Download of the competition kernel a(x)

A reexamination of the MacArthur-May
theory of species packing
Olof Leimar
Stockholm University
• Historical background to species packing
• The MacArthur-May theory
• MacArthur's minimization principle
• The importance of waste in competition
• Reexamination using Fourier analysis
•
Is a continuum of types possible?
Some history
• Why are organisms apportioned into clusters
separated by gaps? (Coyne and Orr 2005)
– "The manifest tendency of life toward
formation of discrete arrays is not deducible
from any a priori considerations. It is simply a
fact to be reckoned with." (Dobzhansky 1935)
– "Homage to Santa Rosalia or Why are there so
many kinds of animals?" (Hutchinson 1959)
Hutchinson noted a ratio of 1:1.28 in size of
resource extracting body parts in mammals
and birds that occupied “neighboring niches”
•
What are the limits to similarity
in coexistence?
Limiting similarity and character
displacement
Interspecific competition
causes divergence of
character in sympatry
(Brown and Wilson 1956)
Character displacement
is evidence in favor of
limiting similarity
If species are too similar
they cannot coexist
Darwin’s finches
MacArthur-May theory of species packing
Robert H MacArthur
Robert M May
Geographical ecology, 1972
Stability and complexity in
model ecosystems, 1973
MacArthur-May theory of species packing
Based on Lotka-Volterra
competition equation
m


dN i
 N i 
ki   ij N j 


dt

j1

Competition coefficient

The competition coefficients are determined by overlap in resource utilization
Closely packed species must have very similar carrying capacities to coexist

May-MacArthur community matrix analysis
They assumed a large number m
of equidistant species, each with
the same carrying capacity and
density, and with competition
given by the overlap of Gaussian
utilization kernels
Linearization around the
equilibrium gives a community
matrix proportional to minus the
competition matrix ij.
 ij  c(i j ) ,
2
c  exp[d 2 (4w 2 )]
The ij matrix is symmetric and positive definite and thus has positive eigenvalues
May and MacArthur (1972) approximated the smallest eigenvalue as
min  4 (w d)exp[ 2 w 2 d 2 ]
1
2
For large overlap (small d/w) this eigenvalue is very close to zero. They referred
to this near-neutrality of the community stability as “an essential singularity”
They concluded that the inter-species gaps d need to be a bit larger than w for
robust coexistence (e.g. in the face of environmental fluctuations)
May-MacArthur community matrix analysis
Lotka-Volterra dynamics:
 x 2 
f (x) 
exp 2 
2
 2w 
2w

C(x)   f (y  x) f (y)dy 


dN i
 N i 
k   ij N j 

dt

j

1
 ij 
2
C((i  j)d)
 c (i j ) ,
C(0)
 x 2 
exp
2 
2
 4w 
4 w
 d 2 
c  exp
2 
 4w 
1
Linearization around equilibrium community: Nj = N* + Uj

dU j
 N    ijU k
j
dt

The equilibrium community is stable if the eigenvalues of the
community matrix are negative
May-MacArthur community matrix analysis
May and MacArthur studied matrices like
1

c
A   4
c
 9
c
 ij 
These are symmetric and positive definite:

1
C(0)
c
1
c
c4

 zi ij z j 
ij
c4
c
1
c
c 9 

c 2 
c 

1 
f (y  id) f (y  jd)dy
1
C(0)
 

2
i
f (y  id) zi dy  0
For large matrices, May and MacArthur claimed that the smallest eigenvalue was
min  1 2c  2c 4  2c 9  2c16  
•
They also claimed that this was approximated (for small d/w) by

min  4 (w d)exp[ 2 w 2 d 2 ]
1
2
The checked this numerically for different matrices A

