Pest insect population dynamics

From a life table we can calculate life expectancy as follows. Assume the stages ${\displaystyle x}$ are uniformly spaced. The average proportion ${\displaystyle L_{x}}$ of organisms alive at stage ${\displaystyle x}$ between beginning and end is[3]

${\displaystyle L_{x}={\frac {l_{x}+l_{x+1}}{2}}}$.

The total number ${\displaystyle T_{x}}$ of future stages to be lived by individuals at age ${\displaystyle x}$ and older is[3]

${\displaystyle T_{x}=L_{x}+L_{x+1}+L_{x+2}+...}$ .

Then the life expectancy ${\displaystyle e_{x}}$ at age ${\displaystyle x}$ is[3]

${\displaystyle e_{x}={\frac {T_{x}}{l_{x}}}}$.

We could have done the same computation with raw numbers of individuals rather than proportions.

If we further know the number ${\displaystyle F_{x}}$ of eggs produced (fecundity) at age ${\displaystyle x}$, we can calculate the eggs produced per surviving individual ${\displaystyle m_{x}}$ as

${\displaystyle m_{x}={\frac {F_{x}}{a_{x}}}}$,

where ${\displaystyle a_{x}}$ is the number of individuals alive at that stage.

The basic reproductive rate ${\displaystyle R_{0}}$,[3] also known as the replacement rate of a population, is the ratio of daughters to mothers. If it's greater than 1, the population is increasing. In a stable population the replacement rate should hover close to 1.[2] We can calculate it from life-table data as[3]

${\displaystyle R_{0}=\sum _{x}l_{x}m_{x}}$.

This is because each ${\displaystyle l_{x}m_{x}}$ product computes (first-generation parents at age ${\displaystyle x}$)/(first-generation eggs) times (second-generation eggs produced by age-${\displaystyle x}$ parents)/(first-generation parents at age ${\displaystyle x}$).

If ${\displaystyle N_{0}}$ is the initial population size and ${\displaystyle N_{T}}$ is the population size after a generation, then[3]

${\displaystyle R_{0}={\frac {N_{T}}{N_{0}}}}$.

The cohort generation time ${\displaystyle T_{c}}$ is the average duration between when a parent is born and when its child is born. If ${\displaystyle x}$ is measured in years, then[3]

${\displaystyle T_{c}={\frac {\sum _{x}xl_{x}m_{x}}{\sum _{x}l_{x}m_{x}}}}$.

If ${\displaystyle R_{0}}$ remains relatively stable over generations, we can use it to approximate the intrinsic rate of increase ${\displaystyle r}$ for the population:[3]

${\displaystyle r\approx {\frac {\ln R_{0}}{T_{c}}}}$.

This is because

${\displaystyle \ln R_{0}=\ln {\frac {N_{T}}{N_{0}}}=\ln {\frac {N_{0}+\Delta N}{N_{0}}}=\ln \left(1+{\frac {\Delta N}{N_{0}}}\right)\approx {\frac {\Delta N}{N_{0}}}}$,

where the approximation follows from the Mercator series. ${\displaystyle T_{c}}$ is a change in time, ${\displaystyle \Delta t}$. Then we have

${\displaystyle r\approx {\frac {\Delta N}{\Delta t}}{\frac {1}{N_{0}}}}$,

which is the discrete definition of the intrinsic rate of increase.

Because insects are ectothermic, "temperature is probably the single most important environmental factor influencing insect behavior, distribution, development, survival, and reproduction."[6] As a result, growing degree-days are commonly used to estimate insect development, often relative to a biofix point,[6] i.e., a biological milestone, such as when the insect comes out of pupation in spring.[7] Degree-days can help with pest control.[8]

Stenotus binotatus, a plant bug in the group Heteroptera

Yamamura and Kiritani approximated the development rate ${\displaystyle r}$ as[9]

${\displaystyle r={\begin{cases}{\frac {T-T_{0}}{K}},&T\geq T_{0}\\0,&T<T_{0}\end{cases}}}$,

with ${\displaystyle T}$ being the current temperature, ${\displaystyle T_{0}}$ being the base temperature for the species, and ${\displaystyle K}$ being a thermal constant for the species. A generation is defined as the duration required for the time-integral of ${\displaystyle r}$ to equal 1. Using linear approximations, the authors estimate that if the temperature increased by ${\displaystyle \Delta T}$ (for instance, maybe ${\displaystyle \Delta T}$ = 2 °C for climate change by 2100 relative to 1990), then the increase in number of generations per year ${\displaystyle \Delta N}$ would be[9]

