Materials and methods

Study area. The LMNR, located in the central part of Qiandongnan, Guizhou Province, China, is made up of steep slopes, which lead into narrow valleys and small waterfalls. It spans the four counties of Leishan, Taijiang, Jianhe, and Rongjiang, and is a watershed between the Yangtze River and the Pearl River (26° 20' 25″-26° 25' 00″ N, 108° 12' 00″-108° 20' 00″ E, 2,178 m asl). LMNR is characterized by a typical subtropical humid climate, with abundant rainfall and less daylight time (^{Chen et al．2012}). The mean annual temperature is about 10 °C; July is the hottest month, with an average temperature of less than 25 °C. The soil is an acidic mountain yellow soil. The mountain forests are structurally and floristically heterogeneous (^{Tang et al. 2007}, ^{Tang & Ohsawa 2009}), with a distinctive vertical distribution of vegetation. The flora is mainly divided into evergreen broad-leaved forests at lower elevations (below 1,350 m asl), evergreen and deciduous broad-leaved mixed forests at middle altitudes (1,350-2,100 m asl), and deciduous broad-leaved forests at higher altitudes (over 2,100 m asl). The T. sinense populations in the LMNR were mainly scattered throughout evergreen and deciduous broad-leaved mixed forests. We investigated the distribution of T. sinense throughout the China and found that the distributions of the population of T. sinense were patchy. In LMNR, we found that there were four patches which were the most typical, and they could well represent the distributions of the natural population of T. sinense. Therefore, the four representative patches were chosen as experimental plots rather than artificial plots (population 1: 7 ha; population 2: 1.905 ha; population 3: 7.4 ha; population 4: 7.02 ha).

Age structure. In order to minimize the damage to T. sinense, 19 individuals with intact growth and different diameter grades were randomly selected in these four patches and their cores were taken to determine their ages. Then the relationship between the age and the DBH of the T. sinense were modeled to obtain the fitting curve (^{Wang et al. 1995}, ^{Harper 1977}, ^{Zhang et al. 2007}). According to the fitting curve, the age of all individuals in the four populations were calculated (^{Hett et al. 1976}). According to the life history characteristics of T. sinense and the methods of ^{Brodie et al. (1995)} and ^{Guedje et al. (2003)}, these populations were grouped into 11 age classes. Pre-reproductive and juvenile trees were classified as seedlings and saplings (I, 0~20 ages), or juveniles (II, 20~40 ages; III, 40~60 ages; IV, 60~80 ages; V, 80~100 ages; VI, 100~120 ages; VII, 120~140 ages). Adult trees were grouped into four age classes (VIII, 140~160 ages; IX, 160~180 ages; X, 180~200 ages; XI, 200~220 ages). The number of T. sinense in each age class was counted, and then the age structure of the populations was analyzed.

Population dynamics analysis. The dynamic change in number of individuals between adjacent age classes (V _n ) was analyzed according to the method of ^{Chen (1998)}, using the following formula:

where S _n and S _n+1 are the number of individuals in the n ^th age class and the next age class, respectively. Max (... ...) represents the maximum value in parentheses; V _n ∈ [-1, 1] When V _n > 0, this means the number of individuals is increasing in a dynamic relationship between adjacent age classes; when V _n < 0, this means there is a decline in the dynamic relationship; when V _n = 0, the dynamic relationship is stable.

The quantity dynamic index (V _pi ) of the age structure of these populations was obtained by weighting the number of individuals (S _n ) of each age class by V _n . Because there is no V _n for the maximum age class (K), the K value was excluded (^{Chen 1998}):

Because the external environmental effect on the population age structure is not considered in equation (2), the quantity dynamic index of population age structure (V _pi ) should be corrected accordingly, that is:

where min (... ...) means the minimum value of the sequence in parentheses, and K is the age class.

