The paper presents an example of a mathematical model for algal cultures. The model presented is based on an elementary growth equation (more generally, evolution or dynamic, because it includes the phenomenon of decreasing population), represented by a first-order differential equation, over which are grafted (according to the influence parameters model, for example used for the USLE modelling of soil erosion is used), as factors of influence, the parameters that influence the evolution of the algae. We have given some examples of simulation of algal evolution processes and the possible applications, advantages and disadvantages of the model are suggested. References are made and conclusions are drawn on the extremely important role played by experiments both in the construction and in the validation of the mathematical models of the living systems (bio-systems) evolution phenomena. The model is easy to transfer and generalize to crop plants, as well as other bio-systems. REZUMAT Lucrarea prezintă un exemplu de model matematic pentru culturile de alge. Modelul prezentat se bazează pe o ecuație elementară de creștere (mai general, evoluție sau dinamică, deoarece include fenomenul de scădere a populației), reprezentat de o ecuație diferențială de ordinul întâi, peste care sunt altoiți (conform modelului parametrilor de influență, folosit spre exemplu, in modelul USLE al eroziunii solului), ca factori de influență, parametrii care influențează evoluția algelor. S-au prezentat câteva exemple de simulare a proceselor de evoluție a algelor și s-au sugerat posibilele aplicații, avantaje și dezavantaje ale modelului. S-a făcut referire și s-au tras concluzii cu privire la rolul extrem de important pe care îl au experimentele atât în construcția, cât și în validarea modelelor matematice ale fenomenelor evoluției sistemelor vii (biosisteme). Modelul este ușor de transferat și generalizat la plantele de cultură, precum și la alte bio-sisteme. INTRODUCTION The mathematical modelling of algal growth process is generally approached very often in the literature. Under these conditions, it's both hard and easy to be original. It is very difficult to fundamentally modify a mathematical model or build a new one. It is easy to operate a simple change, thus obtaining a new model, starting from an old one. In both cases, validation is required. However, the most difficult experimental stage is the previous one, and simultaneously the mathematical model. In the dynamic mathematical modelling of algal crops development, a fairly clear distinction can be made between models that work with a single differential equation of growth, (Stemkovski et.al., 2016; Thornton et al. 2010; Yang JSh et al. 2011; Yang Zh et al., 2017), the rest of the parameters being loads or control functions, and respectively, the models working with a system of differential equations, (Concas et al., 2013; Davidson and Gurney, 1999; Mardlijah et.al., 2017). In the latter case, some of the process's influence parameters, or all, become unknown functions of the differential equations system, but they also have charging and / or elimination components. From an experimental point of view, it is simpler to construct and validate a mathematical model in the first category. When experimental possibilities are lower or at an early stage, a simple model from the first category is therefore more appropriate. For these reasons, the experimental model presented in this paper is part of the first category. The development can be done after validation, gradually, taking into account the large variety of mathematical models existing in the literature. Vol. 57, No. 1 / 2019 92 MATERIALS AND METHODS The results presented in this article are based on hypotheses formulated using data and experimental conclusions by Blinova et.al. (2015) and Nedelcu et.al. (2018 a, b). 1) General Algal Evolution Model The mathematical model presented in this paper is an evolutionary model of the Chlorella vulgaris algae crop. We call it a model of evolution because it is not only a growth mathematical model; it is a model that can also describe the decline of the algae population, including the death of the algae colony. The evolution of the algae population is described by a single ordinary differential equation (Chen Sh et al., 2009; Surendhiran et.al., 2015 and Thornton et al., 2010): ) , ( ) , ( x t x t dt dx (1) Mathematical models of type (1) are frequent in the literature dedicated to the evolution of living matter and originate in the classical growth model that only shows the growth rate in the right side. If the growth rate is strictly positive, then the population grows monotonous. The origin of this model is found in (Malthus, 1798), in the finite form. The logistic form of growth is formulated by Verhulst (1838) and McKendricka & Kesava (1912), for example. Many contemporary mathematical models retain the fundamental forms of growth or modify them, more or less Stemkovski et.al. (2016) and Thornton et al. (20100, for example. The names, meanings, and measurement units of all model parameters are given in Table 1. Table 1 Notation, meanings and units of the mathematical model Notation Meanings Units x algae concentration in solution g/l t time hours 0 x initial value of algae concentration in solution g/l 0 t initial time hours the growth rate of the algae population 1/hours=hours the decline rate of the algae population 1/hours=hours -1 0 model parameter to be determined experimentally 1/hours=hours -1 0 model parameter to be determined experimentally 1/hours=hours -1 average temperature in the algae crop solution Celsius degree co2 time dependence in carbon dioxide concentration in the solution % ph time dependence of pH in the solution cfIL minimum degree of illumination lux cfILb degree of illumination with blue light lux cfILr degree of illumination with red light lux pl wavelength of light nm s time dependence of the solution salinity g/l the function of the temperature influence CO2 the function of influence of the dissolved carbon dioxide concentration in the solution pH the function of the algae crop solution pH L the function of the degree of illumination influence lux N the function of the nutrient concentration in the algae crop solution S the function of the algae crop salinity influence lim natural factor of algae growth moderation at reaching a limit concentration max x maximum algal concentration in solution g/l 1 model parameter to be determined experimentally dimensionless Obviously, the model (1) may become a growth and decrease model, even if only the growth rate is considered, assuming that it can move from negative values to positive values, and possibly return to positive values. 2) The modelling of the process influences by multiplication of separate influences According to (Blinova et.al., 2015 and Nedelcu et.al., 2018a, b), is considered the hypothesis that the physical parameters which influenced the algal evolution, are the following: solution temperature variation, Vol. 57, No. 1 / 2019 93 variation in the amount of dissolved carbon dioxide in the solution, pH variation of the solution, degree of crop illumination, the amount of nutrients introduced into the solution, the salinity of the solution. A second hypothesis we introduce is the structure by of product, of the growth rate: S N L pH CO x x t 2 0 1 ) , ( (2) and of the decrease rate: ) ( 1 ) , ( lim 0 x x pH x t (3) where the form of the moderation factor, by hypothesis, is the following:
MATHEMATICAL MODEL FOR THE EVOLUTION OF Chlorella Algae
P. Cardei,A. Nedelcu,R. Ciupercă,Installations Designed to Agriculture,Romania
Published 2019 in INMATEH Agricultural Engineering
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2019
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INMATEH Agricultural Engineering
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2019-04-30
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Biology, Mathematics, Chemistry, Environmental Science
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