The evolution of phenotypic traits in a predator-prey system subject to Allee effect

This paper considers the evolution of phenotypic traits in a community comprising the populations of predators and prey subject to Allee effect. The evolutionary model is constructed from a deterministic approximation of the stochastic process of mutation and selection. Firstly, we investigate the ecological and evolutionary conditions that allow for continuously stable strategy and evolutionary branching. We find that the strong Allee effect of prey facilitates the formation of continuously stable strategy in the case that prey population undergoes evolutionary branching if the Allee effect of prey is not strong enough. Secondly, we show that evolutionary suicide is impossible for prey population when the intraspecific competition of prey is symmetric about the origin. However, evolutionary suicide can occur deterministically on prey population if prey individuals undergo strong asymmetric competition and are subject to Allee effect. Thirdly, we show that the evolutionary model with symmetric interactions admits a stable limit cycle if the Allee effect of prey is weak. Evolutionary cycle is a likely outcome of the process, which depends on the strength of Allee effect and the mutation rates of predators and prey.     

Adaptive evolution of foraging-related traits in a predator-prey community

In this paper, with the method of adaptive dynamics and geometric technique, we investigate the adaptive evolution of foraging-related phenotypic traits in a predator-prey community with trade-off structure. Specialization on one prey type is assumed to go at the expense of specialization on another. First, we identify the ecological and evolutionary conditions that allow for evolutionary branching in predator phenotype. Generally, if there is a small switching cost near the singular strategy, then this singular strategy is an evolutionary branching point, in which predator population will change from monomorphism to dimorphism. Second, we find that if the trade-off curve is globally convex, predator population eventually branches into two extreme specialists, each completely specializing on a particular prey species. However, if the trade-off curve is concave-convex-concave, after branching in predator phenotype, the two predator species will evolve to an evolutionarily stable dimorphism at which they can continue to coexist. The analysis reveals that an attractive dimorphism will always be evolutionarily stable and that no further branching is possible under this model.     

Emergence of asymmetry in evolution

We investigate symmetry-breaking bifurcation patterns in evolution in the framework of adaptive dynamics (AD). We define weak and strong symmetry. The former applies for populations where only the simultaneous reflection of all individuals is an invariant transformation. The symmetry is strong in populations where reflection of some, but not all, individuals leaves the situation unchanged. We show that in case of weak symmetry evolutionary branching can lead to the emergence of two asymmetric variants, which are mirror images of each other, and the loss of the symmetric ancestor. We also show that in case of strong symmetry, evolutionary branching can occur into a symmetric and an asymmetric variant, both of which survive. The latter, asymmetric branching differs from the generic branching patterns of AD, which is always symmetric. We discuss biological examples for weak and strong symmetries and a specific model producing the new kind of branching.     

The adaptive dynamics of function-valued traits

This study extends the framework of adaptive dynamics to function-valued traits. Such adaptive traits naturally arise in a great variety of settings: variable or heterogeneous environments, age-structured populations, phenotypic plasticity, patterns of growth and form, resource gradients, and in many other areas of evolutionary ecology. Adaptive dynamics theory allows analysing the long-term evolution of such traits under the density-dependent and frequency-dependent selection pressures resulting from feedback between evolving populations and their ecological environment. Starting from individual-based considerations, we derive equations describing the expected dynamics of a function-valued trait in asexually reproducing populations under mutation-limited evolution, thus generalizing the canonical equation of adaptive dynamics to function-valued traits. We explain in detail how to account for various kinds of evolutionary constraints on the adaptive dynamics of function-valued traits. To illustrate the utility of our approach, we present applications to two specific examples that address, respectively, the evolution of metabolic investment strategies along resource gradients, and the evolution of seasonal flowering schedules in temporally varying environments.     

Long-term evolution of polygenic traits under frequency-dependent intraspecific competition

We analytically investigate the long-term evolution of a continuously varying quantitative character in a diploid population that is determined additively by a finite number of loci. The trait is under a mixture of frequency-dependent disruptive selection induced by intraspecific competition and frequency-independent stabilizing selection. Moreover, the trait is restricted to a finite range by constraints on the particular loci. Our investigations are based on explicit analytical results (provided by Bürger [2005. A multilocus analysis of intraspecific competition and stabilizing selection on a quantitative trait. J. Math. Biol. 50, 355-396]; Schneider [2006. A multilocus-multiallele analysis of frequency-dependent selection induced by intraspecific competition. J. Math. Biol. 52, 483-523]) on the short-term dynamics under the assumption of linkage equilibrium. We show that the population always reaches a long-term equilibrium (LTE), i.e., an equilibrium that is resistant against perturbations of mutations of sufficiently small effect. In general, several LTEs can coexist. They can be calculated explicitly, and we provide necessary and sufficient conditions for their existence. In the case that more than one LTE exists, we exemplify numerically that the evolutionary outcome depends crucially on the initial genetic architecture, on the joint distribution of mutational effects across loci, and on the particular realization of the mutation process. Therefore, long-term evolution cannot be predicted from the ecology alone. We further show that a partial order exists for the LTEs. The set of LTEs has a 'largest' element, an LTE which is reached during long-term evolution if the effects of the occurring mutant alleles are sufficiently large.     

