Analysis of random discrete time logistic model

2.1. Random Difference Equations

Basically, using the deterministic difference equations, random difference equations can be obtained in the following three ways. Definition: Let 𝐼 be any sub-interval of real numbers and 𝑓: 𝐼 × 𝐼 → 𝐼 be a function that can be continuously differentiated. For each 𝑥−𝑘 , 𝑥−(𝑘−1) , … , 𝑥0 ∈ 𝐼 initial conditions 𝑥𝑛+1 = 𝑓(𝑥𝑛 , 𝑥𝑛−1 , … , 𝑥𝑛−𝑘 ), 𝑛 = 0,1, … shapes a n-th order difference equation. Using this difference equation, a random difference equation can be constructed i.

With random initial values 𝑥𝑛+1 = 𝑓(𝑥𝑛 , 𝑥𝑛−1 , … , 𝑥𝑛−𝑘 , 𝑛); 𝑛 = 0,1, … ; 𝑥−𝑘 , 𝑥−(𝑘−1) , … , 𝑥0 ∈ 𝐼

Here 𝑥−𝑘 , 𝑥−(𝑘−1) , … , 𝑥0 are random variables. ii. With random non-homogeneous term 𝑥𝑛+1 = 𝑓(𝑥𝑛 , 𝑥𝑛−1 , … , 𝑥𝑛−𝑘 , 𝑛) + 𝑌(𝑛); 𝑛 = 0,1, … ; 𝑥−𝑘 , 𝑥−(𝑘−1) , … , 𝑥0 ∈ 𝐼 using a random 𝑌(𝑡) process and iii. With random coefficients 𝑥𝑛+1 = 𝐴(𝑛)𝑓(𝑥𝑛 , 𝑥𝑛−1 , … , 𝑥𝑛−𝑘 , 𝑛) + 𝑌(𝑛); 𝑛 = 0,1, … ; 𝑥−𝑘 , 𝑥−(𝑘−1) , … , 𝑥0 ∈ 𝐼. Hence, we can make a random difference equation using a random process 𝐴(𝑛) [10-19].

2.2. Random Stochastic Difference Equations

Consider the first order difference equation 𝑥𝑛+1 = 𝐹(𝑛, 𝑥𝑛 ), 𝑛 ∈ 𝑁

It can be used to describe facts that develop in a separate time, where the size of each generation is a function of the previous one. However, real life problems cannot usually be expressed with such a proper mathematical model. Unpredictable effects are included in the model as random variables {𝜉𝑛 } resulting in a stochastic difference equation. 𝑥𝑛+1 = 𝐹(𝑛, 𝑥𝑛 ) + 𝐺(𝑥, 𝑥𝑛 )𝜉𝑛+1 , 𝑛 ∈ 𝑁 (3) The solution of (3) is the discrete time stochastic process of {𝜉𝑛 } compatible with natural filters. Although stochastic difference equations also occur as discretization of stochastic differential equations, analysis of asymptotic properties may be more difficult. Here, both the modeling developed in discrete time and the numerical method analysis for stochastic differential equations has effects [13]. Stochastic difference equations appear in numerical analysis because they are the final product of discretization of stochastic differential equations 𝑑𝑋𝑛 = 𝑓(𝑋(𝑛))𝑑𝑛 + 𝑔(𝑋(𝑛))𝑑𝑊𝑛

where 𝑊𝑛 is a standard Brownian motion. The (4) equation can be discretized by the one-step Euler-Maruyama numerical order. This stochastic difference equation 𝑥𝑛+1 = 𝑥𝑛 + ℎ𝑓(𝑥𝑛 ) + √ℎ𝑔(𝑥𝑛 )𝜉𝑛+1 , 𝑛 = 0,1, …

where {𝜉𝑛 } is the standard normal sequence of random variables, while ℎ is the network dimension [3-9].

3.1. Discrete Uniform Distribution

Definition. Let 𝑘 be a positive bit integer. A random variable 𝑋 with probability function 1⁄𝑘 , 𝑥 = 1,2,3, … , 𝑘 𝑃(𝑥, 𝑘) = { 0, 𝑜𝑡ℎ𝑒𝑟 is called a discrete uniform chance variable [1-2]. Theorem. If 𝑋 has a discrete uniform distribution, then a. 𝐸(𝑋) = b. 𝑉(𝑋) =

3.2. Bernoulli Distribution

Definition. If there are only two results for an 𝑋 random variable, 𝑋 is called a Bernoulli random variable [15]. Bernoulli variables are obtained with the probability function 𝑓(𝑥, 𝑝) = 𝑝 𝑥 (1 − 𝑝)1−𝑥 𝑥 = 0,1 Theorem. If 𝑋 has a Bernoulli distribution, a. 𝐸(𝑋) = 𝑝, b. 𝑉(𝑋) = 𝑝(1 − 𝑝), c. 𝑀𝑥 (𝑡) = 𝑒 𝑡 𝑝 + (1 − 𝑝).

