Thomas Löffler

It is perplexing when species-rich ecosystems change abruptly and, for example, dominant or economically interesting species populations collapse. Although various aspects of such ecosystem regime shift at tipping points have been studied, little attention has been paid to the possible dependence of community stability on the intrinsic growth rates of their species. Intrinsic growth rates of species can vary, e.g., due to evolution, environmental changes or fluctuations, disturbances, or human influences such as exploitation of certain species. We show theoretically and computationally that under certain conditions changing relative intrinsic growth rates of competing species have a strong effect on community stability. In addition, we are investigating which species characteristics favour such changes of communities from stable to unstable and vice versa.

One way to explore assembly of extant and novel communities from species pools, and by that biodiversity and species ranges, is to study the equilibrium behavior of dynamic competition models such as the Lotka–Volterra competition (LVC) model. We present a novel method (COMMUSTIX) to determine all stable fixpoints of the general LVC model with abundances x from a given pool of n species. To that purpose, we split the species in potentially surviving species (x_i > 0) and in others going extinct (x_i = 0). We derived criteria for the stability of x_i = 0 and for the equilibrium of x_i > 0 to determine possible combinations of extinct and surviving species by iteratively applying a mixed binary linear optimization algorithm.
We tested this new method against (a) the numerical solution at equilibrium of the LVC ordinary differential equations (ODEs) and (b) the fixpoints of all combinations of surviving and extinct species (possible only for small n), tested for stability and non-negativity. The tests revealed that COMMUSTIX is reliable, it detects all multiple stable fixpoints (SFPs), which is not guaranteed by solving the ODEs, and more efficient than the combinations method. With COMMUSTIX, we studied the dependence of the fixpoint behavior on the competition strengths relative to the intra-specific competition. If inter-specific competition was considerably lower than intraspecific competition, only globally SFPs occurred. In contrast, if all inter-specific was higher than intraspecific competition, multiple SFPs consisting of only one species occurred. If competition strengths in the species pool ranged from below to above the intra-specific competition, either global or multiple SFPs strongly differing in species composition occurred. The species richness over all SFPs was high for pools of species with similar, either weak or strong competition, and lower for species with dissimilar or close to intra-specific competition strengths.
The new approach is a reliable and efficient tool for further extensive examinations of the dependence of community compositions on parameter settings of the LVC model.

Up-scaling of ecological models, i.e. translating a model from a source to a larger target scale is a topic of growing importance in landscape research. It is required for adapting the scales of applications, data and models and for simplifying models in a controlled way, in order to decrease simulation times and model uncertainties. Up-scaling from lower scales can corroborate existing models, yield reliable forecast models and enhance ecological understanding by identifying relevant source scale processes and explaining how target scale phenomena are constituted from source scale mechanisms.
This chapter is an overview of up-scaling techniques based on hierarchy theory as an ideal framework for a successful up-scaling. Hierarchy theory leads to a general formulation of the up-scaling process, which consists of (a) aggregating source scale variables to target scale variables and (b) deriving the associated target scale model functions.
Different up-scaling approaches are presented and categorized according to model nonlinearity, variable heterogeneity and the used approximations. The methods range from direct evaluation, to implicit up-scaling, heuristic up-scaling, meta-modeling, mean-field approaches, scale transition theory, moment methods, and to scale separation and adaptive mesh approaches.
Examples include the scaling of a) two logistically growing subpopulations to their metapopulation b) photosynthesis from chloroplasts to continental vegetation canopies, c) individual tree dynamics to the stand level, and d) small-scale active seed dispersal behavior of animals to a general seed dispersal kernel and plant migration model.

Including explicitly the link between parent trees and recruitment is indispensable for many forest model applications, e.g. for simulating tree species migrations. However this positive feedback can result in unrealistic dominance of single species. We examine in which way this feedback can be modelled to maintain species diversity and species composition similar to observed forests. A submodel of the spatio-temporal forest model TreeMig for simulating seed production, seed dispersal, seed bank dynamics and sapling development is presented. With the extended model, interactions of different factors that can influence species diversity in a forest are examined. In simulations under different bioclimatic conditions of Switzerland differential hypotheses for maintaining species diversity are tested. The simulated species diversity and dominant species are compared to data compiled from the Swiss National Forest Inventory. Intra-specific density regulation is necessary to produce in the simulations a species diversity and species composition similar to data. Without this intra-specific density dependence, strong seed producers, such as birch (Betula pendula) dominate the stand, in contrast to the data, and decrease species diversity. With intra-specific density regulation, species diversity is further increased by spatial or temporal variability. Temporal variability in environmental factors alone can enhance species diversity if the factors fluctuate between the favourable and unfavourable range of the strong seed producers.

