Model fitting of the seasonal population dynamics of the soybean aphid,Aphis glycines Matsumura,in the field

XU Lei ,ZHAO Tong-hua ,XING Xing ,XU Guo-qing# ,XU Biao ,ZHAO Ji-qiu

1 Institute of Plant Protection, Liaoning Academy of Agricultural Sciences, Shenyang 110161, P.R.China

2 Agricultural Technology Extension Center of Xiuyan Manchu Autonomous County, Anshan 114300, P.R.China

Abstract The soybean aphid,Aphis glycines Matsumura (Hemiptera: Aphididae),is one of the greatest threats to soybean production,and both trend analysis and periodic analysis of its population dynamics are important for integrated pest management (IPM).Based on systematically investigating soybean aphid populations in the field from 2018 to 2020,this study adopted the inverse logistic model for the first time,and combined it with the classical logistic model to describe the changes in seasonal population abundance from colonization to extinction in the field.Then,the increasing and decreasing phases of the population fluctuation were divided by calculating the inflection points of the models,which exhibited distinct seasonal trends of the soybean aphid populations in each year.In addition,multifactor logistic models were then established for the first time,in which the abundance of soybean aphids in the field changed with time and relevant environmental conditions.This model enabled the prediction of instantaneous aphid abundance at a given time based on relevant meteorological data.Taken as a whole,the successful approaches implemented in this study could be used to build a theoretical framework for practical IPM strategies for controlling soybean aphids.

Keywords: soybean aphid,population dynamics,logistic model,inverse logistic model,multifactor logistic model

1.lntroduction

There are approximately 4 400 species of aphids in the world,and they cause severe damage to a variety of economically important crops (Blackman and Eastop 2000).The soybean aphid,AphisglycinesMatsumura(Hemiptera: Aphididae),has been one of the most serious threats to soybean production for many years (Wuet al.2004).As a native of Asia,outbreaks of the soybean aphid have been rampant in North Korea,South Korea,Indonesia,Russia,the Philippines,and Thailand (Wuet al.2009).Since it was first detected in Wisconsin,USA,in 2000 (Ragsdaleet al.2004),the soybean aphid has been a serious threat to U.S.soybean productivity (Johnsonet al.2009),and it has spread to other countries,such as Canada (Venette and Ragsdale 2004).The soybean aphid not only hinders photosynthesis in soybeans(Macedoet al.2003),causing symptoms such as growth retardation and root dysplasia,but it also transmits various plant viruses (Wanget al.2014;Xuet al.2022),so it severely threatens the yield and quality of soybeans in many primary production areas worldwide (Wanget al.2014).

Insect population dynamics are the core of population ecological research,and the simulation of population dynamics is an important basis of integrated pest management (IPM) (Ju and Shen 2005).The models used to simulate the population dynamic development for a single species are mainly logistic models (S-curves),exponential models (J-curves),and matrix models.Without considering predation or intraspecific competition,the simple exponential model can provide an adequate approximation for simulating population growth in the initial stage (Tsoularis 2001).Costamagnaet al.(2007)were the first to attempt to incorporate the linear decrease in the intrinsic rate of increase (r) of the population into the exponential growth model (i.e.,the decreasingrmodel) under predator exclusion conditions in order to comprehensively describe the population dynamics of soybean aphids throughout the whole season.However,fitting the population dynamics in the field by artificially isolating external interference does not seem realistic(Tsoularis 2001).The matrix model is a mathematical model based on insect age-specific life tables (Leslie 1945).Leblanc and Brodeur (2018) and Miksanek and Heimpel (2019) incorporated the decreasingrinto a matrix framework to model the population dynamics of soybean aphids impacted by parasitoids,and they also parameterized the matrix model for host-parasitoid systems through a series of developmental and behavioral bioassays.However,the transfer matrix requires the time intervals dividing the age groups of the insect population to be exactly equal,which puts enormous limitations on the application of this model (Panget al.1980;Xu 1985).For example,when calculating beyond the 2nd unit time interval,there will be deviations in the developmental duration of the insects (Vandermeer 1975).Based on the stage-dependent survival and fecundity of soybean aphid,Lin and Ives (2003) established the stage-structured matrix model for population growth,but that model is only applicable to one single stage for the reproduction of unparasitized adult aphids.

