Interspecific gene flow may profoundly impact genome integrity and adaptive evolution in hybridizing species, leading to novelties such as transgressive traits, supergenes, or, sometimes, the emergence of asexually reproducing lineages. Conventionally, introgression is thought to proceed between reproductively interacting species, mediated by recombining interspecific hybrids, while asexual lineages are considered evolutionary deadlock for genomes trapped in them. Our study on Cobitis loaches in the western Balkan watersheds demonstrates an alternative mechanism where a mix of asexuality and polyploidy facilitates significant introgression from a long-extinct species. Through extensive sampling and cytogenetic and phylogenomic analyses, we identified a sexual species, C. ohridana (OO) coexisting with its asexual hybrid form (OX) originating from hybridization with an extinct species (XX). The diploid OX hybrids pass both parental subgenomes mostly clonally with occasional gene conversions, while triploid OOX hybrids reproduce through meiotic hybridogenesis, producing O-like gametes with localized gene conversions by X alleles. Their mating with the locally dominant sexual species consequently leads to over 4% admixture in both nuclear and mitochondrial genomes. Our findings challenge the view of hybrid asexual lineages as evolutionary dead ends, revealing their significant role as reservoirs of genetic diversity and agents of interspecific gene exchange, even after the extinction of one parental taxon. ### Competing Interest Statement The authors have declared no competing interest.

Hybridization and polyploidy are powerful evolutionary forces, inducing a range of phenotypic outcomes, including non-additive expression, subgenome dominance, deviations in genomic dosage, and transcriptome downsizing. The reasons for these patterns and whether they are universal adaptive responses to genome merging and doubling remain debated. To address this, we develop a thermodynamic model of gene expression based on transcription factor (TF)-promoter binding. Applied to hybridization between species with divergent gene expression levels, cell volumes, or euchromatic ratios, this model distinguishes the effects of hybridization from those of polyploidy. Our results align with empirical observations, suggesting that gene regulation patterns in hybrids and polyploids often stem from the constrained interplay between inherited diverged regulatory networks rather than from subsequent adaptive evolution. In addition, occurrence of certain phenotypic traits depend on specific assumptions about promoter-TF coevolution and their distribution within the hybrid’s nucleoplasm, offering new research avenues to understand the underlying mechanisms. In summary, our model explains how the legacy of divergent species directly influences the phenotypic traits of hybrids and allopolyploids. Hybridization and polyploidization influence gene expression with several prominent trends. This study uses a thermodynamic model to show that many observed gene expression patterns in hybrids and allopolyploids can be explained by regulatory divergence between parents, revealing the complex interplay of admixed regulatory networks.

We show the location of so called critical points, i.e., couples of dif-fusion coefficients for which a non-trivial solution of a linear reaction-diffusion system of activator-inhibitor type on an interval with Neumann boundary conditions and with additional non-linear unilateral condition at one or two points on the boundary and/or in the interior exists. Simultaneously, we show the profile of such solutions.

We study two types of profiles describing the buckling of the spring or beam which is supported from below by an elastic obstacle. We focus on the dependence of a contract set on a strength and an elasticity property of the obstacle as well as on the external force stressing the beam.

We study a simple model of a spring or beam in ID given by a second order ODE and homogeneous Dirichlet boundary conditions. We assume that the spring/beam is unilaterally supported from below on a part of the domain by a soft/elastic obstacle. We focus on the dependence of the critical forces (which stress the beam and cause its buckling) on the strength/elasticity parameter of the obstacle. We present both analytical and numerical results.

Hybridization and genome duplication have played crucial roles in the evolution of many animal and plant taxa. The subgenomes of parental species undergo considerable changes in hybrids and polyploids, which often selectively eliminate segments of one subgenome. However, the mechanisms underlying these changes are not well understood, particularly when the hybridization is linked with asexual reproduction that opens up unexpected evolutionary pathways. To elucidate this problem, we compared published cytogenetic and RNAseq data with exome sequences of asexual diploid and polyploid hybrids between three fish species; Cobitis elongatoides, C taenia, and C tanaitica. Clonal genomes remained generally static at chromosome-scale levels but their heterozygosity gradually deteriorated at the level of individual genes owing to allelic deletions and conversions. Interestingly, the impact of both processes varies among animals and genomic regions depending on ploidy level and the properties of affected genes. Namely, polyploids were more tolerant to deletions than diploid asexuals where conversions prevailed, and genomic restructuring events accumulated preferentially in genes characterized by high transcription levels and GC-content, strong purifying selection and specific functions like interacting with intracellular membranes. Although hybrids were phenotypically more similar to C taenia, we found that they preferentially retained C elongatoides alleles. This demonstrates that favored subgenome is not necessarily the transcriptionally dominant one. This study demonstrated that subgenomes in asexual hybrids and polyploids evolve under a complex interplay of selection and several molecular mechanisms whose efficiency depends on the organism's ploidy level, as well as functional properties and parental ancestry of the genomic region.

