Research School of Biological Sciences **Machine** VP

**Co-Investigator** Sandra Lavorel and Peter Chesson

C.N.R.S., Montpellier, France and Research School of Biological Sciences

**A Spatial Model of Community Dynamics in Patchy Landscapes**

This study focuses on how the interaction between biological characteristics, habitat pattern, and disturbance regime can promote diversity in plant communities. The general aim is to propose mechanisms for the theory of the regeneration niche which proposes that species coexistence can be maintained through interspecific differences in regeneration characteristics. A simulation model of dynamics of two species assemblages in habitats with various spatial patterns and local disturbance frequencies is used to examine specific relationships between species dispersal and germination strategies and their coexistence in those environments. The first step was to identify robust variables to describe various components of the interaction independently. The second step has been to obtain simulated values for these variables under a factorial design of values for the biological and environmental parameters, and to analyze their relative contributions to species coexistence. The third step, presently in progress, is to obtain simulated estimates for variables derived from an analytical resolution of the model.

**What are the basic questions addressed?**

The main goals are to improve the theoretical understanding of the mechanisms of maintenance of species diversity in spatio-temporally variable environments, and to provide measurable criteria that will allow diagnosis and prediction of patterns of species diversity. Simulations have been used to investigate the relationships between population growth rate and the biological and environmental characteristics. Specific questions are: Do species need to be different and how much to coexist in variable environments? What are the relative role of spatial pattern and disturbance regimes in promoting species diversity? What sort of demographic mechanisms determine species coexistence?

**What are the results to date and the future of the work?**

The invasion of a resident population with short dispersal distance by a species with longer-range dispersal was simulated for combinations of habitat pattern, disturbance frequency and germination strategies. A germination strategy was defined by the type of response to disturbance ("disturbance-broken" when disturbances trigger germination, "risk-spreading" when germination is insensitive to disturbance) and the dormancy fraction at dispersal. Simulations estimated the long-term low-density growth rate of the invader, the mean local crowding (number of competing seeds per invader seed at each site) and the effective fecundity of each species (the mean number of seeds successfully dispersed per adult plant). Crowding increased with habitat suitability and decreased with increasing dormancy fractions for the resident. Effective fecundity in a landscape can be taken as a measure of competitive ability. The short-dispersing resident invariably had higher effective fecundity, but this difference decreased with increasing suitability, i.e. competitive differences decreased.

Coexistence depended on both habitat suitability and disturbance frequency. Maximum coexistence was obtained for habitats of intermediate suitability with moderately frequent disturbances. General linear modelling of the long-term low-density growth rate showed that coexistence results from a reduction in local crowding. This growth rate also increased for increasing habitat suitability and connectivity, and for a higher dormancy fractions of the resident species. The effects of disturbance frequency and of invader's dormancy fraction depended on the type of dormancy of the resident species. The analysis showed that 2 different mechanisms are involved in the coexistence of species with different niches. Differences in regeneration niches permit coexistence through competitive equivalency with trade-offs between dispersal and germination traits, but for a limited range of habitat pattern and disturbance conditions. On the other hand, coexistence through density fluctuations of a disturbance-broken species and storage effects can be achieved for a broad range of environmental conditions and species germination strategies. Species coexistence thus results from the combination of two mechanisms.

Our results also demonstrate the importance of detailed attention to spatial patterns and dispersal because of the complexity of spatial effects. Further, spatial pattern and disturbance frequencies need to be considered jointly to understand the dynamics of diversity.

An analytical resolution of the model dynamics was calculated. We obtain a decomposition of the population low-density growth rate according to environmental and competition variables and their covariances. Species coexsitence in the model appears mainly determined by the specific effects of spatial heterogenity and disturbance. Simulations have been estimating the different terms of our equations to validate the analytical results.

The model will next be extended to communities with more than two species, looking at groups of species with similar dispersal and germination strategies..

**What computational techniques are used and why is a supercomputer
required?**

The model simulates the demography of a two-species community. The dynamics is modelled at the individual level, for a landscape consisting of a patterned lattice of habitat sites. First, a landscape pattern is generated using a hierarchical dichotomy for suitable vs unsuitable spatial units. At each time step, the algorithm sweeps across the landscape to calculate the destinations of all seeds produced by established individuals, updates the composition of the seed bank at each site, and draws the adults for the next generation. Calculations are mainly convolutions of basic operations and functions on large arrays; the algorithm also involves numerous random draws. Population maps after a fixed number of generation are stored and analyzed using algorithms of point pattern analysis. The investigation of stochastic processes requires replicated experiments. Hence, a run for a given parameter set consists of a set of 5 simulations for a given set of parameters. Inter-simulation mean and variance for a number of synthetic descriptors are calculated to analyze the dynamics of species coexistence.

The program is written in Fortran 77. Since the algorithm relies on convolution of operations on large arrays (100 x 100), vectorizing is particularly well suited to reducing calculation time. Investigating stochastic processes requires that seeds are dispersed individually and independently. Such a procedure, when in scalar form, is extremely time-expensive (several thousands calculations per time step!), and in fact takes over 2/3 of total simulation time. After reprogramming for maximum vectorization and flexibility to further sophistications, vectorization is still only partial (about 45% at present), due to unavoidable recursivity in the seed bank dynamics. The version of the program for the estimation of the analytical terms, on the other hand, achieves 60% vectorisation.

**Publications**

*Spatial heterogeneity and species coexistence in herbaceous
communities*, S Lavorel, Invited speaker at the International Botanical
Congress, Yokohama, Japan (1993).

*Mechanisms of maintenance of herbaceous species richness in patchy habitats
with local disturbances*, S Lavorel, Annual Meeting of the Ecological
Society of America, Knoxville, U S A (1994).

*How species with different regeneration niches coexist in patterned
landscapes: a simulation study*, P Chesson and S Lavorel, Submitted to
Oikos.