Evaluation of a Mallard Productivity Model

Introduction

The mallard (Anas platyrhynchos) has the most extensive breeding range, and is the most common and most heavily harvested of North American waterfowl. There has been widespread concern in recent years about the status of the mallard population, which seems to be declining. Despite considerable study, however, a thorough understanding of mallard population dynamics has eluded waterfowl biologists, possibly because their investigations have been either extensive but superficial or detailed but narrowly focused.

The model we describe originated in 1972 when a team of researchers at Northern Prairie Wildlife Research Center recognized the potential value of coordinating their research activities. They identified simulation modeling as a powerful tool for integrating their findings and identifying areas in which further research would be most productive. This stochastic model is now a central component of a modeling system designed to aid decision making for management and acquisition of habitat for mallard production (Cowardin et al. 1983). Other components are various data-handling and display routines. The system has recently been applied to a number of actual management situations, such as evaluating land-acquisition options, comparing habitat-management practices, and projecting future impacts of wetland drainage (Cowardin et al. 1983). This model differs from many others presented in this volume in that it describes the process of mallard recruitment, as influenced by habitat and other variables, rather than merely predicting the presence/absence or the abundance of the species. In this sense, it more closely characterizes the quality of the habitat for survival and reproduction of mallards.

The Basic Model

General Overview

In the model, mallards are followed from arrival on the breeding grounds, in the spring, through the summer reproductive season, until the onset of hunting in the fall. The survival and reproduction of individual females are formulated as functions of wetland habitat conditions, nesting habitats available (their area, vegetative features, and the security from destruction of nests), length of breeding season, population age structure, and physical condition of individual birds. Many of these modeled features vary dynamically throughout the season, and all can be manipulated by the investigator. A detailed description of the model is forthcoming, but the following will provide an overview. Some parameter values were based on field investigations, others were developed during the modeling process.

Each simulated hen follows the flow-chart path shown in Figure 1 on each day of a 120-day breeding season. A non-nesting hen can begin nesting with a probability that depends on (1) her physical condition as reflected by her body weight, which initially is a normal random variate but declines after repeated nesting efforts; (2) wetland availability, which normally declines during the season; and (3) the date within the season (a mallard hen being unlikely to initiate a nest late in the season regardless of her condition or the quality of the habitat).

A nesting hen selects a habitat from those available; usually we restrict her to a 4-legal-section block (10.36 km²). The probability that a particular habitat is selected is a function of the area of that habitat multiplied by its attractiveness. Attractiveness to mallards is indexed by a measurement of the height and density of nesting cover (Robel et al. 1970; Kirsch et al. 1978). After nesting commences, the hen's weight is reduced by 10 gm for each egg laid and by 2 gm for each day spent incubating her clutch.

The clutch must survive about 34 days of egg laying and incubation to hatch. On each of those days, the clutch and the hen are at risk to a variety or predators. Certain habitats pose additional hazards, e.g., tillage of cropland and mowing of hayland and roadsides. Survival of the nest varies among habitats and can also be altered by certain management practices, such as predator reduction or habitat manipulation.

If a nest is destroyed, the hen may or may not suffer the same fate, with probability of death equal to 0.06. In addition, hens have a small chance (0.001) each day of dying in other ways, such as predation away from the nest, accidents, and disease. Once a simulated hen dies, her death is tallied and she no longer plays a role.

Should the simulated clutch survive the requisite 34 days, the eggs hatch. The number of eggs hatched equals the number laid, which was a random variable dependent on the hen's condition and the initiation date.

Not all ducklings survive to fledging. Some broods succumb completely, and nearly half the ducklings are lost from remaining broods. Those ducklings that survive all the hazards are tallied as fledged recruits, the ultimate product of the reproductive season.

Model Output

The standard FORTRAN listing resulting from the model includes values of the key input variables, the random number seed used, and the available nesting habitats and their characteristics, including changes that occur during the breeding season. Results of the simulation are succinctly summarized on one page showing the percentage of clutches that hatch, the percentage of hens that are successful in hatching a clutch, the average number of nests per hen, the summer mortality rate of hens, and the recruitment rate (number of fledged females per adult female in the spring population). For each type of nesting habitat, the program lists the number of nesting attempts initiated, the number of successful attempts, and the number of recruits that were hatched in that habitat.

