Page 1 of 31 1 2 3 Ttle: Lfe Tables of Bactrocera cucurbtae (Coqullett) (Dptera: Tephrtdae): wth a Mathematcal Invaldaton for Applyng the Jackknfe Technque to the Net Reproductve Rate 4 5 Runnng ttle: Lfe table and jackknfe technque 6 7 YU-BING HUANG and HSIN CHI 8 9 1 11 12 13 14 15 16 Laboratory of Theoretcal and Appled Ecology, Department of Entomology, Natonal Chung Hsng Unversty, Tachung, Tawan, Republc of Chna Address all correspondence to: Hsn Ch P.O. Box 17-25, Tachung, Tawan Tel.: 886-4-932633124 E-mal: hsnch@dragon.nchu.edu.tw
Page 2 of 31 17 18 19 2 21 22 23 24 25 26 27 28 29 3 31 32 33 34 35 36 37 38 Abstract 1. Lfe table data for the melon fly, Bactrocera cucurbtae (Coqullett), reared on cucumber (Cucums satvus L.) were collected under laboratory and smulated feld condtons. 2. Means and standard errors of lfe table parameters were estmated for two replcates usng the jackknfe technque. 3. At 25ºC, the ntrnsc rates of ncrease (r) found for the two replcates were.1354 and.12 day -1, and the net reproductve rates (R ) were 26.3 and 66. offsprng, respectvely. 4. When the cucumbers kept under smulated feld condtons were covered wth leaves, the r and R for the two replcates were.935 and.99 day -1, 17.5 and 11.4 offsprng, respectvely. However, when smlar cucumbers were left uncovered, the r and R for the two replcates were.143 and.94 day -1, and 27.7 and 1.1 offsprng, respectvely. 5. Our results revealed that consderable varablty between replcates n both laboratory and feld condtons s possble; ths varablty should be taken nto consderaton n data collecton and applcaton of lfe tables. 6. Mathematcal analyss has demonstrated that applyng the jackknfe technque results n unrealstc pseudo-r and overestmaton of ts varance. 7. We suggest that the jackknfe technque should not be used for the estmaton of varablty of R. 39 Key words. Bactrocera cucurbtae, Cucums satvus, lfe table, net reproductve 4 rate, jackknfe method.
Page 3 of 31 41 42 43 44 45 46 47 Introducton The melon fly, Bactrocera cucurbtae (Coqullett) (Dptera: Tephrtdae), has been one of the most mportant pests n Tawan (Huang & Ch, 211), and n many other regons n Asa (Koyama et al., 24; Dhllon et al., 25) for several decades. Although the agrcultural agences have nvested heavly n research, workshops, and control measures related to the fly, t remans a major pest n Tawan (Huang & Ch, 211). For sustanable pest management n organc farmng, t s crucal to develop a 48 comprehensve understandng of the populaton ecology of the target pests. Lfe 49 5 51 52 53 54 55 56 57 58 table studes should be the frst prorty n ecologcally sound pest management programs because only lfe tables can provde the most detaled and correct descrptons of the survval, stage dfferentaton, and reproducton of populatons. Age-specfc female lfe tables of B. cucurbtae were developed by Vargas et al. (1996, 1997, 2) and Yang et al. (1994). However, the theores relatng to female age-specfc lfe tables (Lews, 1942; Lesle, 1945; Brch, 1948) address only female populatons and gnore male populatons. Ch & Lu (1985) and Ch (1988) observed that female age-specfc lfe tables cannot correctly descrbe the growth and stage dfferentaton of nsect and mte populatons. Thus, although numerous female lfe tables have been publshed for many nsect speces, ther practcal 59 applcatons are qute lmted. Huang & Ch (211) reported the frst age-stage, 6 two-sex lfe table for B. cucurbtae under laboratory condtons wth cucumber slces 61 as the rearng medum. They demonstrated that an erroneous relatonshp s obtaned 62 f an age-specfc female lfe table s appled to a two-sex populaton. Furthermore, 63 64 65 they ndcated that the study of lfe tables constructed under feld condtons can be helpful by revealng dfferences between the values of populaton parameters n the feld and n the laboratory.
