Developing, testing and application of rodent ...

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International Journal of Biomathematics Vol. 7, No. 3 (2014) 1450024 (23 pages) c World Scientific Publishing Company DOI: 10.1142/S1793524514500247

Int. J. Biomath. 2014.07. Downloaded from www.worldscientific.com by ZHEJIANG UNIVERSITY on 05/08/15. For personal use only.

Developing, testing and application of rodent population dynamics and capture models based on an adjusted Leslie matrix-based population approach

Liyong Fu∗,§ , Shouzheng Tang∗,¶ , Yingan Liu† , Ram P. Sharma‡ , Huiru Zhang∗ , Yuancai Lei∗ , Hong Wang∗ and Xinyu Song∗ ∗Research

Institute of Forest Resource Information Techniques Chinese Academy of Forestry Beijing 100091, P. R. China †Department

of Mathematics Nanjing Forest University Nanjing, Jiangsu Province 210037, P. R. China ‡Department

of Ecology and Natural Resource Management Norwegian University of Life Sciences ˚ As 1432, Norway §[email protected] ¶[email protected] Received 29 August 2013 Accepted 7 March 2014 Published 30 April 2014

Small rodents in general and the multimammate rat Apodemus agrarius in particular, damage crops and cause major economic losses in China. Therefore, accurate predictions of the population size of A. agrarius and an efficient control strategy are urgently needed. We developed a population dynamics model by applying a Leslie matrix method, and a capture model based on optimal harvesting theory for A. agrarius. Our models were parametrized using demographic estimates from a capture–mark–recapture (CMR) study conducted on the Qinshui Forest Farm in Northwestern China. The population dynamics model incorporated 12 equally balanced age groups and included immigration and emigration parameters. The model was evaluated by assessing the predictions for four years based on the known starting population in 2004 from the 2004–2007 CMR data. The capture model incorporated two functional age categories (juvenile and adult) and used density-dependent and density-independent factors. The models were used to assess the effect of rodent control measures between 2004 and 2023 on population dynamics and the resulting numbers of rats. Three control measures affecting survival rates were considered. We found that the predicted population dynamics of A. agrarius between 2004 and 2007 compared favorably with the observed population dynamics. The models predicted that the population sizes of A. agrarius in the period between 2004 and 2023 under the control measure applied in August 2004 were very similar to the optimal population sizes, and no significant difference was found between the two population sizes. We recommend using the population dynamics and capture models based on CMR-estimated demographic schedules for rodent, provided these data are available. 1450024-1


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L. Fu et al. The models that we have developed have the potential to play an important role in predicting the effects of rodent management and in evaluating different control strategies. Keywords: Apodemus agrarius; capture model; Leslie matrix; optimal population density; population dynamics model; capture–mark–recapture. Mathematics Subject Classification 2010: 92D25


1. Introduction Crop damage by small herbivorous mammals has a long history worldwide [1–7]. Rodents are major pests in tree plantations in North America [1, 7–9], Europe [10, 11], Africa [12–14] and Asia [15–17], where they cause substantial economic losses and disturb ecosystems. The population dynamics of rodents and their relationships to anthropogenic, ecological and environmental factors have been studied throughout the world [14, 18–24]. To understand and predict rodent population dynamics, various models have been developed [18, 21, 25–27]. Regression models describe patterns and are often good at predicting phenomena; however, they do not provide insights into the underlying mechanisms [28]. Early population dynamics models for rodents simply provided a concept, either described in words or presented graphically [18–21, 28]. Such models gave a schematic representation of the different factors that could be important from the perspective of population dynamics, but were often too vague and imprecise to have any practical applications [29]. In a small mammal population, survival and other demographic processes have been reported to differ between functional categories (e.g. subadults and adults) [30]. Stenseth et al. [29] demonstrated that Leslie matrix-based population models [31] combined with capture–mark–recapture (CMR) data would be a useful approach for modeling the population dynamics of rodents. Their model incorporated three functional age categories (juveniles, subadults and adults) of both sexes and used density-dependent and density-independent factors to assess the effect of rodent control on the population dynamics and resulting numbers of rats (Mastomys natalensis) in Africa. Although the predictions followed the observed population dynamics in general, the fit of the model was not perfect. The inaccuracies in the predictions were likely the result of immigration which was not incorporated into the model, but which has a strong impact on population dynamics [32–35]. To reduce rodent damage and the unnecessary use of rodenticides in ecosystems where high rodent populations are likely to occur, the predictive ability of population dynamics models needs to be improved [25, 36, 37]. More accurate models will allow more timely interventions with more appropriate control measures [29]. Therefore, in this study, we developed an adjusted Leslie matrix-based population dynamics model to address this problem. Anthropogenic interference can influence the population dynamics of rodents [14]. Rodents are known to exhibit irregular eruptive population dynamics [28] and extensive damage can be expected during the outbreaks. Rodents also cause 1450024-2


