Non-damped Nonlinear Wave in Continual Hypercycle Replicated Systems

An important class of replicator models involves systems of nonlinear ordinary differential equations with dynamics restrained by the standard simplex in the state space and describes macromolecular interactions in various problems of population genetics and evolutionary game theory [1], as well as in theories of the origin of life [2]. Of special interest is the hypercycle model that was proposed by M. Eigen and P. Shuster classical hypercycle is a finite closed network of self-replicating macromolecules (species) which are connected so that each of them catalyzes the replication of the successor, with the last molecule reinforcing the first one. From the sociological perspective, the catalytic support for the replication of other molecules resembles altruistic behavior, in contrast to conventional autocatalysis. However, the actual number of macromolecules in a hypercycle may be huge, and this may significantly complicate the numerical analysis of the associated dynamical system. It may therefore be reasonable to represent the macromolecules as points in some line segment (of cardinality continuum) and to construct an appropriate distributed model of hypercyclic replication. Such a methodology was previously implemented for Crow–Kimura and Eigen quasispecies models, with a single integra-differential equation replacing a large number of ordinary differential equations [3,4]. Since the model represents an idealized process of replication continuous species in the form of integra-differential equation with space delay in integral simplex. The existence and uniqueness of positive solution are proved. The solutions represent sequence of non-damped nonlinear wave. It is proved existence of Andronov-Hopf bifurcation in steady state position [5]. The results of numerical modelling are presented.
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[3] Bratus A.S., I. Yegorov, A. Novozhilov. Open quasispecies models: Stability, optimization, and distributed extension (печатный). Journal of Mathematical Analysis and Application, https://doi.org/10.1016/j.jmaa.2019.123477 2019.
[4] Братусь А.С., Дрожжин С. В., Якушкина Т. С. Математические модели эволюции и динамики репликаторных систем. Москва, УРСС, 2022, 265 с.
[5] Marsden J. E., McCracken M. The Hopf Bifurcation and its Applications. Springer, New York, 1976.