Purple Martin Population Model

Based on the North American Breeding Bird Survey, the North American Landbird Conservation Plan [13] estimated that the global abundance of Purple Martins is 10 million individuals. Rich et al. [13] provide confidence intervals around this estimate of +/- 50%, suggesting that the true number of Purple Martins is likely to be between 5 and 15 million birds. This estimate reflects the abundance of adult birds because it is derived from breeding bird surveys conducted early in the breeding season.

By mid-summer the total number of Purple Martins in North America includes adults plus their young-of-the-year. Tarof and Brown [10] reviewed estimates of offspring production per breeding adult and reported a range of 3 to 4.4 young per nest, except during a cold, wet year when production was only 0.3 young per nest [14]. Several studies reported that most adults of both sexes breed each year [15] with the highest reported percentage of non-breeding floaters being 13% of males [16]. There are also reports of females as non-breeding floaters, although the percentage is low (0.6% [10]).

Based on results of these studies, we made the simplifying assumption that all adult Purple Martins breed each year and produce 2 young. We assumed a 1:1 sex ratio. In the model, Purple Martins begin nesting on 1 May (day 1 of the model) with most clutches being completed on 12 May. Based on this timing, the average hatch date is 28 May (SD = 10 d), which follows a 16-day incubation period. Nestlings fledge and become independent, on average, 26 d later on 23 June (SD = 20 d). This timing of the nesting stages is typical of Purple Martins throughout much of their range [10]. We modeled the timing of nestling and fledgling production using a normal probability distribution function through the breeding season (Figure 1).

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Figure 1. Phenology of energy need of adult Purple Martins (solid line), nestling production per adult (dotted line), and fledgling production per adult (dashed line) across one year.

Background shading indicates phase of the annual cycle as breeding season (stippling), migration (gray), and winter (no shading).

https://doi.org/10.1371/journal.pone.0076616.g001

Stutchbury et al. [12] estimated annual survival rates of one-year-old, two-to-four-year-old, and older female Purple Martins to be 0.48, 0.62, and 0.51, respectively, or 0.58 across all adult age groups. We estimated daily survival of adult Purple Martins to be 0.9985, which yields an annual survival rate of 0.58. Even though there are clearly age-based differences in reproduction and survival of adult Purple Martins, estimating parameters for each adult age and sex class of Purple Martins was unnecessary for the purpose of estimating energy flow and inclusion of these extra model parameters would unnecessarily increase error propagation in the model [17].

Survival of hatch-year birds during their 1^st year is estimated to be 0.27 in Purple Martins [18]. We used a daily survival probability of 0.9964 for hatch-year birds (i.e., young-of-the year), which yields an annual survival rate of 0.27. A model using demographic parameters drawn directly from the literature (adult survival = 0.58*y^-1, 2 young * adult^-1, and juvenile survival = 0.27*y^-1), resulted in an annual population growth rate (λ) of about 17% (λ = 1.17). This growth rate is considerably higher than is sustainable or expected based on empirical data [5]. We suspect that the primary difference between empirical evidence for recent population declines and our base model population growth of 17% per year is due to elevated reproductive output among Purple Martins that are the subject of intensive studies reported in the literature relative to the average Purple Martin. The quality and maintenance of housing by people can strongly impact the reproductive output of birds and we suspect that the housing and protection of martins that are the subject of intensive study is likely to be better than average.

Rather than use this model with an unrealistically high growth rate as the basis for all of our comparisons, we chose to compare among models with minimal annual population change (λ =1). We argue that these models are optimal for comparisons of energy dynamics because (1) the long-term average λ for Purple Martin populations must be close to 1 and (2) based on breeding bird surveys, this value is within 1% of the value for recent Purple Martin population dynamics, which indicate an average decline over the past few decades of about 1% per year range wide (λ =0.99). Therefore, models with λ=1 have the diagnostic advantage of being both currently and historically relevant when compared to models that estimate unsustainable population increases or declines. To achieve a population growth rate with a long-term average λ of 1.0, we generated three additional models in which we reduced either adult survival (0.43*y^-1 = 0.9977*d^-1), production of young (1.45*adult^-1), or juvenile survival (0.19*y^-1 = 0.9954*d^-1). In each of these three models the reduced parameter value resulted in a stable population.