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considerable damage even at relatively low population densities [12]. It is well known that the management of rodents based on the ad hoc use of rodenticides [13] is not a good strategy; economic, ecologically friendly and sustainable rodent management strategies need to be adopted to reduce rodent damage [26, 29, 38]. In their study of M. natalensis, Vibe-Petersen et al. [14] pointed out that an understanding of the role of predation on population dynamics is not only of scientific value, but is also useful for biological control. In many countries, rodents are considered as major pests and various control measures are used to annihilate them. Forest farmers, for example, often employ massive chemical measures to wipe out local rodent populations; however, this may cause large negative effects on crops and make the ecological system unstable [38–41]. Population diversity is a premise of ecological stability, and rodents play an important role in the stability of local biological chains. In this study, we derived the optimal population density of rodents from capture models using optimal harvesting theory [26, 42, 43]. The models of optimal population density for rodents would be very useful for formulating future ecologically friendly control strategies. The objectives of this study were: (i) to develop an adjusted Leslie model that includes migration to predict the population dynamics of rodents, (ii) to derive the optimal population density of rodents from capture models, and (iii) to analyze and evaluate the effects of rodent control measures on the population dynamics using an adjusted Leslie model and the optimal population density. To achieve these objectives, CMR data from Qinshui Forest Farm in Northwestern China were used. 2. Development of the Two Models 2.1. The population dynamics model It is assumed that all individuals in a population have a maximal age L and the population is divided into m equal age groups each containing T = L/m years. The number in each age group at time t is defined by the vector Nt = (n1,t , n2,t , . . . , nm,t )T , where ()T is the transpose of the matrix, and nk,t (k = 1, . . . , m) is the number of individuals in the age group k [age between (k − 1)T and kT )] at time t. The total m population size at time t is given by nt = k=1 nk,t . Assuming that the time interval t to t + 1 is equal to T , then bk (k = 1, . . . , m) is the average number of offspring born per female in the age group k during unit time T , dk (k = 1, . . . , m) is the proportion of females in the age group k at time t, and qk (k = 1, . . . , m − 1) is the survival rate from the k age group to the next k + 1 during unit time T . Closed and isolated regions are rare in the wild, and individuals in different populations may be the result of mutual flow among different systems. Immigration and emigration to and from geographically open populations are important demographic components to population growth [32–34, 44]. Therefore, in this study, αk is the 1450024-3


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proportion that the number of individuals from an outside system coming into age group k during the time interval t to t + 1 account for in the total number of individuals in age group k in the system. βk is the proportion that the number of individuals in age group k exiting the system during the time interval t to t + 1 account for in the total number of individuals in age group kin the system. For a given k = 1, . . . , m, αk and βk meet the 0 ≤ αk < 1 and the 0 ≤ βk < 1 criteria, respectively. We assumed that the individuals in the first age group always follow their mother in entering or exiting the system. Thus, we get the following equations [45] n1,t+1 =

m

bk dk {1 + (αk − βk )}nk,t + (α1 − β1 )n1,t+1

(2.1)


k=1

and nk,t+1 = qk−1 nk−1,t + (αk − βk )nk,t+1 ,

k = 2, 3, . . . , m,

(2.2)

where nk,t+1 (k = 1, . . . , m) is the total number of individuals in age group k at time t + 1. According to (2.1) and (2.2), n1,t+1 and nk,t+1 (k = 2, 3, . . . , m) can be given by [45] n1,t+1 =

m 1 bk dk {1 + (αk − βk )}nk,t 1 − α1 + β1

(2.3)

k=1

and nk,t+1 = Set



qk−1 nk−1,t , 1 − αk + βk

f1  p1   M=0 . . . 0

k = 2, 3, . . . , m.

f2 0

f3 0

··· ···

fm−1 0

p2 .. . 0

0 .. . 0

···

0 .. .

· · · pm−1

 fm  0  0  ..   .  0

(2.4)

,

m×m

where, fk =

bk dk (1 + αk − βk ) , 1 − α1 + β1

k = 1, 2, . . . , m

(2.5)

and pk =

qk 1 − αk+1 + βk+1

,

k = 1, 2, . . . , m − 1.

(2.6)

Therefore, (2.3) and (2.4) can be expressed in the following matrix form [45] Nt+1 = M Nt , m×1

m×m m×1

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(2.7)


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where M is the typical Leslie matrix [31, 46]. The parameters for bk , dk , αk , βk (k = 1, 2, . . . , m) and qk (k = 1, 2, . . . , m − 1) were obtained from the CMR data. (The formulae for each of the parameters are listed in table in Appendix A.) The dominant eigenvalue λ of the matrix M defines the population growth rate. When λ > 1, λ = 1 or 0 < λ < 1, the population size is either growing, stable or decreasing, respectively [47]. Similar to the assumptions of Smith and Trout [47] in their study, we also suppose that, in addition to being dependent on age, fk (k = 1, 2, . . . , m) and pk (k = 1, 2, . . . , m − 1) in matrix M are themselves dependent on the time of year and follow an annual cycle. If each year is assumed to be divided into s equal units of time of duration equal to t to t + 1, then the population size of the jth (j = 2, . . . , s) unit of time is given by Nj = Mj−1 , . . . , M1 N1

