Int. J. Aquat. Biol. (2021) 9(3): 214-216 ISSN: 2322-5270; P-ISSN: 2383-0956 Journal homepage: www.ij-aquaticbiology.com © 2021 Iranian Society of Ichthyology Original Article Generalized modified Levakovic growth function for aquatic species Deniz Ünal* 1 Department of Statistics, Faculty of Arts and Sciences, Çukurova University, 01330 Adana, Turkey. Article history: Received 24 April 2021 Accepted 22 June 2021 Available online 2 5 June 2021 Keywords: Aquatic Species, Curve fitting, Growth function, Fisheries, Lag time. Abstract: Growth is a concept that includes social, economic and physical sub-processes and it also shows itself in the field of fisheries. It is extremely important to obtain the appropriate mathematical model for growth for aquatic species. The growth modelling for the aquatic species contains the phenomena of bio-diversity, formation and population dynamics of aquatic species or stock estimation for the creature. And this study aims to design and implement a new mathematical model for growth process for aquatic species as Generalized Modified Levakovic Growth (GLM) model. Introduction Mathematical growth models are used to predict the growth of biological systems and intraspecific population dynamics. Growth functions are simple, nonlinear mathematical equations that describe the change in the size of an individual or population over time. In some cases, simple exponential growth functions can model the growth. Growth curves serve to express the development of growth parameters over time mathematically. It is unthinkable to show an ever-increasing trend for the concept of growth; because nothing can be expected to increase continuously (Bertalanffy, 1938). In the simplest terms, even the population growth has a certain saturation level, so called "carrying capacity". This value assumes a numerical upper bound role for growth size. Von Bertalanffy (1938) proposed the first important research on the mathematical expressions of the size of the organism. In addition, the Richards model (Richards, 1959), obtained by the modification of the von Bertalanffy model, has found a wide range of application. Lifeng et al. (1998) was implemented Richards model to various species such as Amoeba proteus, rice, fish, cattle and deer. Later, Brich (1999) defined a new generalized logistic model and *Correspondence: Deniz Ünal DOI: https://doi.org/10.22034/ijab.v9i3.1271 E-mail: dunal@cu.edu.tr compared it to Richards model. To compare and fit the models, Brich (1999) applied the real data set which was also used in the Richards study. The use of the growth models is not only limited to the field of biology. They can be applied to many other fields such as determining the market volume in economics (Fisher and Fry, 1971; Fingleton and López‐Bazo, 2006). Levakovic (1935) model, one of the functions used in the modeling of growth, is actually the generalized form of the Hossfeld (Woollons et al, 1990) model (Lee, 2000). Since the Levakovic model is not very useful in