Introduction

Most of the growth curves are described by linear, power, parabolic, power-exponential, logistic, log-logistic, von Bertalanffy, Gompertz, and Richards curves; see [1], [2], [3], [4], [5], and [6] for the tumor, [7] for the human fetus, [8] for the human life. A recent research article [8] related with a human life modeled by Gompertz and Mirror Gompertz differential equations are (1) and (2) respectively, where β is a positive parameter. The well-known logistic equation in the literature, see [9] is given by (3) where k is the proportionality constant and K is the carrying capacity.

In this study, mathematical modeling is applied to the Pseudomonas putida and mammary tumor datas given in [10, 11], respectively. Note that Pseudomonas putida is a bacterium found in most soil and water habitats, and is significant to the environment due to its complex metabolism and ability to control pollution, [12] and [13]. We model their growth patterns by continuous and discrete Gompertz and Logistic curves. To achieve our goal, we derive 4-parameter and 3-parameter Gompertz and Logistic dynamic equations. We first propose two types of Gompertz dynamic equations: The first type Gompertz dynamic equations are motivated by [14]. We contribute two first type continuous Gompertz curves to the literature. All of the discrete Gompertz curves in this type are new. 4-parameter second type Gompertz dynamic equations are motivated by [2] in which only 3-parameter discrete Gompertz curves are considered. 3-parameter second type continuous Gompertz are investigated earlier in [10]. Inspired by [15], we come up with 4-parameter Logistic dynamic equations while 3-parameter Logistic dynamic equations are constructed earlier in [16]. 4-parameter Logistic discrete curves are new. To establish both dynamic equations, we use the variation of constant formulas together with the circle dot multiplication and the circle minus substraction on time scales. We refer readers to [17] and [18] by Bohner and Peterson for the theory of time scales calculus.

The parameters of these models are estimated by NonlinearModelFit function of Wolfram Mathematica 11.0 applying Monte Carlo simulation and our comparison is based on outputs following from the p-values of parameters, adjusted R-squared, and RMSE (root mean square error), RRMSE (Relative Root Mean Square Error), MAPE (Mean Absolute Percent Error), MAE (mean absolute error), U1 (Theil inequality coefficient, Theil’s U1), and U2 (Theil inequality coefficient, Theil’s U2). We use the Mathematica 11 program for the goodness of fit test of the models. Having at least three small values of each determined statistical criterion, the p value less than 0.05 for each parameter, and adjusted R-squared value close to 1 show better performance in terms of goodness of fit.

Outline of this paper is as follows: In Section 2, we introduce the time scales calculus together with some preliminary results. Sections 3 and 4 are related with first and second type Gompertz dynamic equations. In each section we obtain 4-parameter and 3-parameter continuous and discrete Gompertz curves. In Section 6, Logistic dynamic equations are introduced and we explicitly calculate 4-parameter and 3-parameter continuous and discrete Logistic curves. In the last section, we discuss how Gompertz and Logistic curves fit the growth of Pseudomonas putida and mammary tumor and include our conclusion.

Preliminary results

A time scale, , is an arbitrary nonempty closed subset of the real numbers . The theory of time scales is to introduce a new calculus so that we can unify the continuous and discrete analysis. Here, we give basic definitions and some essential results without proofs. Nevertheless, we mainly refer readers two books [17] and [18] by Bohner and Peterson and the manuscript [16] by Akin-Bohner and Bohner.

The forward jump operator σ on is defined as for all . For this definition we also have . The backward jump operator is defined by for all . Here, we have . If σ(t) > t, we say t is right-scattered, while if ρ(t) < t we say t is left-scattered. If σ(t) = t, we say t is right-dense, while if ρ(t) = t we say t is left-dense. The graininess function is defined by μ(t): = σ(t) − t. It is apparent that for σ(t) = t + 1, ρ(t) = t − 1 and for σ(t) = t, ρ(t) = t. The set is derived from If has left-scattered maximum m, then . Otherwise, . The following notations are also useful: f^σ(t) = f(σ(t)). Note that .

