the equilibrium between

species arrival and extinction– Large A large populations low prob. extinction

Niche-based hypotheses• Niche hypotheses - presume that that

biological and ecological differences between species are the primary determinants of diversity

• Niche differences enable species to coexist stably

• Does not require equilibrium between extinction and arrival

Habitat heterogeneity

• Niche-based hypothesis• Larger islands more habitats

– Why?• More habitats more species

– does not require competition– does not require equilibrium– does not exclude competition or equilibrium

Passive sampling• Larger islands bigger “target”

• Neutral hypothesis • More immigration

more species– competition &

equilibrium not necessary (but possible)

– under this hypothesis E is not related to area

S

RATE

S*small

E

Ilarge

S*large

Ismall

Processes

• Interspecific competition

Competition • Competition occurs when:–a number of organisms use and

deplete shared resources that are in short supply

–when organisms harm each other directly, regardless of resources

– interspecific, intraspecific

Resource competition

competitor #1

competitor #2-

-

competitor #1

competitor #2

resource- -

+

+

Interference competition

Competition• Interference

–Direct attack–Murder–Toxic chemicals –Excretion

• Resource–Food, Nutrients–Light–Space –Water

• Depletable, beneficial, & necessary

Competition & population• Exponential

growth• dN / dt = r N

– r = exponential growth rate

–unlimited growth• Nt = N0 ert

N

t

Competition & population• Logistic growth:

[ K - N ]dN / dt = r N K

• r = intrinsic rate of increase

• K = carrying capacity

N

t

K

Carrying capacity

• Intraspecific competition– among members of the same species

• As density goes up, realized growth rate (dN / dt) goes down

• What about interspecific competition?– between two different species

Lotka-Volterra CompetitionN1 N2 r1 r2 K1 K2

[ K1 - N1 - a2 N2 ]dN1 / dt = r1 N1 K1

[ K2 - N2 - a1 N1 ]dN2 / dt = r2 N2 K2

Lotka-Volterra Competition

• a1 = competition coefficient–Relative effect of species 1 on species 2

• a2 = competition coefficient–Relative effect of species 2 on species 1

• equivalence of N1 and N2

Effects of Ni & Ni’ on growth [ K1 - N1 - a2 N2 ]dN1 / dt = r1 N1 K1

¨ In the numerator, a single individual of N2

has a equivalent effect on dN1 / dt to a2

individuals of N1

Competition coefficients: a’s• Proportional constants relating the effect

of one species on the growth of a 2nd species to the effect of the 2nd species on its own growth– a2 > 1 impact of sp. 2 on sp. 1 greater than

the impact of sp. 1 on itself– a2 < 1 impact of sp. 2 on sp. 1 less than

the impact of sp. 1 on itself– a2 = 1 impact of sp. 2 on sp. 1 equals the

impact of sp. 1 on itself

• total population growth

• dNi / dt = riNi [Ki-Ni-ai’Ni’]/Ki

Notation

• per capita population growth

• dNi / Nidt = ri [Ki-Ni-ai’Ni’]/Ki

dNi / dt vs. dNi / Nidt

Lotka-Volterra equilibrium

• at equilibrium – dN1 / N1dt = 0 & dN2 / N2dt = 0– also implies dN1 / dt = dN2 / dt = 0, so...

• 0 = r1N1 [ (K1-N1-a2N2)/ K1]• 0 = r2N2 [ (K2-N2-a1N1)/ K2]• true if N1 = 0 or N2 = 0 or r1= 0 or r2 = 0

• for 0 = r1N1 [ (K1-N1-a2N2)/ K1]• true if 0 = (K1-N1-a2N2)/ K1

• if N2 = 0, implies N1 = K1 (logistic equilibrium)• as N1 0, implies a2N2=K1 or N2 = K1 / a2

• plot as graph of N2 vs. N1

Lotka-Volterra equilibrium

Equilibrium• dNi / dt = 0 for both species• K1 - N1 -a2N2 = 0 and K2 - N2 -a1N1 = 0

N2

K1/a2

dN1/dt<0

N1

K1

dN1/dt>0

Zero Growth Isocline(ZGI) for species 1

Zero growth isoclinefor sp. 2

N2

N10

K2/a1

K2

dN2 /N2 dt > 0

dN2 /N2 dt < 0

Zero Growth Isocline (ZGI)dN2 /N2 dt = 0

Zero growth isocline for sp. 1

N2

N10

K1

K1 / a2

dN1 /N1 dt > 0

dN1 / N1 dt < 0

Zero Growth Isocline (ZGI)dN1/N1dt = 0

Isocline in 3 dimensions

N2

N1

0 K1

K1 / a2 Zero Growth Isocline ...dN1/N1dt = 0

r1

dN1 / N1dt

Isocline in 3 dimensions

N2

K1 / a2

N1

0K1

Zero Growth Isocline ...dN1/N1dt = 0

IsoclineN2

K1 / a2

N1

0 K1

Zero Growth Isocline ...dN1/N1dt = 0

Two Isoclines on same graph

• May or may not cross• Indicates whether two competitors can coexist• For equilibrium coexistence, both must have

– Ni > 0 – dNi / Ni dt = 0

Lotka-Volterra Zero Growth Isoclines• K1 / a2 > K2 • K1 > K2 / a1

• Region dN1/N1dt>0 & dN2/N2dt>0

• Region dN1/N1dt>0 & dN2/N2dt<0

• Region dN1/N1dt<0 & dN2/N2dt<0

N2

N10

K2/a1

K2dN

2 / N2 dt = 0

K1/a2

K1

dN1 / N

1 dt = 0

Species 1 “wins”