(j = 2, . . . , s),

(2.8)

where N1 is the population size of the first unit of time in the year, and Mj−1 is the corresponding Leslie matrix in the (j − 1)th unit of time in the year (j = 2, . . . , s). Therefore, after one year (s units of time) a new population will be represented by [46] (l+1)

N1

m×1

(l+1)

(l)

= Ms · · · Mj · · · M1 N1 , m×m

m×m

m×m m×1

(2.9)

(l)

and N1 are population sizes of the first unit of time in the (l + 1)th where N1 year and lth year, respectively. We assume that at some time during the year a control measure which removes a proportion h of age group k denoted by hk (k = 1, . . . , m) is implemented. A matrix H with m × m size can be constructed where all elements are zero except for the main diagonal which contains the (1 − hk ) values. If Hj denotes a control measure which is implemented in the jth (j = 1, . . . , s) unit of time in the lth year, then (2.9) becomes (l+1)

N1

(l)

= Ms · · · Mj+1 Hj Mj · · · M1 N1

(2.10)

and λ, the dominant eigenvalue of the matrix, ˜ = Ms · · · Mj+1 Hj Mj · · · M1 , M

(2.11)

defines the population growth rate from year l to l + 1, including any effects of the implemented control measures [47].

2.2. The capture model We assume that the rodents are divided into two age categories: juvenile and adult, with each category containing one or more groups. Without the need for capture, 1450024-5


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a single-species model of rodent with this age distribution can be described by (2.12) as x˙ = γy − r1 x − ρx − ηx2 , (2.12) y˙ = ρx − r2 y, where x and y are the population densities of juvenile and adult rodents, respectively, γ is the birth rate of juvenile rodents, ρ is the transition rate for juvenile rodents aging to adult rodents, η is the rate of self-regulation by population density (in this study, we let η = (x + y)/[xy ln(A)] where A is the age of plantation in years), and r1 and r2 are the mortality rates of the juvenile and adult rodents, respectively. All the parameters in (2.12) are obtained from the CMR data and all are positive numbers. (The formulae for each of the parameters are listed in table in Appendix A.) To simplify the calculations, we set r = r1 + ρ. Then the capture model for juvenile rodents would be: x˙ = γy − rx − ηx2 − E1 x, (2.13) y˙ = ρx − r2 y, where E1 is capturing effort. Similarly, the capture model for adult rodents would be: x˙ = γy − rx − ηx2 , (2.14) y˙ = ρx − r2 y − E2 y, where E2 is capturing effort. Based on reproductive characteristics, current numbers of rodents, local ecological status, and the capture principle of population coexistence, an optimal capture strategy [26, 42, 43], are applied to derive the optimal numbers of rodents. The stability of the equilibrium point for the juvenile capture model (2.13) is first analyzed to capture juvenile rodents. An equilibrium point exists in the solution to the following equation: γy − rx − ηx2 − E1 x = 0, (2.15) ρx − r2 y = 0. The origin (0, 0) is an equilibrium point in the juvenile capture model (2.13). When γρ/r2 − r > E1 , (2.13) has only one positive equilibrium point (x1 , y1 ), where x1 and y1 are given by (2.16) and (2.17), respectively.

1 γρ x1 = − r − E1 , (2.16) η r2

γρ ρ − r − E1 . y1 = (2.17) ηr2 r2 The point (x1 , y1 ) is also known as a stable node (see Appendix A). 1450024-6


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A corresponding capturing effort E1 has the following equilibrium harvesting function:

E1 γρ − r − E1 . f (E1 ) = E1 x1 = (2.18) η r2 To obtain the maximum for f (E1 ), partial differentiation with respect to E1 is required as: E1 γρ − r − E ∂ 1 η r2 2E1 γρ r ∂f (E1 ) = 0. = = − − ∂E1 ∂E1 ηr2 η η


Thus, the optimal capturing effort is E1 = (γρ − rr2 )/(2r2 ).

(2.19)

The optimal population density for the juvenile rodents (D1 ) can be derived as D1 = (γρ − rr2 )/(2ηr2 ).

(2.20)

Similarly, the optimal capturing effort (E2 ) and population density for the adult rodents (D2 ) is E2 = r2 (γρ − rr2 )/(γρ + rr2 )

(2.21)

D2 = (γρ − rr2 )(γρ + rr2 )/4γηr1 r2 .