Landscape patterns result from complex endogenous dynamics and heterogeneity of environmental drivers. Landscape models are appropriate tools for analysing such patterns. We present the dynamic, spatially explicit, grid-based and spatially linked forest-landscape model TreeMig. In each grid cell, the forest dynamics is simulated with a multi-species, height structured forest model, based on growth, competition and death of the trees in each height class. Within-cell heterogeneity is accounted for by assuming that trees are randomly distributed resulting in Poisson distributions of tree densities and light. Reproduction is modelled explicitly by seed production, seed bank dynamics, germination and sapling development. The forests in the different cells interact spatially through seed dispersal. The model is flexible and can be applied on a range of spatial scales, from single stands to entire regions.
The model's ability to generate patterns was tested in two case studies with different spatial resolutions. The first case study shows simulations of pattern formation by endogenous dynamics on a small spatial scale. The simulations are conducted under spatially and temporally homogenous environmental conditions, initialized with seeds of all species in the Centre cell. The simulation shows transiently several types of patterns in the species biomasses: circular standing waves, patch structures and homogenous spread of dominant species. In the second, large-scale case study, the tree species' spread since the last Ice Age in the Alpine region of the Valais is simulated, under temporally and spatially heterogeneous environmental conditions. The simulated spatio-temporal pattern consists of immigration waves of new species into empty or already forested areas, fast die-backs after sharp, strong temperature decreases, slower recolonization after temperature increases and spatial separation of the species according to the environmental conditions. The simulation indicates that the environment forms the basis for the endogenous dynamics, primarily migration and competition, which play a particular role during the transient phases after drastic changes in the boundary conditions (immigration and climate change).
We conclude that the forest-landscape model TreeMig is able to produce landscape patterns resulting from both, endogenous dynamics and exogenous drivers and is suitable for a range of different applications.

Variability influences ecological processes at various scales and is incorporated in different ways in forest models. The forest model DisCForM scales an individual based, stochastic forest patch model up to a height structured tree population model. To describe the variability arising from stochastic processes in the patch model, DisCForM uses theoretical random dispersions of trees in each height class over all patches. This yields a spatial distribution of light and consequently of light dependent process rates. Three major influences of variability on simulations are examined: site condition, patch to patch, and temporal environmental variability. Simulation studies and comparison with forest compositions from the Swiss National Forest Inventory reveal that these influences affect simulated forest dynamics, species composition, and biodiversity, depending on climatic boundary conditions and hence have to be taken into account in modeling.

Individual based, stochastic forest patch models have the potential to realistically describe forest dynamics. However, they are mathematically intransparent and need long computing times. We simplified such a forest patch model by aggregating the individual trees on many patches to height-structured tree populations with theoretical random dispersions over the whole simulated forest area. The resulting distribution-based model produced results similar to those of the patch model under a wide range of conditions. We concluded that the height- structured tree dispersion is an adequate population descriptor to capture the stochastic variability in a forest and that the new approach is generally applicable to any patch model. The simplified model required only 4.1% of the computing time needed by the patch model. Hence, this new model type is well-suited for applications where a large number of dynamic forest simulations is required.

Nonlinear temperature dependence plays a major role in a large variety of ecological models. For the sake of simplicity and efficiency, the temperature dependence functions in many models are calculated for monthly or yearly time intervals, using temperature means or interpolations between means as input. As a consequence, information about the variability of the temperature input data is lost, which leads to a bias in the temperature dependence function and to errors in the model results. We tested the performance of a range of methods against this common approach for calculating temperature dependence on a larger time scale, i.e. for a temporal aggregation. The methods estimate the expected value of the dependence function in different ways, using the mean or standard deviation of temperature variables in different temporal resolutions as input. In our tests we used temperature dependence functions from four different ecological fields; hourly temperature data sets from various climatically differing sites were used as input. The precision of the tested methods increased with the resolution of the input data, although computing time increased. The mean errors ranged from less than 1% to about 8% for the aggregation to 1 month and from about 1% to over 30% for the aggregation to 10 years. The most precise and efficient method is the explicit calculation of the expected value for the dependence function, which is based on the mean and standard deviation of hourly temperatures. The least precise but most efficient method is the common application of the dependence function to mean values. The quality of these methods is mainly determined by the quality of the approximation of the temperature variability. Condensing highly resolved input data into means is only appropriate if either the dependence functions are linear in the observed temperature range, or low precision but very high computing efficiency is required. Given a certain requirement on precision or computing efficiency, we are now able to indicate for a number of input data resolutions the appropriate method to calculate temperature dependence over long time periods.

Rates of ecological processes are usually influenced by temperature. For simplicity and efficiency of ecosystem models it is often necessary to summarize information about temperature dependence from short, e.g. hourly, time intervals over longer, e.g. monthly, time periods, i.e. to calculate long term expected values of dependence functions. This aim can seldom be achieved by applying the temperature function to the mean temperature, because temperature dependencies are in many cases nonlinear. Therefore, we derived seven new, general methods for a temporal aggregation of temperature dependence. The methods determine the expected value interpreting either hourly temperature, daily temperature mean, or daily temperature mean and amplitude as random variables. The dependence function is approximated by a piecewise linear function. Some methods use a triangle as approximation for the daily temperature course, some a parabola as approximation for the density function of the normal distribution. The resulting methods cover a range of temperature data resolutions: monthly mean and standard deviation of hourly temperatures; daily temperature extrema; daily temperature means; and amplitudes, or daily temperature means alone. The methods can be applied to all types of dependence functions, in particular to nonlinear ones.