A stable population is usually characterized by a saturation level,called carrying capacityK,which forms a numerical upper limit on the scale of growth.The logistic equation derived from this limit concept is the foundation of many subsequent extended models (Verhulst 1838).A field investigation in China showed that the population growth of soybean aphids over time was consistent with a logistic model (Zheng and Yao 2006).Field experiments conducted in eastern South Dakota in the USA from 2003 to 2004 showed that the logistic curve perfectly fitted the population dynamics of soybean aphidday accumulation over time,and the inflection points of the curve were calculated to determine the fastest time for the rate of cumulative aphid-days (Beckendorfet al.2008;Catanguiet al.2009).Rhaindset al.(2010) used logistic regressions to model the cumulative abundance of aphids captured in suction traps and aphid densities on soybean plants in Indiana and Illinois in the USA,and the results exhibited distinct seasonal trends of development and declination.However,in reality,Kis not a fixed constant (Tang 1996).Insect populations usually have seasonal multiplication and extinction phases,during which the population size fluctuates due to the influences of various environmental ecological or internal population factors,andKgenerally shows an irregular upward or downward trend (Zhao 1996).When modeling the population dynamics of aphids such asAphisgossypii(Cui and Wang 2000),Pseudoregma(Hormaphidinae:Cerataphidini) (Aoki and Kurosu 2003;Aoki and Imai 2005),and wheat aphids (Zhanget al.2018),Kwas used as an exogenous variable of the logistic equation.Therefore,for any single population,the numerical dynamics in the whole season should not be described by only one logistic equation,and the variations inKmust be taken into account,so an inverse logistic equation can be used to describe the decreasing phases of the population(Zhenget al.2008;Pankovi?et al.2009).Inverse logistic models have been applied in many research fields using animals such asSchizuraconcinna(Brandet al.1973)andHaliotisrubra(Haddonet al.2008;Helidoniotiset al.2011),plants (Wanget al.2008),irrigation (Shanget al.1999) and other subjects.To date,there have been no reports on their use for other aphids,except for a study on the decreasing phases ofAphisgossypiipopulations(Zhenget al.2008;Zhu and Gao 2013).

Multifactor logistic models have been applied in the study of growth prediction,such as for winter wheat(Alexandrov and Hoogenboom 2000),corn (Wang and Zhang 2005),trees (Mendeset al.2006;Semenzatoet al.2011),and quinoa (Liet al.2017).The only application of multifactor logistic model related to aphids was for the epizootiological modeling of aphid mycosis transmission.In a study on the transmission and epidemic mechanism ofPandoraneoaphidisin populations ofSitobion avenaeandMyzuspersicae,Chen and Feng (2005,2006) identified representative variables that were significantly related to the simulated flight,colonization and transmission potential ofP.neoaphidis-infected alate aphids by a stepwise regression analysis.These variables were included in logistic models,and then compound multifactor logistic models were established with high goodness of fit.These successful applications provide evidence that the compound logistic model may serve as an important new strategy for the analysis of population dynamics in aphid-based systems as well.

Through a systematic investigation of soybean aphid field populations from 2018 to 2020,the following piecewise analysis was conducted on seasonal population dynamics.This study adopted the inverse logistic model for the first time,and combined it with the classical logistic model to describe the changes in aphid population abundance from colonization to extinction in the field.Then,the increasing and decreasing phases of aphid population fluctuations were divided by calculating the inflection points of the logistic models.Furthermore,based on the relevant meteorological data during the experiment,new multifactor logistic models were established in which the abundance of soybean aphids in the field changed with time and environmental conditions.These analysis approaches revealed an important new strategy for the simulation of population dynamics,and thus provide an important theoretical basis for developing soybean aphid IPM strategies.