practice, this function is generally used unified with other growth models (Bontemps and Duplat, 2012). The model known as Levakovic I is known as the best model among 4-parameter models (Zeide, 1993). Based on above-mentioned background, in the first part of this study, construction of a new growth model is defined by inspiring Levakovic function called as Generalized Modified Levakovic Growth (GLM) model to apply for aquatics. In the second section, the prominent features of GLM model is discussed and finally conclusion section is given. (1) Construction of generalized modified Levakovic growth function 215 Int. J. Aquat. Biol. (2021) 9(3): 214-216 Statistical modeling with growth data creates a more efficient way to understand biological progress. In this section, a new growth model is constructed as Generalized Modified Levakovic (GLM) growth model. The relative growth rate: The relative growth rate is calculated as: 𝑧𝑧(𝑡𝑡) = 𝑑𝑑𝑑𝑑(𝑡𝑡) 𝑑𝑑𝑡𝑡 𝑦𝑦(𝑡𝑡) = d dt lnt, 𝑧𝑧(𝑡𝑡) > 0, and we define a new relative rate for growth as follows: 𝑧𝑧(𝑡𝑡) = c a b 𝑒𝑒−𝑎𝑎𝑡𝑡 (1 − 𝑏𝑏 𝑒𝑒−𝑎𝑎𝑡𝑡)−1 … … … … … (1) Where 𝑦𝑦∞ = lim𝑡𝑡→∞ 𝑦𝑦(𝑡𝑡) and a, b, c are any real numbers with a, b, c ≥ 0, t is time variable, 𝑡𝑡 ≥ 0 and 𝑦𝑦(𝑡𝑡) is current size at time 𝑡𝑡. Thus: 𝑑𝑑𝑦𝑦(𝑡𝑡) 𝑑𝑑𝑡𝑡 𝑦𝑦(𝑡𝑡) = c a b 𝑒𝑒−𝑎𝑎𝑡𝑡 (1 − 𝑏𝑏 𝑒𝑒−𝑎𝑎𝑡𝑡)−1 … … … … (2) From now on, it is easy to define the instant rate 𝑦𝑦(𝑡𝑡). To obtain the 𝑦𝑦(𝑡𝑡) function which gives the current size at time 𝑡𝑡, the integration of the Equation 2 should be find. � 𝑑𝑑𝑦𝑦(𝑡𝑡) 𝑑𝑑𝑡𝑡 𝑦𝑦(𝑡𝑡) 𝑑𝑑𝑡𝑡 = � c a b 𝑒𝑒−𝑎𝑎𝑡𝑡 (1 − 𝑏𝑏 𝑒𝑒−𝑎𝑎𝑡𝑡)−1 𝑑𝑑𝑡𝑡 … (3) By using the change of variable technique, the integral in the Equation 3 can be evaluated as: 𝑢𝑢 = 1 − 𝑏𝑏 𝑒𝑒−𝑎𝑎𝑡𝑡 → 𝑑𝑑𝑢𝑢 = 𝑎𝑎 𝑏𝑏 𝑒𝑒−𝑎𝑎𝑡𝑡 𝑑𝑑𝑡𝑡 ln 𝑦𝑦(𝑡𝑡) = 𝑙𝑙𝑙𝑙(1 − 𝑏𝑏 𝑒𝑒−𝑎𝑎𝑡𝑡)𝑐𝑐 + 𝑙𝑙𝑙𝑙 𝑐𝑐1 Where 𝑐𝑐1 is an integration constant. Therefore, the formula for GLM growth function is: 𝑦𝑦(𝑡𝑡) = 𝑐𝑐1 (1 − 𝑏𝑏 𝑒𝑒−𝑎𝑎𝑡𝑡)𝑐𝑐 … … … … … (4) To see the 𝑐𝑐1, let us analyze this equation as 𝑡𝑡 tends to infinity: 𝑌𝑌∞ = lim𝑡𝑡→∞ 𝑦𝑦(𝑡𝑡)= 𝑐𝑐1 is an asymptote, i.e. this number gives the carrying capacity. Since 𝑐𝑐1 = 𝑌𝑌∞, the Equation 4 should be arranged as in Equation 5. Then, the general formula for GLM growth function would be: 𝑦𝑦(𝑡𝑡) = 𝑌𝑌∞ (1 − 𝑏𝑏 𝑒𝑒−𝑎𝑎𝑡𝑡)𝑐𝑐 … … … … … (5) Growth rate: Using Equations 2 and 5, we define growth rate for the GLM growth function as follows: 𝑑𝑑𝑦𝑦(𝑡𝑡) 𝑑𝑑𝑡𝑡 = 𝑦𝑦∞c a b 𝑒𝑒−𝑎𝑎𝑡𝑡(1 − 𝑏𝑏 𝑒𝑒−𝑎𝑎𝑡𝑡)𝑐𝑐−1 … (6) Where 𝑦𝑦∞ = lim𝑡𝑡→∞ 𝑦𝑦(𝑡𝑡), a, b, c are any real numbers with a, b, c, t≥0 , where t is time variable and 𝑦𝑦(𝑡𝑡) is current size at time 𝑡𝑡. (2) Properties of generalized modified Levakovic growth function To obtain some identifier features of the GLM model, the initial value of GLM model is: 𝑦𝑦(0) = 𝑌𝑌∞ (1 − 𝑏𝑏 )𝑐𝑐 Initial behavior of the growth rate can be reached by taking zero in the first derivative of GLM model that is: 𝑦𝑦′(𝑡𝑡) = 𝑑𝑑𝑦𝑦(𝑡𝑡) 𝑑𝑑𝑡𝑡 � 𝑡𝑡=0 = 𝑦𝑦∞c a b (1 − 𝑏𝑏 )𝑐𝑐−1 as the initial behavior of the GLM function. Another important feature for a growth curve is the critical points that can be obtained by reaching the points where the first derivative is equal to zero. Therefore, we should take dy(t)/dt = 0. 