Mathematics of Darwin’s Diagram

Bessonov, Nikolai; Reinberg, Natalia; Volpert, Vitaly

doi:10.1051/mmnp/20149302

Cited by 33 publications

(33 citation statements)

References 33 publications

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“…Nonlocal reaction-diffusion equations represent an appropriate framework to describe evolution of biological species [25,27,39]. These equations take into account nonlocal consumption of resources characterizing intraspecific competition and possibly leading to the emergence of multi-modal population density distributions.…”

Section: Introductionmentioning

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: Introductionmentioning

confidence: 99%

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

Bessonov¹,

Bocharov²,

Meyerhans³

et al. 2019

Preprint

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“…Hence, the point where the stability boundary is not smooth correspond to the sudden change of the location of the feasible root of Equation (27). Finally, it is important to mention that the choice of parameters (24) leads to an interesting scenario for which the spatial Hopf and Turing bifurcation curves intersect.…”

Section: Turing Pattern For Nonlocal Prey-predator Modelmentioning

confidence: 99%

“…Whereas nonlocal consumption of resources is more general since it incorporates the interspecific competition for food [24][25][26]. Such modifications enables the explanation of emergence and evolution of biological species as well as speciation in a more appropriate manner [27][28][29][30][31]. The models with nonlocal consumption of resources present complex dynamics for the single species models [28,29,[32][33][34][35] as well as for competition models including two or more species [32,[36][37][38].…”

Section: Introductionmentioning

confidence: 99%

Prey-Predator Model with a Nonlocal Bistable Dynamics of Prey

2018

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Spatiotemporal pattern formation in integro-differential equation models of interacting populations is an active area of research, which has emerged through the introduction of nonlocal intra-and inter-specific interactions. Stationary patterns are reported for nonlocal interactions in prey and predator populations for models with prey-dependent functional response, specialist predator and linear intrinsic death rate for predator species. The primary goal of our present work is to consider nonlocal consumption of resources in a spatiotemporal prey-predator model with bistable reaction kinetics for prey growth in the absence of predators. We derive the conditions of the Turing and of the spatial Hopf bifurcation around the coexisting homogeneous steady-state and verify the analytical results through extensive numerical simulations. Bifurcations of spatial patterns are also explored numerically.

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“…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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Mathematics of Darwin’s Diagram

Cited by 33 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

Prey-Predator Model with a Nonlocal Bistable Dynamics of Prey

Preface to the Issue Nonlocal Reaction-Diffusion Equations

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