2.Materials and methods

2.1.Field survey on the population dynamics of soybean aphids

Soybean aphids were surveyed in a conventional field with consistent field management during 2018–2020 in Xinglong Town,Xiuyan Manchu Autonomous County,Liaoning Province,northeastern China (40°20′N(xiāo),123°22′E).The local soybean Dandou 14,which is a midlate cultivar with a growth period of 131 d in the spring,was planted in early May each year.The field was handweeded periodically during the soybean growing season,and there were no insecticide applications.Beginning in late May,10 plants from 10 points were randomly labeled,and 100 soybean plants were counted.Surveys were conducted every three days until the end of September,when there were no soybean aphids in the field.The soybean growth stages were noted during each survey,whether vegetative or reproductive (Fehret al.1971).

2.2.Single-factor growth model of soybean aphid populations

and the integral form of which is:

The decreasing phases of the population can be described by one inverse logistic equation,the differential form of which is (Zhenget al.2008;Zhu and Gao 2013):

and the integral form of which is:

Simplifying eq.(4) according to Shanget al.(1999):

whereNis the soybean aphid abundance,tis the number of days since May 1,r1andr2are the population growth rates,K1andK2are constants,which are called the carrying capacity,eis the base of the natural logarithm,anda1anda2are the integral constants.

The division points for the acceleration and deceleration of the logistic curve are revealed by the differential transformation of the mathematical model(Chen and Zhou 2005).The second derivative of eq.(2)is solved and set equal to 0,that is:

What a miser4 she was!Here rests a young lady, of a good family, who would always make her voice heard in society, and when she sang Mi manca la voce, *it was the only true thing she ever said in her life

The solutions of eq.(6) are:

Similarly,when solving the second derivative of eq.(5)and setting it to 0,the solutions are:

wheret1andt2are the two inflection points in the increasing process of the soybean aphid population in the field,which divide the increasing process into the gradual increasing (t=0–t1),rapid increasing (t=t1–t2),and tempered increasing (t=t2–∞) phases.Similarly,t3andt4are the two inflection points in the decreasing process,which divide the decreasing process into the gradual decreasing (t=∞–t3),rapid decreasing (t=t3–t4),and tempered decreasing (t=t4–∞) phases.

2.3.Multifactor growth model of the soybean aphid population

Derivation of the multifactor logistic growth modelThe multifactor logistic growth model was established by function derivation.Assuming that the multifactor growth function isN=F(t,A,B,…,factors),whereA,Band other factors all change with time (t),thenN=F[t,A(t),B(t),…,factors(t)].For simplicity,settingK=1 in eq.(1),the expression is thendN/dt=rN(1–N).Solving the total derivative oft,the partial differential equation obtained isdN/dt=?N/?t+(?N/?A)×(dA/dt)+(?N/?B)×(dB/dt)+…,and the characteristic equation isdt/1=dA/(?N/?A)=dB/(?N/?B)=…=dN/[rN(1–N)].Assuming thatA(t),B(t),and other factors(t) are linear functions oft,then the expressions can be stated as:A(t)=a1t+b1,B(t)=a2t+b2,and thendt/1=dA/a1=dB/a2=…=dN/[rN(1–N)].The solutions for this system of partial differential equations are:C1=(K–N)/N×Exp(b1t),C2=(K–N)/N×Exp(b2A),andC3=(K–N)/N×Exp(b3B).Because the form ofN=F[t,A(t),B(t),…,factors(t)] is related to the expressions ofC1,C2,andC3,the simplest assumption isN=C1×C2×C3=[(K–N)/N×Exp(b1t)]×[(K–N)/N×Exp(b2A)]×[(K–N)/N×Exp(b3B)]=[(K–N)/N]3×Exp(b1t+b2A+b3B),and thenN=K/[1+aExp(c1t+c2A+c3B)],which still fits the form of the logistic model.

