Niche theory in ecology and evolution: A mathematical ...
Transcript of 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
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
Introduction
Outline
1 Introduction
2 Theory
3 Controversies
4 Conclusion
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
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”
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
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
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
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
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
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
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
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
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
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
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
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
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
Introduction
ESA14 Ignite: Theory vs. Empiricism
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?
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
Introduction
ESA14 Ignite: Theory vs. Empiricism
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
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
Introduction
ESA14 Ignite: Theory vs. Empiricism
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?
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
Introduction
ESA14 Ignite: Theory vs. Empiricism
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
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
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.
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
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.
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
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.
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
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.
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
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
Regulating variables
(i.e. resource concentrations)
( )R
Populationgrowth rates
, ,...,r r r1 2 L
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
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
Any model can be linearized!Lotka-Volterra competition:
ri = r0i −∑
j
aijnj
Generalized competitioncoefficient:
aij = −∂ri
∂nj
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
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
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
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
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
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!
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
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
Larger similarity inImpact or Sensitivity
⇓
Weaker robustnessof coexistence
[No absolute limit of similarity!]
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
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)
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
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!
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
Theory
Complications abound
What about
population structure?
spatial structure?
temporal structure?
facilitation?
etc.
We have papers about them...
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
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!
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
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!
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
Controversies
Self-organised similarity?Scheffer & Nes, PNAS, 2006
Original model Degeneracies removed
No degeneracies ⇒ No coexistence of similars at t = ∞!
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
Controversies
Niche-neutrality continuum?
No!
Neutral
Non-generic
Niche-segregated
EITHER strictly neutral OR sufficiently niche-segregated!
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
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.
Niche theory in ecology and evolution: A mathematical exercise, or help in biology?
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!