scholarly journals Reaction networks reveal new links between Gompertz and Verhulst growth functions

BIOMATH ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 1904167 ◽  
Author(s):  
Svetoslav Marinov Markov
Keyword(s):  
Growth Models ◽  
Logistic Models ◽  
Reaction Network ◽  
Measured Data ◽  
Reaction Networks ◽  
Logistic Growth ◽  
Additional Species ◽  
One Dimensional ◽  
Growth Functions ◽  
Intrinsic Mechanism

New reaction network realizations of the Gompertz and logistic growth models are proposed. The proposed reaction networks involve an additional species interpreted as environmental resource. Some natural generalizations and modifications of the Gompertz and the logistic models, induced by the proposed networks, are formulated and discussed. In particular, it is shown that the induced dynamical systems can be reduced to one dimensional differential equations for the growth (resp. decay) species. The reaction network formulation of the proposed models suggest hints for the intrinsic mechanism of the modeled growth process and can be used for analyzing evolutionary measured data when testing various appropriate models, especially when studying growth processes in life sciences.

scholarly journals Mathematical Analysis of Some Reaction Networks Inducing Biological Growth/Decay Functions.

2020 ◽  
Vol 7 (1) ◽  
pp. 14
Author(s):  
Ksenia Ivova Tsocheva
Keyword(s):  
Dynamical Systems ◽  
Growth Models ◽  
Reaction Network ◽  
Gompertz Model ◽  
Reaction Networks ◽  
Additional Species ◽  
Intrinsic Mechanism ◽  
Gompertz Growth Model ◽  
Sigmoidal Growth ◽  
Oscillating Processes

In this work, we study some characteristics of sigmoidal growth/decay functions that are solutions of dynamical systems. In addition, the studied dynamical systems have a realization in terms of reaction networks that are closely related to the Gompertzian and logistic type growth models. Apart from the growing species, the studied reaction networks involve an additional species interpreted as an environmental resource. The reaction network formulation of the proposed models hints for the intrinsic mechanism of the modeled growth process and can be used for analyzing evolutionary measured data when testing various appropriate models, especially when studying growth processes in life sciences. The proposed reaction network realization of Gompertz growth model can be interpreted from the perspective of demographic and socio-economic sciences. The reaction network approach clearly explains the intimate links between the Gompertz model and the Verhulst logistic model. There are shown reversible reactions which complete the already known non-reversible ones. It is also demonstrated that the proposed approach can be applied in oscillating processes and social-science events. The paper is richly illustrated with numerical computations and computer simulations performed by algorithms using the computer algebra system  Mathematica.

scholarly journals A New Class of “Growth Functions” with Polynomial Variable Transfer Generated by Real Reaction Networks

2020 ◽  
Vol 20 (6) ◽  
pp. 74-81
Author(s):  
Nikolay Kyurkchiev
Keyword(s):  
Growth Models ◽  
Reaction Network ◽  
Initial Time ◽  
Reaction Networks ◽  
Time Interval ◽  
Adequate Model ◽  
Specific Data ◽  
Numerical Examples ◽  
New Class ◽  
Growth Functions

AbstractIn [4, 5], two classes of growth models with “exponentially variable transfer” and “correcting amendments of Bateman-Gompertz-Makeham-type” based on a specific extended reaction network have been studied [1]. In this article we will look at the new scheme with “polynomial variable transfer”. The consideration of such a dynamic model in the present article is dictated by our passionate desire to offer an adequate model with which to well approximate specific data in the field of computer viruses propagation, characterized by rapid growth in the initial time interval. Some numerical examples, using CAS Mathematica illustrating our results are given.

scholarly journals Building reaction kinetic models for amiloid fibril growth

BIOMATH ◽  
2016 ◽  
Vol 5 (1) ◽  
pp. 1607311 ◽  
Author(s):  
Svetoslav Marinov Markov
Keyword(s):  
Computer Algebra ◽  
Computer Algebra System ◽  
Reaction Kinetic ◽  
Growth Models ◽  
Reaction Network ◽  
Network Models ◽  
Point Of View ◽  
Logistic Growth ◽  
Sigmoidal Functions ◽  
Methodological Aspects

In this work we  discuss some methodological aspects of the creation and formulation of mathematical  models describing the growth of species from the point of view of reaction kinetics. Our discussion is based on familiar examples of growth models such as logistic growth and enzyme kinetics. We   propose several reaction network  models  for  the amiloid fibrillation processes in the citoplasm. The solutions of the models are sigmoidal functions graphically visualized using  the computer algebra system   Mathematica.