3.3. Binomial Distribution

Definition. Let the total number of those who succeeded in 𝑛 independent Bernoulli trials be the random variable 𝑋. For a single experiment, the probability of success is denoted by 𝑝, and the probability of failure is (1 − 𝑝). The binomial random variable 𝑋 has the following probability function 𝑓(𝑥; 𝑛, 𝑝) = (𝑛𝑥)𝑝 𝑥 (1 − 𝑝)𝑛−𝑥 ; 𝑥 = 0,1,2, … , 𝑛. Calculation of consecutive binomial probabilities, (𝑛−𝑥)𝑝

𝑓(𝑥 + 1; 𝑛, 𝑝) = (𝑥+1)(1−𝑝) 𝑓(𝑥; 𝑛, 𝑝); 𝑥 = 0,1, … , 𝑛 − 1. Theorem. If 𝑋 has a binomial distribution, a. 𝐸(𝑋) = 𝑛𝑝, b. 𝑉(𝑋) = 𝑛𝑝(1 − 𝑝), c. 𝑀𝑥 (𝑡) = [𝑒 𝑡 𝑝 + (1 − 𝑝)]𝑛 .

3.4. Negative Binomial (Pascal) Distribution

Definition. Let 𝑋 be the random variable of the number of trials required to achieve success 𝐾 ≥ 1, with the probability of success 𝑝 in each experiment for independent Bernoulli trials. In this case, 𝑋 is called a negative binomial random variable and its probability function [1-2]: 𝑥−1 𝑓(𝑥) = (𝐾−1 )𝑝𝐾 (1 − 𝑝)𝑥−𝐾 ; 𝑥 = 𝐾, 𝐾 + 1, …

3.5. Geometric Distribution

Definition. The number of experiments done to obtain the first desired result (success or unsuccessful) in a Bernoulli experiment repeated 𝑛 times in succession is called a geometric random variable 𝑋. The distribution of this variable is called the geometric distribution and the probability function of the geometric random variable 𝑋, with probability of unsuccessfulness 𝑞 = 1 − 𝑝 and probability of success 𝑝 in a single experiment [1-2]; 𝑓(𝑥) = 𝑃(𝑋 = 𝑥) = 𝑞 𝑥−1 𝑝; 𝑥 = 1,2,3, … Theorem. If 𝑋 has a geometric distribution, 1

3.6. Hypergeometric Distribution

Definition. Let 𝑎 be the number of elements of a given Type 𝐴 in a mass consisting of a finite number of 𝑁 elements. Let 𝑋 be the number of elements of its type in a sample of 𝑛 units that are randomly drawn without replacing them again. 𝑋 is a random hypergeometric variable and the hypergeometric probability mass function is given as [1-2]; (𝑀)(𝑁−𝑀)

Theorem. If 𝑋 has a hypergeometric distribution, a. 𝐸(𝑋) = b. 𝑉(𝑋) = c.

3.7. Poisson Distribution

𝑥! 𝑦𝑖 function 𝑒 𝑦 and the probability function gives (𝑒 𝑦 = ∑∞ 𝑖=0 𝑖! ):

Theorem. If 𝑋 has a Poisson distribution, a. 𝐸(𝑋) = 𝜆, b. 𝑉(𝑋) = 𝜆, 𝑡 c. 𝑀𝑥 = 𝑒 𝜆(𝑒 −1) .

4.1. Discrete Time Probability Distribution

4.1.1. Uniform Distribution In the random logistic difference equation defined as 𝑋(𝑛 + 1) = 𝑋(𝑛) + 𝑟 ∗ 𝑋(𝑛) ∗ (1 −

if 𝑟 is a random variable with a parameterized uniform distribution, K=1000 and 𝑁 = 50, then the probability characteristics obtained from 105 simulations are given below. Expected Value 1400 E(X) 1200

Figure 1. Expected value of random model In the logistics model process (𝑡 ∈ [0,50]), it is observed that the variability increases in the beginning and then remains stable. The extreme values are shown in the Table (Table 1 and Fig 1). Table 1. Expected value, extreme values and times in random model Variable 𝐸(𝑥(𝑡))

It appears that the expected population reached its highest level at the time of 𝑡 = 50. Therefore, the results obtained from the deterministic model are more likely to be observed differently in an experiment that takes place randomly at these moments. In addition, 𝐸 (𝑥(50)) = 999.834 was obtained for the expected value at the end of the process (𝑡 = 50). Similarly, variations of variances for the model (1) are also seen below (Figure 2).

