Branching and aggregation in self-reproducing systems

Volpert, Vitaly

doi:10.1051/proc/201447007

Cited by 10 publications

(7 citation statements)

References 33 publications

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“…Speciation is considered as a general property of the living matter [35]. It manifests itself in the emergence of biological species and in a variety of other systems [46]. In the framework of mathematical modelling, speciation appears due to a non-homogeneous density distributions u(x, t).…”

Section: Discussionmentioning

confidence: 99%

Nonlocal Reaction-Diffusion Model of Viral Evolution: Emergence of Virus Strains

Bessonov¹,

Bocharov²,

Meyerhans³

et al. 2019

Preprint

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The work is devoted to the investigation of virus quasispecies evolution and diversification due to mutations, competition for host cells, and cross-reactive immune responses. The model consists of a nonlocal reaction-diffusion equation for the virus density depending on the genotype considered as a continuous variable and on time. This equation contains two integral terms corresponding to the nonlocal effects of virus interaction with host cells and with immune cells. In the model, a virus strain is represented by a localized solution concentrated around some given genotype. Emergence of new strains corresponds to a periodic wave propagating in the space of genotypes. The conditions of appearance of such waves and their dynamics are described.

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Section: Discussionmentioning

confidence: 99%

Nonlocal Reaction-Diffusion Model of Viral Evolution: Emergence of Virus Strains

Bessonov¹,

Bocharov²,

Meyerhans³

et al. 2019

Preprint

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“…Equation (1.4) has stationary solutions in the form of pulses for all h 2 sufficiently large (r(h 2 ) = 1) [33]. Numerical simulations show that these pulses are stable [13,28,29], though their stability is not proved analytically.…”

Section: Nonlocal Equations In Population Dynamicsmentioning

confidence: 99%

Doubly nonlocal reaction–diffusion equations and the emergence of species

Banerjee

Vougalter

Volpert

2017

Applied Mathematical Modelling

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The paper is devoted to a reaction-diffusion equation with doubly nonlocal nonlinearity arising in various applications in population dynamics. One of the integral terms corresponds to the nonlocal consumption of resources while another one describes reproduction with different phenotypes. Linear stability analysis of the homogeneous in space stationary solution is carried out. Existence of travelling waves is proved in the case of narrow kernels of the integrals. Periodic travelling waves are observed in numerical simulations. Existence of stationary solutions in the form of pulses is shown, and transition from periodic waves to pulses is studied. In the applications to the speciation theory, the results of this work signify that new species can emerge only if they do not have common offsprings. Thus, it is shown how Darwin's definition of species as groups of morphologically similar individuals is related to Mayr's definition as groups of individuals that can breed only among themselves.

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“…The minimal model of speciation includes intraspecific competition (nonlocal consumption), reproduction and mutations (diffusion). Speciation is a general property of the living matter observed in many different contexts [35]. Other nonlocal models of natural selection, intra-and interspecific competition are discussed in [12].…”

Section: Nonlocal Reaction-diffusion Equations In Population Dynamicsmentioning

confidence: 99%

Preface to the Issue Nonlocal Reaction-Diffusion Equations

Alfaro

Apreutesei

Davidson

et al. 2015

Math. Model. Nat. Phenom.

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Nonlocal reaction-diffusion equations in population dynamicsNonlocal reaction-diffusion equations are intensively studied during the last decade in relation with problems in population dynamics and other applications. In comparison with traditional reaction-diffusion equations they possess new mathematical properties and richer nonlinear dynamics. Many studies are devoted to the nonlocal reaction-diffusion equationwherewhich describes the distribution of population density in the case of nonlocal consumption of resources.Here k is a positive integer, k = 1 corresponds to asexual and k = 2 to sexual reproduction. If the kernel φ of the integral is replaced by the δ-function, then we obtain conventional reaction-diffusion equation.In the case k = 1, it is the logistic equation with the reproduction term au(1 − u) proportional to the population density u and to available resources (1 − u). In the case of nonlocal consumption of resources, the integral J(u) describes consumption of resources at the space point x by individuals located in some area around this point. The function φ(x − y) determines the efficiency of such consumption. Introduction of nonlocal consumption of resources changes the properties of solutions of this equation. Consider for certainty the case where k = 1 and σ = 0. Suppose that ∞ −∞ φ(y)dy = 1. Then u = 1 is a stationary solution of this equation. It is stable in the case of the local equation but it can lose its stability for the nonlocal equation. If it becomes unstable, then a periodic in space stationary solution bifurcates from it [13], [19], [21]. This simple result of the linear stability analysis has important consequences from the mathematical point of view and for the applications.

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Branching and aggregation in self-reproducing systems

Cited by 10 publications

References 33 publications

Nonlocal Reaction-Diffusion Model of Viral Evolution: Emergence of Virus Strains

Nonlocal Reaction-Diffusion Model of Viral Evolution: Emergence of Virus Strains

Doubly nonlocal reaction–diffusion equations and the emergence of species

Preface to the Issue Nonlocal Reaction-Diffusion Equations

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