An optional SAS (SAS Institute 1982a) routine produces several detailed tables and figures that facilitate a close inspection of the model's operation. One listing is of all nests, indicating the hen involved, her age, her weight at nest initiation, the habitat, dates on which the nest was begun and terminated, the size of the clutch, and the fate of the nest. Summary tables display the total number of nests according to habitat and fate by age of hen. Another insightful display (Fig. 2) shows the history of each hen throughout the nesting season, including the dates, habitat, and fate of each nest. The success of nests according to date initiated is shown in both a table and a figure (Table 1 and Fig. 3). These data are also presented for each habitat type. Final tables (Tables 2 and 3) give the necessary statistics for the calculation of Mayfield (1961) estimates of nest success rates (Johnson 1979).

The variety of outputs and the flexibility of viewing the simulation process have been valuable aids in debugging the program and in modifying it to conform to biological findings. In addition, many of the displays were used to analyze biases in an intensive field study of radio-telemetered mallards (Cowardin et al. 1985).

Validation

Complete validation of the model would require the comparison of results from the model with measurements of those parameters determined from field studies. This poses a difficult problem for mallard production, because there is no known way to actually count the young fledged from a breeding population. In practice, only indices of mallard production are obtained. It is possible, however, to measure many of the intermediate parameters calculated by the model and compare them with field estimates. Although we have not done studies specifically designed to validate the model, we did apply the model to one area where detailed field studies were done (Cowardin et al. 1983). Model predictions compared with actual estimates were 13.2 vs 13.8 for number of successful nests, 68.9 vs 74.3 for number of nests initiated, 1.78 vs 1.55 for average number of initiated nests per hen, and 0.22 vs 0.19 for the nest success rate.

We also found that this model gave results consistent with other models. Cowardin and Johnson (1979) presented a simple deterministic model relating hen success to nest success. Although this model is completely independent of the stochastic model described here, results have been consistently similar. We also used output from the stochastic model as input to a different deterministic model (Cowardin and Johnson 1979) designed to predict the change in population size from one year to the next. The results suggested a slowly declining population, a conclusion that agrees with continental counts of mallards. The consistency of these results suggests that the model is sufficiently realistic to serve as a tool to assist in the making of management decisions.

Modifications of the Model

Wetland Conditions

Wetland conditions markedly influence the number of mallards attracted to breeding areas; they also affect the probability of nesting, the length of the nesting season, and the number of nests attempted (Krapu et al. 1983; Cowardin et al. 1985). The relation between wetland conditions and reproductive effort has been recognized for some time (Crissey 1969) and is currently incorporated in a model used by the U.S. Fish and Wildlife Service for predicting the size of the fall flight and for setting hunting regulations (Martin et al. 1979). Models currently in use are simple and do not account for the complexity of wetland conditions, which vary in time and space, or for the interactions of wetland conditions with other factors that determine reproductive success.

Our model allows for daily changes in wetland conditions throughout the breeding season and for interactions among factors that influence recruitment rate. We can simulate the impact of various wetland conditions that occur in nature and predict reproductive parameters such as nest success, hen success, summer hen survival, and recruitment.

We simulated dry, average, and wet water conditions during the breeding season. For each set of conditions, we used a data set containing an index to water conditions for each day of the season. The index value approximates the percentage of semipermanent wetland basins (defined by Stewart and Kantrud 1971) containing water on that day. These data sets reflect the normal drying of wetland basins during the season; differences in percentages at the beginning of the breeding season and rates of loss distinguish the dry, average, and wet years. These indices can be entered as data without modifying the structure of the model.

Simulated dry conditions had considerable influence on recruitment, reducing the total number of nests and the nesting activity. Nests per hen declined from 1.97 under average situations to 1.18 in dry years (Table 1). Nesting activity was of diminished intensity and reduced duration (Fig. 3). Nest survival rates were similar under all wetland conditions. Reduced nesting affected the recruitment rate, which dropped from 0.42 under average conditions to 0.24 under dry conditions.