scholarly journals Modeling the Growth of Sudangrass Cultivars at Sowing Times

2019 ◽  
Vol 11 (14) ◽  
pp. 83 ◽  
Author(s):  
Rafael Vieira Pezzini ◽  
Alberto Cargnelutti Filho ◽  
Fernanda Carini ◽  
Cirineu Tolfo Bandeira ◽  
Jéssica Andiara Kleinpaul ◽  
...  
Keyword(s):  
Solar Radiation ◽  
Growth Models ◽  
Logistic Models ◽  
Logistic Growth ◽  
Maximum Acceleration ◽  
Intrinsic Nonlinearity ◽  
Thermal Sum ◽  
Plant Emergence ◽  
The Masses ◽  
Uniformity Trials

This study aimed to fit the Gompertz and Logistic growth models to evaluate the description of fresh and dry masses of shoot as a function of accumulated thermal sum and accumulated solar radiation, to compare the fittings, and to indicate which one best describes the growth of two sudangrass cultivars at four sowing times. Eight uniformity trials were conducted with the sudangrass crop. Five plants were collected from each trial for weighing of fresh and dry shoot masses. These evaluations were carried out three times a week starting from 15 days after plant emergence. The Gompertz and Logistic models were fitted to the masses as a function of accumulated thermal sum and accumulated solar radiation. The parameters and their confidence intervals were estimated. The points of maximum acceleration, inflection, maximum deceleration and asymptotic deceleration, and fit quality indicators were calculated. The intrinsic nonlinearity and the parameter-effects nonlinearity were quantified. The independent variables accumulated thermal sum and accumulated solar radiation can be used to fit the models. Both models satisfactorily describe the growth of fresh and dry shoot masses of cultivars BRS Estribo and CG Farrapo. The Logistic model is more accurate.

scholarly journals Gompertz and Logistic models for morphological traits of sudangrass cultivars during sowing seasons

2019 ◽  
Vol 40 (6Supl3) ◽  
pp. 3399
Author(s):  
Rafael Vieira Pezzini ◽  
Alberto Cargnelutti Filho ◽  
Cláudia Marques de Bem ◽  
Jéssica Maronez de Souza ◽  
Gabriela Görgen Chaves ◽  
...  
Keyword(s):  
Solar Radiation ◽  
Plant Height ◽  
Growth Models ◽  
Logistic Models ◽  
Growth Curves ◽  
Stem Length ◽  
Logistic Growth ◽  
Thermal Sum ◽  
Independent Variable ◽  
Uniformity Trials

The use of mathematical models in the study plant growth allows the identification of phases important to the cultivars and comparison between cultivars of the same species. The objectives of this work were to fit the Gompertz and Logistic growth models for the traits of plant height and stem length as a function of the accumulated thermal sum and accumulated solar radiation, to compare the fittings and the behavior of the sudangrass cultivars and indicate the model that best describes the growth of the cultivars during four sowing seasons. Were conducted eight uniformity trials with sudangrass. At 15 days after emergence, were began the collect and evaluation of five plants from each trial. Were measured plant height and stem length. The models were fitted using the values obtained for the traits of the five plants in each evaluation as a function of the accumulated thermal sum and accumulated solar radiation. Were estimated the parameters, determined their interval of confidence, critical points in the growth curves and quality indicators of the fit. The intrinsic nonlinearities and the parameter effect were also quantified. The accumulated thermal sum and accumulated solar radiation are adequate for the use as an independent variable in the model fitted. Both models were adequate to describe the growth of the traits plant height and stem length of cultivars BRS Estribo and CG Farrapo. However, the Logistic model is more accurate.

scholarly journals Small-Scale Spectrum of a Scalar Field in Water: The Batchelor and Kraichnan Models

2011 ◽  
Vol 41 (11) ◽  
pp. 2155-2167 ◽  
Author(s):  
Xavier Sanchez ◽  
Elena Roget ◽  
Jesus Planella ◽  
Francesc Forcat
Keyword(s):  
Scalar Field ◽  
Thermal Gradient ◽  
Theoretical Models ◽  
Measured Data ◽  
Small Scale ◽  
Temperature Profiles ◽  
One Dimensional ◽  
Kraichnan Model ◽  
The One ◽  
Rate Of Dissipation