Page 4 of 31 66 67 68 Lqudo (1991) demonstrated that fallen fruts on the ground act as a reservor for melon fly populatons. To construct precse predctons of the dynamcs of populatons n the feld, t s necessary to dentfy the dfferences between lfe tables 69 collected n the laboratory and those actual lfe tables under feld condtons. On the 7 71 72 73 74 75 76 77 78 79 8 81 82 83 other hand, due to the tedous and tme-consumng work of lfe table studes, most lfe table studes are carred out by usng sngle cohort wthout replcaton. To estmate the means and varances of populaton parameters obtaned from a sngle cohort, jackknfe technque s wdely used. Meyer et al. (1986) used jackknfe and bootstrap technques n estmatng uncertanty n ntrnsc rate and concluded that jackknfe was more cost-effectve based on smulaton. Efron & Tbshran (1993) dscussed the falure of jackknfe. Ch & Yang (23) ponted out that applcaton of jackknfe wll result n some degree of dscrepancy between the estmated means of populaton parameters and ther theoretcal defnton. When we use the jackknfe method to estmate the mean value of the net reproductve rate, we often obtan some pseudo-r value of zero. An mathematcal explanaton s needed to justfy or falsfy the use of jackknfe technque. In ths study, eggs of melon fles were artfcally ntroduced nto whole cucumbers (Cucums satvus L.), then kept at 25ºC and under feld wth replcatons. Lfe tables were constructed and the populaton parameters were 84 measured for replcates. Furthermore, we derved a mathematcal proof to 85 86 demonstrate the problem of the jackknfe method for the estmaton of the mean and standard error of the net reproductve rate. 87 88 89 9 Materals and methods Lfe Table Study Melon fles were collected n a feld used to grow vegetables and subsequently
Page 5 of 31 91 reared on cucumber (Cucums satvus L.). The colony was mantaned n the 92 93 94 laboratory of the Department of Entomology, Natonal Chung Hsng Unversty (Tachung, Tawan) for two generatons before the begnnng of the lfe table study. For the lfe table study, eggs lad wthn 24 h were collected usng pled cucumber 95 slces followng the method of Huang & Ch (211). For mplantng eggs nto the 96 97 cucumber, a pyramd-shaped hole wth a rectangular base (1.5 cm each sde, 1.5 cm heght) was cut wth an arrowhead-shaped knfe. Twenty eggs were placed n the 98 hole wth a fne wrtng brush. Before the pyramd-shaped cucumber pece was 99 1 11 12 13 14 15 16 17 replaced, ts tp was removed to leave a space for the eggs. To study the cohort lfe tables at 25 C, fve cucumbers wth eggs were kept n a plastc jar (26 cm heght, 23 cm dameter) wth loamy sol. The mouth of the jar was covered wth fne mesh net and kept at a constant temperature of 25ºC n a growth chamber under a photoperod of 12:12 (L:D) h. To study the lfe table under feld condtons, fve cucumbers wth eggs were placed n a jar, kept n a shaded area and covered wth dred mango leaves. Another fve cucumbers wth eggs were placed n a jar and kept under drect sunlght n the feld wth no leaf cover. The feld study was conducted from 5 June to 9 September 26. The average feld temperature was 28.1ºC. Two replcates were 18 used for each treatment. The numbers of emerged adults were observed, and pars of 19 adults were formed. The eggs lad daly by the melon fles were collected on slced 11 cucumber as descrbed n Huang & Ch (211). 111 112 113 114 115 Demographc Analyss The lfe hstory data were analyzed accordng to the age-stage, two-sex lfe table theory (Ch & Lu, 1985) and the method descrbed by Ch (1988). The means and standard errors of the lfe table parameters were estmated wth the jackknfe method
Page 6 of 31 116 (Sokal & Rohlf, 1995). The populaton parameters estmated were the ntrnsc rate 117 of ncrease (r), the fnte rate of ncrease (λ), the gross reproductve rate (GRR), the 118 net reproductve rate (R ) and the mean generaton tme (T). In ths paper, the 119 12 ntrnsc rate of ncrease s estmated wth the teratve bsecton method from the Euler-Lotka formula 121 122 123 124 125 126 127 128 129 13 131 132 133 134 135 x -r x1 e l m 1 (1) wth age ndexed from (Goodman, 1982). The mean generaton tme s defned as the length of tme that a populaton needs to ncrease to R -fold of ts sze (.e., e rt = R or λ T = R ) at the stable age-stage dstrbuton and s calculated as T ln R )/r. x x ( The age-stage lfe expectancy (e xj ) s calculated accordng to Ch & Su (26). To facltate the tedous process of raw data analyss, a computer program TWOSEX-MSChart for the age-stage, two-sex lfe table analyss (Ch, 21) n Vsual BASIC (verson 6, servce pack 6) for the Wndows system s avalable at http://14.12.197.173/ Ecology/ (Chung Hsng Unversty) and at http://nhsbg.nhs.uuc.edu.tw/www/ch.html (Illnos Natural Hstory Survey). We used a Tukey-Kramer procedure (Dunnett, 198) to compare the dfference among treatments followng the descrpton of Sokal & Rohlf (1995). Results Lfe Table of B. cucurbtae 136 The developmental tmes for each stage are lsted n Table 1. At 25 ºC, the 137 138 139 duraton of the preadult stage n whole cucumber was 17.8 and 18.5 d (two replcates). Ths value was much greater than the correspondng value for growth n cucumber kept under feld condtons wth or wthout leaf coverage. The adult
