Global dynamic scenarios in a discrete-time model of renewable resource exploitation: a mathematical study
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ORIGINAL PAPER
Global dynamic scenarios in a discrete-time model of renewable resource exploitation: a mathematical study Lorenzo Cerboni Baiardi · Anastasiia Panchuk
Received: 1 February 2020 / Accepted: 13 August 2020 © Springer Nature B.V. 2020
Abstract We consider the two-dimensional map introduced in Bischi et al. (J Differ Equ Appl 21(10):954– 973, 2015) formulated as a model for a renewable resource exploitation process in an evolutionary setting. The global dynamic scenarios displayed by the model are not so often encountered in smooth twodimensional dynamical systems. We explain the occurrence of such scenarios at the light of the theory of noninvertible maps. Moreover, complex structures of basins of attraction of coexisting invariant sets are observed. We analyze such structures by examining stability properties of chaotic sets, in the case in which a non-topological Milnor attractor is present. Stability changes of a chaotic set occur through global bifurcations (such as riddling and blowout) and are detected by means of the study of the spectrum of Lyapunov exponents associated with the set. Keywords Chaotic dynamics · Noninvertible maps · Milnor attractor · Global bifurcations · Complexity
L. Cerboni Baiardi Department of Economics, Statistics and Finance, University of Calabria, Rende, Italy e-mail: [email protected] A. Panchuk (B) Institute of Mathematics, NAS of Ukraine, Kiev, Ukraine e-mail: [email protected]
1 Introduction In the recent decades, environmental problems arising from human activity have been drawing great attention of scientists from different research areas (e.g., [5,11,16] to cite a few). In particular, models describing exploitation of renewable natural resources have become rather popular (see e.g., [10,26,36] and references therein). Within this stream of literature, the contribution provided in [9] is placed. In that work, the authors consider a two-dimensional map that is the discrete-time counterpart of the continuous-time dynamical system formulated in [26]. The map is a model for a renewable resource exploitation process (fishery), where the resource is renewed according to a logistic-type growth and where players choose between two harvesting strategies (an intensive one and an environmentally friendly one) according to a profit-driven evolutionary selection rule known as replicator dynamics (see [14,22]). In the current paper, we continue considering the nonlinear model proposed in [9]. In comparison with the previous work, we provide wider overview of asymptotic behaviors arising when the internal fixed point loses its stability due to a flip or a Neimark– Sacker bifurcation. In particular, we describe a certain dynamic scenario that is not typically observed in the neighborhood of the flip bifurcation curve. Inter alia, this scenario is characterized by coexistence of two internal attractors, for one of which environmen-
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tally friendly strategy prevails to the greater extent than for the other. In addition
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