Abstract The theoretical models of Batchelor and Kraichnan, which account for the smallest scales of a scalar field passively advected by a turbulent fluid (Prandtl > 1), have been validated using shear and temperature profiles measured with a microstructure profiler in a lake. The value of the rate of dissipation of turbulent kinetic energy ɛ has been computed by fitting the shear spectra to the Panchev and Kesich theoretical model and the one-dimensional spectra of the temperature gradient, once ɛ is known, to the Batchelor and Kraichnan models and from it determining the value of the turbulent parameter q. The goodness of the fit between the spectra corresponding to these models and the measured data shows a very clear dependence on the degree of isotropy, which is estimated by the Cox number. The Kraichnan model adjusts better to the measured data than the Batchelor model, and the values of the turbulent parameter that better fit the experimental data are qB = 4.4 ± 0.8 and qK = 7.9 ± 2.5 for Batchelor and Kraichnan, respectively, when Cox ≥ 50. Once the turbulent parameter is fixed, a comparison of the value of ɛ determined from fitting the thermal gradient spectra to the value obtained after fitting the shear spectra shows that the Kraichnan model gives a very good estimate of the dissipation, which the Batchelor model underestimates.

scholarly journals Uncertainty quantification for quantum chemical models of complex reaction networks

2016 ◽  
Vol 195 ◽  
pp. 497-520 ◽  
Author(s):  
Jonny Proppe ◽  
Tamara Husch ◽  
Gregor N. Simm ◽  
Markus Reiher
Keyword(s):  
Free Energy ◽  
Electronic Structure ◽  
State Space ◽  
Time Scales ◽  
Reaction Network ◽  
Reaction Networks ◽  
Multiple Time Scales ◽  
Complex Reaction ◽  
Discrete State ◽  
Continuous State

For the quantitative understanding of complex chemical reaction mechanisms, it is, in general, necessary to accurately determine the corresponding free energy surface and to solve the resulting continuous-time reaction rate equations for a continuous state space. For a general (complex) reaction network, it is computationally hard to fulfill these two requirements. However, it is possible to approximately address these challenges in a physically consistent way. On the one hand, it may be sufficient to consider approximate free energies if a reliable uncertainty measure can be provided. On the other hand, a highly resolved time evolution may not be necessary to still determine quantitative fluxes in a reaction network if one is interested in specific time scales. In this paper, we present discrete-time kinetic simulations in discrete state space taking free energy uncertainties into account. The method builds upon thermo-chemical data obtained from electronic structure calculations in a condensed-phase model. Our kinetic approach supports the analysis of general reaction networks spanning multiple time scales, which is here demonstrated for the example of the formose reaction. An important application of our approach is the detection of regions in a reaction network which require further investigation, given the uncertainties introduced by both approximate electronic structure methods and kinetic models. Such cases can then be studied in greater detail with more sophisticated first-principles calculations and kinetic simulations.

scholarly journals The shape of density dependence and the relationship between population growth, intraspecific competition and equilibrium population density

2018 ◽  
Author(s):  
Emanuel A. Fronhofer ◽  
Lynn Govaert ◽  
Mary I. O’Connor ◽  
Sebastian J. Schreiber ◽  
Florian Altermatt
Keyword(s):  
Population Growth ◽  
Growth Models ◽  
Population Level ◽  
Logistic Growth ◽  
Population Densities ◽  
Density Regulation ◽  
Logistic Growth Model ◽  
Regulation Functions ◽  
Consumer Resource ◽  
Population Growth Models

AbstractThe logistic growth model is one of the most frequently used formalizations of density dependence affecting population growth, persistence and evolution. Ecological and evolutionary theory and applications to understand population change over time often include this model. However, the assumptions and limitations of this popular model are often not well appreciated.Here, we briefly review past use of the logistic growth model and highlight limitations by deriving population growth models from underlying consumer-resource dynamics. We show that the logistic equation likely is not applicable to many biological systems. Rather, density-regulation functions are usually non-linear and may exhibit convex or both concave and convex curvatures depending on the biology of resources and consumers. In simple cases, the dynamics can be fully described by the continuous-time Beverton-Holt model. More complex consumer dynamics show similarities to a Maynard Smith-Slatkin model.Importantly, we show how population-level parameters, such as intrinsic rates of increase and equilibrium population densities are not independent, as often assumed. Rather, they are functions of the same underlying parameters. The commonly assumed positive relationship between equilibrium population density and competitive ability is typically invalid. As a solution, we propose simple and general relationships between intrinsic rates of increase and equilibrium population densities that capture the essence of different consumer-resource systems.Relating population level models to underlying mechanisms allows us to discuss applications to evolutionary outcomes and how these models depend on environmental conditions, like temperature via metabolic scaling. Finally, we use time-series from microbial food chains to fit population growth models and validate theoretical predictions.Our results show that density-regulation functions need to be chosen carefully as their shapes will depend on the study system’s biology. Importantly, we provide a mechanistic understanding of relationships between model parameters, which has implications for theory and for formulating biologically sound and empirically testable predictions.