•
•
•
A circular
niche-space
•
•
•
•
Examples (Lack, MacArthur)
North American tits and their European counterparts
Foraging height in antbirds (Formicariidae)
MacArthur’s minimization principle
• MacArthur (1969) “Species packing, and what interspecies competition minimizes”
• MacArthur (1970) “Species packing and competitive equilibrium for many species”
“It has always been interesting to some scientists to construct minimum
principles for their science. … Here I attempt an ecological minimum principle”
The principle is, roughly, to obtain the best fit of total resource utilization to
available production
Lotka-Volterra competition equations


dN i
 N i 
k   ij N j 


dt

j

Q should be minimized by the dynamics Q  k
N
i
dN i
Q
 N i
dt
N i

i 
1
 ij N i N j
2 i, j
 dQ  Q dN i   N Q   0
 N dt  iN 
dt
 i 
i
i
i

2
The principle only works when the competition coefficients form a symmetric
matrix. The principle is nicest when this matrix is positive definite (with a close
connection to a positive Fourier transform of the competition kernel), in which

case there is a single local minimum (related to close-packing)
There are, of course, many different
species number explanations
Suggestions mentioned by MacArthur (1972)
There are more species where
•
•
•
•
•
•
•
•
•
there are more opportunities for speciation
there are fewer hazards and catastrophes
more competitors can be packed closely
climate is benign
climate is more stable
the environment is more complex (more readily subdivided)
the environment is more productive
there is heavy predation (giving low abundance of each species)
predators ‘sweep an area clean’ (leaving it ripe for colonization)
“Some of these are almost meaningless, but most are plausible”
Reexamination of the MacArthur-May
theory of species packing
Collaborators:
Ulf Dieckmann, Michael Doebeli, Géza Meszéna, Akira Sasaki
Situation to be studied
Types of organisms (species) characterized by a one-dimensional trait x
Nj is the population density of the type with trait xj
Lotka-Volterra dynamics

dN j
 N j 1   jk N k
k
dt


with  jk  a(x j  x k ) and
x j  jd
Questions to investigate:
How does the shape of the
 competition kernel a(x)
 affect
• the (population dynamical) stability of an equilibrium community
• the uninvadability of an equilibrium community
Competition kernels
MacArthur and May investigated
competition kernels given by
the overlap of the utilization
functions of two species
These competition kernels are
'positive definite'
We also investigated competition
kernels given by the overlap of
the beneficial utilization function
of one species and the total
(including waste) utilization
function of another species
This gives rise to more general
forms of competition kernels
The importance of waste
The minimization principle allows
arbitrarily close packing (arbitrarily
good fit to the available resource
spectrum) if the resources that are
removed by one species correspond
to the resources that are beneficial
to that species
Total ‘utilization’ (including waste)
If part of the resource spectrum is
‘wasted’ by members of the
community, a good fit to available
resources might be unachievable
Beneficial part
The resulting competition kernels
are given by the overlap of the
beneficial part for one species and
the total utilization by a competitor
species
Resulting competition kernel
These competition kernels may set
limits to species packing
Examples of waste in competition
• Birds ‘dropping seeds’ that are too small or too
large to be optimal for their beak; the dropped
seeds are eaten by mice rather than by
competitors
• Predators scaring prey (or inducing defenses
in prey) that are outside of their hunting range
• Different forms of ‘excessive’ territoriality
• Mammal herbivores trampling plants that
might be suitable for competitors
So-called trait-mediated interactions between predators
and prey have been studied a lot in recent years


Competition kernel shape
Competition kernels given by the overlap of
the beneficial utilization function of one
species and the total utilization function of
another species
Beneficial-total overlap (convolution)
a(x) 

f e (y  x) f t (y)dy
aˆ ( )  fˆe ( ) fˆt ( )
Beneficial-beneficial overlap
a0 (x) 

f e (y  x) f e (y)dy
aˆ 0 ( )  fˆe2 ( )
Convention for Fourier transform:
fˆ () 
 f (x)expi2xdx
Three limiting similarity results
• First result on species packing
A version of our problem
The stability of an equilibrium community given by the distribution n(x)
n(x)
 n(x) 1  a(x  x')n(x')dx'
t


(the previous formulation corresponds to n(x) 

Assumptions about the competition kernel
Equilibrium community (infinitely
close-packed)

Fourier transform of (generalized) function f(x)
Result 1

j
N j (x  x j ) with x j  jd )
a(x)  a(x), a(x)  0,
 a(x)dx  

n(x)   1  a(x')dx'
fˆ () 
 f (x)expi2xdx

The equilibrium community n(x) =  is stable if the Fourier transform â() of
the competition kernel a(x) is positive
and unstable if â() changes sign
Verification of Result 1
The equation is
n(x)
 n(x) 1  a(x  x')n(x')dx'
t


