Date Published: June 7, 2018
Publisher: Public Library of Science
Author(s): Shujuan Xia, Takashi Yamakawa, Andrea Belgrano.
Several types of size-based models have been developed to model the size spectra of marine communities, in which abundance scales strongly with body size, using continuous differential equations. In this study, we develop a size-structured matrix model, which can be seen as a discretization of the Mckendrick-von Foerster equation, to simulate the dynamics of marine communities. The developed model uses a series of simple body size power functions to describe the basic processes of predator–prey interactions, reproduction, metabolism, and non-predation mortality based on the principle of mass balance. Linear size spectra with slopes of approximately –1 are successfully reproduced by this model. Several examples of numerical simulations are provided to demonstrate the model’s behavior. Size spectra with cut-offs and/or waves are also found for certain parameter values. The matrix model is flexible and can be freely manipulated by users to answer different questions and is executed over a shorter numerical calculation running time than continuous models.
Since the pioneering work by Sheldon et al. , the marine community size spectrum, i.e., the distribution of abundance or biomass against individuals’ body size on a log–log scale, has attracted significant interest from the scientific community. Classifying individuals over the size range from bacteria to whales into different size groups regardless of taxonomy results in a linear size spectrum with a slope of approximately –1 in unexploited communities [1, 2]. The slopes and intercepts of these size spectra have been proposed as useful indicators in revealing the nature of ecosystems and the effects of anthropogenic perturbation on the structure of marine communities .
We developed a size-structured matrix model to express community dynamics in which each process is modeled by simple body size power functions. The model can be seen as a discretization of the Mckendrick-von Foerster partial differential equation that produces a diffusion of growth, i.e., during each time step, several individuals grow up to the next-higher size class while other individuals remain in the same size class. Thus, the basic difference between our matrix model and the continuous Mckendrick-von Foerster equation lies in its ability to incorporate growth variation in individuals within a size class; in a real process, this diffusion is appropriate because individuals with the same size in a given population often differ in growth rate . The main advantage of our model lies in its flexibility in modeling each process. In addition, the model can be run on a daily or larger time step Δt by increasing the mass ratio δ of successive size classes. In this case, the running time for numerical simulations required by our matrix model will be much shorter than that of PDE models as a result of the reduced number of size classes and loop iterations.
In this paper, we demonstrated the construction of a community matrix model with several body size power functions from individual-level processes. The model can produce stable size spectra under various conditions. Growth diffusion, which plays a role in stabilizing size spectra, is allowed in our model. When parameter values change, the model produces static waves at equilibrium states as a result of top-down cascades or cut-offs at large size ranges caused by reduction in energy allocated to growth or high fishing mortality.