(2.22)

and

The values for the optimal population density of the juvenile and adult rodents (D1 and D2 ) can be used to evaluate and formulate control measures for practical application. After one or more months of implementation, any efficient control measure should be adjusted to bring the population density closer to the optimal population density for the rodents. 3. Data Data were acquired from four typical one-hectare sample plots established in the young plantations in Qinshui Forest Farm in Northwestern China. Trees in this area had been damaged by rodents. The four sample plots were measured in February, April, June, August, October and December of each year between 2004 and 2007. The CMR method [29, 48] was adopted. Naujila’s traps were used to capture the rodents at each measuring during the study period. All rodent holes on the sample plots were numbered. The traps were placed in front of the holes before sunrise and packed up after sunset and inspected every 30 minutes for three consecutive days in each of the months. Toe clipping with specific number coding was used to identify individual animals. The weight, sex, reproductive status, age (determined by carcass weight) and corresponding hole identity number of each captured rodent were recorded. Animals were later released at the place of capture. 1450024-7


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During the study period 51,637 individual rodents were captured between 2004 and 2007 in the four sample plots. The relative proportions of the different rodent species that were captured in the plots during the study period were as follows: A. agrarius 95.4% (49,262 individuals), Rattus norvegicus 3.7% (1,911 individuals) and Rattus lossea Swinhoe 0.9% (464 individuals). No significant differences (Kolmogorov–Smirnov [KS-test], p > 0.05) in species composition among the four sample plots and no interaction effects (p > 0.05) of plot and time on species composition were observed during the study period. Therefore, in this study, we focused wholly on the A. agrarius population dynamics. The statistics of the A. agrarius data collected between 2004 and 2007 from each sample plot are summarized in Table 1. We assumed that no A. agrarius survive beyond two years of age (no A. agrarius older than two years of age were found in the practical survey). We, therefore, set the maximum age of A. agrarius to two years (L = 2). A. agrarius were categorized into 12 equal age groups (m = 12, T = 2 months). Before using the Leslie model to simulate the A. agrarius population dynamics, the model was evaluated by making predictions for the four years for which we had collected the data. We started the predictions using the known population data from 2004. The parameters in the population dynamics model were calculated from the 2004 data that was collected Table 1. Statistics of the A. agrarius data collected between 2004 and 2007 from the four sample plots in Qinshui Forest Farm. Sample plots

Population size

2004

2005

2006

2007

Sample plot 1

Total Female Offspring Immigration Emigration Death

570 ± 365 282 ± 180 162 ± 133 180 ± 87 57 ± 37 104 ± 93

481 ± 273 246 ± 183 125 ± 104 97 ± 51 48 ± 27 84 ± 73

531 ± 297 267 ± 151 147 ± 121 168 ± 75 53 ± 30 97 ± 86

507 ± 287 252 ± 141 133 ± 111 131 ± 67 51 ± 29 91 ± 80

Sample plot 2


502 ± 304 247 ± 150 135 ± 95 97 ± 69 50 ± 30 83 ± 84

513 ± 296 259 ± 144 146 ± 99 106 ± 74 51 ± 30 97 ± 81

491 ± 292 244 ± 193 128 ± 91 87 ± 58 49 ± 29 94 ± 85

518 ± 284 263 ± 147 153 ± 102 123 ± 89 52 ± 28 103 ± 94

Sample plot 3


474 ± 340 237 ± 167 144 ± 132 86 ± 74 47 ± 34 90 ± 96

528 ± 285 267 ± 144 172 ± 145 121 ± 87 53 ± 29 113 ± 105

511 ± 294 259 ± 151 165 ± 140 116 ± 76 51 ± 29 99 ± 87

548 ± 312 278 ± 161 185 ± 153 147 ± 118 55 ± 31 107 ± 101

Sample plot 4


510 ± 313 248 ± 150 129 ± 114 92 ± 72 51 ± 31 94 ± 91

531 ± 275 269 ± 141 143 ± 125 108 ± 81 53 ± 28 104 ± 98

468 ± 230 241 ± 119 122 ± 109 69 ± 47 47 ± 23 86 ± 75

529 ± 316 267 ± 162 145 ± 123 112 ± 93 53 ± 32 107 ± 103

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in the six months, February, April, June, August, October and December in each of the four sample plots. The parameters were then used as starting values for the model prediction runs. The overall accuracy measure (δ) (combined bias and precision) given by (3.1) δ = e¯2 + SD2 , was used to evaluate the models, where 2007 2007 6 6 e¯ = eti 24 = (nti − n ˆ ti ) 24,


t=2004 i=1

(3.2)

t=2004 i=1

is the mean prediction bias for the population size of A. agrarius, and 2007 6 SD = (eti − e¯)2 /(N − 1),