${\displaystyle \Delta N\approx {\frac {\Delta T}{K}}\left(206.7+12.46(m-T_{0})\right)}$,

where ${\displaystyle m}$ is the current annual mean temperature of a location. In particular, the authors suggest that 2 °C warming might lead to, for example, about one extra generation for Lepidoptera, Hemiptera, two extra generations for Diptera, almost three generations for Hymenoptera, and almost five generations for Aphidoidea. These changes in voltinism might happen through biological dispersal and/or natural selection; the authors point to prior examples of each in Japan.[9]

Sunding and Zivin model population growth of insect pests as a geometric Brownian motion (GBM) process.[10] The model is stochastic in order to account for the variability of growth rates as a function of external conditions like weather. In particular, if ${\displaystyle X}$ is the current insect population, ${\displaystyle \alpha }$ is the intrinsic growth rate, and ${\displaystyle \sigma }$ is a variance coefficient, the authors assume that

${\displaystyle dX=\alpha Xdt+\sigma Xdz}$,

where ${\displaystyle dt}$ is an increment of time, and ${\displaystyle dz=\xi _{t}{\sqrt {dt}}}$ is an increment of a Wiener process, with ${\displaystyle \xi _{t}}$ being standard-normal distributed. In this model, short-run population changes are dominated by the stochastic term, ${\displaystyle \sigma Xdz}$, but long-run changes are dominated by the trend term, ${\displaystyle \alpha Xdt}$.[10]

After solving this equation, we find that the population at time ${\displaystyle t}$, ${\displaystyle X_{t}}$, is log-normally distributed:

${\displaystyle X_{t}\sim \operatorname {Log-{\mathcal {N}}} \left(X_{0}e^{\alpha t},X_{0}^{2}e^{2\alpha t}(e^{\sigma ^{2}t}-1)\right)}$,

where ${\displaystyle X_{0}}$ is the initial population.[10]

Lettuce aphid

As a case study, the authors consider mevinphos application on leaf lettuce in Salinas Valley, California for the purpose of controlling aphids. Previous research by other authors found that daily percentage growth of the green peach aphid could be modeled as an increasing linear function of average daily temperature. Combined with the fact that temperature is normally distributed, this agreed with the GBM equations described above, and the authors derived that ${\displaystyle \alpha =0.1199}$ and ${\displaystyle \sigma =0.1152}$.[10] Since the expected population based on the log-normal distribution grows with ${\displaystyle e^{\alpha t}}$, this implies an aphid doubling time of ${\displaystyle \ln 2/0.1199=5.8}$ days. Note that other literature has found aphid generation times to lie roughly in the range of 4.7 to 5.8 days.[11]

A 2013 study analyzed population dynamics of the smaller tea tortrix, a moth pest that infests tea plantations, especially in Japan. The data consisted of counts of adult moths captured with light traps every 5–6 days at the Kagoshima tea station in Japan from 1961 to 2012. Peak populations were 100 to 4000 times higher than at their lowest levels. A wavelet decomposition showed a clear, relatively stationary annual cycle in the populations, as well as non-stationary punctuations between late April and early October, representing 4-6 outbreaks per year of this multivoltine species.[12] The cycles result from population overshoot.[13]

These moths have stage-structured development life cycles, and a traditional hypothesis suggests that these cycles should be most synchronized across the population in the spring due to the preceding effects of cold winter months, and as the summer progresses, the life stages become more randomly assorted.[12] This is often what's observed in North America.[13] However, this study observed instead that populations were more correlated as the season progressed, perhaps because temperature fluctuations enforced synchrony. The authors found that when temperatures first increased above ~15 °C in the spring, the population dynamics crossed a Hopf bifurcation from stability to repeated outbreak cycles, until stabilization again in the fall. Above the Hopf threshold, population-cycle amplitude increased roughly linearly with temperature. This study affirmed the classic concept of temperature as a "pacemaker of all vital rates."[12]

Understanding life-cycle dynamics is relevant for pest control because some insecticides only work at one or two life stages of the insect.[13]