Page 7 of 31 14 141 142 pre-ovpostonal perods (APOP) n the dfferent treatments ranged from 7. to 9.1 d. There were no sgnfcant dfferences among these values. The total pre-ovpostonal perod (TPOP) at 25 C was, however, sgnfcantly longer than those 143 found n the feld. The adult longevtes of both male and female adults at 25 C are 144 also longer than those observed under feld condtons. The total fecundty vared 145 sgnfcantly among treatments (Table 2). Sgnfcantly hgher fecundtes (859 and 146 147 148 149 15 151 152 153 154 155 156 157 158 159 66 eggs/female) were observed n females reared at 25ºC than n females emerged under feld condtons. The hgh coeffcents of varaton (CV) of mean fecundtes showed the hgh reproductve varablty among ndvduals. The detaled age-stage survval rates (s xj ) of B. cucurbtae for the dfferent treatments are plotted n Fg. 1. The parameter s xj s the probablty that a newborn wll survve to age x and stage j. The survval rate curves of B. cucurbtae cohorts vary sgnfcantly between replcates for populatons reared n whole cucumbers. In general, the survval rate n the laboratory s hgher than n the other treatments. At 25 C, cohorts n the laboratory survved longer than those n the feld. Ths dfference s also evdent from the longer developmental tme of the preadult stage and from the adult longevtes (Table 1). The daly mean number of offsprng produced by ndvdual B. cucurbtae of age x and stage j per day s shown wth the age-stage fecundty (f xj ) n Fg. 2. Because only adult females produce offsprng, there s only a sngle curve f x2 (.e., the 16 adult female s the second lfe hstory stage). The age-specfc survval rate (l x ) and 161 the age-specfc fecundty (m x ) are also plotted n Fg. 2. The l x curve descrbes the 162 change n the survval rate of the cohort wth age. Sgnfcant varablty can be 163 164 observed between the two replcates. In one replcate at 25ºC, more than 4% B. cucurbtae survved to the adult stage, but the correspondng value n another
Page 8 of 31 165 replcate was much smaller, approxmately 2%. However, at 25 C, the survval 166 rates n the laboratory are hgher than those n the feld (Fg. 2). 167 168 169 Populaton Parameters The means and standard errors of populaton parameters of B. cucurbtae n the 17 dfferent treatments nvestgated are lsted n Table 2. For the eggs artfcally 171 172 173 174 175 176 177 178 179 18 181 182 183 184 185 186 187 188 189 placed n cucumber and kept at 25ºC, the ntrnsc rates of ncrease (r) found for the two replcates were.1354 and.12 day -1, the net reproductve rates (R ) were 26.3 and 66. offsprng, and the mean generaton tmes (T) were 39.5 and 42.6 days, respectvely. For the cucumbers kept n the feld and covered wth leaves, the populaton parameters (r, R and T) were.935 and.99 day -1, 17.5 and 11.4 offsprng, and 34. and 35. days, respectvely. However, for the cucumbers kept n the feld wthout leaves, the populaton parameters (r, R and T) were.143 and.94 day -1, 27.7 and 1.1 offsprng, and 32.8 and 27.2 days, respectvely. The maxmum ntrnsc rate of ncrease (.1354 d -1 ) was obtaned at 25ºC n the laboratory. All parameters have very hgh values of CV. The age-stage specfc lfe expectancy (e xj ) (Fg. 3) s the lfespan remanng for an ndvdual of age x and stage j. The contrbuton of an ndvdual of age x and stage j to the future populaton s descrbed by the age-stage reproductve value (v xj ) (Fg. 4). The reproductve value of a newborn (v 1 ) s exactly equal to the fnte rate of ncrease. Dscusson Lfe Table of B. cucurbtae The shorter preadult stage n the treatment under feld condtons wth leaf
Page 9 of 31 19 coverage mght be due to the hgher temperature and the hgher humdty. These 191 condtons can promote the decay of cucumber and thereby generate condtons 192 favorable for fles. Vayssères et al. (28) reported that the total preadult 193 194 195 development tme of B. cucurbtae on cucumber at 25 and 3ºC was 17.2 and 13.2 days, respectvely. Huang & Ch (211) reported that the total preadult development tme of B. cucurbtae was 15.1 days at 25ºC. These studes show that the preadult 196 development tme of B. cucurbtae decreases as the temperature ncreases. Under 197 198 199 2 21 22 23 24 25 26 27 28 feld condtons, melon fles n dfferent fallen fruts may experence dfferent mcro-envronments and may result n hgher varatons n developmental rate, survval and reproducton. Because the varable developmental rate among ndvduals s ncorporated n the age-stage, two-sex lfe table, the overlap between stages can be observed n Fg. 1. If the survval curves were constructed based on the means of each stage or adult age (e.g., Marcc, 23, 25; Legasp, 24; Legasp & Legasp, 25; Ln & Ren, 25; Lu, 25; Kvan & Klc, 26; Kontodmas & Stathas, 25; Tsoukanas et al., 26), the stage overlap would not have been observed and would have resulted n errors n the survval curves as well as the fecundty curves. Lu (25) notced the overlap of the stages of Delphastus catalnae (Coleoptera: Coccnelldae). Nevertheless, he gnored the varable developmental rate and constructed 29 age-specfc fecundty schedules based on adult age. Yu et al. (25) and Ch & Su 21 211 212 213 214 (26) gave detaled explanatons and a mathematcal proof to address the errors n lfe tables based on adult age. In Vargas et al. (1997), the fecundty of B. cucurbtae at 24ºC was 578.6 eggs. In Huang & Ch (211), the mean fecundty of melon fles reared on cucumber at 25ºC was 341 eggs. Jang et al. (26) reported that the mean fecundty of melon
Page 1 of 31 215 216 217 218 219 22 221 222 fles reared on cucumber at 3ºC was 895.65 eggs. In ths study, the mean fecundty of B. cucurbtae reared on whole cucumber at 25ºC was hgher than the fecundty gven n Huang & Ch (211). If the survval rate and fecundty are constructed based solely on the adult age, the dfferences n preadult development are gnored, and t s assumed that all adults emerge on the same day. These artfcal manpulatons and assumptons wll not only falsely dmnsh the real varablty among ndvduals, but also consequently result n errors n the survval and fecundty curves (Ch, 1988; Yu et al., 25; Ch & Su, 26; Huang & Ch, 211). 223 224 225 226 227 228 229 23 231 232 233 234 235 Populaton Parameters Due to the problems assocated wth the female age-specfc lfe table (Huang & Ch, 211), we used the age-stage, two-sex lfe table to calculate the populaton parameters of B. cucurbtae. The ntrnsc rate of ncrease (r) ranged from.94 to.1354 days -1. The treatments dd not dffer sgnfcantly based on the estmated means and standard errors obtaned by usng the jackknfe technque and Tukey-Kramer procedure. The net reproductve rate (R ) of melon fles reared n the laboratory at 25ºC was hgher than the correspondng rate under feld condtons. The relatonshp between the net reproductve rate R and the mean female fecundty F was gven by Ch (1988) for the two-sex lfe table as N f R F (2) N where N s the total number of eggs used for the lfe table study at the begnnng and 236 N f s the number of female adults emerged. Yu et al. (25) gave the relatonshp 237 238 239 among the gross reproductve rate (GRR), the net reproductve rate (R ) and the preadult survvorshp (l a ) as GRR la GRR (3) R
