Competitive exclusion and limiting similarity: A unified theory

Robustness of coexistence against changes of parameters is investigated in a model-independent manner by analyzing the feedback loop of population regulation. We define coexistence as a 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 are defined as the differential impact on, and the differential sensitivity towards, the regulating variables, respectively. Either the similarity of the impact niches or the similarity of the sensitivity niches results 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 niche 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.

Adaptive Dynamics in a 2-Patch Environment: A Toy Model for Allopatric and Parapatric Speciation

Adaptation to an environment consisting of two patches (each with different optimal strategy) is investigated. The patches have independent density regulation ('soft selection'). If the patches are similar enough and migration between them is strong, then evolution ends up with a generalist ESS. If either the difference between the patches increases or migration weakens, then the generalist strategy represents a branching singularity: The initially monomorphic population first evolves towards the generalist strategy, there it undergoes branching, and finally two specialist strategies form an evolutionarily stable coalition. Further increasing the between-patch difference or decreasing migration causes the generalist to lose its convergence stability as well, and an initially monomorphic population evolves towards one of the specialists optimally adapted to one of the two patches. Bifurcation pattern of the singularities is presented as a function of patch difference and migration rate.

Connection to speciation theory is discussed. The transition from the generalist ESS to the coexisting pair of specialist strategies is regarded as a clonal prototype of parapatric (if the between-patch difference increases) or allopatric (if the migration decreases) speciation. We conclude that the geographic and the competitive speciation modes are not distinct classes.

Adaptive dynamics for physiologically structured population models

We develop a systematic toolbox for analyzing the adaptive dynamics of multidimensional traits in physiologically structured population models with point equilibria (sensu Dieckmann et al. in Theor. Popul. Biol. 63:309-338, 2003). Firstly, we show how the canonical equation of adaptive dynamics (Dieckmann and Law in J. Math. Biol. 34:579-612, 1996), an approximation for the rate of evolutionary change in characters under directional selection, can be extended so as to apply to general physiologically structured population models with multiple birth states. Secondly, we show that the invasion fitness function (up to and including second order terms, in the distances of the trait vectors to the singularity) for a community of N coexisting types near an evolutionarily singular point has a rational form, which is model-independent in the following sense: the form depends on the strategies of the residents and the invader, and on the second order partial derivatives of the one-resident fitness function at the singular point. This normal form holds for Lotka-Volterra models as well as for physiologically structured population models with multiple birth states, in discrete as well as continuous time and can thus be considered universal for the evolutionary dynamics in the neighbourhood of singular points. Only in the case of one-dimensional trait spaces or when N = 1 can the normal form be reduced to a Taylor polynomial. Lastly we show, in the form of a stylized recipe, how these results can be combined into a systematic approach for the analysis of the (large) class of evolutionary models that satisfy the above restrictions.

On the impossibility of coexistence of infinitely many strategies

We investigate the possibility of coexistence of pure, inherited strategies belonging to a large set of potential strategies. We prove that under biologically relevant conditions every model allowing for coexistence of infinitely many strategies is structurally unstable. In particular, this is the case when the "interaction operator" which determines how the growth rate of a strategy depends on the strategy distribution of the population is compact. The interaction operator is not assumed to be linear. We investigate a Lotka-Volterra competition model with a linear interaction operator of convolution type separately because the convolution operator is not compact. For this model, we exclude the possibility of robust coexistence supported on the whole real line, or even on a set containing a limit point. Moreover, we exclude coexistence of an infinite set of equidistant strategies when the total population size is finite. On the other hand, for infinite populations it is possible to have robust coexistence in this case. These results are in line with the ecological concept of "limiting similarity" of coexisting species. We conclude that the mathematical structure of the ecological coexistence problem itself dictates the discreteness of the species.

Link between population dynamics and dynamics of Darwinian 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. Here "evolutionary branching" may occur. The diversification of life forms thus is demonstrated to be a natural consequence of the Darwinian process.