(3.3)

t=2004 i=1

ˆ ti are the where e¯ is the standard deviation for prediction bias, and nti and n observed and predicted population sizes of A. agrarius for the ith (i = 1, . . . , 6) measurement in year t (t = 2004, . . . , 2007), respectively. All calculations were performed using the R2.4.1 software [49]. Unless otherwise specified, the level of significance was 0.05 (α = 5%) in this study. 4. Results 4.1. The Leslie model The parameter estimates that were used in the predictions are listed in Table 2. Because February falls well outside the breeding season, the starting number of 2-month old A. agrarius in the four sample plots was zero. The population growth rates (λ − 1) for the four sample plots decreased continuously from February to December 2004 (Fig. 1). From February to August, the rate of population growth was positive, indicating that there was an increase in the population size following these months. Conversely, in October and December, the rate of population growth Table 2. Demographic parameter values that were used in the population dynamics model. The values were calculated from the actual 2004 data using Eqs. (1)–(3), (8) and (9). b is the average number of offspring born per female; d is the proportion of females in the population; q is the survival rate; f and p are both elements of the Leslie matrix. These values were used as the starting values for the parameters in the model runs. Sample plots

Parameter

February

April

June

August

Sample plot 1

b d q f p

0±0 0.35 ± 0.22 0.42 ± 0.31 0±0 0.45 ± 0.30

2.30 ± 1.48 0.50 ± 0.09 0.54 ± 0.24 1.38 ± 1.18 0.86 ± 1.20

2.34 ± 1.36 0.53 ± 0.15 0.54 ± 0.26 1.10 ± 0.85 0.57 ± 0.36

1.64 ± 1.09 0.50 ± 0.06 0.52 ± 0.26 0.81 ± 0.56 0.51 ± 0.31

October

December

1.75 ± 1.16 0.11 ± 0.20 0.45 ± 0.15 0.46 ± 0.07 0.52 ± 0.24 0.47 ± 0.18 0.89 ± 0.67 0.019 ± 0.03 0.61 ± 0.37 0.33 ± 0.14 (Continued )

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Sample plots

Parameter

February

April

June

August

October

December

Sample plot 2

b d q f p

0±0 0.38 ± 0.22 0±0 0±0 0.32 ± 0.24

2.14 ± 1.28 0.44 ± 0.17 0.58 ± 0.25 1.22 ± 0.84 1.21 ± 1.78

2.31 ± 1.44 0.52 ± 0.15 0.53 ± 0.28 1.10 ± 0.89 0.52 ± 0.32

1.75 ± 1.20 0.45 ± 0.15 0.60 ± 0.24 1.07 ± 0.88 0.59 ± 0.27

2.11 ± 1.34 0.53 ± 0.16 0.56 ± 0.25 0.83 ± 0.46 0.49 ± 0.25

0.12 ± 0.21 0.45 ± 0.14 0.41 ± 0.20 0.02 ± 0.04 0.28 ± 0.16

Sample plot 3

b d q f p

0±0 0.39 ± 0.24 0.58 ± 0.29 0±0 0.42 ± 0.26

1.77 ± 1.37 0.53 ± 0.22 0.74 ± 0.30 1.37 ± 1.09 1.01 ± 0.98

2.19 ± 1.39 0.53 ± 0.15 0.57 ± 0.31 1.11 ± 0.79 0.55 ± 0.31

1.52 ± 1.04 0.44 ± 0.14 0.57 ± 0.30 0.81 ± 0.73 0.62 ± 0.44

2.44 ± 1.40 0.47 ± 0.08 0.51 ± 0.24 0.89 ± 0.55 0.49 ± 0.30

0.18 ± 0.28 0.50 ± 0.07 0.45 ± 0.17 0.04 ± 0.07 0.31 ± 0.12

Sample

b d q f p

0±0 0.38 ± 0.23 0.31 ± 0.22 0±0 0.20 ± 0.14

1.74 ± 1.48 0.42 ± 0.20 0.69 ± 0.28 1.23 ± 1.07 1.00 ± 0.66

1.49 ± 1.29 0.43 ± 0.15 0.62 ± 0.31 0.88 ± 0.89 0.64 ± 0.38

1.74 ± 1.44 0.42 ± 0.16 0.64 ± 0.26 0.78 ± 0.69 0.65 ± 0.36

1.65 ± 1.42 0.48 ± 0.034 0.50 ± 0.33 0.64 ± 0.62 0.45 ± 0.34

0.11 ± 0.24 0.48 ± 0.22 0.46 ± 0.24 0.02 ± 0.044 0.31 ± 0.14

1.5 Sample plot 1 Sample plot 2 Sample plot 3 Sample plot 4

1.0

Population growth rate


Table 2.

.5

0.0

-.5

-1.0 February

April

June

August

October

December

Time of year ( 2-month period) Fig. 1.

