Barbara Drossel

Biological evolution and statistical physics

This review is an introduction to theoretical models and mathematical calculations for biological evolution, aimed at physicists. The methods in the field are naturally very similar to those used in statistical physics, although the majority of publications have appeared in biology journals. The review has three parts, which can be read independently. The first part deals with evolution in fitness landscapes and includes Fishers theorem, adaptive walks, quasispecies models, effects of finite population sizes, and neutral evolution. The second part studies models of coevolution, including evolutionary game theory, kin selection, group selection, sexual selection, speciation, and coevolution of hosts and parasites. The third part discusses models for networks of interacting species and their extinction avalanches. Throughout the review, attention is paid to giving the necessary biological information, and to pointing out the assumptions underlying the models, and their limits of validity.

Forest fires and other examples of self-organized criticality

We review the properties of the self-organized critical (SOC) forest-fire model. The paradigm of self-organized criticality refers to the tendency of certain large dissipative systems to drive themselves into a critical state independent of the initial conditions and without fine tuning of the parameters. After an introduction, we define the rules of the model and discuss various large-scale structures which may appear in this system. The origin of the critical behaviour is explained, critical exponents are introduced and scaling relations between the exponents are derived. Results of computer simulations and analytical calculations are summarized. The existence of an upper critical dimension and the universality of the critical behaviour under changes of lattice symmetry or the introduction of immunity are discussed. A survey of interesting modifications of the forest-fire model is given. Finally, several other important SOC models are briefly described.

Dynamics of critical kauffman networks under asynchronous stochastic update

We show that the mean number of attractors in a critical Boolean network under asynchronous stochastic update grows like a power law and that the mean size of the attractors increases as a stretched exponential with the system size. This is in strong contrast to the synchronous case, where the number of attractors grows faster than any power law.

Interactive effects of body-size structure and adaptive foraging on food-web stability

Body-size structure of food webs and adaptive foraging of consumers are two of the dominant concepts of our understanding how natural ecosystems maintain their stability and diversity. The interplay of these two processes, however, is a critically important yet unresolved issue. To fill this gap in our knowledge of ecosystem stability, we investigate dynamic random and niche model food webs to evaluate the proportion of persistent species. We show that stronger body-size structures and faster adaptation stabilise these food webs. Body-size structures yield stabilising configurations of interaction strength distributions across food webs, and adaptive foraging emphasises links to resources closer to the base. Moreover, both mechanisms combined have a cumulative effect. Most importantly, unstructured random webs evolve via adaptive foraging into stable size-structured food webs. This offers a mechanistic explanation of how size structure adaptively emerges in complex food webs, thus building a novel bridge between these two important stabilising mechanisms.

Scaling laws and simulation results for the self-organized critical forest-fire model.

We discuss the properties of a self--organized critical forest--fire model which has been introduced recently. We derive scaling laws and define critical exponents. The values of these critical exponents are determined by computer simulations in 1 to 8 dimensions. The simulations suggest a critical dimension

Self-organized criticality in a forest-fire model

d_c=6

Sexual reproduction prevails in a world of structured resources in short supply

above which the critical exponents assume their mean--field values. Changing the lattice symmetry and allowing trees to be immune against fire, we show that the critical exponents are universal.

Number of attractors in random Boolean networks

A forest-fire model is introduced which contains a lightning probability f. This leads to a self-organized critical state in the limit f→0 provided that the time scales of free growth and burning down of forest clusters are separated. We derive scaling laws and calculate all critical exponents. The values of the critical exponents are confirmed by computer simulations. For a two-dimensional system, there is evidence that the forest density in the critical state assumes its minimum possible value, i.e. that energy dissipation is maximum.

Evidence for the Droplet Picture of Spin Glasses

We present a model for the maintenance of sexual reproduction based on the availability of resources, which is the strongest factor determining the growth of populations. The model compares completely asexual species to species that switch between asexual and sexual reproduction (sexual species). Key features of the model are that sexual reproduction sets in when resources become scarce, and that at a given place only a few genotypes can be present at the same time. We show that under a wide range of conditions the sexual species outcompete the asexual ones. The asexual species win only when survival conditions are harsh and death rates are high, or when resources are so little structured or consumer genotypes are so manifold that all resources are exploited to the same extent. These conditions, largely represent the conditions in which sexuals predominate over asexuals in the field.

Evolution of canalizing Boolean networks

The evaluation of the number of attractors in Kauffman networks by Samuelsson and Troein is generalized to critical networks with one input per node and to networks with two inputs per node and different probability distributions for update functions. A connection is made between the terms occurring in the calculation and between the more graphic concepts of frozen, nonfrozen, and relevant nodes, and relevant components. Based on this understanding, a phenomenological argument is given that reproduces the dependence of the attractor numbers on system size.