Border Collision Bifurcations in the Evolution of Mutualistic Interactions

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

Remarks on Branching-Extinction Evolutionary Cycles

We show in this paper that the evolution of cannibalistic consumer populations can be a never-ending story involving alternating levels of polymorphism. More precisely, we show that a monomorphic population can evolve toward high levels of cannibalism until it reaches a so-called branching point, where the population splits into two sub-populations characterized by different, but initially very close, cannibalistic traits. Then, the two traits coevolve until the more cannibalistic sub-population undergoes evolutionary extinction. Finally, the remaining population evolves back to the branching point, thus closing an evolutionary cycle. The model on which the study is made is purely deterministic and derived through the adaptive dynamics approach. Evolutionary dynamics are investigated through numerical bifurcation analysis, applied both to the ecological (resident-mutant) model and to the evolutionary model. The general conclusion emerging from this study is that branching- extinction evolutionary cycles can be present in wide ranges of environmental and demographic parameters, so that their detection is of crucial importance when studying evolutionary dynamics

Discontinuity induced bifurcations of non-hyperbolic cycles in nonsmooth systems

We analyse three codimension-two bifurcations occurring in nonsmooth systems, when a non-hyperbolic cycle (fold, flip, and Neimark-Sacker cases, both in continuous- and discrete-time) interacts with one of the discontinuity boundaries characterising the system's dynamics. Rather than aiming at a complete unfolding of the three cases, which would require specific assumptions on both the class of nonsmooth system and the geometry of the involved boundary, we concentrate on the geometric features that are common to all scenarios. We show that, at a generic intersection between the smooth and discontinuity induced bifurcation curves, a third curve generically emanates tangentially to the former. This is the discontinuity induced bifurcation curve of the secondary invariant set (the other cycle, the double-period cycle, or the torus, respectively) involved in the smooth bifurcation. The result can be explained intuitively, but its validity is proven here rigorously under very general conditions. Three examples from different fields of science and engineering are also reported

The social diversification of fashion

We propose a model to investigate the dynamics of fashion traits purely driven by social interactions. We assume that people adapt their style to maximize social success, and we describe the interaction as a repeated group game in which the payoffs reflect the social norms dictated by fashion. On one hand, the tendency to imitate the trendy stereotypes opposed to the tendency to diverge from them to proclaim identity; on the other hand, the exploitation of sex appeal for dating success opposed to the moral principles of the society. These opposing forces promote diversity in fashion traits, as predicted by the modeling framework of adaptive dynamics. Our results link the so-called horizontal dynamics—the primary driver of fashion evolution, compared with the vertical dynamics accounting for interclass and economic drivers—to style variety

Detection and Continuation of a Border Collision Bifurcation in a Forest Fire Model

The behavior of the simplest forest fire model is studied in this paper through bifurcation analysis. The model is a second-order continuous-time impact model where vegetational growth is described as a continuous and slow dynamic process, while fires are modeled as instantaneous and disruptive events. The transition from Mediterranean forests (characterized by wild chaotic fire regimes) to savannas and boreal forests (where fires are almost periodic) is recognized to be a catastrophic transition known as border collision bifurcation in the context of discrete-tine systems. In the present case such a bifurcation can be easily detected numerically and then continued by solving a standard boundary-value problem. The result of the analysis complements previous simulation studies and are consistent with biological intuition

Bifurcation Analysis of Filippov's Ecological Models

The aim of this paper is the study of the long-term behavior of population communities described by a class of discontinuous models known as Filippov systems. The analysis is carried out by performing the bifurcation analysis of the model with respect to two parameters. A relatively simple method, called the puzzle method, is proposed to construct the complete bifurcation diagram step-by-step. The method is illustrated through four examples concerning the exploitation and protection of interacting populations

Ecological Bistability and Evolutionary Reversals under Asymmetrical Competition

How does the process of life-history evolution interplay with population dynamics? Almost all models that have addressed this question assume that any combination of phenotypic traits uniquely determine the ecological population state. Here we show that if multiple ecological equilibria can exist, the evolution of a trait that relates to competitive performance can undergo adaptive reversals that drive cyclic alternation between population equilibria. The occurrence of evolutionary reversals require neither environmentally-driven changes in selective forces, nor the co-evolution of interactions with other species. The mechanism including evolutionary reversals is two-fold. First, there exist phenotypes near which mutants can invade and yet fail to become fixed; although these mutants are eventually eliminated, their transitory growth causes the resident population to switch to an alternative ecological equilibrium. Second, asymmetrical competition causes the direction of selection to revert between high and low density. When ecological conditions for evolutionary reversals are not satisfied, the population evolves toward a steady state of either low or high abundance, depending on the degree of competitive asymmetry and environmental parameters. A sharp evolutionary transition between evolutionary stasis and evolutionary reversals and cycling can occur in response to a smooth change in ecological parameters, and this may have implications for our understanding of size-abundance patterns

The branching bifurcation of Adaptive Dynamics

We unfold the bifurcation involving the loss of evolutionary stability of an equilibrium of the canoncal 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 equilibium 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 or 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