Speciation in multidimensional evolutionary space

Adaptive dynamics in two-dimensional phenotype space is investigated by computer simulation. The model assumes Lotka-Voltera-type competition and a stochastic mutation process. The carrying capacity has a single maximum in the origin of the strategy space and the competition coefficient decreases with strategy difference. Evolutionary branching, an asexual analog of adaptive speciation, is observed with suitable parameters. The branching at the singular point, which is a fixed point of the directional evolution, may occur into two or three, but not more, directions. Further branchings may occur after the initial separation. The probability of three-branching is studied as a function of several parameters. We conclude that the two-way branching is the predominant mode of adaptive speciation.

Electrodichroism of purple membrane: ionic strength dependence

The dichroism of purple membrane suspension was measured in dc and ac electric fields. From these measurements three parameters can be obtained: the permanent dipole moment, mu, the electrical polarizability, alpha, and the retinal angle, delta, (relative to the membrane normal). The functional dependence of the dichroism on the electric field is analyzed. There is a small decrease ( approximately 2 degrees ) in retinal angle going from dark adapted to the light adapted form. No measurable difference in mu, alpha, and delta was found under the photocycle. The dichroism was measured in two different salt solutions (KCl and CaCl(2)) in the range 0-10 mM. The retinal angle increases from 64 degrees to 68 degrees with increasing ionic strength going through a minimum. This is attributed to the changing (decreasing) inner electric field in the membrane. The polarizability, alpha, consists of two parts. One component is related to the polarization of the purple membrane and the second component to the ionic cloud. The second component decreases with ion concentration approximately as kappa(-3) (kappa is the Debye parameter) in agreement with a model calculation for the polarization of the ionic cloud. The origin of the slightly ionic strength dependent permanent dipole moment is not well understood.

Trapping magnetically oriented chloroplast thylakoid membranes in gels for electric measurements

Electrochromic absorbance change and light gradient photovoltage measurements were carried out in chloroplast thylakoid membranes embedded in different compositions of gels. The goal was to find a system suitable for determining the dependence of the amplitude of the anomalous light gradient photovoltage signal, with opposite sign with respect to the ordinary signal, on the alignment of membranes. We found that polyacrylamide gel drastically increased the permeability of membranes which rendered electric measurements impossible in this gel. Agarose gel was successfully used in electric measurements. We will show that the light gradient photovoltage induced by a 580 nm laser flash depends strongly on the alignment of membranes with respect to the direction of the excitation beam. With face-aligned membranes the amplitude of the anomalous (negative) signal was drastically diminished, and the ordinary (positive) signal became clearly discernible. The observed differences in the decay of the two signals are tentatively explained by differences in the rates of ionic equilibration between different subcompartments of the inner aqueous phase of the thylakoid membrane system.

Coexistence in a fluctuating environment by the effect of relative nonlinearity: a minimal model

The minimal model of the "relative nonlinearity" type fluctuation-maintained coexistence is investigated. The competing populations are affected by an environmental white noise. With quadratic density dependence, the long-term growth rates of the populations are determined by the average and the variance of the (fluctuating) total density. At most two species can coexist on these two "regulating" variables; competitive exclusion would ensue in a constant environment. A numerical study of the expected time until extinction of any of the two species reveals that the criterion of mutual invasibility predicts the parameter range of long-term coexistence correctly in the limit of zero extinction threshold. However, any extinction threshold consistent with a realistic population size will allow only short-term coexistence. Therefore, our simulations question the biological relevance of mutual invasibility, as a sufficient condition of coexistence, for large density fluctuations. We calculate the average and the variance of the fluctuating density of the coexisting populations analytically via the moment-closure approximation; the results are reasonably close to the simulated behavior. Based on this treatment, robustness of coexistence is studied in the limit of infinite population size. We interpret the results of this analysis in the context of necessity of niche segregation with respect to the regulating variables using a framework theory published earlier.