88.38. max deger

Figure 2. Variance of random model In the logistics model process (𝑡 ∈ [0,50]), it is observed that variability increases first and decreases later. The extreme values are shown in the Table (Table 2 ). Table 2. Extreme values and times of variance in random model Variable 𝑉𝑎𝑟(𝑥(𝑡))

It is observed that the population has reached its highest level of deviation from the average at the time of 𝑡 = 15. Therefore, the results obtained from the deterministic model are more likely to be observed differently in an experiment that takes place randomly at these moments. In addition, at the end of the process, 𝑉𝑎𝑟 (𝑥(15)) = 88.38 was obtained for variance, (𝑡 = 15). Similar to variance, variations in the standard deviation for the (1) model are also seen below (Figure 3). Standard Deviation 9.40106 max deger

As the standard deviation is the square root of the variance, the result is that these two numerical characteristics are expected to exhibit similar behavior. The extreme values for standard deviations are shown below (Table 3). Table 3. Extreme values and times of standard deviation in random model Variable 𝑆𝑡𝑑(𝑥(𝑡))

It is observed that the population has reached its highest level of deviation from the average at the time of 𝑡 = 15. Therefore, the results obtained from the deterministic model are more likely to be observed differently in an experiment that takes place randomly at these moments. In addition, 𝑆𝑡𝑑 (𝑥(15)) = 9.40106 was obtained for variance (𝑡 = 15) at the end of the process. Using the results obtained for the standard deviations and expected values, the variation coefficients for the variables 𝑥 (𝑡) in the random model (1) were also calculated as follows (Figure 4). Coefficient Variation 1.6 VK(X) 1.4 1.2

Figure 4. Variation coefficient of the random model Coefficient of Variation (CV) is calculated by definition as 100 × 𝑠𝑡𝑑 (𝑥 (𝑡)) / 𝐸 (𝑥 (𝑡)) and random r parameters for the installation of model (1) are defined to have %5 coefficient of variation. However, as a result of examining the model, it is seen that the coefficient of variation of 𝑥 (𝑡) variables increased to higher rates. The extreme values of the variation coefficients are given in the table below (Table 4). Table 4. Extreme values and times of the coefficient of variation in the random model Variable 𝐶𝑉(𝑥(𝑡))

Despite the %5 coefficient of variation in the parameters, it is observed that the variation rate of 𝑥 (𝑡) is constantly increasing and reaches %1.5 at 𝑡 = 11. Therefore, it can be interpreted that the variability in random results increases as logistic model and then decreases. The results obtained for the expected values of the model (1) are given below (Figure 5).

The confidence intervals given in the figure are calculated as 𝐶𝐼 = (𝐸 (𝑥 (𝑡)) − 3. 𝑠𝑡𝑑 (𝑥 (𝑡)), 𝐸 (𝑥 (𝑡)) + 3. 𝑠𝑡𝑑 (𝑥 (𝑡))) and three gives the range of variation within the standard deviation. This range includes about %99 of the random variable's values for uniform distribution. Therefore, the extreme values obtained for the expected values in these ranges are given below (Table 5). Table 5. Extreme values and times of confidence in the random model Variable 𝐸(𝑥(𝑡))

At the end of the process, three standard deviation intervals for 𝑥(𝑡) variables are obtained as follows: 𝐸 (𝑥(50)) ∈ (971.63082,1028.03718). Model (1) states that the expectation for this value is 𝐸 (𝑥 (50)) = 999.92, that is, approximately %9.9992, and the expected approximate population is in the range of %99 probability (971.63082,1028.03718) at time 𝑡 = 50. 4.1.2. Binomial Distribution In the random logistic difference equation defined as 𝑋(𝑛 + 1) = 𝑋(𝑛) + 𝑟 ∗ 𝑋(𝑛) ∗ (1 −

if 𝑟 is a random variable with a parameterized binomial distribution (𝑛 = 6) and 𝑁 = 50, then the probability characteristics obtained from 105 simulations are given below.