Page 11 of 31 24 All of our results for B. cucurbtae at dfferent treatments are consstent wth the 241 relatonshps gven by equatons 2 and 3. If a lfe table s constructed based on 242 adult age and gnores the preadult mortalty, an erroneous relatonshp between the 243 244 mean fecundty and the net reproductve rate wll be obtaned. Ch & Su (26) dscussed ths problem n detal. Yu et al. (25) and 245 246 247 248 249 25 251 252 253 254 255 256 257 The shorter preovposton perod wll cause a hgher ntrnsc rate of ncrease f fecundty remans the same (Lewontn, 1965). In the study of Huang & Ch (211), the TPOP of B. cucurbtae reared on cucumber at 25ºC was 23.1 d. In our study, the TPOP,.e., the duraton from egg to frst ovposton, of melon fles reared n the laboratory at 25ºC was longer than that under feld condtons. Ths result mght be explaned by the hgher feld temperature (28ºC) and humdty. At 25ºC, the age-stage lfe expectancy gradually decreases wth age because no other adverse effects occur n the laboratory. Under feld condtons, however, the lfe expectances were lower and vared sgnfcantly due to the varable abotc factors. The lfe expectancy s calculated usng the age-stage specfc survval rate (s xj ) wthout assumng that the populaton reaches the stable age-stage dstrbuton (Ch & Su, 26). Thus, t can be used to predct the survval of a populaton under those condtons. For example, at 25ºC both newly emerged female and male adults can 258 be expected to reman alve, on average, more than two months. The lfe 259 26 261 262 263 264 expectancy based on the age-stage, two-sex lfe table reveals the dfference among ndvduals of the same age but of dfferent stages or dfferent sexes. Ch (1988), Ch & Yang (23) and Ch & Su (26) dscussed n detal the dfferences between the tradtonal female age-specfc lfe table and the age-stage, two-sex lfe table and dentfed possble errors n the survval and fecundty curves based on the adult age. Fsher (193) defned the reproductve value as the contrbuton of an ndvdual
Page 12 of 31 265 to the future populaton. The reproductve value sgnfcantly ncreases at the tme 266 of emergence of the adult females. For example, when a female adult emerges at 267 268 269 27 271 age 15 d at 25ºC (Fg. 1), the reproductve value ncreases from a value of less than 1 for a nymph to 36 for a female (Fg. 4). The contrbuton of males to the future populaton s not defned by Fsher (193), and there s no curve for males. The research reported here demonstrates that only lfe table study can completely depct the development, stage dfferentaton, and reproducton of B. 272 cucurbtae and the varablty of these processes n whole cucumber. Moreover, t 273 274 275 276 277 278 279 28 281 282 283 284 285 286 287 288 289 revealed sgnfcant dfferences between lfe tables collected n the laboratory and the feld. Thus, computer smulatons of the growth of feld populatons should ncorporate consderatons of these dfferences. Ch (199) noted that a smulaton based on the age-stage, two-sex lfe table can be used to tme pest management by takng the stage-specfc susceptblty to pestcde applcatons nto consderaton. Ch & Getz (1988) constructed a mass-rearng model based on the age-stage, two-sex lfe table. For an ecology-orented ntegrated pest management of B. cucurbtae, lfe tables collected under dfferent condtons should play mportant roles n the future. However, because a varety of wld cucurbts serve as a host for the melon fly and form a reservor for ths fly (Uchda et al., 199), t mght be necessary to understand the lfe table of the fly on the major wld cucurbts. Usng the Jackknfe Method to Estmate of the Net Reproductve Rate Our results showed hgh values of CV n female mean fecundty and populaton parameters. The hgh CV n mean fecundty s calculated by usng basc descrptve statstcal method and they reflect the dfferences among female ndvduals. The hgh CVs of populaton parameters are, however, estmated by usng the jackknfe
Page 13 of 31 29 technque. The jackknfe technque s a resamplng method whch s usually used 291 when replcaton s mpossble or dffcult. Because lfe table studes are tme- and 292 labor-consumng, replcaton s n general mpractcal n most cases. The jackknfe 293 294 295 method s thus used to estmate the means and standard errors of populaton parameters (Ch & Getz, 1988; Maa et al., 2; Huang & Ch, 211). In the jackknfe method, we frst use data on all ndvduals (n) to calculate the ntrnsc rate 296 of ncrease of the whole cohort (r all ). We then calculate the ntrnsc rate r by 297 298 299 3 31 32 33 34 35 36 37 omttng ndvdual. The pseudo-value r -pseudo s then calculated as: pseudo n r all n r (4) r 1 where n s the total number of ndvduals used at the begnnng of the lfe table study. The mean value of all r -pseudo s the estmated mean value of the ntrnsc rate of ncrease of the cohort: n r pseudo r 1 n (5) Smlarly, f we use the jackknfe method to calculate the mean value of the net reproductve rate, we frst use data on all ndvduals n the cohort to calculate R,all : R, all lxmx. (6) If the total number of eggs lad by all survvng ndvduals at age x s F x, the total eggs lad by the whole cohort from brth to death s F total and can be calculated as x 38 x F. Then, the R,all can also be calculated as x 39 R, all l m nx Fx n n Fx n 1 n x x x x x x x F x F n total (7) 31 311 where n x s the number of survvng ndvduals at age x. Equaton 7 shows that the net reproductve rate s F total dvded by the total number of ndvduals n used at the
