dorsal/arxiv
View SchemaAn evolution equation of the population genetics: relation to the density-matrix theory of quasiparticles with general dispersion laws
| Authors | V. Bezak |
|---|---|
| Categories | |
| ArXiv ID | physics/0209076 |
| URL | https://arxiv.org/abs/physics/0209076 |
| DOI | 10.1103/PhysRevE.67.021913 |
Abstract
The Waxman-Peck theory of the population genetics is discussed in regard of soil bacteria. Each bacterium is understood as a carrier of a phenotypic parameter p. The central aim is the calculation of the probability density with respect to p of the carriers living at time t>0. The theory involves two small parameters: the mutation probability $\mu$ and a parameter $\gamma$ involved in a function w(p) defining the fitness of the bacteria to survive the generation time $\tau$ and give birth to offspring. The mutation from a state p to a state q is defined by a Gaussian. The author focuses attention on an equation generalizing Waxman's equation. The author solves this equation in the standard style of a perturbation theory and discusses how the solution depends on the choice of the fitness function w(p). In a sense, the function $c(p)=1-w(p)/w(0)$ is analogous to the dispersion function E(p) of fictitious quasiparticles. With a general function c(p), the distribution function ${\mathit\Phi}(p,t;0)$ is composed of a delta-function component, $N(t)\delta(p)$, and a blurred component. The author shows that asymptotically N(t) may tend to a positive value, in contrast with zero resulting from Waxman's approximation where $c(p)\sim p^2$.
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"abstract": "The Waxman-Peck theory of the population genetics is discussed in regard of\nsoil bacteria. Each bacterium is understood as a carrier of a phenotypic\nparameter p. The central aim is the calculation of the probability density with\nrespect to p of the carriers living at time t\u003e0. The theory involves two small\nparameters: the mutation probability $\\mu$ and a parameter $\\gamma$ involved in\na function w(p) defining the fitness of the bacteria to survive the generation\ntime $\\tau$ and give birth to offspring. The mutation from a state p to a state\nq is defined by a Gaussian. The author focuses attention on an equation\ngeneralizing Waxman\u0027s equation. The author solves this equation in the standard\nstyle of a perturbation theory and discusses how the solution depends on the\nchoice of the fitness function w(p). In a sense, the function\n$c(p)=1-w(p)/w(0)$ is analogous to the dispersion function E(p) of fictitious\nquasiparticles. With a general function c(p), the distribution function\n${\\mathit\\Phi}(p,t;0)$ is composed of a delta-function component,\n$N(t)\\delta(p)$, and a blurred component. The author shows that asymptotically\nN(t) may tend to a positive value, in contrast with zero resulting from\nWaxman\u0027s approximation where $c(p)\\sim p^2$.",
"arxiv_id": "physics/0209076",
"authors": [
"V. Bezak"
],
"categories": [
"physics.bio-ph",
"q-bio"
],
"doi": "10.1103/PhysRevE.67.021913",
"title": "An evolution equation of the population genetics: relation to the density-matrix theory of quasiparticles with general dispersion laws",
"url": "https://arxiv.org/abs/physics/0209076"
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