Figure 6. Expected value of random model In the logistics model process (𝑡 ∈ [0,50]), it is observed that the variability increases first and then remains stable. The extreme values are shown in the Table (Table 6 and Fig 6). Table 6. Expected value, extreme values and times in random model Variable 𝐸(𝑥(𝑡))

It appears that the expected population reached its highest level at the time of 𝑡 = 50. Therefore, the results obtained from the deterministic model are more likely to be observed differently in an experiment that takes place randomly at these moments. In addition, 𝐸 (𝑥(50)) = 999.862 was obtained for the expected value at the end of the process (𝑡 = 50). Similarly, variations of variances for the model (1) are also seen below (Figure 7). Variance 900 Var(X) 800.313 max deger

In the logistics model process (𝑡 ∈ [0,50]), it is observed that variability increases first and decreases later. The extreme values are shown in the Table (Table 7). Table 7. Extreme values and times of variance in random model Variable 𝑉𝑎𝑟(𝑥(𝑡))

It is observed that the population has reached its highest level of deviation from the average at the time of 𝑡 = 15. Therefore, the results obtained from the deterministic model are more likely to be observed differently in an experiment that takes place randomly at these moments. In addition, at the end of the process, 𝑉𝑎𝑟 (𝑥(15)) = 800.313 was obtained for variance (𝑡 = 15). Similar to variance, variations in the standard deviation for the model (1) are also seen below (Figure 8). As the standard deviation is the square root of the variance, the result is that these two numerical characteristics are expected to exhibit similar behavior. The extreme values for standard deviations are shown below (Table 8). Standard Deviation 30 28.2898 max deger

Figure 8. Standard deviation of the random model Table 8. Extreme values and times of standard deviation in random model Variable 𝑆𝑡𝑑(𝑥(𝑡))

It is observed that the population has reached its highest level of deviation from the average at the time of 𝑡 = 15. Therefore, the results obtained from the deterministic model are more likely to be observed differently in an experiment that takes place randomly at these moments. In addition, 𝑆𝑡𝑑 (𝑥(15)) = 28.2898 was obtained for variance (𝑡 = 15) at the end of the process. Using the results obtained for the standard deviations and expected values, the variation coefficients for the variables 𝑥 (𝑡) in the random model (1) were also calculated as follows (Figure 9).

4.74339 max deger of Variation Coefficient 5 VK(X) 4.5 4 3.5

Figure 9. Variation coefficient of the random model Coefficient of Variation (CV) is calculated by definition as 100 × 𝑠𝑡𝑑 (𝑥 (𝑡)) / 𝐸 (𝑥 (𝑡)) and random 𝑟 parameters for the installation of model (1) are defined to have %5 coefficient of variation. However, as a result of examining the model, it is seen that the coefficient of variation of 𝑥 (𝑡) variables increased to higher rates. The extreme values of the variation coefficients are given in the table below (Table 9). Table 9. Extreme values and times of the coefficient of variation in the random model Variable 𝐶𝑉(𝑥(𝑡))

Despite the %5 coefficient of variation in the parameters, it is observed that the variation rate of 𝑥 (𝑡) is constantly increasing and reaches %0.0474 at 𝑡 = 11. Therefore, it can be interpreted that the variability in random results increases as logistic and then decreases. The results obtained for the expected values of the model (1) are given below (Figure 10). Confidence interval 1100 1000.18 max deger

Figure 10. Confidence interval of the expected values of the random model 1280

The confidence intervals given in the figure are calculated as 𝐶𝐼 = (𝐸 (𝑥 (𝑡)) − 3. 𝑠𝑡𝑑 (𝑥 (𝑡)), 𝐸 (𝑥 (𝑡)) + 3. 𝑠𝑡𝑑 (𝑥 (𝑡))) and three gives the range of variation within the standard deviation. This range includes about %99 of the random variable's values for binom distribution. Therefore, the extreme values obtained for the expected values in these ranges are given below (Table 10). Table 10. Extreme values and times of confidence in the random model Variable 𝐸(𝑥(𝑡))

At the end of the process, three standard deviation intervals for 𝑥(𝑡) variables are obtained as follows: 𝐸 (𝑥(39)) ∈ (914.9926,1084.7314). Model (1) states that the expectation for this value is 𝐸 (𝑥 (39)) = 1000.18, that is, approximately %10.0018, and the expected approximate population is in the range of %99 probability (914.9926,1084.7314) at time 𝑡 = 39. 4.1.3. Geometric Distribution In the random logistic difference equation defined as 𝑋(𝑛 + 1) = 𝑋(𝑛) + 𝑟 ∗ 𝑋(𝑛) ∗ (1 −

if 𝑟 is a random variable with a parameterized geometric distribution and 𝑁 = 50, then the probability characteristics obtained from 105 simulations are given below. Expected Value 1400 E(X) 1200

