138 research outputs found
Dynamical Systems and Their Bifurcations
No full textIn this chapter we summarize the basic definitions and tools of analysis of dynamical systems, with particular
emphasis on the asymptotic behavior of continuous-time autonomous systems. In particular, the possible
structural changes of the asymptotic behavior of the system under parameter variation, called bifurcations,
are presented together with their analytical characterization and hints on their numerical analysis. The
literature on dynamical systems is huge and we do not attempt to survey it here. Most of the results on
bifurcations of continuous-time systems are due to Andronov and Leontovich [see Andronov et al., 1973].
More recent expositions can be found in Guckenheimer & Holmes [1997] and Kuznetsov [2004], while less
formal but didactically very effective treatments, rich in interesting examples and applications, are given
in Strogatz [1994] and Alligood et al. [1996]. Numerical aspects are well described in Allgower & Georg
[1990] and in the fundamental papers by Keller [1977] and Doedel et al. [1991a,b], but see also Beyn et al.
[2002] and Kuznetsov [2004]. This chapter mainly combines material from two previous contributions of the
authors, the first part of the book Biosystems and Complexity [Rinaldi, 1993, in Italian] and the Appendix A
of a recent book on evolutionary dynamics [Dercole & Rinaldi, 2008]
Analysis of Evolutionary Processes: The Adaptive Dynamics Approach and Its Applications
No full textQuantitative approaches to evolutionary biology traditionally consider evolutionary change in isolation from an important pressure in natural selection: the demography of coevolving populations. In this volume, the authors have written the first comprehensive book on Adaptive Dynamics (AD), a quantitative modeling approach that explicitly links evolutionary changes to demographic ones. The book shows how the so-called AD canonical equation can answer questions of paramount interest in biology, engineering, and the social sciences, especially economics.
After introducing the basics of evolutionary processes and classifying available modeling approaches, Dercole and Rinaldi give a detailed presentation of the derivation of the AD canonical equation, an ordinary differential equation that focuses on evolutionary processes driven by rare and small innovations. The authors then look at important features of evolutionary dynamics as viewed through the lens of AD. They present their discovery of the first chaotic evolutionary attractor, which calls into question the common view that coevolution produces exquisitely harmonious adaptations between species. And, opening up potential new lines of research by providing the first application of AD to economics, they show how AD can explain the emergence of technological variety.
"Analysis of Evolutionary Processes" will interest anyone looking for a self-contained treatment of AD for self-study or teaching, including graduate students and researchers in mathematical and theoretical biology, applied mathematics, and theoretical economics
Border Collision Bifurcations in the Evolution of Mutualistic Interactions
No full textThe 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
No full textWe 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 based 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
Generalized boundary equilibria in n-dimensional Filippov systems: The transition between persistence and nonsmooth-fold scenarios
No full textThe transition from persistence to nonsmooth-fold scenarios in relay control system
No full textIn many applications in different fields of science and engineering only two
extreme values of the control variable can be easily applied, and a threshold on a suitable
output variable is used to discriminate among the two control actions. This relay control
introduces a discontinuity, so the closed-loop system is discontinuous piecewise smooth (called
Filippov system). When a standard equilibrium attains the threshold, as a system or control
parameter is varied, two generic scenarios are possible: the standard equilibrium turns into
a pseudo-equilibrium on the discontinuity boundary (persistence), so a stationary solution
persists trough the bifurcation; the collision and disappearance of the standard equilibrium
and a coexisting pseudo-equilibrium (nonsmooth-fold). In this paper we analyze the degenerate
situation separating these two scenarios, and we apply our results to a four-dimensional SISO
system describing the ecological dynamics of a protected natural resource (a resource that cannot
be harvested when below threshold). We show that while profitable exploitation is guaranteed
(though often at the threshold) in the persistence scenario, the food chain collapses after a
nonsmooth-fold
The ecology of asexual pairwise interactions: The generalized law of mass action
No full textA general procedure to formulate asexual (unstructured, deterministic) population dynamical models resulting from individual pairwise interactions is proposed. Individuals are characterized by a continuous strategy that is constant during life and represents their behavioral, morphological, and functional traits. Populations group conspecific individuals with identical strategy and are measured by densities in space. Species can be monomorphic, if only a single value of the strategy is present, or polymorphic otherwise. The procedure highlights the structural properties fulfilled by the population per-capita growth rates. In particular, the effect on the growth rate of jointly perturbing a set of similar strategies is proportional to the product of the corresponding densities, with a proportionality coefficient that can be density-dependent only through the sum of the densities. This generalizes the law of mass action, which traditionally refers to the case in which the per-capita growth rates are linearly density-dependent and insensitive to joint strategy perturbations. Being underpinned by individual strategies, the proposed procedure is most useful for evolutionary considerations, in the case strategies are inheritable. The developed body of theory is exemplified on a Holling-type-II many-prey-one-predator system and on a model of cannibalism
BPcont: An Auto driver for the continuation of branch points of algebraic and boundary-value problems
No full text
- …
