Olof Leimar Stockholm University

A reexamination of the MacArthur-May theory of species packing

• Is a continuum of types possible?

• Historical background to species packing

• The MacArthur-May theory

• MacArthur's minimization principle

• The importance of waste in competition

• Reexamination using Fourier analysis

Some history

• What are the limits to similarity in coexistence?

• 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”

Character displacement is evidence in favor oflimiting similarity

If species are too similarthey cannot coexist

Limiting similarity and character displacement

Interspecific competition causes divergence of character in sympatry(Brown and Wilson 1956)

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

€

dN i

dt= N i ki − α ijN j

j=1

m

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Based on Lotka-Volterracompetition equation

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 analysisThey 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[−d2 (4w2)]

The ij matrix is symmetric and positive definite and thus has positive eigenvaluesMay and MacArthur (1972) approximated the smallest eigenvalue as

€

λmin ≈ 4π12 (w d)exp[−π 2 w2 d2]

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 forrobust coexistence (e.g. in the face of environmental fluctuations)

May-MacArthur community matrix analysis

€

f (x) =1

2πw2exp −

x 2

2w2

⎛

⎝ ⎜

⎞

⎠ ⎟

C(x) = f (y − x) f (y)dy∫ =1

4πw2exp −

x 2

4w2

⎛

⎝ ⎜

⎞

⎠ ⎟

α ij =C((i − j)d)

C(0)= c( i− j )2

, c = exp −d2

4w2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

dU j

dt= −N∗ α ijUkj

∑

€

dN i

dt= N i k − α ijN j

j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Linearization around equilibrium community: Nj = N* + Uj

Lotka-Volterra dynamics:

The equilibrium community is stable if the eigenvalues of the community matrix are negative

May-MacArthur community matrix analysis

€

λmin ≈ 4π12 (w d)exp[−π 2 w2 d2]

€

A =

1 c c 4 c 9

c 1 c c 2

c 4 c 1 c

c 9 c 4 c 1

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

May and MacArthur studied matrices like

These are symmetric and positive definite:

€

ij =1

C(0)f (y − id) f (y − jd)dy∫

ziα ij z j

ij

∑ =1

C(0)f (y − id)

i∑ zi( )

2

∫ dy > 0

€

λmin ≈1− 2c + 2c 4 − 2c 9 + 2c16 −L

For large matrices, May and MacArthur claimed that the smallest eigenvalue was

They also claimed that this was approximated (for small d/w) by

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 toavailable production

€

dN i

dt= N i k − α ijN j

j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟Lotka-Volterra competition equations

€

Q = −k N i

i

∑ +1

2α ijN iN j

i, j

∑Q should be minimized by the dynamics

The principle only works when the competition coefficients form a symmetricmatrix. The principle is nicest when this matrix is positive definite (with a closeconnection to a positive Fourier transform of the competition kernel), in whichcase there is a single local minimum (related to close-packing)€

dN i

dt= −N i

∂Q

∂N i

€

dQ

dt=

∂Q

∂N i

dN i

dt=

i

∑ − N i

∂Q

∂N i

⎛

⎝ ⎜

⎞

⎠ ⎟

2

i

∑ ≤ 0

• 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)

There are more species where

Suggestions mentioned by MacArthur (1972)

“Some of these are almost meaningless, but most are plausible”

There are, of course, many different species number explanations

Reexamination of the MacArthur-May theory of species packing

Collaborators: Ulf Dieckmann, Michael Doebeli, Géza Meszéna, Akira Sasaki

Situation to be studiedTypes of organisms (species) characterized by a one-dimensional trait xNj is the population density of the type with trait xj Lotka-Volterra dynamics

€

dN j

dt= N j 1− α jkNkk

∑( ) with

€

jk = a(x j − xk ) 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 bythe overlap of the utilizationfunctions of two species

These competition kernels are'positive definite'

We also investigated competition kernels given by the overlap ofthe beneficial utilization functionof one species and the total (including waste) utilization function of another species

This gives rise to more generalforms of competition kernels

The importance of waste

Total ‘utilization’ (including waste)

Beneficial part

Resulting competition kernel

The minimization principle allowsarbitrarily close packing (arbitrarilygood fit to the available resourcespectrum) if the resources that are removed by one species correspond to the resources that are beneficial to that species