Changes due to wet-year conditions were noticeable in the number of nests per hen (Table 1), which reflected increased nesting persistence due to improved wetlands. The net results showed slightly higher rates of hen success and recruitment.

Catastrophic Events

Reproductive success is affected by catastrophic weather events, which can be either local or widespread across the breeding range. The net effect of these events on recruitment rate is difficult to isolate in field studies because a hen may renest after her nest is destroyed. The probability of renesting depends on the hen's physical condition, wetland conditions, and the time of the season when nest destruction occurs. Our model furnished an opportunity to investigate the impact of catastrophic events that would be difficult, if not impossible, to assess through field study.

We simulated three events that are fairly common in the northern prairies. The first two were snowstorms, one on 3 May and a later one on 13 May. Storms of this type bury nests under deep snow and cause their destruction or abandonment. They also blanket the nesting habitat and temporarily prevent further nest initiation. Such storms, although frequently severe, are of short duration, and nest initiation resumes as soon as habitat becomes available.

The simulations were accomplished by setting the daily nest mortality rate at 0.90 on the day of the simulated storm. The input data for wetlands were then set to zero to prevent any nest initiations for a period of 3 days.

The third catastrophic event simulated was a midseason flood. Mallards nest primarily in uplands, but some individuals nest in emergent vegetation over water and in wet meadows surrounding wetlands. These nests may be destroyed by rapid rises in water level occurring after heavy summer rainstorms. The simulation was conducted by setting the daily nest mortality rate to 0.5 on the day of the simulated rainstorm for all nests located in wetland. We assumed that half the wetland nests would survive the flood. We also reduced by half the height-density measurement for wetland nests during the remainder of the season. This change simulated the reduced attractiveness of wetland sites for nest initiations following the flood. Also, the data representing wetland conditions were modified to simulate improved water levels, which occur after storms and extend the length of the nesting season.

Both the 3 May and the 13 May snowstorms substantially reduced the survival rate of simulated nests (Table 1). Virtually all nests were destroyed during the 13 May storm. The reduction in nest survival was somewhat compensated for by increased renesting, as new nests were initiated a few days after the storm (Fig. 2). The average number of nests per hen increased by about 0.35, because destruction of many nests early in the season (Fig. 3) allowed the birds to renest. Nests destroyed by the storm did not result in the associated loss of the hen, which we assume occurs in 6% of the nests destroyed by predators. The simulated hens were precluded from nesting for a period of time, which decreased their risk to predation and resulted in a slight increase in summer survival (Table 1). The 13 May storm was disastrous and resulted in the lowest recruitment rate and smallest index to population change of all simulated events. This storm caused a greater reduction in nest success because more birds were nesting at that time. Moreover, these birds had expended more resources and had a lower probability of renesting. Inclement weather could also affect real populations by reducing the food supply available to nesting hens, thereby further reducing the probability of renesting.

The simulated flood had little impact on the population because relatively few nests were affected. The decrease in number of successful nests in wetland was from 10.8% to 4.9% (Table 2). The loss of wetland nests was compensated for by increased nesting associated with improved wetland conditions. Most renests were in habitats more secure than wetland. The percentage of nests initiated in planted cover, for example, increased from 21.9 to 23.4.

Homing

Migrational homing, the predilection of an adult bird to return to the area where it spent the previous nesting season or a yearling bird to come back to the area where it was reared, is recognized in several species of ducks, including mallards (Bellrose 1976). This characteristic may be a critical determinant in certain nesting areas, such as islands, where densities of nesting birds grow far larger than would be anticipated on the basis of wetland habitat. Such population trajectories may be fueled by homing, which is particularly common among adults that successfully produce young (Doty and Lee 1974). Successful hens are hypothesized to favor the same habitat for subsequent nesting.

One way to assess the importance of homing is to follow individual birds from one year to the next. Our current model does not identify birds in subsequent years, however, so we approximated a solution. First, we modified the probability of selecting a particular habitat for nesting so that it depended on the nest success rate, as well as the area and attractiveness of the habitat. This change reflected the propensity of successful hens to select the same habitat for subsequent nests. In the original model, the probability of a hen choosing a habitat for nesting was proportional to the product of the area of that habitat times the height and density measurement of vegetation in that habitat. To these factors we added a function of the nest success rate (P) in the habitat. The function was 1 at P = 0, increased linearly to 3 at P = 0.5, and was constant thereafter.

