This study was conducted on six Date Palm cultivars commercially cultivated in five provinces [Khuzestan (Sayer and Berhee), Hormozgan (Pyarom), Kerman (Mazafati), Bushehr (Kabkab), and Sistan and Baluchestan (Rabi)] of Iran. Sampling was repeated every seven days from late May to early September 2019–2020. To this end, we used a stratified random sampling method to estimate population density because environmental factors affecting population changes may not only be related to environmental conditions of the macro-climate but may also be due to uniformity in the micro-climate (Young and Young, 1998). In these models, the main classes consisted of different trees in the Date Palm gardens, and the sub-classes included bunches in four main geographical directions on each tree. Twenty fruits were randomly selected from each bunch and checked to determine DPSM infestation. The samples were then transferred to the laboratory, and the number of mites were counted and recorded separately. To determine the Date spider mite’s density at the sampling intervals in the field conditions, indicating the relationship of their population densities with Date fruit injury, the parameter Mite-day (Md) was calculated to estimate this index as follows (Machlitt 1998).

$$MD=7\times \left(\frac{M1+M2}{2}\right)$$

1

In this equation M1 and M2 were the mite densities in the current and previous sampling period, respectively.

## Stratified Sampling Standardization

The mite environment was classified into K classes. For example, if the mite density of class I was Ni, then:

If ni samples are taken from class i, and their means are ̄\(\stackrel{̄}{y}1......,\stackrel{̄}{yn}\), then the variances are calculated as Eq. 3.

$$\stackrel{̄}{{var}y}i=\left(\raisebox{1ex}{${S}_{i}^{2}$}\!\left/ \!\raisebox{-1ex}{$ni$}\right.\right)(1-\raisebox{1ex}{$ni$}\!\left/ \!\raisebox{-1ex}{$Ni$}\right.)$$

3

In Eq. 2, and Ni is the variance of each class and sample size, respectively. The mean population in each region is estimating as Eq. 4.

$$\stackrel{̄}{y}=\sum {\frac{Nix\stackrel{̄}{y}i}{N}}_{i}^{k}$$

4

Because individual ratios from all classes were sampled equally. Then, instead of Ni, ni and instead of N, n were used in Eq. 5.

$$SE\left(\stackrel{̄}{y}\right)=[\sum {w}_{i}^{2}\times \frac{{S}_{i}^{2}}{{n}_{i}}{]}^{\frac{1}{2}}$$

5

When the standard error was determined, interval confidence was calculated as Eq. 6.

$${\stackrel{̄}{\mu }}_{st}\pm {t}_{\alpha /2}\left[SE\right(\stackrel{̄}{y}\left)\right]$$

6

The mean and variance of spider mite population were estimated in four main classes of pest, including north, south, east, and west of tree to fit the population density estimation equation in the stratified sampling method, based on equations 2 to 4. Then the standard error value was calculated by using Eq. 5. Finally, the mean population density models were fitted for studied Date Palm cultivars, based on Eq. 6. The confidence level is often considered at 5 and 1 percent. The n or sample sizes (actual mean population change with 95% probability) was calculated by using Eq. 7. Appropriate and applicable sample sizes were calculated by using population confidence interval (Eq. 7).

$$n=\frac{{S}^{2}}{{L}^{2}\raisebox{1ex}{$N$}\!\left/ \!\raisebox{-1ex}{$4$}\right.+{S}^{2}}$$

7

In Eq. 7, S2, L, N, and n are sample variance, confidence limit, total sample size, and sample size, respectively (Padilla et al. 2017).

## Population dispersion

The data included 32 series in each province and on each cultivar. In total, 192 data series were collected. Different indices were used to determine the population dispersion pattern. Sara et al. 2020). The Patchiness index was calculated by using Eq. 8 (Lloyd, 1967).

$$ID=\frac{{S}^{2}}{\stackrel{̄}{x}}(n-1)$$

8

Taylor’s power law is based on an exponential relation between variance and means, as expressed in Eq. 9.