Assume and let , then we define f^Δ(t) to be the number (provided it exists) with the property that given any ϵ > 0, there is a neighborhood U of t such that for all s ∈ U. f^Δ(t) is called the delta derivative of f(t) at t. Note that the delta-derivative turns out to be the usual derivative when while it is the forward difference operator when . If f is differentiable at t, then f is continuous at t. If f is continuous at t and t is right- scattered, then f is differentiable at t with If f is differentiable and t is right-dense, then If f is differentiable at t, then (4) If are differentiable at , then the product is also differentiable at t with

If f is continuous at each right-dense point and whenever is left-dense exists as a finite number, then we say that is rd-continuous. A function is called an antiderivative of provided F^Δ(t) = f(t) holds for all . In this case, we define the integral of f by (5) If 1 + μ(t)p(t) ≠ 0 for all , is called regressive. The set of all regressive and rd-continuous functions is denoted by R. If 1 + μ(t)p(t) > 0 for all , is called positively regressive. The set of all positively regressive and rd-continuous functions is denoted by R⁺.

If p, q ∈ R and α is a constant, then we define (6) and for all . Finding a simple formula of the derivative of any power of a function yields to the introduction of a circle dot multiplication. A circle dot multiplication ⊙ is defined in [16] as

Note that ⊖p = −p, p ⊕ q = p + q and α ⊙ p = αp for the continuous case. If p is regressive, then we define the exponential function by (7) where is the cylinder transformation such that ξ₀(z) = z. If is rd-continuous and regressive, then the exponential function e_p(t, t₀) is the unique solution of the IVP on for each fixed For data analysis we need to calculate exponential functions (8) (9) for a regressive constant β, see Table 2.4 in [17].

We use the following properties of the exponential function e_p(t, s), .

Theorem 0.1. If p, q are regressive and , then

1. e_p(t, t) ≡ and e₀(t, s) ≡ 1;
2. e_p(σ(t), s) = (1 + μ(t)p(t))e_p(t, s);
3. ;
4. ;
5. e_p(t, s)e_q(t, s) = e_p⊕q(t, s);
6. if p > 0 for all , then e_p(t, t₀) > 0 for all ;
7. if p ∈ R⁺, then e_p(t, t₀) > 0 for all .

In addition, two of the useful formulas for a circle dot are (10) and (11) where p is a regressive function and α is a constant, see [16].

The followings are the variation of constants formulas, see Theorems 2.74 and 2.77 in [17]. The equation (12) is called regressive if x^Δ = p(t)x is regressive (i.e., p is regressive) and is rd-continuous.

Theorem 0.2. Suppose (12) is regressive. Let and . The unique solution of the IVP is given by

Theorem 0.3. Suppose (12) is regressive. Let and . The unique solution of the IVP is given by

First type gompertz dynamic equations

In this section, we will introduce Gompertz dynamic curves motivated by the 4-parameter Gompertz curve (13) given in [19] for the growth curve analyses of bacterial counts. Here, K can be found as the growth rate coefficient, t₀ is the initial time, A + B is the carrying capacity of the environment for the population. To explain the carrying capacity notion we can say that every environment has its own limits, therefore it is impossible for species to grow up infinitely. Thus, the number of the population should be finite.

In order to obtain the Gompertz model in the continuous case, we differentiate Eq (13) and obtain In addition, note that we have (14) on , where we use Theorem 0.1 and (10). Since e_⊖K(t, t₀) = e^{−K(t−t₀)} for , and (14) holds, then we obtain (15) Motivated by the calculation above, we have the following initial value problem modeling 4-parameter Gompertz curve on time scales.

Theorem 0.4. The initial value problem (16) has the solution of the form , where K is the growth rate and t₀ is the initial time, ω₀ is the value of the function at the initial time and B is the coefficient that has an impact on carrying capacity.

Proof. We notice that the positivity of K implies the positivity of by Theorem 0.1. Since , we have the positively regressivity of ⊖e_⊖K. Since 1 + μ(K⊙⊖e_⊖K) = (1 + μ(⊖e_⊖K))^K > 0 by (11), the dynamic equation in the IVP (16) is regressive. Therefore, we apply Theorem 0.3 and obtain the unique solution for (17) where we use (14) and Theorem 0.1.

Example 0.5. Let . Then the continuous Gompertz curve (18) is obtained from (17) for by using Eqs (8) and (15). This is compatible with the continuous Gompertz growth curve (13) by taking A = e(ω₀ − B) in (13).

Example 0.6. Let . Since e_⊖K(t, t₀) = (1 + K)^{−(t−t₀)} for by (9), (14) yields and so (19) for . Thus, the discrete Gompertz growth curve (20) again follows from (17) for .

Motivated by the first variation of constant formula, Theorem 0.2, we derive another Gompertz curve on time scales.

Theorem 0.7. The initial value problem (21) has the solution of the form (22) for , where K is the decay rate coefficient and regeressive, t₀ is the initial time, ω₀ is the value of the function at the initial time and B is the coefficient that has an impact on carrying capacity.