Page 14 of 31 312 313 314 315 316 317 318 319 32 321 322 323 324 325 326 327 328 begnnng of the lfe table study. If the omtted ndvdual s type N (those dyng at mmature stages) or M (male), we defne the total eggs lad by n-1 ndvduals at age x as F x,. It s clear that F x, = F x for all ages, because types N and M do not lay eggs. The net reproductve rate wth ndvdual omtted,.e., R,, can be calculated as n F F F 1 R F (8), x, x, x, x x n 1 nx, x n 1 x n 1 n 1 x where n x, s the number of survvng ndvduals at age x f ndvdual s omtted. The pseudo-value for the omsson of ndvdual s calculated analogously to Equaton 4: n R R, pseudo n R, all 1, (9) Replacng R, accordng to the proofs of Equaton 7 and 8, we fnd 1 1 R, pseudo n Fx n 1 Fx (1) n x n 1 x Consequently, we obtan, pseudo x x x x R F F (11) Thus, we prove that f the omtted ndvdual s type N or M, the pseudo-value R,-pseudo wll always be zero. If the omtted ndvdual s a female and can produce b x, eggs at age x, the total number of eggs lad by ths female durng ts lfe span can be calculated as x 329 33 331 332 B b x, (12) x If ndvdual s omtted, then the total eggs produced by the remanng ndvduals n cohort at age x s F x,t. It s clear that F x, Fx bx, or F x Fx, bx, (13) 333 Accordng to Equaton 8, we have
Page 15 of 31 334 335 336 337 338 1 1 R, Fx, Fx bx, (14) n 1 n 1 x The pseudo-value for the omsson of ndvdual s x n R R, pseudo n R, all 1, (15) Replacng R, of Equaton 15 wth ts value n Equaton 14, we can smplfy Equaton 15 to 16. 339 R, pseudo 1 n n x F x 1 n 1 n 1 Fx bx, x 34 341 342 343 344 345 346 347 348, pseudo Fx Fx bx, x x x x R b B (16) It s clear that f the omtted ndvdual s a female, the pseudo-value of the net reproductve rate s exactly the total fecundty of ndvdual tself, R pseudo, bx, B (17) x Ths analyss shows that f the jackknfe method s used, the pseudo-value of the net reproductve rate obtaned by omttng ndvdual s exactly the total number of eggs lad by ndvdual. It s exactly the fecundty of ndvdual. The mean of all pseudo-values s the total number of eggs lad by all ndvduals dvded by n: n n R, pseudo B Rˆ 1 1 (18) n n x, 349 By defnton, t s clear that n B F. 1 x x 35 The mean of all R,-pseudo s then 351 352 n B F. (19) n n R ˆ 1 R, all The above proof can be concluded by makng the followng four observatons: 1)
Page 16 of 31 353 354 355 356 357 358 359 36 361 362 363 364 365 366 367 368 369 37 371 372 373 374 375 376 377 the mean value of the net reproductve rate estmated wth the jackknfe method s exactly the same as the R,all wthout the use of the jackknfe method; 2) the net reproductve rate equals the total eggs of the cohort dvded by n,.e., the total number of newborns used for the lfe table study; 3) f the omtted ndvdual s one of the males or one of those that ded at an mmature stage, the pseudo-value s zero; and 4) f the omtted ndvdual s female, the pseudo-value s the fecundty of that omtted female. In Fg. 5, the frequency dstrbutons of pseudo-r values of three treatments showed the zeros obtaned by usng the jackknfe technque. It s clear that the omsson of a sngle ndvdual of type N or M wll generate a pseudo-r of zero. The hgher the preadult mortalty or proporton of male, the hgher the zero pseudo-r bar. Because there s generally preadult mortalty, the bar of zero pseudo-r wll be an mportant factor determnng the frequency dstrbuton of all lfe table data. Ths s also the reason why statstcal software shows the pseudo-r faled the normalty test and nstead suggests Mann-Whtney Rank Sum test or others. The omsson of a sngle ndvdual of type N or M caused the pseudo-r of the resampled populaton to zero. If we carry out a true replcaton of lfe table study as we dd n ths study, however, we wll generally not get a populaton wth zero net reproductve rate,.e., all ndvduals are ether type N or M. Ths shows the jackknfe technque wll generate bologcally unrealstc pseudo-r, whch results n an overestmaton of varances and standard errors of the net reproductve rates. The overestmaton of varances and standard errors consequently make sgnfcant dfferences between treatments undetectable by usng statstcal tests. Varance analyss s mportant for revealng the varablty of expermental results. The queston of the sutablty of the jackknfe method for the estmaton of
Page 17 of 31 378 379 38 381 382 383 384 385 386 387 388 389 39 391 392 393 394 395 396 397 398 399 the mean and standard errors of the net reproductve rate s not the only dffculty assocated wth lfe table analyss. The sample sze must be suffcently large to prevent naccurate estmaton of the standard errors. Because there are many problems assocated wth female lfe tables and analyses based on adult age (Ch & Lu, 1985; Ch, 1988; Yu et al., 25; Ch & Su, 26; Huang & Ch, 211), the applcaton of the jackknfe method to female lfe tables (Lesle, 1945; Brch, 1948; Maa et al., 2) or n analyses based on female populaton and adult age (Maa et al., 2) wll not produce correct estmates. The sgnfcant dfferences between replcates n ths study showed, however, that the varablty n developmental rate, survval, and reproducton of a lfe table could not be properly descrbed and estmated wth the jackknfe method. For ths reason and many others, the predcton of populaton dynamcs under feld condtons s dffcult. In ths paper, we lmt our dscusson to the applcaton of jackknfe method to the net reproductve rate. There are other resamplng methods, e.g., bootstrappng, permutaton test, cross valdaton, etc. Smlar analyss s requred to re-evaluate ther applcaton n the estmaton of means and varances of populaton parameters. Despte these dffcultes and problems, the lfe table s the only sold theory whch can correctly descrbe the survval, stage dfferentaton, and reproducton n detal. The necessty and the dffcultes assocated wth lfe table study demonstrate that we need to draw the attenton of scentsts to lfe table theory and data analyss n nsect ecology, ntegrated pest management, as well as bologcal control.
