Our study

We are in the process of conducting an analysis of lead-poisoned bald eagles for Northeast states and provinces. The goal is to determine if lead is impacting eagles at a population level despite increasing overall numbers. Current estimates indicate that 17% of annual mortalities are due to lead poisoning. We have secured funding from Morris Animal Foundation to fund a post-doctoral fellow to conduct more intensive population modeling to assess potential impacts.

Our method

Population matrix models are foundational quantitative​ tools in ecology, wildlife management, and conservation biology. Matrices enhance understanding of the dynamics, structure, and demographic consequences to changes within populations (Caswell 2001). We integrated empirical adult time series data and the life history skeleton of bald eagles into the combinatorial optimization algorithm to obtain values of matrix elements and migration parameters for several states in the Northeastern United States. 

Population matrix models hinge on stable population theory and a few consequent theorems (Lebreton 2005). Stable population theory links a fixed set of life history characteristics of an age-structured (Leslie, 1945) or stage-structured matrix (Lefkovitch, 1965) to the population-level dynamics they generate (Tuljapurkar, 2008). Stable population theory gives rise to the Fundamental Theorem of Demography and the Strong and Weak Theorems of Ergodicity. The Fundamental Theorem of Demography states that a matrix can be spectrally decomposed into eigenvalues and vectors, which specify asymptotic (“long term”) or transient ("short term") population behavior. The Strong Ergodic Theorem states that if survival and fertility remain static, the long-term population growth will stabilize, and the population dynamics will become independent of initial conditions, while the Weak Ergodic Theorem states when two populations are subject to the same conditions, they will converge to have identical asymptotic dynamics (Arthur, 1982). 

It is through these theorems that we are able to link updated average cohort survival estimates with the population dynamics that they generate. 

A population matrix model is structured according to the transitions in a life cycle (Caswell, 2001). A life history includes not only the stage-specific fecundity and mortality rates, but the entire sequence of changes through which an organism passes in its development from conception to death (Lande, 1982). Changes to the way individuals move through their life history results in changes to the configuration of the matrix, while changes in average survival or fertility result in changes to the magnitudes of the matrix elements. 

The matrix elements are compound functions of “vital rates” and "transition probabilities" (Doak, Kareiva, & Klepetka, 1994). Vital rates are the survival and fertility parameters, while transition probabilities represent graduation (or demotion) from one stage to another. 

A population matrix model is parameterized according to the functions of a life history (Caswell, 2001), so they reflect specific life history configurations. The columns represent the stages; column one represents the hatchling stage, column two represents immature or non-breeding adult stage, and column three represents breeding adult stage. 

Reproduction events line the top row, while compound (survival and transition) elements appear in matrix rows two and three.

A reproductive event is parameterized as birth pulse ("census occurs at a single time per year, and after survival is tabulated"), with units equal to fecundity ("maximum reproductive potential per female per time unit") or fertility ("actual reproduction per female per time unit"; Caswell, 2001).

The trace ("diagonal elements") of the matrix contains three self-loops; matrix elements that represent the probability that an individual will survive one discrete time unit yet fail to transition out of that stage. The off-diagonal elements represent the probability that an individual will survive one discrete time unit and transition into another stage.

There is extensive literature on both parameterizing Leslie matrices and demonstrating the equivalence of Leslie matrix and life table approaches (e.g., Taylor and Carley 1988; Noon and Sauer, 1992). For additional information on how to parameterize a general model matrix, see Caswell (2001).

Our data

Data have been provided from New York, Pennsylvania, Maine, Massachusetts, Quebec, Prince Edward Island, and Newfoundland, beginning with samples collected as early as 1994. 

Our funding source

Financial support was provided in part by the Morris Animal Foundation under grant # D18ZO-103: Northeast regional meta-analysis of lead toxicosis impacts on bald eagles (Haliaeetus leucocephalus).

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