After the above derivation,the expression is:

whereyis the number of aphids per 100 plants;tis the number of days since May 1;x1,…,xnare the meteorological factors that significantly affect the dependent variable;Kis the carrying capacity;anda,b,b1,…,bnare all parameters.

Selection of meteorological factorsAmong the previously proposed meteorological factors that may significantly affect soybean aphid abundance (Xuet al.2016),both the daily average relative humidity and the daily average rainfall can be correlated with the environmental humidity,while the daily average temperature,daily average maximum temperature,and daily average minimum temperature can be correlated with the environmental temperature.To avoid collinearity,these groups of related factors were not included together in the multifactor regression analysis.Based on a comparison of the correlation coefficients of meteorological factors (Xuet al.2016),the daily average rainfall (x1),the daily average wind speed (x2),and the daily average temperature (x3) were selected for inclusion in the multifactor logistic growth model.Thex1,x2,andx3data were converted into daily averages for the respective fluctuation phases and combined withtto create independent variables for predicting the multifactor logistic models of both the increasing and decreasing phases.Microclimatic data were collected using WatchDog 2000 Series Weather Stations (Spectrum Technologies Inc.,Aurora,IL,USA).

2.4.Statistical analysis

In view of the large fluctuations in the survey data of soybean aphids in different population development periods,the cubic root of aphids per 100 plants was used as the dependent variable in the single-factor and multifactor logistic growth models.The initial estimate ofKin the logistic model was calculated by linear regression using Microsoft Excel 2007 (Microsoft Corp.,Redmond,WA,USA) to achieve fast convergence.For the singlefactor growth model,two groups of data were substituted into the expressions,and the estimates of parametersaandrwere found by solving the binary linear equation group.For the multifactor growth model,five groups of data were substituted into the expressions,and the estimates of parametersa,b,b1,b2,andb3were found by solving the linear equation group with five unknowns.The estimates of the above parameters were assigned as the initial values then the nonlinear regression analysis from IBM SPSS Software version 24.0 (IBM Inc.,Armonk,NY,USA) was used to generate the model expressions and plot fitting curves.

3.Results

3.1.Population dynamics of soybean aphids in the field

According to the field survey for three consecutive years (Fig.1),the initial appearance phase of soybean aphids generally occurred in late May.After June,as the temperature increased,the population density increased rapidly,and an obvious peak appeared in mid-July.The peaks reached 15 851,14 208,and 31 765 aphids per 100 plants in 2018,2019,and 2020,respectively,and then the aphid population declined sharply and tended to disappear in late September.

Fig.1 Population dynamics of soybean aphids in a soybean field in Xiuyan,Liaoning Province,China during 2018–2020.

3.2.Single-factor growth model of soybean aphid population

The seasonal soybean aphid population growth was divided into two periods of increasing and decreasing.The alatae migrated from the overwintering host ofRhamnusdavuricaPall.to the summer host of soybean for colonization and then spreading,entering the peak period which lasted until mid-July (Fig.1).The population growth of soybean aphids showed a typical singlepeak S-type curve,which could be described by the classical logistic equation.Since late July,because of the disadvantageous conditions of the field environment and host plant nutrition,soybean aphid development slowed,and the aphid abundance began to decline.By the end of August,the field conditions were no longer suitable for the late reproduction of soybean aphids,and the aphid abundance gradually decreased until the aphids migrated back toR.davuricafor overwintering (Fig.1),which could be described by the inverse logistic equation.Therefore,both classical logistic and inverse logistic equations were established for the population fluctuation phases,both in each year and in general,by nonlinear regression analysis (Table 1).The significance test showed that the logistic model provided an excellent simulation of the soybean aphid population dynamics (Table 1;Fig.2),and both increasing and decreasing phases exhibited a typical single-peaked S-type curve.

Table 1 Single-factor growth models and tests of the soybean aphid population dynamics1)

Fig.2 Temporal dynamics curves and inflection points of the soybean aphid population in Xiuyan,Liaoning Province,China in each year.

