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Population genetics models for the statistics of DNA samples under different demographic scenarios - Maximum likelihood versus approximate methods

Andrzej Polański, Marek Kimmel (2003)

International Journal of Applied Mathematics and Computer Science

The paper reviews the basic mathematical methodology of modeling neutral genetic evolution, including the statistics of the Fisher-Wright process, models of mutation and the coalescence method under various demographic scenarios. The basic approach is the use of maximum likelihood techniques. However, due to computational problems, intuitive or approximate methods are also of great importance.

Population Growth and Persistence in a Heterogeneous Environment: the Role of Diffusion and Advection

A. B. Ryabov, B. Blasius (2008)

Mathematical Modelling of Natural Phenomena

The spatio-temporal dynamics of a population present one of the most fascinating aspects and challenges for ecological modelling. In this article we review some simple mathematical models, based on one dimensional reaction-diffusion-advection equations, for the growth of a population on a heterogeneous habitat. Considering a number of models of increasing complexity we investigate the often contrary roles of advection and diffusion for the persistence of the population. When it is possible we demonstrate...

Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies

Alexander Lorz, Tommaso Lorenzi, Michael E. Hochberg, Jean Clairambault, Benoît Perthame (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Resistance to chemotherapies, particularly to anticancer treatments, is an increasing medical concern. Among the many mechanisms at work in cancers, one of the most important is the selection of tumor cells expressing resistance genes or phenotypes. Motivated by the theory of mutation-selection in adaptive evolution, we propose a model based on a continuous variable that represents the expression level of a resistance gene (or genes, yielding a phenotype) influencing in healthy and tumor cells birth/death...

Positive and Negative Feedback in Engineering and Biology

E. S. Zeron (2008)

Mathematical Modelling of Natural Phenomena

No other concepts have shaken so deeply the bases of engineering like those of positive and negative feedback. They have played a most prominent role in engineering since the beginning of the previous century. The birth certificate of positive feedback can be traced back to a pair of patents by Edwin H. Armstrong in 1914 and 1922, whereas that of negative feedback is already lost in time. We present in this paper a short review on the feedback's origins in the fields of engineering and biology....

Positive fixed point theorems arising from seeking steady states of neural networks

Gen Qiang Wang, Sui-Sun Cheng (2010)

Mathematica Bohemica

Biological systems are able to switch their neural systems into inhibitory states and it is therefore important to build mathematical models that can explain such phenomena. If we interpret such inhibitory modes as `positive' or `negative' steady states of neural networks, then we will need to find the corresponding fixed points. This paper shows positive fixed point theorems for a particular class of cellular neural networks whose neuron units are placed at the vertices of a regular polygon. The...

Positive periodic solution for ratio-dependent n -species discrete time system

Mei-Lan Tang, Xin-Ge Liu (2011)

Applications of Mathematics

In this paper, sharp a priori estimate of the periodic solutions is obtained for the discrete analogue of the continuous time ratio-dependent predator-prey system, which is governed by nonautonomous difference equations, modelling the dynamics of the n - 1 competing preys and one predator having nonoverlapping generations. Based on more precise a priori estimate and the continuation theorem of the coincidence degree, an easily verifiable sufficient criterion of the existence of positive periodic solutions...

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