Figure 11. Expected value of random model In the logistics model process (𝑡 ∈ [0,50]), it is observed that the variability increases first and then remains stable. The extreme values are shown in the Table (Table 11 and Fig 11). Table 11. Expected value, extreme values and times in random model Variable 𝐸(𝑥(𝑡))

It appears that the expected population reached its highest level at the time of 𝑡 = 50. Therefore, the results obtained from the deterministic model are more likely to be observed differently in an experiment that takes place randomly at these moments. In addition, 𝐸 (𝑥(50)) = 999.956 was obtained for the expected value at the end of the process(𝑡 = 50). Similarly, variations of variances for the model (1) are also seen below (Figure 12). In the logistics model process (𝑡 ∈ [0,50]), it is observed that variability increases first and decreases later. The extreme values are shown in the Table (Table 12 ). Table 12. Extreme values and times of variance in random model Variable 𝑉𝑎𝑟(𝑥(𝑡))

It is observed that the population has reached its highest level of deviation from the average at the time of 𝑡 = 10. Therefore, the results obtained from the deterministic model are more likely to be observed differently in an experiment that takes place randomly at these moments. In addition, at the end of the process, 𝑉𝑎𝑟 (𝑥(10)) = 25561.9 was obtained for variance (𝑡 = 10). Similar to variance, variations in the standard deviation for the model (1) are also seen below (Figure 13). 4

Figure 13. Standard deviation of the random model As the standard deviation is the square root of the variance, the result is that these two numerical characteristics are expected to exhibit similar behavior. The extreme values for standard deviations are shown below (table 13). Table 13. Extreme values and times of standard deviation in random model Variable 𝑆𝑡𝑑(𝑥(𝑡))

It is observed that the population has reached its highest level of deviation from the average at the time of 𝑡 = 10. Therefore, the results obtained from the deterministic model are more likely to be observed differently in an experiment that takes place randomly at these moments. In addition, 𝑆𝑡𝑑 (𝑥(10)) = 159.881 was obtained for variance (𝑡 = 10) at the end of the process. Using the results obtained for the standard deviations and expected values, the variation coefficients for the variables 𝑥 (𝑡) in the random model (1) were also calculated as follows (Figure 19). Coefficient of Variation (CV) is calculated by definition as 100 × 𝑠𝑡𝑑 (𝑥 (𝑡)) / 𝐸 (𝑥 (𝑡)) and random r parameters for the installation of model (1) are defined to have %5 coefficient of variation. However, as a result of examining the model, it is seen that the coefficient of variation of 𝑥 (𝑡) variables increased to higher rates. The extreme values of the variation coefficients are given in the table below (Table 14).

Figure 14. Variation coefficient of the random model Table 14. Extreme values and times of the coefficient of variation in the random model Variable 𝐶𝑉(𝑥(𝑡))

Despite the %5 coefficient of variation in the parameters, it is observed that the variation rate of 𝑥 (𝑡) is constantly increasing and reaches %0.32 at 𝑡 = 6. Therefore, it can be interpreted that the variability in random results increases as logistic model and then decreases. The results obtained for the expected values of the model (1) are given below (Figure 15). The confidence intervals given in the figure are calculated as 𝐶𝐼 = (𝐸 (𝑥 (𝑡)) − 3. 𝑠𝑡𝑑 (𝑥 (𝑡)), 𝐸 (𝑥 (𝑡)) + 3. 𝑠𝑡𝑑 (𝑥 (𝑡))) and three gives the range of variation within the standard deviation. This range includes about %99 of the random variable's values for geometric distribution. Therefore, the extreme values obtained for the expected values in these ranges are given below (Table 15). Confidence interval 1200

Figure 15. Confidence interval of the expected values of the random model 1284

Table 15. Extreme values and times of confidence in the random model Variable 𝐸(𝑥(𝑡))

At the end of the process, three standard deviation intervals for 𝑥(𝑡) variables are obtained as follows: 𝐸 (𝑥(14)) ∈ (520.313,1479.599). Model (1) states that the expectation for this value is 𝐸 (𝑥 (14)) = 1172.62 that is, approximately %11.7262, and the expected approximate population is in the range of %99 probability (520.313,1479.599) at time 𝑡 = 14. 4.1.4. Hypergeometric Distribution In the random logistic difference equation defined as 𝑋(𝑛 + 1) = 𝑋(𝑛) + 𝑟 ∗ 𝑋(𝑛) ∗ (1 −

if 𝑟 is a random variable with a parameterized hypergeometric distribution and 𝑁 = 50, then the probability characteristics obtained from 105 simulations are given below. In the logistics model process (𝑡 ∈ [0,50]), it is observed that the variability increases first and then remains stable. The end values are shown in the Table (Table 16 and Fig 16). Table 16. Expected value, end values and times in random model Variable 𝐸(𝑥(𝑡))

