A Probabilistic Test of the Neutral Model

C. M. Mutshinda1, R.B. O’Hara1, I.P. Woiwod2

1University of Helsinki, and 2Rothamsted Research, UK.

Plan of the talk

IntroductionModelResultsConclusionSuggestions

INTRODUCTION•There is a long-standing interest in identifying the mechanisms underlying the dynamics of ecological communities

•The list of presumed mechanisms is still growing

•Existing theories can be subdivised in two categories: neutral and non-neutral models

•The debate between the two sides is still very much alive

•Neutral models assume Ecological Equivalence of species, i.e. same demographic properties (birth death immigration speciation rates) for all individuals irrespective of species.

Consequence: Species richness and relative species abundance distributions (SAD) are assumed to be generated entirely by drift between species

An ecological community is a group of trophically similar species that actually or potentially compete in a local area for the same or similar resources.

•Non-neutral models consider that species may differ in their demographic properties, their competitive abilities or their responses to environmental fluctuations

The most documented version of neutral models is the Unified Neutral Theory of Biodiversity and Biogeography (UNTBB) developed by Hubbell in 2001.

From now on, neutral theory refers to Hubbell's model

• The UNTBB considers communities on two scales of communities:

Local Community

Governed by birth, death, immigration (from a

metacommunity)

Dynamics taking place an ecological time scale.

Metacommunity Include an additional mechanism of speciation taking

place on an evolutionary time scale.

•Main Assumptions of the UNTBB:

Ecological Equivalence

Zero-Sum (ZS) assumption : constant

community size (saturated communities)

Consequences of the assumptions

A typical SAD, the zero – sum multinomial

(ZSM).

Relative Species Abundance entirely

genarated by random Drift

•Criticisms of the UNTBB: have concerned both assumptions

Ecological Equivalence (e.g. Mauer &Mc Gill

2004; Poulin 2004; Chase2005) Zero-Sum assumption (e.g. Alder 2003; McGill 2003; Williamson & Gaston 2005 )

The critics of the ZSM have generally assumed equilibrium and have proceeded by comparing the fit of the ZSM to a theoretical distribution mainly the Lognormal

how realistic the parameter estimates are

if the changes in the abundance of the species can be explained by the model with a realistic community size

A sensible way of examining the neutral model would would consist of fitting the model to the data and assessing:

However, over the last 30 years, ecologists have been moving away from equilibrium ideas (e.g. Wallington et al. 2005), but Hubbell leaps straight back in.

A dynamical model such as the UNTBB can be examined without assuming equilibrium.

We Develop and fit a discrete-time neutral model identical to Hubbell's in all other aspects except that

We relax the assumption of constant community size

3 macro-moth (Lepidoptera) time series from the

Rothamsted Insect Survey light-traps network in

the UK: Geescroft I & II (from the Rothamsted farm

in Hertfordshire) and Tregaron (from a Nature

reserve in mid-Wales)

Data

Geescroft I (352, 40); Geescroft II (319, 26); Tregaron

(371, 28).

Number of species and years:

THE MODEL

Immigration rate at time tRelative abundance of sp. i at t-1

•Process Model

, 1 , 1 ,1 * *i t t t i t t i tJ m C m P

)(~1, ii PoisN

)(~ ,, titi PoisN

:community size at time t,t i tiJ N

Nber of ind. of species i at time t

,i t iP P

, , 1 ,1 * *i t t i t t i tm N m JP

, 1 *i t t iJP J P

(time-scale separation)

•Sampling Model

, ,~ *i t i t ty Pois N q

Sampling rate (observed proportion) at time t

The same analyses were carried out on the geometrid

(Geometridae) species alone which are known to

respond in a similar way to light (Taylor and French

1974).

Nber of geometrid species in the 3 datasets: 135, 127

& 135 respectively.

Model Fitting

Bayesian approach

Noninformative priors

: (1,1)tm Beta , : (5,100)i tJP U : 0.1, 0.2tq

We used MCMC via OpenBUGS to fit the model

~ 0.01,0.01i

RESULTS1

97

0

19

80

19

90

20

00

2.0

2.5

3.0

3.5

4.0

Greescroft I A

Lo

g1

0(C

om

mu

nit

y s

ize

)

19

70

19

80

19

90

20

00

2.0

2.5

3.0

3.5

4.0Observed

Expected

19

75

19

80

19

85

19

90

19

95

2.0

2.5

3.0

3.5

4.0

Greescroft II B

19

75

19

80

19

85

19

90

19

95

2.0

2.5

3.0

3.5

4.0

19

75

19

80

19

85

19

90

19

95

20

00

2.0

2.5

3.0

3.5

4.0

TregaronC

19

75

19

80

19

85

19

90

19

95

20

00

2.0

2.5

3.0

3.5

4.0

Fig. 1: Unrealistic Community sizes

1970 1980 1990 2000

05

10

15

20

Sa

mp

lin

g R

ate

Greescroft I

1975 1985 1995

05

10

15

20

Greescroft II

1980 1990 2000

05

10

15

20

Tregaron

Fig. 2: Unrealistic Sampling Rates

The horizontal dashed line is drawn at height 1!