Linearization around equilibrium community; n(x) =  + u(x)

u(x)
   a(x  x')u(x')dx'
t
Fourier transform of the linearized equation

uˆ ( )
  aˆ ( ) uˆ ( )
t
(our assumptions about a(x) imply that â() is real and symmetric and that â(0) > 0)
We see that all small perturbations (i.e. with any frequency ) of the equilibrium
 die down if â() > 0. If â() < 0 for some frequency , the corresponding
perturbation will grow, destabilizing the equilibrium.
Some Fourier analysis we need
fˆ () 
 f (x)expi2xdx
f (x) 
 fˆ ()expi2xd

Poisson’s summation formula

1 
 a(kd)  d  aˆ (k /d)
k
k

where a(x) is a continuous function of bounded variation with
 a(x)dx  

The following two relations are versions of Poisson’s summation formula

1 
 a(kd)exp(i2kd)  d  aˆ (k /d   )
k
k



1 
 a(kd  x)  d  aˆ (k /d)exp(i2xk /d)
k
k
since the transform of g(x)  a(x)exp(i20 x) is gˆ()  aˆ (  0 )
ˆ
ˆ
 and the transform of g(x)  a(x  x0 ) is g()  a()exp(i2x0)


• Second result on species packing
Community stability for equidistantly spaced community on the real line
n(x)   N j (x  x j ) with x j  jd
j



•
•

dN j
 N j 1  a( jd  kd)N k
k
dt
with period md, i.e. N
By requiring periodicity
a “circular trait-space”

Equilibrium community: N j  N 1
= Nj, we get a community on
•
 a(kd)
k
k
k
k
Result 2 
The equilibrium community Nj = N* is stable if the Fourier transform Â() of
the “sampled” kernel A(x) is positive and unstable if Â() changes sign

Note that Â() is periodic with period 1/d, so Â() for 0   < 1/d is enough
For a community on a circle, only frequencies  = k/(md) with 0  k < m apply
Note that Â() approaches â() as d goes to zero and that  > 0 holds if â > 0
•
A circular
niche-space
 a(kd)(x  kd)
Aˆ ( )  d a(kd)exp(i2kd)   aˆ (k /d   )
The “sampled” kernel A(x)  d
has the Fourier
transform
j+m
•
•
•
•
Verification of Result 2
The equation is

dN j
 N j 1  a( jd  kd)N k
k
dt

Linearization around equilibrium community; Nj = N* + Uj
dU
 j  N 
k a( jd  kd)Uk
dt
Writing the deviation as u(x)   j U j (x  jd) the linearized equation becomes

dU j
u(x)

 (x  jd)  N    a( jd  kd) (x  jd)U k
j dt
k
j
t

 N  
 v d

N
l a(ld)(x  kd  ld)Uk   d k A(x  kd)Uk
k
A(x  x')u(x')dx'

where v d  N d ; the Fourier transform of the linearized equation is then


uˆ ()
 v d Aˆ ( ) uˆ ( )
t


• Third result on species packing
Uninvadability for equidistantly spaced equilibrium community on the real line
n(x)   N  (x  jd)  u(x) with u( jd)  0
j
u(x)
 u(x) 1 N   a(x  kd) to first order
k
t
 1 N   a(x  kd) 1  d d a(x  kd)
F(x)
Fitness landscape
k
k


From Poisson’s summation formula F(x)  1  d
 1  d

 aˆ (k /d)exp(i2xk /d)
 2  aˆ (k /d)cos(2xk /d)
Since F(0) = 0, we have F(x)  F(x)  F(0)  2 d
Result 3
k
d

k0
k0
aˆ (k /d)[1 cos(2xk /d)]

(i) The equilibrium community Nj = N* is uninvadable if F(x) is non-positive
(ii) If the competition kernel a(x) has a positive Fourier transform â(), every

equidistantly spaced community is invadable
If â(1/d) is substantially less than zero, there is “a good chance” that the equilibrium
community with spacing d is uninvadable
If â(k/d) < 0 for k > 0 the community is uninvadable


