Pest insect population dynamics
The population dynamics of pest insects is a subject of interest to farmers, agricultural economists, ecologists, and those concerned with animal welfare. Life tables. A life table shows how and how many insects die as they mature from eggs to adults. It helps with pest control by identifying at what life stage pest insects are most vulnerable and how mortality can be increased. A cohort life table tracks organisms through the stages of life, while a static life table shows the distribution of life stages among the population at a single point in time. Following is  an example of a cohort life table based on field data from Vargas and Nishida (1980). The overall mortality rate was 94.8%, but this is probably an underestimate because the study collected the pupae in cups, and these may have protected them from birds, mice, harsh weather, and so on. Life expectancy. From a life table we can calculate life expectancy as follows. Assume the stages formula_1 are uniformly spaced. The average proportion formula_2 of organisms alive at stage formula_1 between beginning and end is The total number formula_5 of future stages to be lived by individuals at age formula_1  and older is Then the life expectancy formula_8 at age formula_1 is We could have done the same computation with raw numbers of individuals rather than proportions. Basic reproductive rate. If we further know the number formula_11 of eggs produced (fecundity) at age formula_1, we can calculate the eggs produced per surviving individual formula_13 as where formula_15 is the number of individuals alive at that stage. The basic reproductive rate formula_16, 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. We can calculate it from life-table data as This is because each formula_18 product computes (first-generation parents at age formula_1)/(first-generation eggs) times (second-generation eggs produced by age-formula_1 parents)/(first-generation parents at age formula_1). If formula_22 is the initial population size and formula_23 is the population size after a generation, then Generation time. The cohort generation time formula_25 is the average duration between when a parent is born and when its child is born. If formula_1 is measured in years, then Intrinsic rate of increase. If formula_16  remains relatively stable over generations, we can use it to approximate the intrinsic rate of increase formula_29 for the population: This is because where the approximation follows from the Mercator series. formula_25 is a change in time, formula_33. Then we have which is the discrete definition of the intrinsic rate of increase. Growth models. In general, population growth roughly follows one of these trends: Insect pest growth rates are heavily influenced by temperature and rainfall, among other variables. Sometimes pest populations grow rapidly and become outbreaks. Degree-day calculations. Because insects are ectothermic, "temperature is probably the single most important  environmental factor influencing insect behavior, distribution, development, survival, and reproduction." As a result, growing degree-days are commonly used to estimate insect development, often relative to a point, i.e., a biological milestone, such as when the insect comes out of pupation in spring. Degree-days can help with pest control. Yamamura and Kiritani approximated the development rate formula_29 as with formula_37 being the current temperature, formula_38 being the base temperature for the species, and formula_39 being a thermal constant for the species. A generation is defined as the duration required for the time-integral of formula_29 to equal 1. Using  linear approximations, the authors estimate that if the temperature increased by formula_41 (for instance, maybe formula_41 = 2 °C for climate change by 2100 relative to 1990), then the increase in number of generations per year formula_43 would be where formula_45 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. Geometric Brownian motion. Sunding and Zivin model population growth of insect pests as a geometric Brownian motion (GBM) process. 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 formula_46 is the current insect population, formula_47 is the intrinsic growth rate, and formula_48 is a variance coefficient, the authors assume that where formula_50 is an increment of time, and formula_51 is an increment of a Wiener process, with formula_52 being standard-normal distributed. In  this model, short-run population changes are dominated by the stochastic term, formula_53, but long-run changes are dominated by the trend term, formula_54. After solving this equation, we find that the population at time formula_55, formula_56, is log-normally distributed: where formula_58 is the initial population. 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 formula_59 and formula_60. Since the expected population based on the log-normal distribution grows with formula_61, this implies an aphid doubling time of formula_62 days. Note that other literature has found aphid generation times to lie roughly in the range of 4.7 to 5.8 days. Repeated outbreak cycles. 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 1961-2012. 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. The cycles result from population overshoot. 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. This is often what's observed in North America. 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." Understanding life-cycle dynamics is relevant for pest control because some insecticides only work at one or two life stages of the insect. Effects of pest control. B. Chaney, a farm advisor in Monterey County, CA, estimates that mevinphos would kill practically all aphids in a field upon application. Wyatt, citing data from various "Arthropod Management Tests", estimates that the percent of lettuce aphids killed is 76.1% for endosulfan and 67.0% for imidacloprid. Insecticides used on gypsy moths in the 1970s had roughly a 90% kill rate. Impact of climate change. Temperature change is argued to be  the biggest direct abiotic impact of climate change on herbivorous insects. In temperate regions, global warming will affect overwintering, and warmer temperatures will extend the summer season, allowing for more growth and reproduction. A 2013 study estimated that on average, crop pests and pathogens have moved to higher latitudes at a rate of about 2.7 km/year since 1960. This is roughly in line with estimates of the rate of climate change in general.