Spatial Niche Packing, Character Displacement and Adaptive Speciation Along an Environmental Gradient

In this study, we examine the ecology and adaptive dynamics of an asexually reproducing population, migrating along an environmental gradient. The living conditions are optimal at the central location and deteriorate outwards. The different strategies are optimized to the ecological conditions of different locations. The control parameters are the migration and the tolerance of the strategies towards the environmental condition (location). Locally, population growth is logistic and selection is frequency-independent, corresponding to the case of a single limiting resource. The behavior of the population is modeled by numerically integrated reaction-diffusion equations as well as by individual-based simulations. Limiting similarity, spatial niche segregation and character displacement are demonstrated, analogous to resource-heterogeneity based niche partitioning. Pairwise invasibility analysis reveals a convergent stable singular strategy optimized to the central, optimal location. It is evolutionary stable if the migration rate and the tolerance are large. Decreasing migration or decreasing tolerance bifurcates the singular strategy to an evolutionary branching point. Individual-based simulation of evolution confirms that, in the case of branching singularity, evolution converges to this singular strategy and branches there. Depending on the environmental tolerance, further branching may occur. The branching evolution in the asexual model is interpreted as a sign that the ecology of an environmental gradient is prone to adaptive geographic speciation

Dynamics of Population on the Verge of Extinction

Theoretical considerations suggest that extinction in dispersal-limited populations is necessarily a threshold-like process that is analogous to a critical phase transition in physics. We use this analogy to find robust, common features in the dynamics of extinctions, and suggest early warning signals which may indicate that a population is endangered. As the critical threshold of extinction is approached, the population spontaneously fragments into discrete subpopulations and, consequently, density regulation fails. The population size declines and its spatial variance diverges according to scaling laws. Therefore, we can make robust predictions exactly in the range where prognosis is vital, on the verge of extinction

The Dynamics of Adaptation and Evolutionary Branching

We present a formal framework for modeling evolutionary dynamics with special emphasis on the generation of diversity through branching of the evolutionary tree. Fitness is defined as the long term growth rate which is influenced by the biotic environment leading to frequency-dependent selection. Evolution can be described as a dynamics in space with variable number of dimensions corresponding to the number of different types present. The dynamics within a subspace is governed by the local fitness gradient. Entering a higher dimensional subspace is possible only at a particular type of attractors where the population undergoes evolutionary branching

Competitive Exclusion and Limiting Similarity: A Unified Theory

Robustness of coexistence against changes of parameters is investigated in a model-independent manner through analyzing the feed-back loop of population regulation. We define coexistence as fixed point of the community dynamics with no population having zero size. It is demonstrated that the parameter range allowing coexistence shrinks and disappears when the Jacobian of the dynamics decreases to zero. A general notion of regulating factors/variables is introduced. For each population, its 'impact' and 'sensitivity' niches a re defined as the differential impact on, and the differential sensitivity towards, the regulating variables, respectively. Either similarity of the impact niches, or similarity of the sensitivity niches, result in a small Jacobian and in a reduced likelihood of coexistence. For the case of a resource continuum, this result reduces to the usual "limited niches overlap" picture for both kinds of niche. As an extension of these ideas to the coexistence of infinitely many species, we demonstrate that Roughgarden's example for coexistence of a 'continuum' of populations is structurally unstable

Dynamics of Similar Populations: The Link Between Population Dynamics and Evolution

We provide the link between population dynamics and the dynamics of Darwinian evolution via studying the joint population dynamics of "similar" populations. Similarity implies that the "relative" dynamics of the populations is slow compared to, and decoupled from, their "aggregated" dynamics. The relative dynamics is simple, and captured by a Taylor expansion in the difference between the populations. The emerging evolution is directional, except at the "singular" points of the evolutionary state space, where "evolutionary branching" may happen

Continuous coexistence or discrete species? A new review of an old question

Question: Is the coexistence of a continuum of species or ecological types possible in real-world communities? Or should one expect distinctly different species? Mathematical methods: We study whether the coexistence of species in a continuum of ecological types is (a) dynamically stable (against changes in population densities) and (b) structurally robust (against changes in population dynamics). Since most of the reviewed investigations are based on Lotka-Volterra models, we carefully explain which of the presented conclusions are model-independent. mathematical conclusions: Seemingly plausible models with dynamically stable continuous- coexistence solutions do exist. However, these models either depend on biologically unrealistic mathematical assumptions (e.g. non-differentiable ingredient functions) or are structurally unstable (i.e. destroyable by arbitrarily small modifications to those ingredient functions). The dynamical stability of a continuous-coexistence solution, if it exists, requires positive definiteness of the model's competition kernel. Biological conclusions: While the classical expectation of fixed limits to similarity is mathematically naive, the fundamental discreteness of species is a natural consequence of the basic structure of ecological interactio

An Analytically Tractable Model for Competitive Speciation

Several recent models have shown that frequency-dependent disruptive selection created by intraspecific competition can lead to the evolution of assortative mating 4 and, thus, to competitive sympatric speciation. However, since most results from these 5 models rely on limited numerical analyses, their generality has been subject to considerable debate. Here, we consider one of the standard models, the so-called Roughgarden model, with a simplified genetics where the selected trait is determined by a single diallelic locus. This model is sufficiently complex to maintain key properties of the general multilocus case, but still simple enough to allow for a comprehensive analytical treatment. By means of invasion fitness analysis, we describe the impact of all model parameters on the evolution of assortative mating. Depending on (1) the strength and (2) shape of stabilizing selection, (3) the strength and (4) shape of pairwise competition, (5) the shape of the mating function, and (6) the type of assortative mating, which may or may not lead to sexual selection, we find five different evolutionary regimes. In one of these regimes, the evolution of complete reproductive isolation is possible through arbitrarily small steps in the strength of assortative mating. Our approach provides a mechanistic understanding of several phenomena that have been found in previous models. The results demonstrate how, even in a simple model of competitive speciation, results depend in a complex way on ecological and genetic parameters

Evolutionary Optimization Models and Matrix Games in the Unified Perspective of Adaptive Dynamics

Matrix game theory and optimization models offer two radically different perspectives on the outcome of evolution. Optimization models consider frequency-independent selection and envisage evolution as a hill-climbing process on a constant fitness landscape, with the optimal strategy corresponding to the fitness maximum. By contrast, in evolutionary matrix games selection is frequency-dependent and leads to fitness equality among alternative strategies once an evolutionarily stable strategy has been established. In this review we demonstrate that both optimization models and matrix games represent special cases within the general framework of adaptive dynamics. Adaptive dynamics theory considers arbitrary nonlinear frequency and density dependence and envisages evolution as proceeding on an adaptive landscape that changes its shape according to which strategies are present in the population. In adaptive dynamics, evolutionarily stable strategies correspond to conditional fitness maxima: the ESS is characterized by the fact that it has the highest fitness if it is the established strategy. In this framework it can also be shown that dynamical attainability, evolutionary stability, and invading potential of strategies are pairwise independent properties. In optimization models, on the other hand, these properties become linked such that the optimal strategy is always attracting, evolutionarily stable and can invade any other strategy. In matrix games fitness is a linear function of the potentially invading strategy and can thus never exhibit an interior maximum: Instead, the fitness landscape is a plane that becomes horizontal once the ESS is established. Due to this degeneracy, invading potential is part of the ESS definition for matrix games and dynamical attainability is a dependent property. We conclude that adaptive dynamics provides a unifying framework for overcoming the traditional divide between evolutionary optimization models and matrix games