𝑑𝑑𝑦𝑦(𝑡𝑡) 𝑑𝑑𝑡𝑡 = 𝑦𝑦∞c a b 𝑒𝑒−𝑎𝑎𝑡𝑡(1 − 𝑏𝑏 𝑒𝑒−𝑎𝑎𝑡𝑡)𝑐𝑐−1 = 0 By exponential transformation, tcritical=ln b1/a can be calculated. Then, the value of the GLM growth model at critical point would be 𝑦𝑦(𝑡𝑡𝑐𝑐𝑐𝑐𝑐𝑐𝑡𝑡𝑐𝑐𝑐𝑐𝑎𝑎𝑐𝑐) = 0. Inflection point: Inflection points are the points where the graph of a function changes curvature i.e. the point where the curve turns to convex from concave or vice versa. Therefore, it is one of the most important properties that gives information about the structure of a curve. Finding these points is possible by setting the second derivative to zero and often requires a rather tedious calculation process. In this section, the inflection point of the GLM function is calculated. 𝑑𝑑2𝑦𝑦(𝑡𝑡) 𝑑𝑑𝑡𝑡2 = −𝑦𝑦∞c 𝑎𝑎2 b 𝑒𝑒−𝑎𝑎𝑡𝑡(1 − 𝑏𝑏 𝑒𝑒−𝑎𝑎𝑡𝑡)𝑐𝑐−1[1 + 𝑏𝑏 (1 − 𝑐𝑐)𝑒𝑒−𝑎𝑎𝑡𝑡(1 − 𝑏𝑏 𝑒𝑒−𝑎𝑎𝑡𝑡)−1] … (7) For 𝑡𝑡1 = 𝑙𝑙𝑙𝑙(𝑏𝑏𝑐𝑐)1/𝑎𝑎 or 𝑡𝑡2 = ln𝑏𝑏1/𝑎𝑎 second derivative given in Equation 7 is equal to 0, i.e. 𝑑𝑑2𝑦𝑦(𝑡𝑡) 𝑑𝑑𝑡𝑡2 =0 for 𝑡𝑡1 and 𝑡𝑡2. Population size at inflection point: Notice that 𝑡𝑡2 = ln𝑏𝑏1/𝑎𝑎 is the critical value for GLM growth function, therefore by consideration of 𝑡𝑡1 = 𝑙𝑙𝑙𝑙(𝑏𝑏𝑐𝑐)1/𝑎𝑎, we can calculate y(t) at inflection point 𝑡𝑡1 = 𝑙𝑙𝑙𝑙(𝑏𝑏𝑐𝑐)1/𝑎𝑎as follows: y(𝑡𝑡1) = 𝑌𝑌∞ (1 − 1 𝑐𝑐 )𝑐𝑐 Which gives the population size at point of inflection. And population at inflection point is: y(𝑡𝑡1) = 𝑌𝑌∞ (1 − 1 𝑐𝑐 )𝑐𝑐 … … … … … (8) 216 Ünal / Generalized modified Levakovic growth function for aquatic species The maximum specific growth rate: Using the value of 𝑡𝑡1 = 𝑙𝑙𝑙𝑙(𝑏𝑏𝑐𝑐)1/𝑎𝑎 in the first derivative of y(𝑡𝑡), that is calculating the following equation, y′(𝑡𝑡1) = 𝑌𝑌∞ 𝑎𝑎 �1 − 1 𝑐𝑐 � 𝑐𝑐−1 … … … … … (9) The slope of the curve at inflection point 𝑡𝑡1 = 𝑙𝑙𝑙𝑙(𝑏𝑏𝑐𝑐)1/𝑎𝑎 can be obtained. Lag Time: The points where the growth reaches its maximum and stops the lag time, denoted by λ. This point is the x-intercept of the tangent curve at the inflection point. Again, by considering 𝑡𝑡1 = 𝑙𝑙𝑙𝑙(𝑏𝑏𝑐𝑐)1/𝑎𝑎 as an inflection point, the lag time λ can be reached by writing tangent equation about (t1, y(t1)) point. Then using the Equations 8 and 9, the tangent equation would be as: y- y(𝑡𝑡𝑐𝑐𝑖𝑖𝑖𝑖)= 𝑦𝑦′(𝑡𝑡𝑐𝑐𝑖𝑖𝑖𝑖). (𝑡𝑡 -𝑡𝑡𝑐𝑐𝑖𝑖𝑖𝑖) At (λ, 0), this equation gives the lag time as: 0 − y(𝑡𝑡1)= 𝑦𝑦′(𝑡𝑡1). (λ -𝑡𝑡1) −𝑌𝑌∞ (1 − 1 𝑐𝑐 )𝑐𝑐 = 𝑌𝑌∞ 𝑎𝑎 (1 − 1 𝑐𝑐 )𝑐𝑐−1 (λ − 𝑙𝑙𝑙𝑙(𝑏𝑏𝑐𝑐)1/𝑎𝑎) And by performing some manipulations, λ − ln(𝑏𝑏𝑐𝑐) 1 𝑎𝑎 = 𝑐𝑐 − 1 𝑐𝑐 𝑎𝑎 equation can be reached. Then the lag time is as follows: λ = 𝑐𝑐 − 1 + ln(𝑏𝑏𝑐𝑐)𝑐𝑐 𝑐𝑐 𝑎𝑎 Conclusion The time-dependent evolution/variation seen in a population or living thing is expressed by growth functions. There is an enormous amount of empirical and theoretical study relating to the growth. Because, as emphasized in this study, the importance of accurate modeling of growth is known by scientists. The model presented in the study will contribute to study of biological systems particularly aquatics. 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