Zhou, Can (2015-08). A Predator-prey System with Seasonal Reproduction: Theoretical and Statistical Developments. Doctoral Dissertation. Thesis uri icon

abstract

  • Predator-prey interaction (PPI) is important in both ecological theory and application. Various theoretical and empirical approaches have been proposed to develop mathematical and statistical tools for studying PPI. The aim of this dissertation is to expand our understanding along these two lines. Using mathematical models, I compare the dynamics of predator-prey models with one specific type of interaction, where both predator and prey exhibit seasonal reproduction and predation is continuous. I show that the use of a continuous-time predator-prey model with an instantaneous approximation of seasonal reproduction can produce unstable dynamics, whereas the use of a discrete-time model, which incorporates seasonal reproduction, always produces locally stable equilibria. Finally, with stage-structured predator-prey models, I demonstrate how demographic parameters affect asymptotic dynamics. I then explore statistical methods to infer the type of PPI from non-stationary time series. Traditional approaches assume that each population in a community is itself regulated, i.e., each time series is stationary. However, complex community structure and a lack of regulation in an individual population alone can result in inappropriate inferences based on traditional approaches. I introduce a statistical framework to analyze non-stationary time series that could be collectively regulated. I also demonstrate the method with time series of selected shrimp and fish populations in the Gulf of Mexico. Finally, I compare the performance of four methods to determine the type of PPI from short and noisy time series. The data-generating model includes nonlinear predation, compensatory density-dependence, and both process and observation errors. To represent our uncertainty about the true form of species interaction, none of the statistical models incorporate all these factors. I show that three methods that account for autocorrelation have better performance in terms of type-I error rates. Among these three methods, only one method produces consistent type-I error rates at all three commonly used significant levels, but it has low power. This method provides a conservative approach to identify PPI. In conclusion, my research identified challenging aspects for research aimed at understanding the dynamical significance of PPI, and explored the potential for mathematical modeling integrated with analysis of empirical time series data to advance knowledge.
  • Predator-prey interaction (PPI) is important in both ecological theory and application. Various theoretical and empirical approaches have been proposed to develop mathematical and statistical tools for studying PPI. The aim of this dissertation is to expand our understanding along these two lines.

    Using mathematical models, I compare the dynamics of predator-prey models with one specific type of interaction, where both predator and prey exhibit seasonal reproduction and predation is continuous. I show that the use of a continuous-time predator-prey model with an instantaneous approximation of seasonal reproduction can produce unstable dynamics, whereas the use of a discrete-time model, which incorporates seasonal reproduction, always produces locally stable equilibria. Finally, with stage-structured predator-prey models, I demonstrate how demographic parameters affect asymptotic dynamics.

    I then explore statistical methods to infer the type of PPI from non-stationary time series. Traditional approaches assume that each population in a community is itself regulated, i.e., each time series is stationary. However, complex community structure and a lack of regulation in an individual population alone can result in inappropriate inferences based on traditional approaches. I introduce a statistical framework to analyze non-stationary time series that could be collectively regulated. I also demonstrate the method with time series of selected shrimp and fish populations in the Gulf of Mexico.

    Finally, I compare the performance of four methods to determine the type of PPI from short and noisy time series. The data-generating model includes nonlinear predation, compensatory density-dependence, and both process and observation errors. To represent our uncertainty about the true form of species interaction, none of the statistical models incorporate all these factors. I show that three methods that account for autocorrelation have better performance in terms of type-I error rates. Among these three methods, only one method produces consistent type-I error rates at all three commonly used significant levels, but it has low power. This method provides a conservative approach to identify PPI.

    In conclusion, my research identified challenging aspects for research aimed at understanding the dynamical significance of PPI, and explored the potential for mathematical modeling integrated with analysis of empirical time series data to advance knowledge.

publication date

  • August 2015