On the changing concept of evolutionary population stability as a reflection of a changing point of view in the quantitative theory of evolution

 Eighteen different terms, currently employed to define various concepts of evolutionary stability in population dynamics are mentioned in this paper. Most of these terms are used in different connotations and even different meanings by different authors. On the other hand, different terms are often employed by different authors to define quite the same concept. Twenty-five years ago there was only one, well-defined, concept of stability, universally recognized in the field. In this paper I will try to relate the recent confusion, concerning concepts of population stability, with a more serious, though not that well-recognized, confusion in the modern analytic approach to population dynamics and quantitative evolution. Concepts of population stability will be examined in relation to each other on the one hand and, on the other hand, in relation to two dichotomies regarding the dynamic processes to which they correspond: Short-term versus long-term processes and processes concerning phenotypic changes versus process concerning genotypic changes. A hopefully more consistent use of the current terminology is suggested. Received 15 August 1993; received in revised form 15 September 1994     

On the non-existence of an optimal migration rate

 We show that an optimal migration rate may not exist in a population distributed over an infinite number of individual living sites if empty sites occur. This is the case when the mean number of offspring per individual μ is finite. We make the assumption of uniform migration to other sites whose rate is determined by the parent’s genotype or the offspring’s genotype at a single locus in a diploid hermaphrodite population undergoing random mating. In both cases, for μ small enough, any population at fixation would go to extinction. Moreover, in the latter case, for intermediate values of μ, the only fixation state that could resist the invasion of any mutant would lead the population to extinction. These are the two conditions for the non-existence of an optimal migration rate. They become less stringent as the cost for migration expressed by a coefficient of selection 1−β becomes larger, that is, closer to 1. The results are obtained assuming that the allele at fixation is either nondominant or dominant. Although the optimal migration rate is the same in both cases when it exists, the optimality properties may differ. Received 14 December 1995; received in revised form 5 April 1996     

The iterated continuous prisoner's dilemma game cannot explain the evolution of interspecific mutualism in unstructured populations

The evolutionionary origin of inter- and intra-specific cooperation among non-related individuals has been a great challenge for biologists for decades. Recently, the continuous prisoner's dilemma game has been introduced to study this problem. In function of previous payoffs, individuals can change their cooperative investment iteratively in this model system. Killingback and Doebeli (Am. Nat. 160 (2002) 421-438) have shown analytically that intra-specific cooperation can emerge in this model system from originally non-cooperating individuals living in a non-structured population. However, it is also known from an earlier numerical work that inter-specific cooperation (mutualism) cannot evolve in a very similar model. The only difference here is that cooperation occurs among individuals of different species. Based on the model framework used by Killingback and Doebeli (2002), this Note proves analytically that mutualism indeed cannot emerge in this model system. Since numerical results have revealed that mutualism can evolve in this model system if individuals interact in a spatially structured manner, our work emphasizes indirectly the role of spatial structure of populations in the origin of mutualism.     

Long-term effects of local disturbance history on mobile stream invertebrates

The patchy distribution of benthic invertebrates in streams and rivers is an important and widely researched phenomenon. Previous studies on reasons for this patchiness have neglected the potential role of local disturbance history, probably because most lotic invertebrates are mobile and any effect of disturbance history was thought to be short-lived. Here we demonstrate for a New Zealand gravel-bed stream that local disturbance history can have long-term effects on the distribution of highly mobile stream invertebrates. Buried scour chains (100 at each of three 20-m sites within a 350-m reach) indicated that a spate with a return period of 5 months caused a mosaic of bed patches with different stabilities. More than 2 months after the spate, we took random, quantitative samples at each site from five patches that had experienced 4 cm or more of scour during the spate, from five patches with 4 cm or more of fill, and from five stable patches. Density of the dominant invertebrate taxon, the highly mobile mayfly Deleatidium spp., and densities of another three of the seven most common taxa differed significantly between patch stability categories. Larvae of Deleatidium, the black fly Austrosimulium spp. and the dipteran Eriopterini were most abundant in fill patches, whereas Isopoda were most abundant in scour patches. Total invertebrate densities and densities of six common taxa also differed between sites, although these were only 95–120 m apart. These results show that local disturbance history can have long-term effects on lotic invertebrates and be an important cause of invertebrate patchiness. The observed effects might have been even stronger had we sampled sooner after the spate or after a large flood. Disturbance history may influence invertebrates both directly (through dislodgement or mortality) and indirectly, through effects on the spatial distribution of their resources. Our results suggest that the role of disturbance in structuring animal communities dominated by mobile species may be more important than previously thought. Received: 25 January 2000 / Accepted: 14 April 2000     