Figure 16. Expected value of random model It appears that the expected population reached its highest level at the time of 𝑡 = 50. Therefore, the results obtained from the deterministic model are more likely to be observed differently in an experiment that takes place randomly at these moments. In addition, 𝐸 (𝑥(50)) = 999.913 was obtained for the expected value at the end of the process(𝑡 = 50). Similarly, variations of variances for the model (1) are also seen below (Figure 17).

Figure 17. variance of random model In the logistics model process (𝑡 ∈ [0,50]), it is observed that variability increases first and decreases later. The extreme values are shown in the Table (Table 17 ). Table 17. Extreme values and times of variance in random model Variable 𝑉𝑎𝑟(𝑥(𝑡))

It is observed that the population has reached its highest level of deviation from the average at the time of 𝑡 = 14. Therefore, the results obtained from the deterministic model are more likely to be observed differently in an experiment that takes place randomly at these moments. In addition, at the end of the process, 𝑉𝑎𝑟 (𝑥(14)) = 1593.05 was obtained for variance (𝑡 = 14).Similar to variance, variations in the standard deviation for the model (1) are also seen below (Figure 18). Standard Deviation 39.913 max deger

As the standard deviation is the square root of the variance, the result is that these two numerical characteristics are expected to exhibit similar behavior. The end values for standard deviations are shown below (table 18). Table 18. Extreme values and times of standard deviation in random model Variable 𝑆𝑡𝑑(𝑥(𝑡))

It is observed that the population has reached its highest level of deviation from the average at the time of 𝑡 = 14. Therefore, the results obtained from the deterministic model are more likely to be observed differently in an experiment that takes place randomly at these moments. In addition, 𝑆𝑡𝑑 (𝑥(14)) = 39.913 was obtained for variance (𝑡 = 14) at the end of the process. Using the results obtained for the standard deviations and expected values, the variation coefficients for the variables 𝑥 (𝑡) in the random model (1) were also calculated as follows (Figure 19). Coefficient of Variation (CV) is calculated by definition as 100 × 𝑠𝑡𝑑 (𝑥 (𝑡)) / 𝐸 (𝑥 (𝑡)) and random r parameters for the installation of model (1) are defined to have %5 coefficient of variation. However, as a result of examining the model, it is seen that the coefficient of variation of 𝑥 (𝑡) variables increased to higher rates. The extreme values of the variation coefficients are given in the table below (Table 19). Coefficient 6.67337 max deger of Variation

Figure 19. Variation coefficient of the random model Table 19. Extreme values and times of the coefficient of variation in the random model Variable 𝐶𝑉(𝑥(𝑡))

Despite the %5 coefficient of variation in the parameters, it is observed that the variation rate of 𝑥 (𝑡) is constantly increasing and reaches %0.0667 at 𝑡 = 10. Therefore, it can be interpreted that the variability in random results increases as logistic model and then decreases. The results obtained for the expected values of the model (1) are given below (Figure 20).

Figure 20. Confidence interval of the expected values of the random model The confidence intervals given in the figure are calculated as 𝐶𝐼 = (𝐸 (𝑥 (𝑡)) − 3. 𝑠𝑡𝑑 (𝑥 (𝑡)), 𝐸 (𝑥 (𝑡)) + 3. 𝑠𝑡𝑑 (𝑥 (𝑡))) and three gives the range of variation within the standard deviation. This range includes about %99 of the random variable's values for uniform distribution. Therefore, the extreme values obtained for the expected values in these ranges are given below (Table 20). Table 20. Extreme values and times of confidence in the random model Variable 𝐸(𝑥(𝑡))

At the end of the process, three standard deviation intervals for 𝑥(𝑡) variables are obtained as follows: 𝐸 (𝑥(29)) ∈ (880.174,1119.652). Model (1) states that the expectation for this value is 𝐸 (𝑥 (29)) = 1002.82 that is, approximately %10.0282, 𝑎𝑛𝑑 the expected approximate population is in the range of %99 probability (880.174,1119.652) at time 𝑡 = 29. 4.1.5. Poisson distribution In the random logistic difference equation defined as 𝑋(𝑛 + 1) = 𝑋(𝑛) + 𝑟 ∗ 𝑋(𝑛) ∗ (1 −

if 𝑟 is a random variable with a parameterized Poisson distribution and 𝑁 = 50, then the probability characteristics obtained from 105 simulations are given below.