Page 18 of 31 4 41 42 43 Acknowledgments Ths research s supported by the Bureau of Anmal and Plant Health Inspecton and Quarantne, Tawan (95AS-13.2.1-BQ-B3(5), 94AS-13.2.1-BQ-B3(5), 93AS-1.8.1-BQ-B3(5)), 92AS-1.8.1-BQ-B3(7)).
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Page 21 of 31 454 455 456 457 458 459 46 461 462 463 464 465 466 467 468 469 47 471 472 473 474 475 476 477 478 Legasp, J.C. (24) Lfe hstory of Podsus maculventrs (Heteroptera: Pentatomdae) adult females under dfferent constant temperatures. Envronmental Entomology, 33, 12-126. Legasp, J.C. & Legasp, B.C. (25) Lfe table analyss for Podsus maculventrs mmatures and female adults under four constant temperatures. Envronmental Entomology, 34, 99-998. Lesle, P.H. (1945) One the use of matrces n certan populaton mathematcs. Bometrka, 33, 183-212. Lews, E.G. (1942) One the generaton and growth of a populaton. Sankhya, 6, 93-96. Lewontn, R.C. (1965) Selecton for colonzng ablty. The Genetc of Colonzng Speces (ed. by H. G. Baker and G. L. Stebbns), pp. 77-94. Academc Press, San Dego, CA. Ln, L. & Ren, S.X. (25) Development and reproducton of B botype Bemsa tabac (Gennadus) (Homoptera: Aleyroddae) on four ornamentals. Insect Scence, 12, 137-142. Lqudo, N.J. (1991) Frut on the ground as a reservor of resdent melon fly (Dptera: Tephrtdae) populatons n papaya orchards. Envronmental Entomology, 2, 62-625. Lu, T.X. (25) Lfe hstory and lfe table analyss of the whtefly predator Delphastus catalnae (Coleoptera: Coccnelldae) on collards. Insect Scence, 12, 129-135. Maa, A.H.N., Luz, A.J.B. & Campanhola, C. (2) Statstcal nference on 455 assocated fertlty lfe table parameters usng jackknfe technque: 456 computatonal aspects. Journal of Economc Entomology, 93, 511-518.
Page 22 of 31 479 48 481 482 483 484 485 486 487 488 489 49 491 492 493 494 495 496 497 498 499 5 51 52 53 Marcc, D. (23) The effects of clofentezne on lfe-table parameters n two-spotted spder mte Tetranychus urtcae. Expermental and Appled Acarology, 3, 249-263. Marcc, D. (25) Sublethal effects of tebufenpyrad on the eggs and mmatures of two-spotted spder mte, Tetranychus urtcae. Expermental and Appled Acarology, 36, 177-185. Meyer, J.S., Ingersoll, C.G., McDonald, L.L. & Boyce, M.S. (1986) Estmatng uncertanty n populaton growth rates: Jackknfe vs. Bootstrap technques. Ecology, 67, 1156-1166. Sokal, R.R. & Rohlf, F.J. (1995) Bometry. 3rd ed. W. H. Freeman, San Francsco, Calforna, USA. Tsoukanas, V.I., Papadopoulos, G.D., Fantnou, A.A. & Papadouls, G.Th. (26) Temperature-dependent development and lfe table of Iphseus degenerans (Acar: Phytosedae). Envronmental Entomology, 35, 212-218. Uchda, G.K., Vargas, R.I., Beardsley, J.W. & Lqudo, N.J. (199) Host sutablty of wld cucurbts for melon fly, Dacus cucurbtae Coqullett, n Hawa, wth notes on ther dstrbuton and taxonomc status. Proceedngs of the Hawaan Entomologcal Socety, 3, 37-52. Vargas, R.I., Walsh, W.A., Jang, E.B., Armstrong, J.W. & Kanehsa, D.T. (1996) Survval and development of mmature stage of four Hawaan frut fles (Dptera: Tephrtdae) reared at fve constant temperatures. Annuals of the Entomologcal Socety of Amerca, 89, 64-69. Vargas, R.I., Walsh, W.A., Kanehsa, D., Jang, E.B. & Armstrong, J.W. (1997) Demography of four Hawaan frut fles (Dptera: Tephrtdae) reared at fve constant temperatures. Annuals of the Entomologcal Socety of Amerca, 9,