σ2 = a µb (9)

where σ2 is the population variance and µ is the population mean; *a* and *b* are constants. This equation is tested on field data by plotting log s2 against log \(\stackrel{-}{\text{X}}\) in Eq. 10

$$Log{S}^{2}=Loga+bLog\stackrel{̄}{X}$$

10

The coefficient of determination (R2) of the regression line indicates how powerful the model is for the mite population. *a* is a sampling parameter and *b* can be used as an index of aggregation. If so, b > 1, b = 1 and b < 1 then the population dispersion will be cumulative, random and uniform, respectively (Reiczigel et al. 2005). The coefficient of explanation ICM (crowding mean index based on LIoyd method) indicates competition intensity by other individuals based (Chao et al. 2015). The parameters \(\stackrel{-}{\text{X}}\) or X* is the mean of the effect of another individual in the sampling unit on any other individual in the population. This index was calculated by using Eq. 10 (Reiczigel et al. 2005). The non-patchiness index (IP) was calculated by using Eq. 12.

$${X}^{*}=\stackrel{̄}{X}+\frac{{S}^{2}}{\stackrel{̄}{X}}-1$$

11

$$IP=\frac{{X}^{*}}{\stackrel{̄}{X}}$$

12

The regression curve was adjusted between X* and \(\stackrel{-}{\text{X}}\) to calculate the Iowa index that is defined by Eq. 13 (Iwao and Kuno, 1968).

$${X}^{*}=\alpha +\beta \stackrel{̄}{X}$$

13

The constant α indicates the species’ crowding. If α was positive, the species tended to clumped, and if α was negative, the species tended to disperse. The constant \(\beta\) indicates that how much population tended to clumped at high densities. If so, \(\beta\) >1, \(\beta\) =1 and \(\beta\) <1 then the population will be cumulative, random and uniform, respectively (Iwao and Kuno, 1968).

## Waldʹs Sequential sampling model

Two population levels are essential in principles of sequential sampling model (Wald, 1947), including HO: population below a certain level and Hi: population above a certain level (Li et al. 2012). This theory has terms defined as follows. CD, OC, and ASN were defined as critical density, operational curve function, and average sample size curve. Three factors affect OC and ASN functions, including sampling stop lines, population, and population density of mites. Thus, there are two decision lines for the sequential sampling model that is likely to be described by equations 14 and 15(Kafeshani et al. 2018.).

Top stop line \(UBn=n*(cd+Z\times \frac{s}{\sqrt{n}})\) (14)

Bottom stop line \(LBn=n*(cd-Z\times \frac{s}{\sqrt{n}})\) (15)

The ASN curve represents the average of sample sizes needed to decide. The sample sizes were varied by population density fluctuation. According to the clumped of the pest, K, \(\stackrel{-}{x}\), \(\stackrel{-}{{x}_{1}}\) and Sd is accumulation constant, the population means (equivalent to 80% of economic injury level), specified population level (equivalent to 1461.6, 201.9, 222.2, 45.5, 58.1 and 102.0 for Sayer, Barhee, Kabkab, Pyarom, Rabi and Mazafati cultivars respectively) (Latifian et al. 2021) that is important for decision making, and standard deviation, respectively (Fowler and Lynch, 1987). Other model parameters were calculated by using Equations 16 to 20.

، \({h}_{2}=\frac{Ln\frac{1-B}{\propto }}{Ln\frac{{p}_{2}{q}_{1}}{{p}_{1}{q}_{2}}}\)، \(b=k\frac{Ln\frac{{q}_{2}}{{q}_{1}}}{Ln\frac{{p}_{2}{q}_{1}}{{p}_{1}{q}_{2}}}\) (16)

*q* *2* *= 1 + p**1* *، q* *1**=1+p* (17)

$$Oc=L\left(A\right)=\frac{{A}^{h\left(M\right)}}{{A}^{h\left(M\right)}-{B}^{h\left(M\right)}}$$

18

$$M=A=K\frac{1-(\frac{{M}_{1}+k}{{M}_{2}+k}{)}^{h\left(M\right)}}{\left(\frac{{M}_{2}\left({M}_{1}+K\right)}{{M}_{1}\left({M}_{2}+K\right)}{)}^{h\left(M\right)}\right)}$$

19

$$ASN=EA\left(N\right)=\frac{b\left(OC\right)+a(1-oc)}{KLn\left(\frac{{M}_{1}+k}{{M}_{2}+k}\right)+M{ln}(\frac{{M}_{2}({M}_{1}+{k}_{1})}{{M}_{1}({M}_{2}+{k}_{2})})}$$

20

In the above equations, b is the slope of the decision lines, and h1 and h2 are the intersections with the y-axis. Suppose the probability of making wrong decisions is above the upper and lower limit lines, respectively. Then, M and P are mite population density and infected to total fruits ratio. In equations 17,19, and 20, K calculated as follows. In Eq. 21, *V* and *m* are the variance and mean of the infected fruits.