Establishment of a static life table. Given that T. sinense is, a long-life cycles tree species, a static life table was established. Static life tables capture discrete periods of the dynamic process of aging where multiple estimate the number of surviving individuals (x) (^{Jiang 1992}, ^{Zhou et al. 1992}), based on the following formulas (^{Silvertown 1982}, ^{Jiang 1992}). According to the assumptions underlying static life tables, the age combination is stable, and the proportion of each age class remains constant (^{Jiang 1992}). Then the data of the four populations in LMNR were corrected by smoothing technique (^{Proctor 1980}, ^{Brodie et al. 1995}, ^{Molles 2002}).

where T: the sum of all the individuals in each group; n: the number of age classes in each group; a_x: individuals of each groups in the x age class; ax¯ : the means of each group, which were considered to be the mid-value. Based on the differences between the maximum and minimum values of number surviving in the two groups and the difference between the means of the two groups, the number of individuals in each age class was corrected by one unit of smoothing (^{Jiang 1992}).

where x: the age class; a _x : number of surviving individuals in age class x; A _x : number of surviving individuals in age class x after smoothing; A ₁ : number of surviving individuals in the I age class after smoothing; l _x : standardized number of surviving individuals at the beginning of age class x (generally converted to 1,000); d _x : standardized number of individuals dying between age classes x and x+1; q _x : mortality rate between age classes x and x+1; L _x : number of surviving individuals between age classes x and x+1; T _x : total number of individuals from age class x; e _x : life expectancy at age class x; K _x : killing power; S _x : survival rate (^{Silvertown 1982}, ^{Jiang 1992}).

Survival curve. Taking the ln ^lx of the number of survivors as the vertical coordinate and each age class as the horizontal coordinate, the static life table was used as a basis for the survival curve, which was fitted using SPSS 23.0 software and was modeled by R language (^{R Core Team 1995}). OriginPro 8.0 software was used to draw the mortality rate (q _x ) and killing power (K _x ) curves (^{Feng 1983}).

According to ^{Hett & Loucks (1976)}, three kinds of mathematical models (linear equation, exponential equation, and power function equation) can be used to test of the best fit of the survival curve on T. sinense data (^{Deevey 1947}). These were:

where x is age class, N _x is the natural logarithm of the standardized number of surviving individuals in age class x, N _x = lnl _x ; N ₀ and b were directly obtained by fitting the mathematical model.

Survival analysis. To better analyze the age structure of the T. sinense populations in each patch of LMNR and further clarify the survival rules for the populations, this study introduced four functions. The survival functions are functions relating to any age class, which are more intuitive than the survival curve, so survival analysis has a greater practical applied value in the analysis of population life tables compared to survival curves (^{Yang et al. 1991}, ^{Guo 2009}). The four functions (population survival rate function S _(i) , cumulative mortality function F _(i) , mortality density function f _(ti) , and hazard rate function λ _(ti) were calculated as follows:

where i is age class; S _i is the survival rate in age class i; S _i = S _x as in equation (2); and h _i is the width of the age class.

Time series analysis. Time series analysis is often used to predict dynamic changes in population size (^{Xiao et al. 2004}, ^{Zhang et al. 2017}). In this paper, the moving average method was applied to the analysis:

where n is the time being predicted (age class period in the study); M ⁽¹⁾ _t is the population size in age class t after n age class periods in the future, and the population size of the current k age class of X _k . The population quantity dynamics in the next 2, 4, 6, 8, and 10 age class time periods were predicted.

Results

Age structure of T. sinense populations. The fitting results showed that the relationship between the ages and DBH was well fitted by the linear regression (R ² = 0.826, P < 0.01). Therefore, in 95 % prediction range, the family had better fitting observation value. The ages of T. sinense in four populations were calculated by the formula (y = 3.15 × x + 21.68) (Figure 1). Then the structures of age were analyzed according to the ages.

The age structures of the T. sinense populations in the LMNR were all close to the pyramid type, although their age structures were all incomplete. The maximum number of individuals in each population was observed in the II or III age class, there were relatively no seedlings and saplings existed in these populations except for Population 2 (Figure 2).

Quantitative dynamic analysis of the T. sinense populations. There were significant fluctuations among different age classes in these populations, with a decreasing trend between some age classes. The most obvious negative trends appeared between adjacent age classes. The dynamic indices of each population in I age class were the negative value. As the age class increased, the indexes fluctuated between positive, zero and negative (Table 1). The maximum V _pi appeared in Population 1, indicating the highest stability, but its value was only 50 %. Although the population dynamic indexes of the four populations were positive, the values were very small. Even though the numbers of medium and young age class individuals in the populations were greater than those of adult individuals, the number of seedlings and saplings was much less, and the fluctuations within populations were very large (Table 2). To some extent, the four populations are all at risk.

Static life tables and survival curves. As age class increased, l _x decreased, the trend of l _x in population 2 was more apparently. While a higher individual e _x was apparent in the I, II age classes. In the X and XI age classes, e _x became minimal. The d _x of populations were decreased sharp in their younger age class, excepted population 1. Its d _x was more stable in I to VII ages classes (Table 3).