Feedback mechanisms in mineral replacement networks: an experimental investigation of the ultramafic model system

2021 ◽  
Author(s):  
Ingvild Aarrestad ◽  
Oliver Plümper ◽  
Desiree Roerdink ◽  
Andreas Beinlich
Keyword(s):  
Reaction Rate ◽  
Energy Transport ◽  
Kinetic Modelling ◽  
Indirect Evidence ◽  
Reaction Network ◽  
Reaction Networks ◽  
Silicate System ◽  
Rock Interaction ◽  
Feedback Mechanisms ◽  
Salinity Effect

<p>The overall rates of multi-component reaction networks are known to be controlled by feedback mechanisms. Feedback mechanisms represent loop systems where the output of the system is conveyed back as input and the system is either accelerated or regulated (positive and negative feedback respectively). In other words, feedback mechanisms control the rate of a reaction network without external influences. Feedback mechanisms are well-studied in a variety of reaction networks (e.g. bio-chemical, atmospheric); however, in fluid-rock interaction systems they are not researched as such. Still, indirect evidence, theoretical considerations and direct observations attest to their existence [e.g. 1, 2, 3]. It remains unknown how mass and energy transport between distinct reaction sites affect the overall reaction rate and outcome through feedback mechanisms. We propose that feedback mechanisms are a missing critical ingredient to understand reaction progress and timescales of fluid-rock interactions. We apply the serpentinization of ultramafic silicates as a relatively simple reaction network to investigate feedback mechanisms during fluid-rock interactions. Recent studies show that theoretical timescale-predictions appear inconsistent with natural observations [e.g. 4, 5]. The ultramafic silicate system is ideal for investigating feedback mechanisms as it is relevant to natural processes, is reactive on timescales that can be explored in the laboratory, and natural peridotite typically consists of less than four phases. Our preliminary observations indicate a feedback between pyroxene dissolution and olivine serpentinization. Olivine serpentinization appears to proceed faster in the presence of pyroxene. Furthermore, the bulk system reaction rate increases with increasing fluid salinity, which is opposite to the salinity effect on the monomineralic olivine system. Dunite (>90% olivine) is rare, which is why it is crucial to explore the more common pyroxene-bearing systems. The salinity effect is important to investigate due to the inevitable increase in fluid salinity from the boiling-induced phase separation and OH-uptake in the formation of serpentine. Here we present preliminary textural and chemical observations, which will subsequently be used for kinetic modelling of feedback.</p><p>[1] Ortoleva P., Merino, E., Moore, C. & Chadam, J. (1987). American Journal of Science <strong>287</strong>, 997-1007.</p><p>[2] Centrella, S., Austrheim, H., & Putnis, A. (2015). Lithos <strong>236–237</strong>, 245–255.</p><p>[3] Nakatani, T. & Nakamura, M. (2016). Geochemistry, Geophysics, Geosystems <strong>17</strong>, 3393-3419.</p><p>[4] Ingebritsen, S. E. & Manning, C. E. (2010). Geofluids <strong>10</strong>, 193-205.</p><p>[5] Beinlich, A., John, T., Vrijmoed, J.C., Tominaga, M., Magna, T. & Podladchikov, Y.Y. (2020). Nature Geoscience <strong>13</strong>, 307–311.</p>

scholarly journals One-dimensional game of life and its growth functions

1992 ◽  
Vol 15 (3) ◽  
pp. 499-508
Author(s):  
Mohammad H. Ahmadi
Keyword(s):  
Power Series ◽  
Rational Function ◽  
Formal Power Series ◽  
Infinite Product ◽  
Growth Function ◽  
The Other ◽  
Arbitrary Integer ◽  
One Dimensional ◽  
Growth Functions ◽  
Formal Power

We start with finitely many1's and possibly some0's in between. Then each entry in the other rows is obtained from the Base2sum of the two numbers diagonally above it in the preceding row. We may formulate the game as follows: Defined1,jrecursively for1, a non-negative integer, andjan arbitrary integer by the rules:d0,j={1     for   j=0,k         (I)0   or   1   for   0<j<kd0,j=0   for   j<0   or   j>k              (II)di+1,j=di,j+1(mod2)   for   i≥0.      (III)Now, if we interpret the number of1's in rowias the coefficientaiof a formal power series, then we obtain a growth function,f(x)=∑i=0∞aixi. It is interesting that there are cases for which this growth function factors into an infinite product of polynomials. Furthermore, we shall show that this power series never represents a rational function.

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