Page 23 of 31 54 55 56 57 58 59 51 511 512 513 514 515 516 517 518 519 52 521 162-168. Vargas, R.I., Walsh, W.A., Kanehsa, D., Stark, J.D. & Nshda, T. (2) Comparatve demography of three Hawaan frut fles (Dptera: Tephrtdae) at alternatng temperatures. Annuals of the Entomologcal Socety of Amerca, 93, 75-81. Vayssères, J.F., Carel, Y., Coubes, M. & Duyck, P.F. (28) Development of mmature stages and comparatve demography of two cucurbt-attackng frut fles n Reunon Island: Bactrocera cucurbtae and Dacus clatus (Dptera Tephrtdae). Envronmental Entomology, 37, 37-314. Yang, P., Carey, J.R. & Dowell, R.V. (1994) Comparatve demography of two cucurbt-attackng frut fles, Bactrocera tau and B. cucurbtae (Dptera: Tephrtdae). Annuals of the Entomologcal Socety of Amerca, 87, 538-545. Yu, J.Z., Ch, H. & Chen, B.H. (25) Lfe table and predaton of Lemna bplagata (Coleoptera: Coccnelldae) fed on Aphs gossyp (Homoptera: Aphddae) wth a proof on relatonshp among gross reproducton rate, net reproducton rate and preadult survvorshp. Annuals of the Entomologcal Socety of Amerca, 98, 475-482.
Page 24 of 31 522 523 524 Table 1. Means and standard errors of the developmental tme, longevty, adult preovposton perod (APOP) and total preovposton perod (TPOP) of Bactrocera cucurbtae for dfferent treatments Feld condtons 25ºC Parameter Stage Wthout leaf coverage Wth leaf coverage Rep. 1 Rep. 2 Rep. 1 Rep. 2 Rep. 1 Rep. 2 Developmental Preadult 17.8 ±.2 a 18.5 ±.2 b 11.4 ±.2 c 11.4 ±.1 c 11. ±. c 11.2 ±.2 c tme (days) Adult longevty Male 74.8 ± 7.1 a 63.2 ± 9.5 a 33.6 ± 13.1 b 34.7 ± 11.8 b 63.9 ± 9.8 a 13.2 ± 8.7 b (days) Female 58.9 ± 6.5 a 45.6 ± 12. a 55.4 ± 11.6 a 16.3 ± 4.7 b 28.5 ± 8.9 a 22.6 ± 14.5 a APOP (days) Female 8.7 ±.3 a 8.9 ±.6 a 9.1 ±.3 a 8.6 ±.4 a 9. ±.6 a 7. ± 2. a TPOP (days) Female 26.7 ±.3 a 28. ±.8 b 2.7 ±.3 c 2. ±.3 c 2. ±.6 c 18.5 ± 1.5 c Means n the same row followed by the same letter are not sgnfcantly dfferent (P >.5) usng the Tukey-Kramer procedure.
Page 25 of 31 525 526 Table 2. Means, standard errors, and coeffcents of varaton (CV) (n parentheses) of the populaton parameters of Bactrocera cucurbtae for dfferent treatments 527 Populaton parameters Mean fecundty (F) (eggs/female) The ntrnsc rate of ncrease r (days -1 ) The fnte rate of ncrease λ (days -1 ) Gross reproductve rate (GRR) (offsprng) The net reproductve rate R (offsprng/ndvdual) The mean generaton tme T (days) 25ºC Wthout leaf coverage Feld condtons Wth leaf coverage Rep. 1 Rep. 2 Rep. 1 Rep. 2 Rep. 1 Rep. 2 859.5 ± 17.8 a (61.4%).1354 ±.6 a (44.%) 1.145 ±.7 a (6%) 636.3 ± 129.6 a (23.6%) 26.3 ± 44.8 a (217.3%) 39.5 ±.8 a (19.1%) 66.1 ± 179.9 a (86.2%).12 ±.116 a (116.2%) 1.15 ±.13 a (11.6%) 426.5 ± 159.5 a (374.6%) 66. ± 26.3 b (398.%) 42.6 ± 1.5 a (34.3%) 345.9 ± 92.5 b (75.6%).143 ±.151 a (145.2%) 1.11 ±.17 a (15.1%) 322.4 ± 12. a (372.1%) 27.7 ± 11.7 b (423.5%) 32.8 ± 1.5 a (45.7%) 112.1 ± 42.8 b (114.4%).94 ±.197 a (217.3%) 1.94 ±.21 a (19.5%) 119.3 ± 56.4 a (473.%) 1.1 ± 4.9 b (482.4%) 27.2 ± 2.3 b (83.2%) 218.5 ± 115.5 b (149.5%).935 ±.12 a (224.3%) 1.98 ±.23 a (2.7%) 146.5 ± 94.7 a (646.3%) 17.5 ± 1.5 b (62.5%) 34. ± 3.9 a (113.4%) Means n the same row followed by the same letter are not sgnfcantly dfferent (P >.5) usng the Tukey-Kramer procedure. 227.6 ± 162.3 b (159.5%).99 ±.38 a (417.3%) 1.94 ±.4 a (36.9%) 868.61 ± 484.84 a (558.2%) 11.4 ± 8.8 b (776.4%) 35. ± 7.4 a (212.2%)
Page 26 of 31 528 529 53 531 532 533 534 535 536 537 538 539 54 Fgure captons Fg. 1. Age-stage specfc survval rate (s xj ) of Bactrocera cucurbtae for dfferent treatments. Fg. 2. Age-specfc survval rate (l x ), female age-specfc fecundty (f x2 ), age-specfc fecundty (m x ) and age-specfc maternty (l x m x ) of Bactrocera cucurbtae for dfferent treatments. Fg. 3. Age-stage specfc lfe expectancy (e xj ) of Bactrocera cucurbtae for dfferent treatments. Fg. 4. Age-stage specfc reproductve value (v xj ) of Bactrocera cucurbtae for dfferent treatments. Fg. 5. Frequency dstrbuton of pseudo-r grouped for dfferent treatments. Each bar represents the number of pseudo-r between two tcks. The bar at zero represents the frequency of pseudo-r zero.