To model the estimated variance in total number of adult Purple Martins, we performed 10,000 model simulations, which were initialized with a number of adult Purple Martins chosen at random from a normal distribution with a mean of 10 million individuals and a standard deviation of 2.5 million (Table 1). We chose a standard deviation of 2.5 million Purple Martins because 2 standard deviations approximates the confidence interval of +/- 5 million Purple Martins published by Rich et al. [13]. To understand how sensitive model outputs were to choice of initial population size we also ran 10,000 simulations with mean initial abundances of 7.5 and 12.5 million Purple Martins.

Purple Martin Energetics

Utter and Lefebvre [19] reported that daily energy expenditures of breeding Purple Martins range from about 137 kJ*d^-1 for males to about 175 kJ*d^-1 for females engaged in feeding nestlings. This difference in energy expenditure is due to females feeding young whereas the males engaged in little parental care. This situation is likely atypical, because males generally provision offspring as frequently as females [20]. Therefore, to estimate the energy expenditure of single independent Purple Martins, we assumed the values for males reported by Utter and Lefebvre [19] were representative of non-breeding birds. To represent the daily energy requirement of a single non-breeding Purple Martin, we used a randomly drawn value from a normal distribution with mean 137 kJ*d^-1 (SD = 1.37kJ* d^-1; 1% of the mean). For breeding Purple Martins we assumed that energy expenditure was 155 kJ*d^-1(SD =1.55kJ* d^-1), that is, intermediate between values for breeding males and females [19]. We used an intermediate value for breeding birds because both parents typically engage in parental care, unlike those birds measured by Utter and Lefebvre [19]. Our model assumed that Purple Martins require this 155 kJ*d^-1 from mid March through mid August, which roughly corresponds to the breeding season (Figure 1). While the duration of the high-energy period may overestimate the length of the breeding period for any particular Purple Martin, the difference is likely offset by other energetically expensive activities (molt and migration) that occur in this time frame but that are not explicitly accounted for in our model.

Purple Martin diets contain many taxonomic orders of insects [21,22,23]; as well as a few non-insect prey. These insects all contain different amounts of energy*g^-1. Brooks et al. [24] tested whether using a mean value for the energy content of insect prey was adequate to explain the ecological energetics of insectivorous vertebrates. They concluded that a constant value of 23J*mg^-1 (=23 kJ*g^-1 of dry mass) was adequate for studies of generalist vertebrate insect predators. Based on this result, we assumed insects consumed by Purple Martins provide 23 kJ*g^-1 (SD = 0.23 kJ) of energy, meaning that a non-breeding Purple Martin needs to eat about 6.0 g of dry mass of insects*d^-1 to meet basic energy requirements (137 kJ*d^-1) and that a breeding bird needs about 6.7 g*d^-1 to reap 155 kJ*d^-1.

To convert from dry mass to numbers of insects consumed we used the relationships in Sage [25]. We use Sage’s [25] dry mass vs. length relationship (R² = 0.86) and wet mass vs. length (R² = 0.87) for Insecta to estimate the number of insects that Purple Martins would need to eat to meet average daily energetic requirements. The relationships were based on data for 153 adult insects from Orthoptera (n =36), Hemiptera (n = 26), Coleoptera (n = 29), Lepidoptera (n = 25), Diptera and Hymenoptera (n = 37). We estimated the mass of insects by drawing at random a mean insect length from a normal distribution with an average of 20 mm (± 2mm). The energy provided by insects of this length was estimated from the equations of Sage [25]. We modeled all of the Purple Martin abundance and energetics relationships using a Matlab script ( [26]; Table 2).

To estimate the number and biomass of insects consumed by Purple martins in different phases of the annual cycle we divided the model estimates for biomass and number of insects consumed into a breeding period (1 April to 31 August), fall migration (1 to 30 September), winter residency (1 October to 28 February), and spring migration (1 to 31 March). We used an existing breeding range map [27] to estimate the land area within the range of the Purple Martin during these 4 phases of the annual cycle. We estimated that the Purple Martin population was spread over 5.72 million km², 2.97 million km², and 11.6 million km² of land area in the breeding, migratory, and winter periods. We made the simplifying assumption that Purple Martins foraged in the lowest 1km of the aerosphere and converted km² of land area to km³ of habitat volume in which the birds forage. We calculated the biomass and number of insects consumed by Purple Martins by dividing the model estimates of number and biomass of insects consumed by estimates of habitat volume for each phase of the annual cycle.