A Quantitative Reasoning Framework and the Importance of Quantitative Modeling in Biology
Robert Mayes,
David Owens,
Joseph Dauer,
Kent Rittschof
Issue:
Volume 11, Issue 1, February 2022
Pages:
1-17
Received:
20 August 2021
Accepted:
9 October 2021
Published:
18 January 2022
Abstract: Biology is becoming more quantitative. If we are to support the future of quantitative biology, then the next generation of biologists must be prepared to consistently integrate quantitative reasoning into subject matter that has traditionally been considered through a qualitative lens. We introduce a quantitative reasoning framework and discuss the importance of quantitative modeling in biology. The framework includes the Quantitative Act as a support for Quantitative Modeling and Quantitative Interpretation. The QM BUGS diagnostic instrument was developed to assesses undergraduate biology students’ abilities to create and apply models employing pre-calculus mathematics. A brief discussion of our research findings based on implementation of the instrument include the lack of student ability to develop quantitative models. We present items from the instrument as examples of the Quantitative Act elements: variable quantification through identifying variable and attributes, measurement, variation, quantitative literacy, and context. We also provide items representing quantitative modeling and quantitative interpretation. We then view quantitative biology from K-12 and collegiate perspectives, including instructional practices for teaching quantitative biology, motivating problem contexts that afford quantification, instructional strategies of repetition, scaffolding, peer teaching and learning, direct instruction and teacher moves on the K-12 level, as well as identifying five competencies for the next generation of biologists which require QA abilities.
Abstract: Biology is becoming more quantitative. If we are to support the future of quantitative biology, then the next generation of biologists must be prepared to consistently integrate quantitative reasoning into subject matter that has traditionally been considered through a qualitative lens. We introduce a quantitative reasoning framework and discuss th...
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Bifurcations and Dynamical Behavior of 2D Coupled Chaotic Sine Maps
Yamina Soula,
Abdel Kaddous Taha,
Daniele Fournier-Prunaret,
Nasr-Eddine Hamri
Issue:
Volume 11, Issue 1, February 2022
Pages:
18-30
Received:
14 January 2022
Accepted:
4 February 2022
Published:
18 February 2022
Abstract: The main characteristics of a dynamical system are determined by the bifurcation theory. In particular, in this paper we examined the properties of the discrete dynamical system of a two coupled maps, i.e. the maps with an invariant unidimensional submanifold. The study of coupled chaotic systems shows rich and complex dynamic behaviors, particularly through structures of bifurcations or chaotic synchronization. A bifurcation is a qualitative change of the system behavior under the influence of control parameters. This change may correspond to the disppearance or appearance of new singularities or a change in the nature of singularities. We can define different kinds of bifurcations for fixed points and period two cycles as, saddle-node, period doubling, transcritical or pitchfork bifurcations. The study of the sequence of bifurcations permits to understand the mechanisms that lead to chaos. The phenomena of synchronization and antisynchronization in coupled chaotic systems is very important because its applications in several areas, such as secure communication or biology. In this paper, we study bifurcation properties of a two-dimensional coupled map T with three parameters. The first objective is to locate the bifurcation curves and their evolution in the parametric plane (a,b), when a third parameter c varies. The equations of some bifurcation curves are given analytically; cusp points and co-dimension two points on these bifurcation curves are determined. The second is related to the study of the chaotic synchronization and antisynchronization in the phase space (x,y).
Abstract: The main characteristics of a dynamical system are determined by the bifurcation theory. In particular, in this paper we examined the properties of the discrete dynamical system of a two coupled maps, i.e. the maps with an invariant unidimensional submanifold. The study of coupled chaotic systems shows rich and complex dynamic behaviors, particular...
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be real Möbius groups. For any 2×2 matrix A in 
where 
. The collection of all real Möbius transformations for which 
takes the values 1 forms a group which can be identified with 
. We write 
to mean that f is a random variable in
. We can g...
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, the probability 
is greater than 0.027282. (2) If f is hyperbolic transformation and g is parabolic transformation, then the probability 
is discrete is greater than 7/12. (3) If f is elliptic of order n and g is elliptic of order 2, then the probability