If part of the resource spectrum is‘wasted’ by members of the community, a good fit to availableresources might be unachievable

The resulting competition kernelsare given by the overlap of the beneficial part for one species andthe total utilization by a competitorspecies

These competition kernels may setlimits 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 shapeCompetition kernels given by the overlap of the beneficial utilization function of one species and the total utilization function of another species

€

a0(x) = fe (y − x) fe (y)dy∫ˆ a 0(φ) = ˆ f e

2(φ)

€

a(x) = fe (y − x) f t (y)dy∫ˆ a (φ) = ˆ f e (φ) ˆ f t (φ)

Beneficial-total overlap (convolution)

Beneficial-beneficial overlap

€

ˆ f (φ) = f (x)exp −i2πφx( )∫ dx

Convention for Fourier transform:

A version of our problemThe stability of an equilibrium community given by the distribution n(x)

€

∂n(x)

∂t= n(x) 1− a(x − x')n(x')dx'∫( )

with

€

x j = jd(the previous formulation corresponds to

€

n(x) = N jδ(x − x j )j∑ )

Assumptions about the competition kernel

€

a(−x) = a(x), a(x) ≥ 0, a(x)dx < ∞∫

Result 1The 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

Fourier transform of (generalized) function f(x)

€

ˆ f (φ) = f (x)exp −i2πφx( )∫ dx

Equilibrium community (infinitely close-packed)

€

n(x) = ν =1 a(x')dx'∫

Three limiting similarity results• First result on species packing

Linearization around equilibrium community; n(x) = + u(x)

€

∂u(x)

∂t= −ν a(x − x ')u(x ')dx '∫

Fourier transform of the linearized equation

€

∂n(x)

∂t= n(x) 1− a(x − x')n(x')dx'∫( )

€

∂ˆ u (φ)

∂t= −ν ˆ a (φ) ˆ u (φ)

(our assumptions about a(x) imply that â() is real and symmetric and that â(0) > 0)

The equation is

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.

Verification of Result 1

Some Fourier analysis we need

The following two relations are versions of Poisson’s summation formula

€

a(kd)exp(−i2πφkd)k=−∞

∞

∑ =1

dˆ a (k /d + φ)

k=−∞

∞

∑

€

ˆ f (φ) = f (x)exp −i2πφx( )∫ dx

€

f (x) = ˆ f (φ)exp i2πφx( )∫ dφ

€

a(kd + x)k=−∞

∞

∑ =1

dˆ a (k /d)exp(i2πxk /d)

k=−∞

∞

∑

Poisson’s summation formula

€

a(x)dx∫ < ∞where a(x) is a continuous function of bounded variation with

€

a(kd)k=−∞

∞

∑ =1

dˆ a (k /d)

k=−∞

∞

∑

since the transform of

€

g(x) = a(x)exp(−i2πφ0x)

€

ˆ g (φ) = ˆ a (φ + φ0)is

and the transform of

€

g(x) = a(x + x0) is

€

ˆ g (φ) = ˆ a (φ)exp(i2πx0φ)

• Second result on species packingCommunity stability forequidistantly spaced community on the real line

with

€

x j = jd

€

n(x) = N jδ(x − x j )j∑

Result 2The 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

Equilibrium community:

€

N j = N∗ =1 a(kd)k

∑€

dN j

dt= N j 1− a( jd − kd)Nkk

∑( )

By requiring periodicity with period md, i.e. Nj+m = Nj, we get a community ona “circular trait-space”

The “sampled” kernel

€

A(x) = d a(kd)δ(x − kd)k

∑

has the Fourier transform

€

ˆ A (φ) = d a(kd)exp(−i2πφkd)k

∑ = ˆ a (k /d + φ)k

∑

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 applyNote that Â() approaches â() as d goes to zero and that  > 0 holds if â > 0

• ••

••

••

•A circular niche-space

Verification of Result 2

Linearization around equilibrium community; Nj = N* + Uj

€

∂u(x)

∂t=

dU j

dtδ(x − jd)

j∑ = −N∗ a( jd − kd)δ(x − jd)Ukj

∑k

∑

= −N∗ a(ld)δ(x − kd − ld)Uk = −N∗

dA(x − kd)Ukk

∑l

∑k

∑

= −vd A(x − x ')u(x ')dx '∫; the Fourier transform of the linearized equation is then