We study a reaction-diffusion system in 1D with various boundary conditions. We focus on the sets of "critical points", i.e. the couples of diffusion rates, for which linearization emanates nontrivial solutions (spatial patterns). We utilize the software AUTO and XPPAUT to numerically obtain continuous branches of these critical points.

Interspecific competition is a fundamental process affecting community structure and evolution of interacting species. Besides direct competition, this process is also mediated by shared enemies, which can change the outcome of competition dramatically. However, previous studies investigating interactions between competing species and their parasites (parasite-mediated competition) completely overlooked the effect of 'sperm' parasites (i.e. sperm-dependent parthenogens or pseudogams) on competition. These organisms originate by interspecific hybridization, produce clonal gametes, but exploit parental species for their own reproduction, being therefore analogous to classical parasites. Here we use the reaction-diffusion model and show that pseudogams alter the outcome of interspecific competition significantly. They may either slow down competitive exclusion of the inferior competitor or even turn the outcome of competition between the species. Asexual organisms may thus have unexpectedly strong impact on community structure, and have more significant evolutionary potential than was previously thought.

Unilateral boundary value problems for second order ODEs with jumping nonlinearities are studied. Problems under consideration can be interpreted as a simple model of a beam which is supported by a pointwise unilateral elastic obstacle. Couples of eigenvalues and eigenfunctionsare found and their dependence on the parameter corresponding to the strength of the obstacle is shown.

For a reaction–diffusion system which is subject to Turing’s diffusion-driven instability and which is equipped with unilateral obstacles of various types, the nonexistence of bifurcation of stationary solutions near certain critical parameter values is proved. The result implies assertions about a related mapping degree which in turn implies for “small” obstacles the existence of a new branch of bifurcation points (spatial patterns) induced by the obstacle.

Given a reaction-diffusion system which exhibits Turing’s diffusion-driven instability, the influence of unilateral obstacles of opposite sign (source and sink) on bifurcation and critical points is studied. In particular, in some cases it is shown that spatially nonhomogeneous stationary solutions (spatial patterns) bifurcate from a basic spatially homogeneous steady state for an arbitrarily small ratio of diffusions of inhibitor and activator, while a sufficiently large ratio is necessary in the classical case without unilateral obstacles. The study is based on a variational approach to a non-variational problem which even after transformation to a variational one has an unusual structure for which usual variational methods do not apply.

The equation Δu+λu+g(λ,u)u=0 is considered in a bounded domain in R2 with a Signorini condition on a straight part of the boundary and with mixed boundary conditions on the rest of the boundary. It is assumed that g(λ,0)=0 for λ∈R, λ is a bifurcation parameter. A given eigenvalue of the linearized equation with the same boundary conditions is considered. A smooth local bifurcation branch of non-trivial solutions emanating at λ0 from trivial solutions is studied. We show that to know a direction of the bifurcating branch it is sufficient to determine the sign of a simple expression involving the corresponding eigenfunction u0. In the case when λ0 is the first eigenvalue and the branch goes to the right, we show that the bifurcating solutions are asymptotically stable in W1,2-norm. The stability of the trivial solution is also studied and an exchange of stability is obtained.

We consider a reaction-diffusion system of activator-inhibitor or substrate-depletion type in one space dimension which is subject to diffusion-driven instability. We determine the change of bifurcation when a pure Neumann condition is supplemented with a Signorini condition. We show that this change differs essentially from the known case when also Dirichlet conditions are assumed.

