A General Unified Niche-Assembly/Dispersal-Assembly Theory
of Forest Species Biodiversity
Keith Rennolls
CMS, University of Greenwich, Park Row, London SE10 9LS
k.rennolls@gre.ac.uk
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Abstract: A generalised niche-assembly/dispersal-assembly theory is outlined. The aim
is an explanatory theory of island/sample species abundance curves, of species-area
curves, and of island biogeography phenomena. A niche space is defined over a spatial
domain in terms of, for example, a stratification obtained from an environmental variable
defined over that domain. Within each niche a neutral individual behaviour model is
adopted (i.e. species-independent). Individual mortality is random and replacement is
stochastic, depending on local species density per niche. A general formulation of a spatial
model of stochastic inter-niche (seed) dispersal is presented; a Markovian inter-niche
dispersal probability matrix is introduced. By stages, a simple spatially symmetric model of
dispersal is defined which is neutral within niches, but not inter-niche neutral. Finally, this
may be reduced to a distance-independent niche-dispersal (1st order) model which is
essentially a simplest “truly unified model” (Hubbell (2001, p319) which generalises
Hubbell’s “unified theory” (the 0th order model).
Keywords: species abundance distributions, species diversity, neutral theory, dispersalassembly, niche assembly.
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Introduction
When we observe complex phenomena in nature, e.g. the spatial occurrence of trees of
different species in a tropical forest, we have a natural tendency to try to understand what
we see by forming simple rules and laws. Simple laws, or models, have a natural beauty to
the human mind. We might look separately at one aspect of the phenomenon at a time,
e.g. the background niche-structure, or the dispersal process. Alternatively we might look,
one at a time, at various relationships in data collected from the phenomenon, e.g. the
species-abundance curve, or the species-area curve. Hence we have niche theories,
dispersal theories, and the log-normal and power “laws” to describe the species-area
relationship. All of these restricted theories or empirical models capture valuable insights
into aspects of the complex multi-dimensional dynamic phenomenon in which we are
interested.
However, there really is little reason for us to expect any of these restricted theories alone
to be able to explain all aspects of the complex multi-dimensional dynamic phenomenon,
or the sample data collected from it. Hence, Hubbell’s unified neutral theory of biodiversity
is interesting for how far it can go to describe and explain the observed data (Hubbell
1997, 2001).
It is likely to be a fruitless to try to build a realistic model of the dynamics of a complex
multi-dimensional process by patching together the descriptive relationships observed
when consideration is restricted to cross-sections of the complexity space. Only if the
processes underlying the cross-sectional laws were independent could such compounding
of cross-sectional laws be expected to lead to a coherent theory of the phenomenon in all
its complexity. However, usually, we find processes are inter-dependent and relationships
are non-linear, so that simple construction of a whole theory from cross-sectional models
is not possible.
In this paper a model of a complex forest is specified, with a tropical forest in mind. In this
model a collection of essentially homogeneous niches are postulated. Within each niche it
is assumed that all species have adapted to the niche, and to each other, and hence to
have the same functional and behavioural phenotype. Hence, the theory is individualneural within each niche. However, we assume that the adaption of a particular species to
a niche implies that an individual from that species have a reduced probability of filling the
gap left by an individual in another niche, (through dispersion), compared with the
replacement probability from a species adapted to the niche in which the gap occurs. This
still allows that a species might prefer an alternative niche to that to which it is adapted, so
that if the opportunity arose, a species would still migrate from its saturation-niche. Hence,
the model includes niche differentiation effects, and is dispersal based. It reduces to
Hubbell’s model when there is a single homogenous niche.
Hubbell (2001, p319) in presenting his unified neutral theory, anticipates that there will be:
“a truly unified theory that at a more fundamental level reconciles these
two apparently conflicting perspectives”, (i.e. the niche-assembly and the
dispersal-assembly approaches).
Hubbell goes on to say (p320) that he believes that such a truly unified theory will:
“integrate ecological drift and demographic and environmental
stochasticity in niche assembly theory, but this will require a deep reevaluation of the current niche-assembly theoretical paradigm”.
See Chave (2004) for a review of more recent work on Hubbell’s Neutral Theory, and
Chace (2005) for a recent critique.
The model presented below is one of a very large family of such “truly unified theories”. It
is a small generalisation of Hubbell’s neutral model, but seems to achieve a status of being
a “truly unified” theory without the need for a fundamental re-evaluation of current nichetheoretical paradigm.
The Model
We use the concepts and notation of Hubbell (1997, 2001) as far as possible, extended to
take into account niche differentiation.
The Island Model
Consider a continuous and connected subset of the plane, I ⊂ R , an island or local
community of area A. In contrast to Hubbell’s model, I may have arbitrary size, so it is
possible that there may be dispersal limitation within I. This will introduce new dispersiongenerated dynamic effects, such as migration waves across I, for example.
2
Suppose that I is sub-divided into KI sub-regions Ik : k=1, …,KI of areas Ak : k=1, … ,KI ,
each of which is assumed to be homogeneous in spatial distribution of resources. Such a
sub-region we call a niche. The niche structure of I consists of the mosaic of niche subregions. The niche structure might be defined as a stratification of the area of I, by using
an environmental variable which is co-related with resource availability. For example,
elevation is such a variable and is commonly used; it defines the topography of I. The
niches defined within such a topography might be regions of high-slope, valleys, and
plateaux, all of these being local environments which are known to sustain
characteristically different mixes of resident species. Alternative niche stratifications might
be based on other environmental variables such as soil-type, geology, or the Cartesian
products of such stratifications.
