niche theory in ecology and evolution: a mathematical

Niche theory in ecology and evolution: A mathematical exercise, or help in biology?

Niche theory in ecology and evolution:

A mathematical exercise, or help in biology?

Géza Meszéna

Eötvös University, Budapest

MBA15:Linking Mathematical Theories with Empirical Realities

Leicester, 2015


Introduction

Outline

1 Introduction

2 Theory

3 Controversies

4 Conclusion


Introduction

Why are there so many kinds of animals?

Different pictures in ecology and evolution:we need a mathematical unification.

Niche space

Niche 1

Species 1

Niche 5

Species 5

Niche 3

Species 3

Niche 2

Species 2

Niche 4

Species 4

Adaptive landscape

Species 1

Species 5

Species 3

Species 2

Species 4

Species occupy differentniches.

Species occupy differentpeaks of landscape.

Tension: “wittest wins” versus “coexistence with reduced competition”


Introduction

ESA14 Ignite: Theory vs. Empiricism

Karen C. AbbottFrederick R. Adler: Go to the ant, thou theoristGreg Dwyer: Models without data can be useful, data analysiswithout mechanistic models often is notAnnette M. Ostling: The case for general theory developmentas a guiding force in ecologyJeremy W. Fox: When does data settle arguments and whendoes it not?Mercedes Pascual: Team Theory: When we can’t experimentand we need to tackle long and large scales


Introduction


Frederick R. Adler: Go to the ant, thou theoristGreg Dwyer: Models without data can be useful, data analysiswithout mechanistic models often is notAnnette M. Ostling: The case for general theory developmentas a guiding force in ecologyJeremy W. Fox: When does data settle arguments and whendoes it not?Mercedes Pascual: Team Theory: When we can’t experimentand we need to tackle long and large scalesBradley J. Cardinale: Team Empiricism: If a model is writtenin a forest, and no one hears it, does it make a sound?


Introduction


Greg Dwyer: Models without data can be useful, data analysiswithout mechanistic models often is notAnnette M. Ostling: The case for general theory developmentas a guiding force in ecologyJeremy W. Fox: When does data settle arguments and whendoes it not?Mercedes Pascual: Team Theory: When we can’t experimentand we need to tackle long and large scalesBradley J. Cardinale: Team Empiricism: If a model is writtenin a forest, and no one hears it, does it make a sound?John J. Stachowicz: Empirical rules. A picture is worth athousand words. . . or equations


Introduction


Annette M. Ostling: The case for general theory developmentas a guiding force in ecology

Jeremy W. Fox: When does data settle arguments and whendoes it not?

Mercedes Pascual: Team Theory: When we can’t experimentand we need to tackle long and large scales

Bradley J. Cardinale: Team Empiricism: If a model is writtenin a forest, and no one hears it, does it make a sound?

John J. Stachowicz: Empirical rules. A picture is worth athousand words. . . or equations

Alison G. Power: Bridging the gap between theory and data:Is empiricism the driver or the passenger?


Introduction


Jeremy W. Fox: When does data settle arguments and whendoes it not?

Mercedes Pascual: Team Theory: When we can’t experimentand we need to tackle long and large scales

Bradley J. Cardinale: Team Empiricism: If a model is writtenin a forest, and no one hears it, does it make a sound?

John J. Stachowicz: Empirical rules. A picture is worth athousand words. . . or equations

Alison G. Power: Bridging the gap between theory and data:Is empiricism the driver or the passenger?

Shahid Naeem: Condensation and ignition in ecologicalresearch: Making sense of biodiversity’s demise


Introduction

Is there such thing, as niche theory?

Once upon a time we have it:

Gause’s principle & limiting similarityLotka-Volterra model & resource utilization functionHutchinson’s niche space

Before long, the picture fell apart:

Ecology became too complicated for Lotka-Volterra.Mechanistic models did not lead general results.No clear conclusion on Gause and limiting similarity.

Since, theoretical ecology has grown up:

Beyond the specific models (eg. Caswell).Renewed interest in coexistence theory (eg. Chesson).Niche has remained controversial.


Introduction

Goal: Clarity to niche theory!

Needed: a comprehensive mathematical theory of ecologicalniche.

It should be applicable to any ecological situation.

It should provide biological insight.