BACKGROUND:Quest for understanding the nature of mechanisms governing the life span of clonal organisms lasts for several decades. Phylogenetic evidence for recent origins of most clones is usually interpreted as proof that clones suffer from gradual age-dependent fitness decay (e.g. Muller's ratchet). However, we have shown that a neutral drift can also qualitatively explain the observed distribution of clonal ages. This finding was followed by several attempts to distinguish the effects of neutral and non-neutral processes. Most recently, Neiman et al. 2009 (Ann N Y Acad Sci.:1168:185-200.) reviewed the distribution of asexual lineage ages estimated from a diverse array of taxa and concluded that neutral processes alone may not explain the observed data. Moreover, the authors inferred that similar types of mechanisms determine maximum asexual lineage ages in all asexual taxa. In this paper we review recent methods for distinguishing the effects of neutral and non-neutral processes and point at methodological problems related with them.RESULTS AND DISCUSSION:We found that contemporary analyses based on phylogenetic data are inadequate to provide any clear-cut answer about the nature and generality of processes affecting evolution of clones. As an alternative approach, we demonstrate that sequence variability in asexual populations is suitable to detect age-dependent selection against clonal lineages. We found that asexual taxa with relatively old clonal lineages are characterised by progressively stronger deviations from neutrality.CONCLUSIONS:Our results demonstrate that some type of age-dependent selection against clones is generally operational in asexual animals, which cover a wide taxonomic range spanning from flatworms to vertebrates. However, we also found a notable difference between the data distribution predicted by available models of sequence evolution and those observed in empirical data. These findings point at the possibility that processes affecting clonal evolution differ from those described in recent studies, suggesting that theoretical models of asexual populations must evolve to address this problem in detail.REVIEWERS:This article was reviewed by Isa Schön (nominated by John Logsdon), Arcady Mushegian and Timothy G. Barraclough (nominated by Laurence Hurst).

Recent modeling studies exploring the effect of consumers' adaptivity in diet composition on food web complexity invariably suggest that adaptivity in foraging decisions of consumers makes food webs more complex. That is, it allows for survival of a higher number of species when compared with non-adaptive food webs. Population-dynamical models in these studies share two features: parameters are chosen uniformly for all species, i.e. they are species-independent, and adaptive foraging is described by the search image model. In this article, we relax both these assumptions. Specifically, we allow parameters to vary among the species and consider the diet choice model as an alternative model of adaptive foraging. Our analysis leads to three important predictions. First, for species-independent parameter values for which the search image model demonstrates a significant effect of adaptive foraging on food web complexity, the diet choice model produces no such effect. Second, the effect of adaptive foraging through the search image model attenuates when parameter values cease to be species-independent. Finally, for the diet choice model we observe no (significant) effect of adaptive foraging on food web complexity. All these observations suggest that adaptive foraging does not always lead to more complex food webs. As a corollary, future studies of food web dynamics should pay careful attention to the choice of type of adaptive foraging model as well as of parameter values.

We study a bifurcation problem for the equation Δu+λu+g(λ,u)u=0 on a rectangle with Signorini boundary conditions on a part of one edge and mixed (zero Dirichlet and Neumann) boundary conditions on the rest of the boundary. Here λ∈R is the bifurcation parameter, and g is a small perturbation. We prove, under certain assumptions concerning an eigenfunction u0 corresponding to an eigenvalue λ0 of the linearized equation with the same nonlinear boundary conditions, the existence of a local smooth branch of nontrivial solutions bifurcating from the trivial solutions at λ0 in the direction of u0. The contact sets of these nontrivial solutions are intervals which change smoothly along the branch. The main tool of the proof is a local equivalence of the unilateral BVP to a system consisting of a corresponding classical BVP and of two scalar equations. To this system classical Crandall–Rabinowitz type local bifurcation techniques (scaling and Implicit Function Theorem) are applied.

The direction of bifurcation of nontrivial solutions to the elliptic boundary value problem involving unilateral nonlocal boundary conditions is shown in a neighbourhood of bifurcation points of a certain type. Moreover, the stability and instability of bifurcating solutions as well as of the trivial solution is described in the sense of minima of the potential. In particular, an exchange of stability is observed.

We consider a reaction–diffusion system of activator-inhibitor or substrate-depletion type which is subject to diffusion-driven instability. We show that obstacles (e.g. a unilateral membrane) for both quantities modeled in terms of inequalities introduce a new bifurcation of spatially non-homogeneous steady states in the domain of stability of the trivial solution of the corresponding classical problem without obstacles.

We prove that the solutions to a 2D Poisson equation with unilateral boundary conditions of Signorini type as well as their contact intervals depend smoothly on the data. The result is based on a certain local equivalence of the unilateral boundary value problem to a smooth abstract equation in a Hilbert space and on an application of the Implicit Function Theorem to that equation.

We consider a reaction-diffusion system exhibiting diffusion driven instability if supplemented by Dirichlet-Neumann boundary conditions. We impose unilateral conditions given by inclusions on this system and prove that global bifurcation of spatially non-homogeneous stationary solutions occurs in the domain of parameters where bifurcation is excluded for the original mixed boundary value problem. Inclusions can be considered in one of the equations itself as well as in boundary conditions. The proof is based on the degree theory for multivalued mappings (jump of the degree implies bifurcation). We show how the degree for a class of multivalued maps including those corresponding to a weak formulation of our problem can be calculated.