Oscillatory behavior in discrete slow power-law models
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ORIGINAL PAPER
Oscillatory behavior in discrete slow power-law models Silvia Jerez · Emilene Pliego · Francisco J. Solis
Received: 23 June 2020 / Accepted: 23 September 2020 © Springer Nature B.V. 2020
Abstract Discrete mathematical slow oscillatory models are proposed to describe biological interactions between two populations by considering power-law functions. Conditions for slow convergence to the equilibrium point are imposed on model parameters. Moreover, to obtain oscillatory solutions we prove that model exponents may be parameterized by only one parameter. As a by-product, we also discover a family of functions that can be regarded as a two-dimensional generalization of the Schwarzian derivative. Diverse particular model cases are analyzed numerically in order to show orbital solutions. Finally, applications for biochemical and population models are presented. Keywords Discrete power-law models · Oscillatory behavior · Slow systems · Schwarzian derivative · Bone remodeling · Lotka–Volterra systems Mathematics Subject Classification 37N25 · 92D25 · 65Z05
This work was supported by CONACyT, Mexico Project CB2016-286437. S. Jerez · E. Pliego · F. J. Solis (B) CIMAT, 36000 Guanajuato, GTO, Mexico e-mail: [email protected] S. Jerez e-mail: [email protected] E. Pliego e-mail: [email protected]
1 Introduction In an ecological community, it is fundamental to understand interaction mechanisms between species. In most cases, the processes that govern such interactions are unknown. Consequently, their descriptions require representations general enough to capture the essence of the observed response and, at the same time, they should be mathematically feasible. One candidate that satisfies these requirements is the power-law interaction [1,2]. Power-law models arise when non-integer kinetic orders are used and by their properties to describe inhibition mechanisms with simplified equations. Such functional interaction has been used to model a diverse type of applications and to provide a general framework for addressing questions about regulation in complex cases, such as paracrine/autocrine signaling systems in cellular environments [3–5] and predator–prey interactions [6–9] and references therein. It is important to remark that power-law functions have also been included in diverse nonlinear population models, see, for example, [10– 13]. Under certain circumstances of species interactions, oscillatory behavior is present and such phenomena have been well documented, see [14–18]. Such solution behavior plays a fundamental role in the coexistence of species since they tend to correlate with each another when coupled, presenting fast growth periods alternating with slow growth periods. Mathematical analysis
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of those oscillatory models is difficult due to their complicated nonlinear terms and also by their variables and parameters of very different orders of magnitude. Theoretical techniques to obtain conditions for periodic or oscillatory solutions usually rely on numerical methods and bifu
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