The l _x of T. sinense decreased rapidly in the I to VIII (Populations 1 and 2) or I to III (Populations 3 and 4) age classes, but more slowly in the IV or VII age classes, and then remained at a relatively low level thereafter (Figure 3). By modeling and comparing the survival curves of four populations, it was found that the fluctuating trend of the four populations was generally the same, and the P value = 0.6, which proved that there was no significant difference among the four populations (Figure 4).

Based on the change trend of the survival curve in the four populations, these survival curves all tended to the Deevey-II and Deevey-III types. The model simulation showed that the R ² and F values from equation (13) were both greater than those from equation (14) did (Table 4), indicating that the survival curves of the T. sinense populations1, 2,4 were a Deevey-II type, but population 3 were a Deevey-III typ.

Mortality rates and killing power curves. The variation patterns of q _x and K _x were similar (Figure 5). In Populations 1 and 2, there was a peak value in the VII age class; in Populations 3 and 4, the peak values appeared in the II and VI or II and VII age classes, respectively. When came to IX and X age classes, the values of two in all populations were stable. By the X and XI age classes, the q _x and K _x had dramatically increased.

Survival analysis. The survival function values were obtained according to the static life tables (Table 5). In the four populations, the S _(i) declined monotonically with increasing age class, while the F _(i) increased monotonically, showing a complementary trend. In the four Populations, the S _(i) and the F _(i) both changed noticeably during the I, II age classes, and then more gradually; while all the Populations in LMNR, a halcyon change trend appeared after the VII age class (Figure 6). The S _(i) of the four populations generally continued to decline throughout the survival process until the individuals died.

In the four populations, the changing trend of f _(ti) was similar to that of λ _(ti) , as both peaked in the II and IX age classes. The changing in population 4 was softer (Figure 7).

Time series analysis. The number of individuals in T. sinense populations in the LMNR is predicted to decline over time. When the 2-age class for new plants was reached, the numbers of surviving individuals in the younger age classes (III and IV age classes) in four Populations were predicted to decrease significantly; conversely, the number of individuals in the middle age classes all numbers of individuals remained relatively stable. When the 4-age class time for new plants was reached, the number of individuals showed a decreasing trend with increasing age class, which was similar to the change law of mortality curve. In the future, the number of individuals decreased sharply over time. When it was time for new plants to reach the 10-age class, the number of individuals predicted to survive in the four populations was no more than ten (Table 6).

Discussion

The population dynamics and changes in population structure of T. sinense are the result of its biological characteristics and interaction with the environment (^{Grubb 1977}). Population structure in plants can be divided into three types, i.e., increasing, stable, and declining (^{Guo 2009}). Relict and endangered plants with long lifespans tend to be of the declining type (^{Zhang et al. 2004}, ^{Zhang et al. 2008}). The reasons for population decline in endangered plants are mainly attributed to two factors: poor regenerative ability and adaptation (including poor seed set and low germination), and anthropogenic factors (over-exploitation, over-grazing, forest floor denudation) (^{Gupta & Chadha 1995}, ^{Jasrai & Wala 2001}). In our study, there was a significant positive correlation between age and DBH， the ages can be well fitted by the DBH in LMNR (^{Hett et al. 1976}). The age structures of the T. sinense populations in the LMNR were all close to the pyramid type, with the survival curve approximating a Deevey-II type and Deevey-III type; in the four populations of LMNR, their survival was similar. The population dynamic indexes of these populations (V _pi ) were all greater than zero. However, the dynamic index between adjacent age classes was negative, especially for the I age class. Furthermore, the seedling and sapling individuals were usually lacking that was consistent with those of Davidia involucrata (^{Liu et al. 2012}). These results indicated that most of the T. sinense populations investigated in the LMNR were relatively stable, but some of they were in early recession (^{Deevey 1947}).

The mortality rate and killing power curve can be seen to reflect the quantity dynamic change of T. sinense populations. The two curves for these populations showed that two peaks appeared in the II and IX age classes, or the VII age class. The peak in the II age class suggested that the juveniles in T. sinense populations were vulnerable to pests and diseases and adverse environments due to lower competitive ability (^{Han et al. 2015}, ^{Li et al. 2018}); these harsh environmental conditions would result in the greater mortality rate of juveniles, this characteristic was common with Abies fanjingshanensis, the rare and endangered plant endemic to Guizhou (^{Li et al. 2011}). In addition, the rapid growth of saplings into juveniles would gradually increase inter- and intra-species competition for limited environmental resources (such as light and mineral nutrition) in the T. sinense community (^{Tian et al. 2018}). This phenomenon might explain the higher mortality rate of juveniles.