Population growth rates for the four sample plots in 2004.

was negative, indicating that there was a decrease of the population size following these months. There were no significant differences among the four sample plots, but there was a significant difference in the population growth rates in the different months during 2004. The population sizes of A. agrarius for each sample plot in February, April, June, August, October and December between 2004 and 2007 were predicted based 1450024-10


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Fig. 2. Observed and predicted population sizes of A. agrarius in the years 2004–2007. The solid line represents the observed population sizes of A. agrarius and the dashed line represents the predicted population sizes of A. agrarius. F, February; A, April; J, June; A, August; O, October; and D, December.

on the observed population sizes in each of the sample plots in February in 2004 (i.e. 99, 95, 53 and 132 for sample plot 1, sample plot 2, sample plot 3 and sample plot 4, respectively), and on the rate of population growth (λ − 1) in the different months for each of the sample plots (see Fig. 1). The simulated trajectories for the posteriori predictions starting from February in 2004 followed the observed population dynamics consistently for each sample plot (Fig. 2) and discrepancies between the numbers was small across the 4-year study period. The overall accuracy 1450024-11


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Table 3. Statistics for the population dynamics model fitting for the actual population size of A. agrarius 2004–2007. The values from each of the six months in 2004 were used as starting values in the model. e¯, SD and δ were calculated from Eqs. (6)–(8), respectively. The population growth rates (λ − 1) for each month used in the fitting are from Fig. 1. Sample plots

Statistics

February

Sample plot 1

e¯ SD δ

36.36 50.54 62.26

Sample plot 2

e¯ SD δ

Sample plot 3

Sample plot 4

April

June

August

October

December

−632.71 628.41 891.75

−38.85 78.77 87.83

−25.21 76.82 80.85

−87.89 96.42 130.47

630.78 348.49 720.64

56.24 37.63 67.67

−109.06 90.18 141.52

−35.61 53.80 64.52

41.53 36.76 55.46

2.21 41.58 41.64

−47.55 61.72 77.91

e¯ SD δ

62.29 36.26 72.08

45.92 50.20 68.03

85.46 52.98 100.55

112.18 55.60 125.21

−17.98 61.74 64.30

−29.33 68.23 74.27

e¯ SD δ

43.71 58.03 72.65

52.89 59.96 79.96

28.66 56.47 63.33

28.83 57.39 64.22

−40.36 62.25 74.19

−41.91 63.08 75.73

measure (δ) varied from 62.26 to 72.65 (Table 3) in the different sample plots, and the correlations between the predicted and actual population sizes for all four sample plots were > 0.97, indicating a perfect model fit. The predicted population sizes of A. agrarius between February 2004 and December 2023 for each sample plot are shown in Fig. 3. 4.2. Effect of control measures Based on the 2004 CMR data, the optimal population sizes of A. agrarius for each sample plot over the 20-year period (February 2004–December 2023) were simulated (Fig. 3). The differences between the predicted population sizes and the optimal population sizes for each of the sample plots were significant. To adjust the population size of A. agrarius closer to the optimal population size for each sample plot, three different control measures were incorporated into the model; namely, increased mortality was implemented: (i) in April, the month with the highest reproductive rate (reproductive rates for each of the sample plots were 0.29, 0.63, 0.72 and 0.29, respectively), (ii) in August, the month with the highest survival rate (survival rates for each of the sample plots were 0.64, 0.57, 0.60 and 0.52, respectively), and (iii) in the six different months of the year: February, April, June, August, October and December. The element of hk (k = 1, . . . , 12) in the matrix H in a month with the control measures applied was determined by hk = 1 − nk,t D/n2t .

(4.1)

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Optimal population size Control measure applied in April

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Control measure applied in August Control measure applied in six months

800 600 400 200


Sample plot 2 (February 2004-December 2023)

1200 1000 800 600 400 200


0 1400 Sample plot 3 (February 2004-December 2023)

1200 1000 800 600 400 200 0 1400

Sample plot 4 (February 2004-December 2023) 1200



0 1400

1000 800 600 400 200 0 2004

2006

2008 2010

20122014 2016

2018 2020 2022

Fig. 3. Effects of the different control measures on predicted population sizes of A. agrarius 2004–2023.

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Fig. 4. Effects of different control measures on predicted population growth rates of A. agrarius in 2004. For each sample plot and each control measure, λ − 1 was calculated using (2.13).

control measures were implemented in April or August, respectively, the population growth rates for each sample plot decreased in the implemented month, but did not vary in the following months when compared with the growth rates without the control measures (Fig. 1). When the control measure was implemented in all six months, the population growth rates for the corresponding month decreased, except for in April. The population growth rates for each of the sample plots in subsequent years (2005–2023) with no further control measures applied were equal to the original population growth rates shown in Fig. 1. The effects of the control measures on population growth for each sample plot in the period between February 2004 and December 2023 are shown in Fig. 3. When the control measure was applied in April 2004, the population sizes in February and