Comparative mapping of QTLs for agronomic traits of rice across environments by using a doubled-haploid population

We report here the RFLP mapping of quantitative trait loci (QTLs) which affect some important agronomic traits in cultivated rice. An anther culture-derived doubled-haploid (DH) population was established from a cross between indica and japonica rice varieties. A molecular linkage map comprising 137 markers was constructed based on this population which covered the rice genome at intervals of 14.8 cM on average. The linkage map was used to locate QTLs for such important agronomic traits as heading date, plant height, number of spikelets per panicle, number of grains per panicle, 1 000-grain weight and the percentage of seed set, by interval mapping. Evidence of genotype-by-environment interaction was found by comparing QTL maps of the same population grown in three diverse environments. A total of 22 QTLs for six agronomic traits was detected which were significant in at least one environment, but only seven were significant in all three environments; seven were significant in two environments and eight could only be detected in a single environment. However, QTLs-by-environment interaction was trait dependent. QTLs for spikelets and grains per panicle were common across environments while traits like heading date and plant height were more sensitive to environment. Received: 22 February 1996 / Accepted: 10 May 1996     

Global stability in consumer-resource cascades

 Models of population growth in consumer-resource cascades (serially arranged containers with a dynamic consumer population, v, receiving a flow of resource, u, from the previous container) with a functional response of the form h(u/v ^b ) are investigated. For b∈[0, 1], it is shown that these models have a globally stable equilibrium. As a result, two conclusions can be drawn: (1) Consumer density dependence in the functional or in the per-capita numerical response can result in persistence of the consumer population in all containers. (2) In the absence of consumer density dependence, the consumer goes extinct in all containers except possibly the first. Several variations of this model are discussed including replacing discrete containers by a spatial continuum and introducing a dynamic resource. Received 25 February 1995 / received in revised form 27 July 1995     

Adaptation versus migration in demographically unstable populations

 We analyze a model of the joint population and evolutionary dynamics of a diploid “island” population that receives a recurrent influx of immigrants from a hypothetical continent population. We derive a criterion for the initial spread of a rare allele when the island population dynamics initially are chaotic. By computing a Liapunov exponent, we show that this criterion depends on a generalization of the geometric mean absolute fitness of individuals heterozygous for the rare allele. This criterion applies regardless of whether the initial population is self-sustaining or not. Received: 26 October 1998     

Asymmetric synthesis of all stereoisomers of 6-methylpipecolic acids

Summary.  Asymmetric synthesis of all four stereoisomers of 6-methylpipecolic acids with high enantiomeric purity via iterative AD reaction, starting from 1,6-heptadiene, has been described. Received March 25, 2002 Accepted June 15, 2002 Published online January 30, 2003 Authors' address: Hiroki Takahata, Faculty of Pharmaceutical Sciences, Tohoku Pharmaceutical University, Sendai 981-8558, Japan, E-mail: takahata@tohoku-pharm.ac.jp RID="*" ID="*"  The stereo-configurations of 8 and 9 in Fig. 3 and 12 in Fig. 4 show only major isomers. RID="**" ID="**"  The absolute configurations of 11 and 14     

Variant evolutionary trees under phenotypic variance

Evolutionary branching, which is a coevolutionary phenomenon of the development of two or more distinctive traits from a single trait in a population, is the issue of recent studies on adaptive dynamics. In previous studies, it was revealed that trait variance is a minimum requirement for evolutionary branching, and that it does not play an important role in the formation of an evolutionary pattern of branching. Here we demonstrate that the trait evolution exhibits various evolutionary branching paths starting from an identical initial trait to different evolutional terminus traits as determined by only changing the assumption of trait variance. The key feature of this phenomenon is the topological configuration of equilibria and the initial point in the manifold of dimorphism from which dimorphic branches develop. This suggests that the existing monomorphic or polymorphic set in a population is not an unique inevitable consequence of an identical initial phenotype.     