Previous studies have showed that a sufficient number of younger individuals are a prerequisite for the successful regeneration of a tree population (^{Pala et al. 2012}, ^{Dutta & Devi 2013}), especially for endangered species (^{Xie & Chen 1999}, ^{Guo 2009}). Therefore, the deficiency of seedlings, saplings and the greater mortality rate of juveniles may cause a bottleneck in the regeneration and recovery of these natural T. sinense populations, which might eventually result in population decline (^{Cao et al. 2012}). Therefore, the apparently relatively stable population of T. sinense might already be in the early stages of decline, similar to that reported for Emmenopterys henryi (^{Kang et al. 2007}) and Aloe peglerae (^{Phama et al. 2014}).

The survival curve showed that the l _x of T. sinense decreased rapidly in the I ~ VIII (Populations 1 and 2) or I ~ III (Populations 3 and 4) age classes, but more slowly in the IV or VII age classes, and then remained at a relatively low level. The S _(i) and the F _(i) both changed noticeably during the I, II age classes, and then more gradually. The changing trend of f _(ti) was similar to that of λ _(ti) , as both peaked in the II and IX age classes. These findings suggested that the T. sinense population slumped during the early stage, and then remained stable during the middle stage, before a decline in the final stage. However, the dynamic trend detected for T. sinense differed from that obtained for Pinus taiwanensis (^{Bi et al. 2001}) and Taxus yunnanensis (^{Su et al. 2005}). The lower fitness of seeds and seedlings leads to a deficiency in the number of seedlings (^{Gan et al. 2013}, ^{Han et al. 2015}, ^{Li et al. 2018}). The juvenile trees of T. sinense that survive this natural environmental screening process will have stronger resistance and competitive ability, leading to a decline (^{Li et al. 2015}, ^{Cao et al. 2012}). As a result, the population size might remain relatively stable in its middle stages. By the time T. sinense individuals enter the older ages, they are in a physiological senescence phase (Silvertown et al. 2001, ^{Xiao et al. 2004}, ^{Zhang et al. 2008}). Therefore, the population decline in its final stages might be attributed to the physiological senescence of T. sinense individuals (^{Han et al. 2015}, ^{Li et al. 2018}).

Time series analysis showed that, over the next 10 age class times, the number of individuals in the T. sinense populations would decrease sharply, and the number of adult individuals would gradually increase, while the number of younger individuals was increasingly insufficient, seriously affecting the survival and development of the T. sinense populations. It can be inferred that as time goes by, there will be fewer and fewer individuals of the T. sinense population in LMNR (^{Li et al. 2015}).

In general, in order to adapt to the abominable environment (the flowering period coincides with the rain period), most of them were inbred, which leaded to the low germination rate of seeds in the field. The survival rate of seedlings was affected by the environment in which they lived (including the associated species and the litter thickness) (^{Cao et al. 2012}, ^{Tian et al. 2018}). In the natural population of T. sinense, the niche of associated species was larger, the natural regeneration of T. sinense population was poor, and there was a danger that it would be replaced by its associated tree species (such as Crecidiphyllum japonicum, Acer pictum subsp. mono) (^{Li 2015}, ^{Tian et al. 2018}). Therefore, the populations of T. sinense in LMNR will gradually decline if the current problem is not resolved.

In order to effectively protect the endangered plants T. sinense, some management strategies should be carried out. First, some of the more competitive tree species should be thinned to reduce interspecific competition and provide a better habitat for the growth of juveniles of T. sinense. In addition, improved conditions for the germination of T. sinense seeds and seedling establishment should be created to accelerate the natural regeneration of the population.

Conclusion

The age structure of T. sinense populations in the LMNR were all close to the pyramid type, with few seedlings and saplings, indicating that the population was in the early stages of decline. The population size of T. sinense slumped during the early age classes, and then remained relatively stable in the middle, followed by a decline in the final stages. The lack of seedlings or saplings in the populations might reflect a bottleneck in the regeneration of the T. sinense populations that were studied here. Through the investigation of T. sinense population in the LMNR, we have confirmed that it is in a dangerous situation. Based on our study, we have suggested several strategies to conserve and manage the T. sinense population.