Proof. Since K is regressive, e_⊖K is also regressive by (7). The dynamic equation in the IVP (21) is regressive. Then, in order to obtain the unique solution (22) we apply Theorem 0.3 for where we use (14) and Theorem 0.1.

Example 0.8. Let . Then the alternative continuous Gompertz curve (23) is obtained from (22) for . It is worth to mention that Eq (23) is a new Gompertz curve in the continuous case.

Example 0.9. Let . Then using (19), we have for . Since the alternative discrete Gompertz growth curve (24) again follows from (22) for .

The Gompertz growth curve (23) is given as 4-parameter Gompertz growth curve in [19]. From this point of view, the 3-parameter Gompertz growth curve on time scales (25) is obtained from (17) when B = 0, and so the 3-parameter continuous and discrete Gompertz curves are (26) for and for (27) respectively. From (22) when B = 0, the alternative 3-parameter continuous and discrete Gompertz curves (28) and (29) are gained to the literature, respectively.

Second type gompertz dynamic equations

It is clear that (13) is not a Gompertz model when the dependent variable is log-transformed. In this subsection, we will derive Gompertz dynamic equations involving logarithmic functions. This idea of the derivation of Gompertz dynamic equations is inspired from the Gompertz difference equation (39) where a is taken as the growth rate and b is taken as the exponential rate of growth deceleration, which was firstly given by Bassukas et. al. [3]. The equivalent form of (39) is given by (40) and so the continuous version becomes (41) Unifying (40) and (41), we end up by (42) By the second variation of constants formula, Theorem 0.3, we get the alternative dynamic equation (43) Notice that Eq (42) turns out to be (39) when while (43) turns out to be (44) for a nonzero constant b and when we obtain the following Gompertz differential equation from the Gompertz dynamic Eq (43) as (45) which is equivalent to Gompertz differential Eq (1) when a = 0, b − 1 = β, and G = x in (45). In [8], Gompertz differential Eq (41) with a = 0 is called the Mirror Gompertz differential equation, and is equivalent to (2) when a = 0, b − 1 = β, and G = 1 − x in (41).

From now on, we take a = α and b − 1 = β in (42) and (43) and assume (46) where g₀ is a real number. The following theorems yield the second type Gompertz dynamic curves.

Theorem 0.12. Suppose that β is regressive and α > 0. Then the solution of the IVP (42)–(46) is given by (47) for .

Proof. If ln G(t) is taken as u(t), then the IVP (42)–(46) becomes u^Δ(t) = α + βu(t), u(t₀) = g₀, . Then by Theorems 0.1 and 0.2, we obtain

Since G = e^u, (47) is obtained as the solution of the IVP (42)–(46).

Theorem 0.13. Suppose β is regressive and α > 0. Then the solution of the IVP (43)–(46) is given by (48) for .

Proof. If ln G(t) is taken as u(t), then the IVP (43)–(46) turns out to be u^Δ(t) = α−βu^σ(t), u(t₀) = g₀, . Again, Theorems 0.1 and 0.3 yield for . Since G = u, (48) is obtained as the solution of the IVP (43)–(46).

Example 0.14. Let . Then the solutions of the IVPs (42)–(46) and (43)–(46) are (49) and (50) respectively for t ∈ [t₀, ∞) and here we use (8).

Example 0.15. Let . Then the solutions of the IVPs (42)–(46) and (43)–(46) are (51) and (52) respectively for and here we use (9).

In (51) by taking t₀ = 0, β = b − 1 and α = a Equation 3.1 is obtained in [11]. Both continuous Gompertz growth curves (18) and (50) are obtained from the IVPs (13) and (43)–(46). If we let B = 0, K = β in (18), and in (50), we observe that which implies Similarly, the continuous Gompertz curves (23) and (49) are the solutions of the IVPs (21) and (42)–(46). If we let B = 0, β = −K in (23), and in (49), we observe that which implies From these observations, we conclude the 3-parameter first type continuous Gompertz curve and the second type continuous Gompertz curve are identical. However, since such an intimate relation among the discrete curves cannot be observed, considering first and second type discrete Gompertz curves contributes to the literature.

Logistic dynamic equations

In this section, we derive 4-parameter and 3-parameter Logistic continuous and discrete curves from Logistic dynamic equations.