CONCLUSION

The neutral model does not fit the data well as it

would need parameter values that are impossible

Thus, random drift alone cannot explain the

variation in species abundances

environmental stochasticityDensity-dependenceSpecies heterogeneity

Effects of species interactions

Possible reasons for the excess of temporal variation:

A number of important mechanisms are simply ignored. These include:

SUGGESTIONS

The model can be extended to include the missing components, this will result in a complex model

Ecological hypotheses such as neutral community structure can be examined from the results

Complex models can be developed and fitted under the hierarchical Bayesian framework

We examined if parameters of such a model may be identifiable, we developed a dynamical model including environmental stochasticity and interaction coefficients

The model was fitted to a dataset comprising 10 among the most abundant species at Geescroft I

All the parameters turned out to be identifiable

Nber Scientific name Common name

1 Selenia dentaria Early Thorn

2 Selenia tetralunaria Purple Thorn

3 Apeira syringaria Lilac beauty

4 Odontopera bidentata Scalloped Hazel

5 Colotois pennaria Feathered Thorn

6 Crocallis elinguaria Scalloped oak

7 Opistograptis luteolata Brimstone moth

8 Ourapteryx sambucaria Swallow-tail

9 Opocheima pilosaria Pale brinbley beauty

10 Lycia hispidaria Brindley beauty

Scientific and common names of the 10 species

Process model

, ,1, 1 , , ,exp 1

S

i j j tji t i t i t i t

i

NE N N r

K

: density-independent per capita growth rate of species i at time t,

:per capita effect of species j on the growth of species i,

:carrying capacity for species i,

: number of species in the community

,i tr

,i j

iK

S

1, ~ ,i t i ir N

, , ,~ *i t i t i ty Pois N q

1, ~ 0,i j N

Sampling model

Parameter model

, 1 ~i iN Pois

, , 1~ , 2i t i tN Pois t

~ (0,0.1)i N ~ (0.001, 0.001)i

, ~ (1,1)i tq Beta ~ (0.0001)iK Exp

~ 0.01,0.01i

Priors

Model fitting by MCMC via OpenBUGS

,

Results

•The posterior estimates of the interaction coefficients

reveal a significant negative effect of the Opistograptis

luteolata (species #7) on the reminder as illustrated in

the following table

• Significant differences in species-specific

environmental variances

The results suggest a non-neutral community structure

Species 1 2 3 4 5 6 7 8 9 10

1 -0.32 0.12 -0.02 0.13 0.13 0.08 0.69 0.05 0.08 0.00

2 0.31 -1.07 -0.05 0.01 0.00 -0.11 0.58 -0.02 -0.11 0.07

3 0.27 0.05 0.00 0.09 0.11 0.13 0.52 0.05 0.05 0.01

4 0.13 0.01 0.00 -0.10 0.03 0.08 0.32 0.04 0.13 0.01

5 0.15 -0.02 0.00 0.08 0.19 0.03 0.53 0.03 0.13 -0.01

6 0.01 -0.09 0.01 0.00 -0.04 0.21 0.51 0.10 0.14 -0.03

7 0.3 0.26 -0.02 0.21 0.02 -0.13 0.94 -0.01 0.12 -0.05

8 0.29 0.05 0.02 0.07 0.10 0.10 0.65 0.03 0.10 0.01

9 0.20 0.04 0.02 0.06 0.07 0.07 0.06 0.04 0.04 0.00

10 0.2 0.06 0.02 0.08 0.09 0.08 0.58 0.05 0.06 -0.02

posterior means of the interaction coefficients

posterior means of the interaction coefficients

Remarks

•Real communities are typically much larger than 10

species. Hence, The dimensionality of the model

may be too large

•Some interaction coefficients are almost zero or

insignificant, it might be worth not estimating them

•Sensible ways of pulling the model's dimensionality

down to a tractable level are needed, and this is where

variable selection comes into play.

We are now working on Bayesian variable selection

methods such as Gibbs Variable Selection, Stochastic

Search Variable Selection or Reversible Jump MCMC

to extend the applicability of the model to large

community datasets.

Work in Progress

THANK YOU

Alder, P. B. (2003) Neutral models fail to reproduce observed species-area and species-time relationships in Kansas grasslands Ecology 85(5), 1265-1272. Chase, J. M. (2005) Towards a really unified theory for metacommunities, Functional Ecology 19, 182-186.Gelman, A., Carlin, J.B, Hal, Stern, H.S. & Rubin, D.B. 2003. Bayesian Data Analysis. Second Edition, Chapman& Hall.Hubbell, S.P. 2001. The unified Neutral Theory of Biodiversity and Biogeography, Princeton University Press.Mauer, B.A. & McGill, B.J. 2004. Neutral and non-neutral macroecology. Basic & Applied Ecology 5, 413 – 422McGill, B.J. 2003. A test of the unified neutral theory. Nature 422, 881-885.Poulin, R. 2004. Parasites and the neutral theory of biodiversity. Ecography 27,1: 119-123.Wallington, T. J., Hobbs, R. & Moore, S.A. (2005) Implications of Current Ecological Thinking for Biodiversity Conservation: a Review of Salient Issues. Ecology and Society 10(1), 15.Williamson, M & Gaston, K.J. 2005. The lognormal is not an appropriate null hypothesis for the species- abundance distribution. Journal of Animal Ecology.Woiwod, I. P. & Harrington, R. 1994. Flying in the face of change: The Rothamsted Insect Survey. In Long- term Experiments in Agricultural and Ecological Sciences (ed. R. A. Leigh & A. E. Johnston), pp. 321-342. Wallingford: CAB International