Lotka-Volterra Zero Growth Isoclines• K2 > K1 / a2

• K2 / a1 > K1

• Region dN1/N1dt>0 & dN2/N2dt>0

• Region dN1/N1dt<0 & dN2/N2dt>0

• Region dN1/N1dt<0 & dN2/N2dt<0

N2

N10

K2/a1

K2 dN2 / N

2 dt = 0K1/a2

K1

dN1 / N

1 dt = 0

Species 2 “wins”

Competitive Asymmetry

• Competitive Exclusion• Suppose K1 K2. What values of a1 and a2

lead to competitive exclusion of sp. 2?• a2 < 1.0 (small) and a1 > 1.0 (large)• effect of sp. 2 on dN1 / N1dt less than effect of

sp. 1 on dN1 / N1dt • effect of sp. 1 on dN2 / N2dt greater than

effect of sp. 2 on dN2 / N2dt

Lotka-Volterra Zero Growth IsoclinesN2

• K1 / a2 > K2

• K2 / a1 > K1

• Region both species increase

• Regions & one species decreases & one species increases

• Region both species decrease

N1

0

K2/a1

K2

dN2 / N

2dt = 0

K1

dN1 / N

1 dt = 0

K1/a2

Stable coexistence

Stable Competitive Equilibrium• Competitive Coexistence• Suppose K1 K2. What values of a1 and a2 lead to

coexistence?• a1 < 1.0 (small) and a2 < 1.0 (small)• effect of each species on dN/Ndt of the other is less

than effect of each species on its own dN/Ndt• Intraspecific competition more intense than

interspecific competition

N1

0

K2/a1

K2

dN2 / N

2 dt = 0

K1

dN1 / N

1dt = 0

Lotka-Volterra Zero Growth Isoclines

K1/a2

N2

• K2 > K1 / a2

• K1 > K2 / a1

• Region both species increase

• Regions & one species decreases & one species increases

• Region both species decrease

Unstable twospecies equilibrium

Unstable Competitive Equilibrium

• Exactly at equilibrium point, both species survive• Anywhere else, either one or the other “wins”• Stable equilibria at:

– (N1 = K1 & N2 = 0) – (N2 = K2 & N1 = 0)

• Which equilibrium depends on initial numbers– Relatively more N1 and species 1 “wins”– Relatively more N2 and species 2 “wins”

Unstable Competitive Equilibrium• Suppose K1 K2. What values of a1 and lead to

coexistence?• a1 > 1.0 (large) and a2 >1.0 (large)• effect of each species on dN/Ndt of the other is

greater than effect of each species on its own dN/Ndt

• Interspecific competition more intense than intraspecific competition

Lotka-Volterra competition• Four circumstances

– Species 1 wins– Species 2 wins– Stable equilibrium coexistence– Unstable equilibrium; winner depends on initial N’s

• Coexistence only when interspecific competition is weak

• Morin, pp. 34-40

Competitive Exclusion Principle• Two competing species cannot coexist

unless interspecific competition is weak relative to intraspecific competition

• What makes interspecific competition weak?– Use different resources– Use different physical spaces– Use exactly the same resources, in the same

place, at the same time Competitve exclusion

Model assumptions

• All models incorporate assumptions• Validity of assumptions determines validity

of the model• Different kinds of assumptions• Consequences of violating different kinds

of assumptions are not all the same

Simplifying environmental assumption

• The environment is, with respect to all properties relevant to the organisms:– uniform or random in space– constant in time

• realistic?• if violated need a better experimental system

Simplifying biological assumption

• All the organisms are, with respect to their impacts on their environment and on each other:– identical throughout the population

• clearly must be literally false• if seriously violated need to build a different

model with more realistic assumptions

Explanatory assumptions• What we propose as an explanation of nature

(our hypothesis)– r1, r2, K1, K2, a1, a2 are constants– competition is expressed as a linear decline in per

capita growth (dN / N dt ) with increasing N1 or N2

– Some proportional relationship exists between the effects of N1 and N2 on per capita growth

• If violated model (our hypothesis) is wrong

Interspecific competition: Paramecium

• George Gause• P. caudatum goes

extinct• Strong

competitors, use the same resource (yeast)

• Competitve asymmetry

• Competitive exclusion

• P. caudatum & P. burseria coexist

• P. burseria is photosynthetic

• Competitive coexistence

• Apparently stable

Interspecific competition: Paramecium

Experiments in the laboratory

• Gause’s work on protozoa• Flour beetles (Tribolium)• Duck weed (Lemna, Wolffia)• Mostly consistent with Lotka Volterra• No clear statement of what causes

interspecific competition to be weak

Alternative Lotka-Volterra competition

• Absolute competition coefficients

dNi / Nidt = ri [1 – bii Ni - bij Nj]equivalent to:

dNi / Nidt = ri [Ki - Ni - aj Nj] / Ki

= ri [Ki/Ki - Ni/Ki - ajNj/Ki] = ri [1- (1/Ki)Ni – (aj/Ki)Nj]

Absolute Lotka-Volterra

N1

0

1/b21

1/b22

dN2 / N

2dt = 0

1/b11dN

1 / N1 dt = 0

1/b12

Stable coexistence

N2

Competitive effect vs. response

• Effect: impact of density of a species– Self density (e.g., b11)– Other species density (e.g., b21)

• Response: how density affects a species– Self density (e.g., b11)– Other species’ density (e.g., b12)

• Theory: effects differ (b11 > b21)• Experiments: responses (b11, b12)

Absolute Lotka-Volterra

N1

0

1/b21

1/b22

dN2 / N

2dt = 0

1/b11dN

1 / N1 dt = 0

1/b12

Stable coexistence

N2

Not ecological models• No mechanisms of competition in the model

– Phenomenological• Environment not explicitly included• Mechanistic models of Resource competition