From eq.(7),the inflection points of the increasing phase of the soybean aphid population (t1andt2),were calculated as 40 and 59 in 2018,respectively,41 and 62 in 2019,respectively,and 35 and 57 in 2020,respectively.Integrating the three-year data,the inflection points of the increasing phase,t1andt2,were 39 and 59 in general,respectively.From eq.(8),the inflection points of the decreasing phase (t3andt4),were calculated as 91 and 113 in 2018,respectively,98 and 115 in 2019,respectively,and 100 and 118 in 2020,respectively.Integrating the three-year data,the inflection points of the decreasing phase,t3andt4,were 96 and 116 in general,respectively.According to the above inflection points,the increasing process of the soybean aphid population in the field was divided into gradual increasing,rapid increasing,and tempered increasing phases and the decreasing process was similarly divided into gradual decreasing,rapid decreasing,and tempered decreasing phases(Table 2;Fig.2).

In the case of the general models (Table 2;Fig.3),from May 25 to June 10 (the VE–V3 stage of the soybean plants),the growth phase of the soybean aphid population was gradually increasing,and the number of aphids per 100 plants increased from 0 to 198.From June 11 to June 30 (V3–R1 stage),the aphid populations entered the rapid increasing phase,and the number of aphids per 100 plants increased to 9 094 in the field.From July 1 to July 19 (R1–R3 stage),the aphid abundance remained at high levels,but the populations entered the tempered increasing phase,and the aphids per 100 plants increased to 17 308 in the field.From July 20 to August 6 (R3–R5 stage),the aphid populations entered the gradual decreasing phase,and the aphids per 100 plants decreased from 18 421 to 9 838 in the field.From August 7 to August 26 (R5–R7 stage),the aphid populations entered the rapid decreasing phase,and the aphids per 100 plants decreased from 9 838 to 207 in the field.After August 27 (R7–R8 stage),during the tempered decreasing phase,the aphids per 100 plants tended to disappear from the initial level of 207 in the field.

Fig.3 Temporal dynamics curves and the inflection points of the soybean aphid population in Xiuyan,Liaoning Province,China across all three years during 2018–2020.

3.3.Multifactor growth model of the soybean aphid population

Multifactor logistic growth models were constructed for the relationship between the meteorological variables and the soybean aphid population density using nonlinear regression analysis (Table 3).The significance testshowed that the highest and lowestR2were 0.947 and 0.793,respectively,and the Sig.Fvalues were all <0.001.Therefore,these multifactor logistic growth models by function derivation could describe the population dynamics of soybean aphids based on the time and meteorological factors,and the instantaneous aphid abundance at a given time in the field could be predicted by substituting the relevant meteorological data.

Table 3 Multifactor growth models and tests of the soybean aphid population dynamics1)

4.Discussion

In the S-curve model of a population,the inflection point is crucial and has specific biological significance,and it determines the curve shape (Duanet al.2015).In Northeast China,the soybean aphid has two migration and two dispersal peaks (Wanget al.1962;Wuet al.2004).The four inflection points in the curves of the seasonal dynamic models in this study are closely related to those four peaks.

Around the flowering time of the overwintering host plantR.davurica,once the soybean aphids migrate into the field,the second generation begins to produce alatae and the population gradually expands in the field,which becomes the primary source for reproduction and damage from the aphids (Wanget al.1962).This process is termed the gradual increasing phase in this study,andt1(V3 stage) in the population increasing model designates the end of the first migration peak (also known as the“spring migration”).The initial colonization patterns of soybean aphid are highly patchy within a field (Wuet al.2004;Xuet al.2022),and the aggregative and nonuniform distribution is one of the main characteristics of this pest in the field (Wanget al.1962;Xuet al.2016,2022).At that point,some plants have not yet been infested,while others may have as many as thousands of aphids which form the typical aphid nests (Liet al.2011) and become the secondary source for damage in the field.Under crowded conditions,the dense alatae are forced to spread to the surrounding healthy plants (Dixon 1998;Foxet al.2004) and may even be transported long distances from field to field by jet streams (Hodgsonet al.2005).These dispersal mechanisms result in a significant increase in the colonized plant rate and cause a gradual transition from the dotted or patchy pattern to infestation of the whole field (Wanget al.1962),forming the first dispersal peak.This process is the rapid increasing phase,andt2(R1 stage) in the population increasing model designates the end of the first dispersal peak.

