other patterns in communities
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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. S. A. Species-area relationships. - PowerPoint PPT PresentationTRANSCRIPT
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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
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Species-area relationships• Islands, either oceanic or habitat• Selected areas within continents• How is number of species related to area?
S
A
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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
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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 )
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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
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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
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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
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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
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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
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Dynamic equilibrium
• equilibrium when E = I
• determines S*
• how are rates related to area?
S
RATE
S*
E
I
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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
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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
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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
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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
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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
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Processes
• Interspecific competition
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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
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Resource competition
competitor #1
competitor #2-
-
competitor #1
competitor #2
resource- -
+
+
Interference competition
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Competition• Interference
–Direct attack–Murder–Toxic chemicals –Excretion
• Resource–Food, Nutrients–Light–Space –Water
• Depletable, beneficial, & necessary
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Competition & population• Exponential
growth• dN / dt = r N
– r = exponential growth rate
–unlimited growth• Nt = N0 ert
N
t
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Competition & population• Logistic growth:
[ K - N ]dN / dt = r N K
• r = intrinsic rate of increase
• K = carrying capacity
N
t
K
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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
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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
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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
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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
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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
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• 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
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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
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• 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
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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
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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
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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
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Isocline in 3 dimensions
N2
N1
0 K1
K1 / a2 Zero Growth Isocline ...dN1/N1dt = 0
r1
dN1 / N1dt
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Isocline in 3 dimensions
N2
K1 / a2
N1
0K1
Zero Growth Isocline ...dN1/N1dt = 0
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IsoclineN2
K1 / a2
N1
0 K1
Zero Growth Isocline ...dN1/N1dt = 0
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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
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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”
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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”
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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
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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
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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
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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
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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”
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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
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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
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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
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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
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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
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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
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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
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Interspecific competition: Paramecium
• George Gause• P. caudatum goes
extinct• Strong
competitors, use the same resource (yeast)
• Competitve asymmetry
• Competitive exclusion
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• P. caudatum & P. burseria coexist
• P. burseria is photosynthetic
• Competitive coexistence
• Apparently stable
Interspecific competition: Paramecium
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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
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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]
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Absolute Lotka-Volterra
N1
0
1/b21
1/b22
dN2 / N
2dt = 0
1/b11dN
1 / N1 dt = 0
1/b12
Stable coexistence
N2
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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)
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Absolute Lotka-Volterra
N1
0
1/b21
1/b22
dN2 / N
2dt = 0
1/b11dN
1 / N1 dt = 0
1/b12
Stable coexistence
N2
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Not ecological models• No mechanisms of competition in the model
– Phenomenological• Environment not explicitly included• Mechanistic models of Resource competition