Page 27 of 31 541 542 1..8.2 25 o C Replcate 1 Preadult Female Male 1..8.2 25 o C Replcate 2. 2 4 6 8 1 12 14. 2 4 6 8 1 12 14 Survval rate (s xj ) 1..95.9.1.5. 1..8.2 Wthout leaf coverage Replcate 1 2 4 6 8 1 12 14 Wth leaf coverage Replcate 1 1..95.9.1.5. 1..96.4 Wthout leaf coverage Replcate 2 2 4 6 8 1 12 14 Wth leaf coverage Replcate 2 543. 2 4 6 8 1 12 14. Age (days) 2 4 6 8 1 12 14 544 545 Fg. 1. Age-stage specfc survval rate (s xj ) of Bactrocera cucurbtae for dfferent treatments.
Page 28 of 31 546 547 1..8 25 o C Replcate 1 l x f x2 8 6 1..8 25 o C Replcate 2 8 6.6.4.2 m x l x m x 4 2.6.4.2 4 2. 2 4 6 8 1 12 14. 2 4 6 8 1 12 14 Survval rate (s xj ) 1..8.6.4.2. 1..8.6.4.2 Wthout leaf coverage Replcate 1 2 4 6 8 1 12 14 Wth leaf coverage Replcate 1 8 6 4 2 8 6 4 2 1..8.6.4.2. 1..8.6.4.2 Wthout leaf coverage Replcate 2 2 4 6 8 1 12 14 Wth leaf coverage Replcate 2 8 6 Fecundty 4 2 8 6 4 2 548. 2 4 6 8 1 12 14. Age (days) 2 4 6 8 1 12 14 549 55 551 Fg. 2. Age-specfc survval rate (l x ), female age-specfc fecundty (f x2 ), age-specfc fecundty (m x ) and age-specfc maternty (l x m x ) of Bactrocera cucurbtae for dfferent treatments.
Page 29 of 31 1 8 6 4 25 o C-1 Preadult Female Male 1 8 6 4 25 o C-2 2 2 2 4 6 8 1 12 14 2 4 6 8 1 12 14 552 553 Lfe expectancy (e xj ) (days) 1 8 6 4 2 1 8 6 4 2 1 Wthout leaf coverage-1 8 Wthout leaf coverage-2 6 4 2 2 4 6 8 1 12 14 2 4 6 8 1 12 14 1 Wth leaf coverage-1 8 Wth leaf coverage-2 6 4 2 2 4 6 8 1 12 14 Age (days) 2 4 6 8 1 12 14 554 555 Fg. 3. Age-stage specfc lfe expectancy (e xj ) of Bactrocera cucurbtae for dfferent treatments.
Page 3 of 31 556 557 25 2 25 o C Replcate 1 25 2 25 o C Replcate 2 15 Preadult Female 15 1 1 5 5 2 4 6 8 1 12 14 2 4 6 8 1 12 14 Reproductve value (v xj ) 25 2 15 1 5 25 2 15 1 5 Wthout leaf coverage Replcate1 2 4 6 8 1 12 14 Wth leaf coverage Replcate 1 25 2 15 1 5 25 2 15 1 5 Wthout leaf coverage Replcate 2 2 4 6 8 1 12 14 Wth leaf coverage Replcate 2 558 2 4 6 8 1 12 14 Age (days) 2 4 6 8 1 12 14 559 56 Fg. 4. Age-stage specfc reproductve value (v xj ) of Bactrocera cucurbtae for dfferent treatments.
Page 31 of 31 1 9 8 7 8 6 4 2 25 o C Replcate 1 Varance = 2922 4 8 12 16 2 1 9 8 7 4 2 25 o C Replcate 2 Varance = 6923 4 8 12 16 2 Frequency 1 9 8 7 4 2 1 9 8 7 4 2 Wthout leaf coverage Replcate 1 Varance = 13732 2 4 6 8 1 Wth leaf coverage Replcate 1 Varance = 1193 2 4 6 8 1 1 9 8 7 4 2 1 9 8 7 4 2 Wthout leaf coverage Replcate 2 Varance = 2369 2 4 6 8 1 Wth leaf coverage Replcate 2 Varance = 787 2 4 6 8 1 561 Pseudo R 562 563 564 Fg. 5. Frequency dstrbuton of pseudo-r grouped for dfferent treatments. Each bar represents the number of pseudo-r between two tcks. The bar at zero represents the frequency of pseudo-r zero.