After the R1 soybean stage,the intensification of crowding prompts large numbers of alatae to spread rapidly throughout the field,and favorable climatic conditions may lead to severe infestation (Wuet al.2004),which forms the second dispersal peak until the R5 soybean stage.Thet3(R5 stage) in the population decreasing model designates the end of the second dispersal peak.During this period (R1–R5 stage),the delayed density dependence of the populations of natural enemies associated with soybean aphid is rapidly revealed and expands to become a dominant controlling power (Liet al.2011).The aphid population in the field experiences the tempered increasing phase and the gradual decreasing phase with the R3 stage as the turning point.During the R5–R7 soybean stage,the growth points of the soybean plants stop growing (Wuet al.2004),and the soybean aphids transfer to the back of the middle and lower leaves of the plant.Poor physiological conditions and crowding of the host plants can led to a decrease in adult body size in the aphids (Martínez and Costamagna 2018),and they show a low growth rate and decreased fecundity (Watt and Hales 1996).This process is the rapid decreasing phase in the population decreasing model.After the R7 soybean stage,with the temperature dropping and the host-plant aging,the field conditions are no longer suitable for the survival of the soybean aphids;so,they produce gynoparae and alate males which migrate back toR.davuricain succession for overwintering (Ragsdaleet al.2004).This process is the tempered decreasing phase,andt4(R7 stage) in the population decreasing model designates the beginning of the second migration peak (also known as the “autumn migration”).

In many crops,aphid pests are characterized by boomand-bust dynamics (Lin and Ives 2003).Soybean aphid reproduction is positively correlated with the nitrogen content of the host plant (Walter and DiFonzo 2007).According to the findings of Riedellet al.(2005) and Osborne and Riedell (2006),the concentration of the nitrogen fixing products in soybean plants (e.g.,ureides)begins to increase rapidly at the R1 stage,peaks at approximately the R5 stage,and decreases sharply to about 0 by the R7 stage.Interestingly,in this study,t2(R1 stage) in the increasing model andt3(R5 stage)in the decreasing model divided the aphid population into the tempered increasing phase and the gradual decreasing phase,respectively,taking the R3 stage as the turning point,whilet4(R7 stage) in the decreasing model indicated that the aphid population had entered the tempered decreasing phase (Table 2;Fig.2).That is,except for the R3–R5 stage of the plants,the developmental trend of the soybean aphid population and soybean plant nitrogen content was consistent.Therefore,one intriguing possibility is that the divergence may be driven by the natural enemies.As the soybean aphid population peaked at the R3 stage (Fig.1),the delayed density dependence of the natural enemy populations associated with soybean aphids was rapidly revealed (Bahlai and Sears 2009;Miksanek and Heimpel 2019) and peaked after 5–10 days in Liaoning (Liet al.2011).As such,it is reasonable to infer that although the nitrogen content of the host plant continued to increase,the suppression of aphid population outbreaks and damage after the R3 stage is dictated largely by the ability of natural enemy pressure to keep up with the aphid populations.This suggestion is also consistent with the viewpoints of Rauwald and Ives (2001),Lin and Ives(2003),and Miksanek and Heimpel (2019).In summary,our data indicate that the decrease inrcannot be attributed solely to plant phenology (Van den Berget al.1997;Costamagnaet al.2007),but seems to be a result of different confounding underlying mechanisms,such as biological control dominated by natural enemies (Leblanc and Brodeur 2018;Miksanek and Heimpel 2019).