Figure 21. Expected value of random model In the logistics model process (𝑡 ∈ [0,50]), it is observed that the variability increases first and then remains stable. The extreme values are shown in the Table (Table 21 and Fig 21). Table 21. Expected value, end values and times in random model Variable 𝐸(𝑥(𝑡))

It appears that the expected population reached its highest level at the time of 𝑡 = 50. Therefore, the results obtained from the deterministic model are more likely to be observed differently in an experiment that takes place randomly at these moments. In addition, 𝐸 (𝑥(50)) = 999.911 was obtained for the expected value at the end of the process(𝑡 = 50). Similarly, variations of variances for the model (1) are also seen below (Figure 22). Variance 1800

In the logistics model process (𝑡 ∈ [0,50]), it is observed that variability increases first and decreases later. The extreme values are shown in the Table (Table 22 ). Table 22. Extreme values and times of variance in random model Variable 𝑉𝑎𝑟(𝑥(𝑡))

It is observed that the population has reached its highest level of deviation from the average at the time of 𝑡 = 14. Therefore, the results obtained from the deterministic model are more likely to be observed differently in an experiment that takes place randomly at these moments. In addition, at the end of the process, 𝑉𝑎𝑟 (𝑥(14)) = 1757.5 was obtained for variance (𝑡 = 14). Similar to variance, variations in the standard deviation for the model (1) are also seen below (Figure 23). As the standard deviation is the square root of the variance, the result is that these two numerical characteristics are expected to exhibit similar behavior. The end values for standard deviations are shown below (Table 23). Table 23. Extreme values and times of standard deviation in random model Variable 𝑆𝑡𝑑(𝑥(𝑡))

Figure 23. standard deviation of the random model It is observed that the population has reached its highest level of deviation from the average at the time of 𝑡 = 14. Therefore, the results obtained from the deterministic model are more likely to be observed differently in an experiment that takes place randomly at these moments. In addition, 𝑆𝑡𝑑 (𝑥(14)) = 41.9225 was obtained for variance (𝑡 = 14) at the end of the process. Using the results obtained for the standard deviations and expected values, the variation coefficients for the variables 𝑥 (𝑡) in the random model (1) were also calculated as follows (Figure 24). Coefficient of Variation (CV) is calculated by definition as 100 × 𝑠𝑡𝑑 (𝑥 (𝑡)) / 𝐸 (𝑥 (𝑡)) and random r parameters for the installation of model (1) are defined to have %5 coefficient of

variation. However, as a result of examining the model, it is seen that the coefficient of variation of 𝑥 (𝑡) variables increased to higher rates. The extreme values of the variation coefficients are given in the table below (Table 24). Coefficient of Variation 8 7.04612 max deger

Figure 24. Variation coefficient of the random model Table 24. Extreme values and times of the coefficient of variation in the random model Variable 𝐶𝑉(𝑥(𝑡))

Despite the %5 coefficient of variation in the parameters, it is observed that the variation rate of 𝑥 (𝑡) is constantly increasing and reaches %0.07 at 𝑡 = 10. Therefore, it can be interpreted that the variability in random results increases as logistic model and then decreases. The results obtained for the expected values of the model are (1) given below (Figure 25). Confidence interval 1100 1003.41 max deger

Figure 25. Confidence interval of the expected values of the random model

The confidence intervals given in the figure are calculated as 𝐶𝐼 = (𝐸 (𝑥 (𝑡)) − 3. 𝑠𝑡𝑑 (𝑥 (𝑡)), 𝐸 (𝑥 (𝑡)) + 3. 𝑠𝑡𝑑 (𝑥 (𝑡))) and three gives the range of variation within the standard deviation. This range includes about %99 of the random variable's values for poisson distribution. Therefore, the extreme values obtained for the expected values in these ranges are given below (Table 25). Table 25. Extreme values and times of confidence in the random model Variable 𝐸(𝑥(𝑡))

At the end of the process, three standard deviation intervals for 𝑥(𝑡) variables are obtained as follows: 𝐸 (𝑥(28)) ∈ (874.1435,1125.6785). Model (1) states that the expectation for this value is 𝐸 (𝑥 (28)) = 1003.41 that is, approximately %10.0341, 𝑎𝑛𝑑 the expected approximate population is in the range of %99 probability (874.1435,1125.6785) at time 𝑡 = 28.