€

∂ˆ u (φ)

∂t= −vd

ˆ A (φ) ˆ u (φ)

The equation is

€

dN j

dt= N j 1− a( jd − kd)Nkk

∑( )

€

dU j

dt= −N∗ a( jd − kd)Ukk

∑

€

u(x) = U jδ(x − jd)j

∑Writing the deviation as the linearized equation becomes

where

€

vd = N∗ d

• Third result on species packingUninvadability forequidistantly spaced equilibrium community on the real line

with

€

n(x) = N∗δ(x − jd)j

∑ + u(x)

Result 3(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

Fitness landscape

€

F(x) =1− N∗ a(x − kd)k

∑ =1−ν d d a(x − kd)k

∑

€

∂u(x)

∂t= u(x) 1− N∗ a(x − kd)

k∑( )

From Poisson’s summation formula

€

F(x) =1−ν dˆ a (k /d)exp(i2πxk /d)

k∑

=1−ν d − 2ν dˆ a (k /d)cos(2πxk /d)

k≥0∑

€

u( jd) = 0

to first order

Since F(0) = 0, we have

€

F(x) = F(x) − F(0) = 2ν dˆ a (k /d)[1− cos(2πxk /d)]

k>0∑

If â(1/d) is substantially less than zero, there is “a good chance” that the equilibriumcommunity with spacing d is uninvadable If â(k/d) < 0 for k > 0 the community is uninvadable

Recap of competition kernel shape and Fourier transformsThe sign structure of the transform of a competition kernel is important

If the Fourier transform changes sign to negative values, the kernel shape can

destabilize close-packed communities

prevent invasion into inter-species gaps

€

a(x) = fe (y − x) f t (y)dy∫ˆ a (φ) = ˆ f e (φ) ˆ f t (φ)

€

a0(x) = fe (y − x) fe (y)dy∫ˆ a 0(φ) = ˆ f e

2(φ)

Community stability

For competition kernels with Gaussian shape (or any other shape that makesthe Fourier transform positive), an equidistantly spaced community is always stable

(cf. May and MacArthur approximation of the smallest eigenvalue)

For a platykurtic competition kernel, thespacing needs to be bigger than somecritical value for community stability (inverse spacing smaller than somecritical value)

This is caused by the negative values of the Fourier transform of the kernel

Illustration of Result 2

€

ˆ A (φ) = ˆ a (k /d + φ)k

∑ > 0

Beneficial-total overlap

Spacing: 1/d = 0.43

Beneficial-total overlap, 1/d = 1

Beneficial-beneficial overlap

Competition landscapes

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

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")

Less mid-gap competition

More mid-gap competition

Less mid-gap competition

Beneficial-beneficial overlap

Beneficial-total overlap

Beneficial-total overlap

€

F(x) − F(0) = 2ν dˆ a (k /d)[1− cos(2πxk /d)]

k>0∑

Illustration of Result 3

Simulation of community dynamics

Simulation: new species with random phenotypes are introduced at low density and species with very low densities (extinct) are removed

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

Limits to species packing

Competition with beneficial-total overlap

Competition with beneficial-beneficial overlap

Close-packing is possible

Copyright ©2006 by the National Academy of Sciences

Scheffer and van Nes (2006) Proc. Natl. Acad. Sci. USA

The general topic is popular

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

Scheffer and van Nes (2006)

Self-organized lumpy patterns in the abundance of competing species along a niche axis

• •

•

• ••

•

•A circular niche-space

Competition functionTruncated!

‘Truncation’ of the shape of the Gaussian competitionkernel may have given rise to theclustering

(since truncation causes the tails to oscillate in the Fourier transform of a competition kernel)

Example of our own simulations

Truncated Gaussian competition kernel

Gaussian competitionkernel (no truncation)

Circular niche space, just like Scheffer and van Nees

It seems like the shape of the competition function is important for clumping

Copyright ©2006 by the National Academy of Sciences

Scheffer and van Nes (2006)

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

(This idea seems OK)

€

dN j

dt= rN j 1−

1

Kα jkNkk

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟

− gN j

2

N j2 + H 2

Top-down control

Copyright ©2006 by the National Academy of Sciences

Scheffer and van Nes (2006)

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 datafor comparison

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