Symmetric competition as a general model for single-species adaptive dynamics

Adaptive dynamics is a widely used framework for modeling long-term evolution of continuous phenotypes. It is based on invasion fitness functions, which determine selection gradients and the canonical equation of adaptive dynamics. Even though the derivation of the adaptive dynamics from a given invasion fitness function is general and model-independent, the derivation of the invasion fitness function itself requires specification of an underlying ecological model. Therefore, evolutionary insights gained from adaptive dynamics models are generally model-dependent. Logistic models for symmetric, frequency-dependent competition are widely used in this context. Such models have the property that the selection gradients derived from them are gradients of scalar functions, which reflects a certain gradient property of the corresponding invasion fitness function. We show that any adaptive dynamics model that is based on an invasion fitness functions with this gradient property can be transformed into a generalized symmetric competition model. This provides a precise delineation of the generality of results derived from competition models. Roughly speaking, to understand the adaptive dynamics of the class of models satisfying a certain gradient condition, one only needs a complete understanding of the adaptive dynamics of symmetric, frequency-dependent competition. We show how this result can be applied to number of basic issues in evolutionary theory.Comment: 26 pages, 1 figur

Lattice Boltzmann method for viscoelastic fluids

Lattice Boltzmann model for viscoelastic flow simulation is proposed; elastic effects are taken into account in the framework of Maxwell model. The following three examples are studied using the proposed approach: a transverse velocity autocorrelation function for free evolving system with random initial velocities, a boundary-driven propagating shear waves, and a resonant enhancement of oscillations in a periodically driven fluid in a capillary. The measured shear wave dispersion relation is found to be in a good agreement with the theoretical one derived for the Navier-Stokes equation with the Maxwell viscoelastic term.Comment: 4 pages, 5 figure

Continuously stable strategies as evolutionary branching points

Evolutionary branching points are a paradigmatic feature of adaptive dynamics, because they are potential starting points for adaptive diversification. The antithesis to evolutionary branching points are Continuously stable strategies (CSS's), which are convergent stable and evolutionarily stable equilibrium points of the adaptive dynamics and hence are thought to represent endpoints of adaptive processes. However, this assessment is based on situations in which the invasion fitness function determining the adaptive dynamics have non-zero second derivatives at a CSS. Here we show that the scope of evolutionary branching can increase if the invasion fitness function vanishes to higher than first order at a CSS. Using a class of classical models for frequency-dependent competition, we show that if the invasion fitness vanishes to higher orders, a CSS may be the starting point for evolutionary branching, with the only additional requirement that mutant types need to reach a certain threshold frequency, which can happen e.g. due to demographic stochasticity. Thus, when invasion fitness functions vanish to higher than first order at equilibrium points of the adaptive dynamics, evolutionary diversification can occur even after convergence to an evolutionarily stable strategy.Comment: 22 pages, 4 figure

A model for the self-organization of vesicular flux and protein distributions in the Golgi apparatus

The generation of two non-identical membrane compartments via exchange of vesicles is considered to require two types of vesicles specified by distinct cytosolic coats that selectively recruit cargo and two membrane-bound SNARE pairs that specify fusion and differ in their affinities for each type of vesicles. The mammalian Golgi complex is composed of 6-8 non-identical cisternae that undergo gradual maturation and replacement yet features only two SNARE pairs. We present a model that explains how the distinct composition of Golgi cisternae can be generated with two and even a single SNARE pair and one vesicle coat. A decay of active SNARE concentration in aging cisternae provides the seed for a cis > trans SNARE gradient that generates the predominantly retrograde vesicle flux which further enhances the gradient. This flux in turn yields the observed inhomogeneous steady-state distribution of Golgi enzymes, which compete with each other and with the SNAREs for incorporation into transport vesicles. We show analytically that the steady state SNARE concentration decays exponentially with the cisterna number. Numerical solutions of rate equations reproduce the experimentally observed SNARE gradients, overlapping enzyme peaks in cis, medial and trans and the reported change in vesicle nature across Golgi: Vesicles originating from younger cisternae mostly contain Golgi enzymes and SNAREs enriched in these cisternae and extensively recycle through the Endoplasmic Reticulum (ER), while the other subpopulation of vesicles contains Golgi proteins prevalent in older cisternae and hardly reaches the ER.Comment: 15 pages, 6 figure

Competition-driven evolution of organismal complexity

Non-uniform rates of morphological evolution and evolutionary increases in organismal complexity, captured in metaphors like "adaptive zones", "punctuated equilibrium" and "blunderbuss patterns", require more elaborate explanations than a simple gradual accumulation of mutations. Here we argue that non-uniform evolutionary increases in phenotypic complexity can be caused by a threshold-like response to growing ecological pressures resulting from evolutionary diversification at a given level of complexity. Acquisition of a new phenotypic feature allows an evolving species to escape this pressure but can typically be expected to carry significant physiological costs. Therefore, the ecological pressure should exceed a certain level to make such an acquisition evolutionarily successful. We present a detailed quantitative description of this process using a microevolutionary competition model as an example. The model exhibits sequential increases in phenotypic complexity driven by diversification at existing levels of complexity and the resulting increase in competitive pressure, which can push an evolving species over the barrier of physiological costs of new phenotypic features.Comment: Open PDF with Acrobat to see movies, 22 pages, 9 figure

Convergent Approximation for the 2-Body Correlation Function in an Interface

A convergent approximation is proposed for a mean field density-density correlation function in a system with a two-phase interface. It is based on a fourth-order expansion of the Hamiltonian in terms of fluctuations around the equilibrium profile. The approach is illustrated by one and three dimensional calculations for systems characterized by the Ginzburg-Landau functional.Comment: 10 pages, 2 figure