Theory

Steps from LV & classical niche to modern theory

1 Resources ⇒

Regulating variables

2 Lotka Volterra ⇒

linearization of dynamics

3 Resource utilization ⇒

impact & sensitivity

4 Limit of similarity ⇒

Robustness of coexistence

Population dynamics

Population regulation

Externalparameters ( )E

Population sizes, ,...,n n n1 2 L


(i.e. resource concentrations)

( )R

Populationgrowth rates

, ,...,r r r1 2 L


Theory


1 Resources ⇒








Any model can be linearized!Lotka-Volterra competition:

ri = r0i −∑

j

aijnj

Generalized competitioncoefficient:

aij = −∂ri

∂nj


Theory


1 Resources ⇒








Classical niche theory (ad hoc):

aij ∼∑

k

uikujk

Resource utilization

Proposed theory (derived):

−aij =∂ri

∂nj

=∑

k

∂ri

∂Rk

∂Rk

∂nj

= S i ·I j

Sensitivity of Species i

Impact of Species j


Theory


1 Resources ⇒








Equilibrium:

r(R(n), E) = 0

Perturbation:

∂n

∂E= a

−1 ∂r

∂E

Robustness:

det a = det(S i I j)

must be large!⇒ Species should be different!


Theory


1 Resources ⇒








Larger similarity inImpact or Sensitivity

⇓

Weaker robustnessof coexistence

[No absolute limit of similarity!]


Theory

Structural instability of continuous coexistence

Population sizes n(x)

Populationgrowth rates

r(x,n)

R

Population dynamics

Population regulation

dn/dt = r(x,n)n

Theorems proven:

Compactness of the operator of regulation: coexistence ofinfinitely many fixed types is structurally unstable.

+ analicity in 1D: the possibility of a coexistence with limitpoint is structurally unstable

(Meszéna & Gyllenberg, JMB, 2005; Barabás et al., 2012, EER)


Theory

Niche space: Ways of niche segregation

Varieties for niche space:

Temperature

Food size

Cold patch Warm patch

Large food

Small food

Fo

od

co

ncen

trati

on

Discreteregulatingvariables

Continuumof regulating

variables

Four niches 2D niche space Circular niche space

Time of year

Niche space: set of regulating variables.Not necessarily an Euclidean space of a few dimension!


Theory

Complications abound

What about

population structure?

spatial structure?

temporal structure?

facilitation?

etc.

We have papers about them...


Controversies

Structural instability of continuous coexistence(Szabó & Meszéna, Oikos, 2006)

Lotka-Volterra competition for a resource continuum;Gaussian competition kernel.

-1 0 10

0.02

0.04

Str

ateg

y D

ensi

ties (a)

σ

-1 0 1Resource Axis

0

0.01

0.02 (b)

σ

-1 0 10

0.005

0.01 (c)

σ

-1 0 10

0.4

0.8

Str

ateg

y D

ensi

ties (a)

σ

-1 0 1Resource Axis

(b)

σ

-1 0 1

(c)

σ

-1 0 10

0.15

0.3

Str

ateg

y D

ensi

ties (a)

σ

-1 0 1Resource Axis

0

0.3

0.6 (b)

σ

-1 0 10

0.3

0.6 (c)

σ

-1 0 10

1

2 (a)

σ

-1 0 1

(b)

σ

-1 0 1

(c)

σ

-1 0 10

0.4

0.8

Str

ateg

y D

ensi

ties (d)

σ

-1 0 1

(e)

σ

-1 0 1

(f)

σ

-1 0 10

0.5

1 (g)

σ

-1 0 1Resource Axis

(h)

σ

-1 0 1

(i)

σ

Except the immediate vicinity of continuous coexistence:Segregation by niche width!


Controversies

Self-organised similarity?Scheffer & Nes, PNAS, 2006

Claim: coexistence of similars!!!!!(Scheffer & Nes: Self-organized similarity, PNAS, 2006)

Model: apparent competition

r(y) = r0

(

1 −

∫

α(y , x)n(x)dx

K (y)

)

− gn(y)

[n(y)]2 + H2(1)

Generalized competition:

δr(y)

δn(x)= −

r0

K (y)α(y , x) − gδ(y − x)... (2)

Degeneracies:

They choose K (y) =const.

Delta function for apparent competition!


Controversies

Self-organised similarity?Scheffer & Nes, PNAS, 2006

Original model Degeneracies removed

No degeneracies ⇒ No coexistence of similars at t = ∞!


Controversies

Niche-neutrality continuum?

No!

Neutral

Non-generic

Niche-segregated

EITHER strictly neutral OR sufficiently niche-segregated!


Conclusion

Conclusion: We have the theory

Coexisting species must differin their way of regulation!

Niche space: set/list of regulating variables.

Niche of a species: impact and sensitivity.

Robustness of coexistence is lost when either the impact, orthe sensitivity niches are too similar.

Intuitive requirement for niche segregation by niche widthprevails as a rule of thumb.

Self-organized similarity and niche-neutrality are wrongconcepts.


Conclusion

Thanks for the coworkers!

György Barabás (University of Chicago)

Mats Gyllenberg (University of Helsinki)

Hans Metz (University of Leiden)

Kalle Parvinen (University of Turku)

Liz Pásztor (Eötvös University)

Péter Szabó (Szent István University)

András Szilágyi (Eötvös University)

Thanks for your attention!

niche theory in ecology and evolution: a mathematical

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