Collecting water quality data across large lakes is often done under regulatory mandate; however, it is difficult to connect nutrient concentration observations to sources of those nutrients and to quantify this relationship. This difficulty arises from the spatial and temporal separation between observations, the impact of hydrodynamic forces, and the cost involved in discrete samples collected aboard vessels. These challenges are typified in Lake Erie, where binational agreements regulate riverine loads of total phosphorus (TP) to address the impacts from annual harmful algal blooms (HABs). While it is known that the Maumee River supplies 50 % of the nutrient load to Lake Erie, the details of how the Maumee River TP load changes Lake Erie TP concentration have not been demonstrated. We developed a hierarchical spatially referenced Bayesian state-space model with an adjacency matrix defined by surface currents. This was applied to a 2 km-by-2 km grid of nodes, to which observed lake and river TP concentrations were joined. The model generated posterior samples describing the unobserved nodes and observed nodes on unobserved days. We quantified the impact plume of the Maumee River by experimentally changing concentration data and tracking the change in in-lake predictions. Our impact plume represents the spatial and temporal variation of how river concentrations correlate with lake concentrations. We used the impact plume to scale the Maumee River spring TP load to an effective Maumee River TP spring load for each node in the lake. By assigning an effective load to each node, the relationship between load and concentration is consistent throughout our sampling locations. A linear model of annual lake node mean TP concentration and effective Maumee River load estimated that, in the absence of the Maumee River load, lake concentrations at the sampled nodes would be 23.1 µg L−1 (±1.75, 95 % CI, credible interval) and that for each 100 t of spring TP effective load delivered to Lake Erie, mean TP concentrations increase by 11 µg L−1 (±1, 95 % CI). Our proposed modeling technique allowed us to establish these quantitative connections between Maumee TP load and Lake Erie TP concentrations which otherwise would be masked by the movement of water through space and time.

The previous two chapters have presented the state-space model as a general framework for modelling population dynamics and discussed alternative ways of fitting SSMs to data. In this chapter, we address model formulation and model evaluation.

There are many reasons why we are interested in how long wild animals survive. A particularly pressing one is so that we can evaluate the effects of climate and anthropomorphic changes. In an early example, North and Morgan (1979) demonstrated a link between winter temperature and the survival of grey herons, Ardea cinerea, using point-process models as well as logistic regression, as in Sect. 5.1. As a further example, one might be interested in calibrating the effect of a change in hunting regulations on survival probability of wild fowl.

In this chapter, we develop a “building block” approach to defining population dynamics models, in which each building block corresponds to one biological process, and is represented by one matrix (Lebreton 1973; Lebreton and Isenmann 1976; Buckland et al. 2004, 2007). Matrix models are usually defined within a deterministic framework, but we will need stochastic models. Thus we will think of the matrix as a mathematical tool for telling us how many animals of each type we expect to have in our population once the process (e.g. survival, birth, movement) has occurred, given the numbers present beforehand. We separately specify the probability distribution associated with the process, which will determine the actual numbers of each type. We refer to the types of animal as states.

In terms of modelling population dynamics, the mark-recapture literature has in recent years been dominated by methods for estimating survival, as described in Chap. 7. In this chapter, we consider open-population mark-recapture methods for estimating abundance, survival and births. We first summarise conventional methods (Seber 1973, 1982).

In Chap. 5, we recommended that formulation of population dynamics models should be guided by aims to answer specific scientific questions or assess or predict the effects of management actions. Management actions might target a specific life stage. For example, we might ask "How does removing wetland plants (such as bulrush or cattail) that have started to cover ponds and reduce the amount of open water in a waterfowl breeding area affect reproductive success?" The consequences of actions, however, typically ripple throughout the entire population life history and effective management requires more detailed ecological study. This in turn requires information about demographic processes and abundances for multiple life stages to characterize the population dynamics.

Australian & New Zealand Journal of StatisticsVolume 48, Issue 3 p. 393-413 INVITED DISCUSSION OF THE PAPER BY DOMIJAN, JORGENSEN & REID Stephen Haslett, Stephen HaslettSearch for more papers by this authorR.B. Millar, R.B. MillarSearch for more papers by this authorK.B. Newman, K.B. NewmanSearch for more papers by this authorMartyn Plummer, Martyn PlummerSearch for more papers by this authorA.H. Welsh, A.H. WelshSearch for more papers by this author Stephen Haslett, Stephen HaslettSearch for more papers by this authorR.B. Millar, R.B. MillarSearch for more papers by this authorK.B. Newman, K.B. NewmanSearch for more papers by this authorMartyn Plummer, Martyn PlummerSearch for more papers by this authorA.H. Welsh, A.H. WelshSearch for more papers by this author First published: 13 October 2006 https://doi.org/10.1111/j.1467-842X.2006.00447.xCitations: 1Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume48, Issue3September 2006Pages 393-413 RelatedInformation

SummaryThis paper proposes a unified framework for defining and fitting stochastic, discrete‐time, discrete‐stage population dynamics models. The biological system is described by a state‐space model, where the true but unknown state of the population is modelled by a state process, and this is linked to survey data by an observation process. All sources of uncertainty in the inputs, including uncertainty about model specification, are readily incorporated. The paper shows how the state process can be represented as a generalization of the standard Leslie or Lefkovitch matrix. By dividing the state process into subprocesses, complex models can be constructed from manageable building blocks. The paper illustrates the approach with a model of the British grey seal metapopulation, using sequential importance sampling with kernel smoothing to fit the model.

We develop a unified framework for jointly defining population dynamics models and measurements taken on a population. The framework is a state-space model where the population processes are modelled by the state process and measurements are modelled by the observation process. In many cases, the expected value for the state process can be represented as a generalisation of the standard population projection matrix: each sub-process within the state process may be modelled by a separate matrix and the product of these matrices is a generalised Leslie matrix. By selecting appropriate matrices and their ordering, a wide range of models may be specified. The method is fully flexible for allowing stochastic variation in the processes. Process parameters may themselves be modelled as functions of covariates. The structure accommodates effects such as density dependence, competition and predator–prey relationships, and metapopulations are readily modelled. Observations on the population enter through an observation process model, and we show how likelihood functions can be built that reflect both demographic stochasticity (which appears in the state process) and stochastic errors in the observations. Parameter estimation and estimation of state process variables can be conducted using sequential Monte Carlo procedures.

A stochastic model for the movement and eventual mortality of an individual animal is formulated as a combination of three components: initial spatial location, survival status at any point in time, and spatial translation between points in time. Alternative theories about survival and migration can be expressed in terms of different models for any of the three components. The model can be extended to groups of animals, spatially and/or temporally aggregated, by appropriate integration. When information about animal counts is partial or inexact, as from mark-recapture or harvest data, state-space models are a natural framework for estimating both unknown parameters and animal abundance. As an example, a multivariate, linear normal state-space model that explicitly incorporates each of the three individual animal components is formulated for the migration and harvest of Pacific coho salmon (Oncorhynchus kisutch). Using recoveries of tagged coho salmon caught in ocean fisheries and associated measures of fishing effort, the Kalman filter and maximum likelihood are used to estimate parameters of the processes, and the Kalman smooth is used to estimate abundances. Given estimated parameters and current harvest and effort data, real-time management of exploited populations could be improved by using the Kalman prediction algorithm.