Second, we allowed the number of breeding hens in an area to depend not only on the number and area of wetlands, but also on the hen success rate of that area in the previous year. This change permitted population growth due to homing by successful hens and their female offspring. We assumed that 75% of successful hens return to the breeding area in the next year, as well as 30% of their female offspring. In addition, a certain number of hens pioneer into the area because of the wetlands there. We assumed that the number of pioneers would be reduced by one-half for each homing hen. Simulations suggested that the population would be stable if 28% of hens are successful, each producing an average of two young females. The population would increase if the hen success rate exceeded 28% and would decline if it fell below 28%.

We explored the effects of these changes by simulating production on a 10.36-km² area typical of fairly good habitat in North Dakota. The field contained a 7-ha plot of planted nesting cover, protected by a predator-resistant fence. This habitat is an attractive one that has a high hen success rate and yields many offspring. Wetlands on the area attracted an estimated 21 mallard hens during the initial year.

Under the original model (based on three replications), 3.5 of the 39.4 nests initiated were in the fenced plot (Table 3). Total production of females was (21)(0.537) = 11.3.

From the model with homing, we projected the breeding population to be 22.6 females, and 5.9 of the 40.6 initiated nests were in the fenced plot. This difference from the original model appears modest, but it caused increases in the hen success rate and in the recruitment rate. Total production of females was (22.6)(0.633) = 14.3, a 27% increase compared with the former model.

Discussion

The mallard has been the subject of several other modeling efforts primarily designed to estimate its fall flight or to develop exploitation strategies (Geis et al. 1969; Hammack and Brown 1974; Anderson 1975a, 1975b; Hochbaum and Caswell 1978; Martin et al. 1979). These models, most of which covered the entire North American range, typically treated reproduction as a function involving only the breeding population size and the number of wetlands containing water. Our model is much more restricted in scope, pertaining to discrete areas within the glaciated prairie. In part because of this restriction, and in part because of the greater biological content of the model, it incorporates far more of the factors now known to influence productivity.

Models such as the one presented here allow interactions among variables to arise naturally. These interactions often account for behavior that appears contrary to intuition. One such example from the present application is the effect of flooding, a weather calamity that destroyed many wetland nests. Despite this loss, however, total productivity was unaffected, because hens that lost nests renested—often in more secure habitats—and improved wetland conditions increased renesting among the population in general.

The extensions of the model described here illustrate several facets of the population dynamics of mallards. The analyses contrasting wet, average, and dry years were consistent with expectations based on field experience and showed the marked effect of drought. Spring storms, another common phenomenon in much of the mallard's breeding range, were also influential, but their severity depended on when they occurred in relation to the breeding season of the species. The destruction of nests by spring flooding was offset by increased renesting, often in more secure habitats. The homing phenomenon was shown to exert considerable influence on productivity, and it should therefore be considered in further research and be incorporated in future management plans.

The value of the model described here as a research tool lies in its ability to incorporate new biological findings and to allow "experimenting" with the simulated population. This may shed light on how investigations of the real population should be most profitably designed. The management applications are diverse, ranging from assisting with the interpretation of operationally gathered productivity indices, to exploring potential results of selected habitat manipulations, and to evaluating strategies for acquisition of further habitat.

Despite these virtues, the model is no more than a collection of equations reflecting our beliefs about mallard population dynamics. Actions taken in response to predictions from the model should be monitored to assess the model's performance. It can be continually improved as we learn more about the processes involved.

Acknowledgments

We are grateful to T. J. Dwyer, R. J. Greenwood, G. L. Krapu, A. B. Sargeant, and G. A. Swanson, all members of the original mallard research team that conceived the model discussed here. R. D. Crawford and D. S. Gilmer also contributed to that effort, as did all biologists who have developed knowledge about mallard reproductive dynamics. A. M. Frank assisted with computer programming. This report benefitted from reviews by W. E. Grenfell, A. T. Klett, G. L. Krapu, M. G. Raphael, T. L. Shaffer, and the editors of the proceedings.