Recap of competition
kernel shape and
Fourier transforms
The sign structure of the transform of
a competition kernel is important
a(x) 

f e (y  x) f t (y)dy
aˆ ( )  fˆe ( ) fˆt ( )
a0 (x) 

f e (y  x) f e (y)dy
aˆ 0 ( )  fˆe2 ( )
If the Fourier transform changes sign to
negative values, the kernel shape can
 destabilize close-packed
communities
 prevent invasion into inter-species
gaps
Beneficial-total overlap
Community stability
Spacing: 1/d = 0.43
Illustration of Result 2
Aˆ ( )   aˆ (k /d   )  0
k
For competition kernels with Gaussian
shape (or any other shape that makes
the Fourier transform positive), an
equidistantly spaced community is
always stable
Beneficial-total overlap, 1/d = 1
(cf. May and MacArthur approximation
of the smallest eigenvalue)
For a platykurtic competition kernel, the
spacing needs to be bigger than some
critical value for community stability
(inverse spacing smaller than some
critical value)
This is caused by the negative values of
the Fourier transform of the kernel
Beneficial-beneficial overlap

Beneficial-total overlap
Competition landscapes
More mid-gap competition
Illustration of Result 3
F(x)  F(0)  2 d 
k0
aˆ (k /d)[1 cos(2xk /d)]
For competition kernels with platykurtic
shape (which makes the Fourier transform
change sign), there is a range of gap sizes
such that competition is most intense for
mid-gap phenotypes
Beneficial-total overlap
Less mid-gap competition
For larger gap-sizes, there is less mid-gap
competition, and new species can invade
For positive definite competition kernels,
there is always less mid-gap competition,
regardless of the size of the gap
However, the competition landscape gets
extremely flat for small gap sizes
("essential singularity")
Beneficial-beneficial overlap
Less mid-gap competition
Simulation of community dynamics
Simulation: new species with
random phenotypes are introduced
at low density and species with very
low densities (extinct) are removed
Competition with beneficial-total overlap
Limits to species packing
After this process continues for a
long time, a characteristic
community pattern develops
For a 'wasteful' competition kernel,
a distinctive gap size in niche space
is maintained
This is related to the negative values
of the Fourier transform of the
kernel
For a kernel with positive Fourier
transform, there is no characteristic
community gap size
Competition with beneficial-beneficial overlap
Close-packing is possible
The general topic is popular
Scheffer and van Nes (2006) Proc. Natl. Acad. Sci. USA
Self-organized similarity, the evolutionary emergence
of groups of similar species
“There are two alternative ways to survive together: being
sufficiently different of being sufficiently similar”
Copyright ©2006 by the National Academy of Sciences
Self-organized lumpy patterns in the abundance of competing
species along a niche axis
•
•
••
A circular
niche-space
‘Truncation’ of
the shape of the
Gaussian
competition
kernel may have
given rise to the
clustering
•
• • •
(since truncation
causes the tails
to oscillate in the
Fourier transform
of a competition
kernel)
Competition function
Truncated!
Scheffer and van Nes (2006)
Copyright ©2006 by the National Academy of Sciences
Example of our own simulations
Circular niche space, just like Scheffer and van Nees
Truncated Gaussian
competition kernel
Gaussian competition
kernel (no truncation)
It seems like the shape of the competition function is important for clumping
Simulated evolution of 100 species (dots in a) that are initially randomly distributed over
the niche axis results in convergence toward self-organized lumps of similar species in
the presence of density-dependent losses
•
•
••
A circular
niche-space
Top-down control from
natural enemies can
prevent species from
becoming very
abundant, reducing the
risk of competitive
exclusion.
•
• • •
This can lead to
permanent coexistence
of groups of similar
species, separated by
gaps
dN j
 1

 rN j 1   jk N k 
 K k

dt
N 2j
g 2
N j  H2
(This idea seems OK)
Top-down control
Scheffer and van Nes (2006)
Copyright ©2006 by the National Academy of Sciences
Size distributions of species in nature often show a lumpy pattern, illustrated
here for European aquatic beetles (a, data compiled by Drost et al. 1992)
Empirical data
for comparison
Scheffer and van Nes (2006)
Copyright ©2006 by the National Academy of Sciences
Summing up
• The shape of competition kernels influences
species packing and limiting similarity
• Shapes such that the Fourier transform of the
kernel changes sign destabilize very close
packing
• These shapes can also, for situations with
intermediate interspecies gaps, prevent
invasion into the gaps
• There can be rather strong selection against
invasion into the gap
• Waste in resource utilization is one possible
cause of such competition kernel shapes