Goodness-of-fit test for gompertz and logistic curves and conclusion

The main aim of this study for the statistical analysis of Gompertz and Logistic curves is to determine whether their equations are able to model Pseudomonas Putita and tumor data sets given in [10] and [11]. In order to achieve our goal, p-values of the parameters, adjusted R-squared values and six types of errors, namely, RMSE (root mean square error), RRMSE (Relative Root Mean Square Error), MAPE (Mean Absolute Percent Error), MAE (mean absolute error), U1 (Theil inequality coefficient, Theil’s U1), U2 (Theil inequality coefficient, Theil’s U2) for each data set calculated, where

Here, e_t shows the error component, y_t the original time series values, the estimated values of the time series, and T the number of observations of the series. The criteria to determine an equation showing better performance in terms of goodness of fit is to have statistically significant coefficients; in other words, meaningful p-values for each coefficient, the larger adjusted R-squared value and smaller errors (see S1 and S2 Figs). The coefficient estimates of these models are obtained by the Mathematica 11.0 and the Wolfram Language uses “ConjugateGradient”, “Gradient”, “LevenbergMarquardt”, “Newton”, “NMinimize”, and “QuasiNewton” methods.

The curves which successfully model Pseudomonas putita data are 4-parameter first type continuous Gompertz curve (18), 3-parameter first type Gompertz curves (26), (27), (29), Zwietering modification of continuous Gompertz curve (31), Gompertz-Liard curves (35), (38), and 2-parameter second type Gompertz curves 49α0, 50α0, 51α0 and 52α0 that are obtained from (49), (50), (51), (52) with t₀ = 1 and α = 0. According to the results in S1 Fig Eq (18) has the best fit among Gompertz curves. Therefore, we observe that the performance of 4-parameter Gompertz curves for bacteria is better than 3 and 2-parameter Gompertz growth curves. In addition to these, Eqs (27), (29) and (38) are the 3- parameter discrete Gompertz growth curves are new, thus, contribute to the literature.

When we concentrate attention on the Logistic type growth curves, from S1 Fig, it is apparent that all of the Logistic type equations are successful in modeling the bacteria data set. In addition, growth curves (70), (78), (80), (81) are the new 4-parameter discrete Logistic curves that are obtained in this study. According the results in S1 Fig, among the Logistic type growth curves, 4-parameter Logistic growth curves are better in modeling when it is compared with 3- parameter ones.

Moreover, we can infer that the 4-parameter continuous Logistic curve (63) and (70), (78) model better than Eq (18). Thus, Pseudomonas Putita data is modeled by Logistic type equations better than Gompertz type equations. In addition, Eq (63) highlighted with orange, (70) and (78) highlighted with green in S1 Fig, have the smallest errors among the other Logistic equations. Eqs (70) and (78) have smaller errors after the eighth decimal when they are compared with Eq (63) highlighted yellow in S1 Fig. Therefore, new discrete growth curves (70) and (78) are the best in modeling bacteria data.

Eq (29) highlighted with yellow, (35) highlighted with green, 49α0, 50α0, 51α0 and 52α0 highlighted with orange in S2 Fig are the curves that successfully model the tumor data among the Gompertz curves. Eq (35) has the best fit in modeling based on S2 Fig. This equation is the Gompertz Liard continuous equation that was developed for tumor modeling in the literature, so our result is compatible with the one in [14]. On the other hand, 3-parameter discrete Gompertz growth curve (29) also models the tumor data set and this equation is developed in this study as well. In addition, 4-parameter continuous and discrete second type Gompertz curves (51) and (52) are also new. 2-parameter version 51a0 of (51) was studied in [8] as Mirror Gompertz curve. Nevertheless, its discrete version 52α0 is a new contribution to the literature. At this point, we declare that 3-parameter Gompertz curves are more successful in modeling tumor data than 4-parameter Gompertz curves. Although the errors of the Logistic type curves are smaller than the errors of Gompertz type curves, all of their parameters are not statistically significant. Therefore, the Gompertz-Liard curve (35) is the best curve in modeling when it is compared with all the other curves and so one can say that Gompertz type curves have better fitting than Logistic type curves for tumor data.

As a result, Logistic curves are better in modeling bacteria data whereas tumor data is modeled better by Gompertz curves. Increasing the number of parameters of Logistic curves give favorable results for bacteria data while decreasing the number of the parameters of Gompertz curves for tumor data turns out to be reasonable. S3 Fig gives us the curve fittings of 4-parameter discrete Logistic curve (70) and 3-parameter discrete Gompertz curve (29) for bacteria data set and also shows the importance of the number of parameters.

Supporting information