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 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 (1 st order) model which is essentially a simplest truly unified model (Hubbell (2001, p319) which generalises Hubbell s unified theory (the 0 th order model). Keywords: species abundance distributions, species diversity, neutral theory, dispersalassembly, niche assembly. 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 2, 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. Suppose that I is sub-divided into K I sub-regions I k : k=1,,k I of areas A k : k=1,,k I, 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 N ik be the abundance of species i in niche k, in I. Note that species i in niches k 1 and k 2 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 J k be the total number of individuals of all species in niche-k of I, and J the total number of individuals on I. J J k = = s k N ik i= 1 K Jk k = 1 (1) where s k is the number of species in I k. 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 j th individual of the i th species in the k th niche be x jik. The J k individuals in I k are uniformly distributed over the area A k. The individual locations { x jik } j, i, k might be regarded as the realization of a Poisson point process, rate λ k, 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 R 2 which contains I. That is, I Ω R 2. 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 S k species ( s k ), with N ik individuals of species i in niche k, of which the j th individual has position x jik. J k and J are the total number of individuals in the enclosing meta-population Ω. We define N' P' ik ik = (2) J ' k 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 ~ th Suppose that the j tree of species i in stratum k on I dies, and leaves a gap at location x~ jik 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~ jik, 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, j j ; p( j, > 0; i' Ik } ~ where p( j, is the transition-probability that individual (j,i, 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, 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 D k 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 θ k times that for any other-niche to niche-k dispersal.
Hence the simplified within-i dispersal model has K 2 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 θ k is equal to unity, then we have Hubbell s within-i dispersal model. We may specify the transition probabilities for N ik 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.