Convergence of multilocus systems under weak epistasis or weak selection

 The convergence of multilocus systems under viability selection with constant fitnesses is investigated. Generations are discrete and nonoverlapping; the monoecious population mates at random. The number of multiallelic loci, the linkage map, dominance, and epistasis are arbitrary. It is proved that if epistasis or selection is sufficiently weak (and satisfies a certain nondegeneracy assumption whose genericity we establish), then there is always convergence to some equilibrium point. In particular, cycling cannot occur. The behavior of the mean fitness and some other aspects of the dynamics are also analyzed. Received: 15 November 1997 / Revised version: 25 May 1998     

Exploring phenotype space through neutral evolution

RNA secondary-structure folding algorithms predict the existence of connected networks of RNA sequences with identical secondary structures. Fitness landscapes that are based on the mapping between RNA sequence and RNA secondary structure hence have many neutral paths. A neutral walk on these fitness landscapes gives access to a virtually unlimited number of secondary structures that are a single point mutation from the neutral path. This shows that neutral evolution explores phenotype space and can play a role in adaptation. Received: 23 December 1995 / Accepted: 17 March 1996     

Secondary structure and patterns of evolution among mammalian mitochondrial 12S rRNA molecules

Forty-nine complete 12S ribosomal RNA (rRNA) gene sequences from a diverse assortment of mammals (one monotreme, 11 marsupials, 37 placentals), including 11 new sequences, were employed to establish a ``core' secondary structure model for mammalian 12S rRNA. Base-pairing interactions were assessed according to the criteria of potential base-pairing as well as evidence for base-pairing in the form of compensatory mutations. In cases where compensatory evidence was not available among mammalian sequences, we evaluated evidence among other vertebrate 12S rRNAs. Our results suggest a core model for secondary structure in mammalian 12S rRNAs with deletions as well as additions to the Gutell (1994: Nucleic Acids Res. 22) models for Bos and Homo. In all, we recognize 40 stems, 34 of which are supported by at least some compensatory evidence within Mammalia. We also investigated the occurrence and conservation in mammalian 12S rRNAs of nucleotide positions that are known to participate in the decoding site in E. coli. Twenty-four nucleotide positions known to participate in the decoding site in E. coli also occur among mammalian 12S rRNAs and 17 are invariant for the same base as in E. coli. Patterns of nucleotide substitution were assessed based on our secondary structure model. Transitions in loops become saturated by approximately 10–20 million years. Transitions in stems, in turn, show partial saturation at 20 million years but divergence continues to increase beyond 100 million years. Transversions accumulate linearly beyond 100 million years in both stems and loops although the rate of accumulation of transversions is three- to fourfold higher in loops. Presumably, this difference results from constraints to maintain pairing in stems. Received: 21 June 1995 / Accepted: 25 March 1996     

The adaptive dynamics of the evolution of host resistance to indirectly transmitted microparasites

We use adaptive dynamics and pairwise invadability plots to examine the evolutionary dynamics of host resistance to microparasitic infection transmitted indirectly via free stages. We investigate trade-offs between pathogen transmission rate and intrinsic growth rate. Adaptive dynamics distinguishes various evolutionary outcomes associated with repellors, attractors or branching points. We find criteria corresponding to these and demonstrate that a major factor deciding the evolutionary outcome is whether trade-offs are acceleratingly or deceleratingly costly. We compare and contrast two models and show how the differences between them lead to different evolutionary outcomes.     

Evolutionary dynamics of predator-prey systems: an ecological perspective

 Evolution takes place in an ecological setting that typically involves interactions with other organisms. To describe such evolution, a structure is needed which incorporates the simultaneous evolution of interacting species. Here a formal framework for this purpose is suggested, extending from the microscopic interactions between individuals – the immediate cause of natural selection, through the mesoscopic population dynamics responsible for driving the replacement of one mutant phenotype by another, to the macroscopic process of phenotypic evolution arising from many such substitutions. The process of coevolution that results from this is illustrated in the context of predator–prey systems. With no more than qualitative information about the evolutionary dynamics, some basic properties of predator–prey coevolution become evident. More detailed understanding requires specification of an evolutionary dynamic; two models for this purpose are outlined, one from our own research on a stochastic process of mutation and selection and the other from quantitative genetics. Much of the interest in coevolution has been to characterize the properties of fixed points at which there is no further phenotypic evolution. Stability analysis of the fixed points of evolutionary dynamical systems is reviewed and leads to conclusions about the asymptotic states of evolution rather different from those of game-theoretic methods. These differences become especially important when evolution involves more than one species. Received 10 November 1993; received in revised form 25 July 1994