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April in the following years (2005–2023) were lower for each sample plot than the corresponding optimal population sizes, while in the other months, June, August, October and December, the population sizes remained higher than the optimal population sizes. When the control measure was applied in August 2004, the population sizes in February, April, June and August in the following years (2005–2023) were lower for each sample plot than the optimal population sizes. In October and December, however, the population sizes were close to the optimal. When the control measure was applied in all six months in 2004, the population sizes in the six months of the following years (2005–2023) were lower for each sample plot than the optimal population sizes. Thus, the population sizes of A. agrarius in the period between 2004 and 2023 under the control measure applied in August 2004 were very similar to the optimal population sizes; no significant differences were found between the measured and optimal population sizes in the four sample plots (KS-test). However, the differences between the optimal population sizes and the population sizes under the control measures applied in April and in all months in 2004 for all the sample plots were significantly different. 5. Discussion Many factors can affect the accuracy and precision of population dynamics models, including distance from the stable stage distribution, the variability of demographic rates, and sample size [50]. A small sample size may produce inaccurate estimates of the population size because of large sampling errors [51]. One possible way to minimize sampling error for matrix model estimates is to focus the greatest sampling effort on the life stages to which population size is most sensitive [52]. However, no relevant prior knowledge of the importance of different demographic transitions to population size was available for this study. Therefore, we used four typical sample plots to reduce sampling errors, and also restricted the interval between any two surveys to two months (T = 2), the time within which each age group in the Leslie matrix was contained. In addition to model accuracy, the precision of the model is also important. Thus, the overall accuracy measure (δ), which combines both bias and precision was used to evaluate the models. For sample plots 2 and 3, the mean prediction bias was larger than the standard deviation of the prediction bias (Table 3), indicating that the models for these two sample plots were biased. For the other two sample plots (1 and 4), the mean prediction bias was smaller than the standard deviation of the prediction bias. Population dynamics models that produce precise but biased estimates can still be useful if the magnitude of the bias is known and can be corrected for [53]. When the population sizes from February, June and October in 2004 were used as the starting values in the models for sample plots 1–4, smaller values of the overall accuracy measure (δ) were obtained (Table 3), indicating a perfect fit of the models to the observed data. We found that the population dynamics model starting from

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250 Sample plot 1 Sample plot 2 Sample plot 3 Sample plot 4

200

150

100

50

0 0

50

100

150

200

250

300

350

The mean age of plantation (Months)

Fig. 5. Optimal population densities of A. agrarius with increasing mean age of plantations 2004–2023. In year 2004, the mean age of plantation for each of the sample plots was five years (60 months).

February for each sample plot was more stable than the model starting from any of the other months. This finding may indicate that the population dynamics model is affected by both density-dependent and density-independent factors [29]. From (2.20) and (2.22), it can be seen that, for the optimal population density of A. agrarius, the remaining numbers of A. agrarius could be related to densitydependent factors (e.g. birth rate, transition rate from juvenile to adults, mortality rate of the juvenile, and adult rodents) and density-independent factors (e.g. the mean age of plantation). For population dynamics models [29], the interactions of nonlinear density-dependent and density-independent factors in the capture model are too complicated to intuitively predict model outcomes. To simplify the capture model, we considered only the effects of the mean age of plantation that affect the parameter for the rate of self-regulation (η) of the optimal population density of rodents. We found that the optimal population density of A. agrarius for each sample plot was positively correlated to the mean age of plantation in the Qinshui Forest Farm (Fig. 5). In an ecological sense, this finding is largely explained by the deceasing impact of A. agrarius on the survival rate of young plantations as the mean age of the plantation increases. The optimal population densities among sample plots 1, 2 and 4 were very similar (Fig. 5), indicating that the densitydependent factors had a similar contributory effect on the population size. The three control measures used in this study were implemented in the models by increasing mortality rates of A. agrarius. A control strategy that continuously lowered survival rates was found to result in a die-out of the population [29]. Furthermore, (2.13) indicates that, if the capturing effort is large [i.e. (E1 ) ≥ 1450024-16


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(αβ/r2 −r)], the A. agrarius population would become extinct as the result of excessive capture [42, 43]. Many other means besides increasing mortality rate could be used as control measures; decreasing fertility is one of them [29]. It has been shown that, unlike increasing mortality rate, reducing fertility does not result in population extinction even at higher efficiency, but outbreaks could be avoided [29]. One limitation of fertility control was that it did not result in persistent low population densities even when applied in the intensive periods of reproduction [29]. Future studies that combine increasing mortality rate with decreasing fertility are needed to identify more efficient rodents-associated control measures that are applicable to their population dynamics. Several limitations are associated with the population dynamics model and capture model. First, these models are deterministic, and so the optimal numbers of age groups in the Leslie matrix for specific rodents is uncertain. Age-structured models of population growth consisting of immature and mature individuals [16, 17, 26, 54] or juveniles, sub-adults and adults [29] have been analyzed previously. To reduce sampling error, we classified age-structured groups of A. agrarius into 12 equal age groups each of which covered two months (the maximum lifetime of A. agrarius is two years). Second, stochastic factors such as density-independent factors were not considered in the models. Biological systems are prone to being stochastic [55–57] and this effect may obscure, prevent or alter asymptotic dynamics, which will certainly affect the population dynamics of A. agrarius. Third, the models are densityindependent; therefore, to take into account the damage caused by A. agrarius on the young forest plantations, the mean age of plantation as a density-independent factor was included in the capture model. Thus, different density-independent factors to be introduced into the capture model according to the specific application that is being modeled. Despite the above limitations, our study shows that it is possible to simulate the population growth of A. agrarius using the population dynamics model, and to estimate the optimal population density of A. agrarius using the capture model. Currently, we are collecting data to further validate and re-calibrate our models. It is worth noting that, because of the relative importance of environmental stochasticity for rodent population dynamics, predictions using these models should not be made very far ahead of the current conditions [25]. Therefore, when using the population dynamics and capture models, sampling surveys for rodents in the area under study should be conducted periodically, and used to recalibrate and validate the existing models. 6. Conclusions Here, we studied the control of rodents in young plantation areas. By integrating information derived from a CMR study, a population dynamics model and capture model were developed using a Leslie matrix and optimal harvesting theory, respectively. We found that, despite its limitations, the population dynamics model could predict population sizes of rodents with reasonable accuracy and precision. 1450024-17