4.2. White Noise

In the random logistic difference equation defined as 𝑋(𝑛 + 1) = 𝑋(𝑛) + 𝑟 ∗ 𝑋(𝑛) ∗ (1 −

if 𝑟 is a random variable with a parameterized uniform distribution (𝑛𝑜𝑖𝑠𝑒 = 1) and 𝑁 = 50, then the probability characteristics obtained from 105 simulations are given below. Expected Value 1400 E(X) 1200

Figure 26. Expected value of random model In the logistics model process (𝑡 ∈ [0,50]), it is observed that the variability increases first and then remains stable. The extreme values are shown in the Table (Table 26 and Fig 26). Table 26. Expected value, end values and times in random model Variable 𝐸(𝑥(𝑡))

It appears that the expected population reached its highest level at the time of 𝑡 = 50. Therefore, the results obtained from the deterministic model are more likely to be observed differently in an experiment that takes place randomly at these moments. In addition, 𝐸 (𝑥(50)) = 999.774 was obtained for the expected value at the end of the process (𝑡 = 50). Similarly, variations of variances for the model (1) are also seen below (Figure 27). In the logistics model process (𝑡 ∈ [0,50]), it is observed that variability increases first and decreases later. The extreme values are shown in the Table (Table 27). Table 27. Extreme values and times of variance in random model Variable 𝑉𝑎𝑟(𝑥(𝑡))

Figure 27. Variance of random model It is observed that the population has reached its highest level of deviation from the average at the time of 𝑡 = 13. Therefore, the results obtained from the deterministic model are more likely to be observed differently in an experiment that takes place randomly at these moments. In addition, at the end of the process, 𝑉𝑎𝑟 (𝑥(13)) = 10.0728 was obtained for variance (𝑡 = 13). Similar to variance, variations in the standard deviation for the (1) model are also seen below (Figure 28).

Figure 28. Standard deviation of the random model As the standard deviation is the square root of the variance, the result is that these two numerical characteristics are expected to exhibit similar behavior. The extreme values for standard deviations are shown below (Table 28). Table 28. Extreme values and times of standard deviation in random model Variable 𝑆𝑡𝑑(𝑥(𝑡))

It is observed that the population has reached its highest level of deviation from the average at the time of 𝑡 = 13. Therefore, the results obtained from the deterministic model are more likely to be observed differently in an experiment that takes place randomly at these moments. In addition, 𝑆𝑡𝑑 (𝑥(13)) = 3.17376 was obtained for variance (𝑡 = 13) at the end of the process. Using the results obtained for the standard deviations and expected values, the variation coefficients for the variables 𝑥 (𝑡) in the random model (1) were also calculated as follows (Figure 29). Coefficient of Variation (CV) is calculated by definition as 100 × 𝑠𝑡𝑑 (𝑥 (𝑡)) / 𝐸 (𝑥 (𝑡)) and random r parameters for the installation of model (1) are defined to have %5 coefficient of variation. However, as a result of examining the model, it is seen that the coefficient of variation of 𝑥 (𝑡) variables increased to higher rates. The extreme values of the variation coefficients are given in the table below (Table 29).

Figure 29. Variation coefficient of the random model Table 29. Extreme values and times of the coefficient of variation in the random model Variable 𝐶𝑉(𝑥(𝑡))

Despite the %5 coefficient of variation in the parameters, it is observed that the variation rate of 𝑥 (𝑡) is constantly increasing and reaches %0.0078 at 𝑡 = 6. Therefore, it can be interpreted that the variability in random results increases as logistic model and then decreases. The results obtained for the expected values of the model (1) are given below (Figure 30). Confidence interval 1100 1002.65 max deger

Figure 30. Confidence interval of the expected values of the random model

The confidence intervals given in the figure are calculated as 𝐶𝐼 = (𝐸 (𝑥 (𝑡)) − 3. 𝑠𝑡𝑑 (𝑥 (𝑡)), 𝐸 (𝑥 (𝑡)) + 3. 𝑠𝑡𝑑 (𝑥 (𝑡))) and three gives the range of variation within the standard deviation. This range includes about %99 of the random variable's values for normal distribution. Therefore, the extreme values obtained for the expected values in these ranges are given below (Table 30). Table 30. Extreme values and times of confidence in the random model Variable 𝐸(𝑥(𝑡))

At the end of the process, three standard deviation intervals for 𝑥(𝑡) variables are obtained as follows: 𝐸 (𝑥(50)) ∈ (990.45272,1009.49528). Model (1) states that the expectation for this value is 𝐸(𝑥(50)) = 1002.67 that is, approximately %10.0265, and the expected approximate population is in the range of %99 probability (990.45272,1009.49528) at time 𝑡 = 50.