Based on the stepwise regression analysis of Xuet al.(2016),the meteorological factors affecting the development of aphid abundance were incorporated into a logistic model to build a polynomial representing the rate of the soybean aphid population fluctuation.A series of factors with important agroecological significance were then incorporated,which provided references for a deeper epidemiological simulation analysis.TheR2values of the multifactor growth models of the soybean aphid population in this study,which were based meteorological factors,were between 0.793 and 0.947,while theR2values of the linear models by the multifactor stepwise regression analysis were between 0.604 and 0.849 (Xuet al.2016).Clearly,the fitting precision of the models was improved.Temperature is known to have the potential to greatly influence the fecundity,generation time,and life expectancy of soybean aphids (McCornacket al.2004;Brosiuset al.2007).In addition,rainfall erosion may lead to substantial aphid population declines(Ragsdale 2001) and even have an effect on withinplant distribution (McCornacket al.2008).In this study,the daily average temperature over 10 days around the R1 soybean stage in 2020 was 3.68 and 2.99°C higher than those in 2018 and 2019,respectively,while the daily average rainfall decreased by 13.5 and 18.2 mm,respectively.The differences in temperature and rainfall among years were among the reasons why the peak aphid abundance in 2020 (around the R1 stage) was much larger than those in the other two years (Fig.1).This finding is consistent with the results reported by Yue and Hao (1990) in Jilin Province and also in line with their theory that the precipitation/temperature ratio (mm/°C) in this period is significantly correlated with aphid densities.The multifactor logistic growth model developed in this study will enable growers to predict the damage of soybean aphids at a certain time according to the meteorological conditions,so it may replace the stepwise regression linear model currently being used for soybean aphid management (Xuet al.2016) and provide a better guide for scientific control in actual production.

In this study,the derivation of the multifactor logistic model assumed that the meteorological factors have the simplest linear relationship with time,and thus,the form of the logistic model obtained was also the simplest.As the relationship between meteorological factors and time could also be a quadratic function,cubic function,and other more complex functions,other revised forms of the logistic model could then be derived.However,the difficulties in doing so lie in integration and derivation step,especially regarding the solution of the equation,which would be challenging.However,if the relational function between the impact factors and time could not be established accurately,it would be difficult to calculate the inflection points of the multifactor logistic models to precisely divide the increasing and decreasing phases of the soybean aphid population.This potentially useful relationship needs further exploration.Furthermore,systematic investigations on natural enemy resources in soybean fields are currently underway.Combined with the methodology of plant physiology,future research for the development of improved models will focus on incorporating these effects into the models by establishing the statistical relationships of soybean aphid population growth with both the nitrogen content of soybean plants(Huet al.1992) and total natural enemies (Hallettet al.2014).All of these efforts can assist growers in managing soybean aphids in a more scientifically valid and reasonable fashion.

5.Conclusion

In the present study,three attempts were made to accurately model the soybean aphid population dynamics in the field.

(1) The classical logistic model and the inverse logistic model were first adopted to describe the changes in population abundance from colonization to extinction in the field,and then the increasing and decreasing phases of population fluctuation were divided by calculating the inflection points of the models,which exhibited distinct seasonal trends for soybean aphids in each of the three study years.

(2) Next,nonlinear regression analysis was used to fit the curve of soybean aphid population dynamics with time,which had higher precision than the simple linear regression.

(3) By function derivation,a multifactor logistic growth model of the changes in the soybean aphid population with time and multiple environmental factors in the field was then established.This model can predict the instantaneous aphid abundance at a given time based on the relevant meteorological data.

The inverse logistic model and multifactor logistic model introduced here for the first time cannot only be used to develop a theoretical framework for IPM strategies of soybean aphids in practice but it can also provide new ways for both trend analysis and periodic analysis of pest populations,while opening new paths for the study of entomological population ecology.

Acknowledgements

This study was supported by the Chinese National Special Fund for Agro-scientific Research in the Public Interest (201003025 and 201103022),the National Key Research and Development Program of China(2018YFD0201004) and the Discipline Construction Project of Liaoning Academy of Agricultural Sciences,China (2019DD082612).

Declaration of competing interest

The authors declare that they have no conflict of interest.