Let Nik be the abundance of species i in niche k, in I. Note that species i in niches k1 and
k2 are different species. The zero-sum assumption of Hubbell’s model for I is adopted, for
convenience. It is likely to be a reasonable approximation if one were considering mature
trees only, or even trees with dbh over 10cms. The zero sum assumption would not hold
between mature trees and sapling replacements: tree growth is not considered in this
model.
Let Jk be the total number of individuals of all species in niche-k of I, and J the total number
of individuals on I.
sk
J k = ∑ Nik
i =1
K
J = ∑ Jk
(1)
k =1
where sk is the number of species in Ik.
Suppose that the locations of individuals of each species in each niche are distributed
(uniformly) randomly over the niche area. Let the location of the jth individual of the ith
species in the kth niche be xjik. The Jk individuals in Ik are uniformly distributed over the
area Ak. The individual locations { xjik }j, i, k might be regarded as the realization of a
Poisson point process, rate λk, restricted to the niche-k in the plane. The spatial distribution
is not an important feature of this model. It could as well be regular, or a mix of regular
and Poisson. It is defined explicitly so that simulated realizations may readily be
generated, and effects of distance dependent dispersal probabilities (in extended neutral
models) might be investigated.
The meta-community model
The inclusive meta-community, Ω, for the island I is defined on a closed and bounded subset of R2 which contains I. That is, I ⊂ Ω ⊂ R2. Ω may be regarded as the “Population” of
which I is an “areal sample”. The meta-community for I, as it is usually defined, is Ω’ =Ω - I,
the enclosing environment of the island or local community I.
Within Ω’ we define the same quantities as defined for I, with a dash used to distinguish.
Hence there are K’ niches, with areas A’k having Sk species (≡ s’k), with N’ik individuals of
species i in niche k, of which the jth individual has position x’jik. J’k and J’ are the total
number of individuals in the enclosing meta-population Ω’. We define
P 'ik =
N 'ik
J 'k
(2)
We assume that the size of Ω’ is much larger than I, so that P’ik would not change
significantly in the time scales need to simulate the dynamic behaviour of I. This
assumption is not necessary, but we adopt it to ensure maximal consistency with the
treatment and notation of Hubbell. We wish to use the results of Hubbell as limiting test
cases for the new more general model.
The dispersal model
~
Suppose that the j th tree of species i in stratum k on I dies, and leaves a gap at location
x~j ik that may be filled by seeding from remaining individuals in I or Ω’.
In reality, dispersal behaviour will be highly dependent on the local environment, be it local
topography, local weather conditions, or local animal movement patterns. However,
consider first the (seed) dispersal properties only of individuals in the same niche as the
gap. Define the recruitment neighbourhood Γof the gap at x ~j ik , in the same niche, to be
the set of individuals which have non-zero probability of seeding the gap with a new
individual of the donor species, that is
~
~
~
Γ( j , i, k ; k ) = {( j , i, k ) | j ≠ j ; p ( j , i ' , k ; j , i, k ) > 0; ∀i '∈ I k }
~
where p ( j , i ' , k ; j , i, k ) is the transition-probability that individual (j,I’,k) fills the gap. Note
this neighbourhood set is over all species in niche k. The neutral assumption that withinniche transition-probability is the same for all species within the niche is adopted. Similarly
~
the neutral assumption for the inter-niche transition-probabilities p( j , i ' , k ' ; j , i, k ) are
common to all species in niche k’ is adopted.
Real world dispersal probabilities between two points in the plane will depend in some way
on all the possible dispersal routes that might exist between the two points. Every
individual would have its own unique gap filling probability. The situation is radically
simplified by considering dispersal to be possible between niches k’ and k only within a
distance Dk’k and that any individual in niche k’ which is within this neighbourhood disc of
the gap location has an equal probability of seeding the gap.
Finally, because of niche differentiation, suppose that the within-niche dispersal probability
for niche k is θk times that for any other-niche to niche-k dispersal.
Hence the simplified within-I dispersal model has K2 dispersal distance parameters, and K
θ-parameters, [the K(K+1) parameter model]. We may simplify further by supposing a
common dispersal distance, D, so that the dispersal model then has K+1 parameters. This
might be reasonable if the gap is vastly more likely to be filled by close neighbouring trees,
and the distance of these is small in comparison with the scale of the environmental
variation, and hence scale of the niche. Such a model is still considerably more complex
than Hubbell’s model, since it is still an explicitly spatial dispersal model. If we suppose
that the maximal diameter of I is less than D, (in which case D is not used explicitly), then
we have a K-parameter niche-dispersal model with no spatial of distance dependent
components. If further we suppose that each of the θk is equal to unity, then we have
Hubbell’s within-I dispersal model.
We may specify the transition probabilities for Nik under the assumed zero-sum model with
K+1 parameters, and also give the simplification to the K-parameter distance-independent
niche-dispersal model. In fact it is this K-parameter niche-dispersal model which is the
“simplest truly unified model” of the niche-assembly and dispersal-assembly approaches,
which Hubbell (2001) anticipated.
Reference
Chase, J. M. (2005) Towards a really unified theory for meta-communities. Functional
Ecology, Volume 19 Issue 1, pp182-186.
Chave, J. (2004) Neutral theory and community ecology. Ecology Letters, 7: 241–253
Hubbell, S. P (1997) A unified theory of biogeography and relative species abundance
and its application to tropical rain forests and coral reefs. Coral Reefs 16, Suppl.: S9-S21.
Hubbell, S. P (2001) The Unified Neutral Theory of Biodiversity and Biogeography
Princeton University Press, 448pp.