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This model can also be used as a tool for providing insights into the possible influence of density-dependent factors. The capture model was developed by considering the importance of ecological stability and the minimization of damage to young plantations by rodents in a forest farm. This model is useful for designing and evaluating pest control strategies. We recommend that these models are used to assess population dynamics and control strategies for rodents in forest farms.


Acknowledgments The authors thank Prof. Yemao Xia, Prof. Xiangdong Lei and other researchers at the Chinese Academy of Forestry for their insightful suggestions. We also thank two anonymous reviewers for useful comments on previous versions of the manuscript. Total expense of field investigation was borne by the Chinese National Natural Science Foundations (Nos. 31300534, 31100476) for financial support for this study.

Appendix A Proposition. When αβ/r2 − r > E,

x˙ = γy − rx − ηx2 − E1 x,

(A.1)

y˙ = ρx − r2 y, have only one positive equilibrium point, also called the stable node.

Proof. (1) The equilibrium points in Eq. (A.1) are obtained by solving the following equations: γy − rx − ηx2 − E1 x = 0, (A.2) ρx − r2 y = 0. Only one set of nonzero solutions (x1 , y1 ) can be obtained from Eqs. (A.1) and (A.2), namely

1 αβ β αβ − r − E , y1 = −r−E . x1 = η r2 ηr2 r2 Thus, when αβ/r2 − r > E, and x1 > 0 and y1 > 0, this solution is the only positive equilibrium point. (2) The Jacobi matrix at point (x1 , y1 ) for Eq. (A.1) is J(x1 , y1 ) =

−r − E − 2ηx1

α

β

−r2

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1 x

+

1 y

”. ln(A)

(F )

k=1

,m 1 ” X nk,t r2

r1

γ

x

k=m1 +1

k=m1 +1

k=m1 +1 ,m m1 h 1 ”i X “ X (L) nk,t − nk+1,t+1 − nk+1,t+1 r1 = nk,t k=1 k=1 , m m “ h ”i X X (L) r2 = nk,t − nk+1,t+1 − nk+1,t+1

(F )

dk = nk,t /nk,t , m1 X x= nk,t S k=1, m X γ = n1,t nk,t

Formula

nk,t

(L)

during the time interval t to t + 1. nk,t is the number of females in the kth age group at time t · nk+1,t+1 is the number of individuals that come from an outside system and belong to the (k + 1)th age group during the time interval t to t + 1. nk+1,t+1 is the total number of individuals in the (k + 1)th age group at time t + 1. S is the size of sample plots (hectare).

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Note: In the capture model, the rodents in the first to the m1 th age groups are assumed to be juveniles. In the (m1 + 1)th to the (O) mth age groups (0 < m1 < m) the rodents are assumed to be adult. nk,t+1 is the total number of offspring born in the kth age group

η=

“ (L) ρ = nm1 +1,t+1 − nm1 +1,t+1

k=m1 +1

“ ”. (L) qk = nk+1,t+1 − nk+1,t+1 nk,t , m X y= nk,t S

dk

Parameters

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bk =

(O) (F ) nk,t+1 /nk,t

Formula

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bk

Parameters

Formulae for the parameters in the population dynamics model and capture model were obtained from the CMR data.


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and the eigenvalue of the matrix satisfies the following equation:

y1 x1 y1 x1 λ2 + α + ηx1 + β − αβ = 0. λ + α + ηx1 β x1 y1 x1 y1 When the two roots are set asλ1 , λ2 , then λ1 + λ2 < 0, λ1 λ2 > 0 and

2 y1 x1 + 4αβ > 0. ∆= α + ηx1 − β x1 y1 Thus, for λ1 < 0 and λ2 < 0, (x1 , y1 ) is a stable node.


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