Narrow Results

The paper investigates the role of selection on evolutionary rescue of population. The statistical mechanics technique is used to model dynamics of a population experiencing a natural selection and an abrupt change in the environment. The paper assesses the selective pressure produced by two different mechanisms: by strength of resistance and by strength of selection (by intraspecific competition). It is shown that both mechanisms are capable of providing an evolutionary rescue of population in particular conditions. However, for a small level of an extinction rate, the population cannot be rescued without intraspecific competition.

We study the properties of a n²-dimensional Lotka–Volterra system describing competition among n different species with adaptive skills, i.e. whose interaction coefficients are time averages of the species level of interaction over their past.

Starting by the case of adaptive competition among species all having the same carrying capacities, we focus our attention on the model obtained on perturbing the carrying capacity of a fixed species, which is made more or less disadvantaged.

We prove the existence of a certain class of invariant subspaces and introduce a seven-dimensional reduced model, where n appears as a parameter, which gives full account of existence and stability of equilibria in the system. The relevance of this reduced model to the complete one has also been found when the time dependent regimes have been investigated.

Ecologically relevant questions, i.e. species survival and the time dependent behavior of the system have also been analyzed focusing on the role of behavioral adaptation. In particular, we have found that competitive exclusion cannot occur but coexistence is possible in various forms (i.e. competitive stable equilibria and different periodic oscillations).

The paper describes the slow evolution of two adaptive traits that regulate the interactions between two mutualistic populations (e.g. a flowering plant and its insect pollinator). For frozen values of the traits, the two populations can either coexist or go extinct. The values of the traits for which populations extinction is guaranteed are therefore of no interest from an evolutionary point of view. In other words, the evolutionary dynamics must be studied only in a viable subset of trait space, which is bounded due to the physiological cost of extreme trait values. Thus, evolutionary dynamics experience so-called border collision bifurcations, when a system invariant in trait space hits the border of the viable subset. The unfolding of standard and border collision bifurcations with respect to two parameters of biological interest is presented. The algebraic and boundary-value problems characterizing the border collision bifurcations are described together with some details concerning their computation.

We present in this paper the first example of chaotic evolutionary dynamics in biology. We consider a Lotka–Volterra tritrophic food chain composed of a resource, its consumer, and a predator species, each characterized by a single adaptive phenotypic trait, and we show that for suitable modeling and parameter choices the evolutionary trajectories approach a strange attractor in the three-dimensional trait space. The study is performed through the bifurcation analysis of the so-called canonical equation of Adaptive Dynamics, the most appropriate modeling approach to long-term evolutionary dynamics.

We unfold the bifurcation involving the loss of evolutionary stability of an equilibrium of the canonical equation of Adaptive Dynamics (AD). The equation deterministically describes the expected long-term evolution of inheritable traits — phenotypes or strategies — of coevolving populations, in the limit of rare and small mutations. In the vicinity of a stable equilibrium of the AD canonical equation, a mutant type can invade and coexist with the present — resident — types, whereas the fittest always win far from equilibrium. After coexistence, residents and mutants effectively diversify, according to the enlarged canonical equation, only if natural selection favors outer rather than intermediate traits — the equilibrium being evolutionarily unstable, rather than stable. Though the conditions for evolutionary branching — the joint effect of resident-mutant coexistence and evolutionary instability — have been known for long, the unfolding of the bifurcation has remained a missing tile of AD, the reason being related to the nonsmoothness of the mutant invasion fitness after branching. In this paper, we develop a methodology that allows the approximation of the invasion fitness after branching in terms of the expansion of the (smooth) fitness before branching. We then derive a canonical model for the branching bifurcation and perform its unfolding around the loss of evolutionary stability. We cast our analysis in the simplest (but classical) setting of asexual, unstructured populations living in an isolated, homogeneous, and constant abiotic environment; individual traits are one-dimensional; intra- as well as inter-specific ecological interactions are described in the vicinity of a stationary regime.

The well-posedness of a non-local advection–selection–mutation problem deriving from adaptive dynamics models is shown for a wide family of initial data. A particle method is then developed, in order to approximate the solution of such problem by a regularized sum of weighted Dirac masses whose characteristics solve a suitably defined ODE system. The convergence of the particle method over any finite interval is shown and an explicit rate of convergence is given. Furthermore, we investigate the asymptotic-preserving properties of the method in large times, providing sufficient conditions for it to hold true as well as examples and counter-examples. Finally, we illustrate the method in two cases taken from the literature.

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.

In this paper, with the method of adaptive dynamics, we investigate the coevolution of phenotypic traits of predator and prey species. The evolutionary model is constructed from a deterministic approximation of the underlying stochastic ecological processes. Firstly, we investigate the ecological and evolutionary conditions that allow for continuously stable strategy and evolutionary branching. We find that evolutionary branching in the prey phenotype will occur when the frequency dependence in the prey carrying capacity is not strong. Furthermore, it is found that if the two prey branches move far away enough, the evolutionary branching in the prey phenotype will induce the secondary branching in the predator phenotype. The final evolutionary outcome contains two prey and two predator species. Secondly, we show that under symmetric interactions the evolutionary model admits a supercritical Hopf bifurcation if the frequency dependence in the prey carrying capacity is very weak. Evolutionary cycle is a likely outcome of the mutation-selection processes. Finally, we find that frequency-dependent selection can drive the predator population to extinction under asymmetric interactions.

In bio-systems, there exist several phenomena breaking the laws of total probability such as the lactose-glucose interference in E. coli growth. We call such phenomena the contextual dependent adaptive systems. Recently we introduced a new mathematical framework to treat the probability in those systems. In this paper, we discuss the essence of this mathematical frame with a simple example “a state change of tongue for sweetness”.

There exist several phenomena breaking the classical probability laws. Such systems are adaptive (context dependent) systems. In this paper, we present a new mathematical formula to compute the probability in those systems by using the concepts of the adaptive dynamics and lifting theory.