Collective behavior of animals: swarming and complex patterns

En esta nota repasamos algunos modelos basados en individuos para describir el movimiento colectivo de agentes, a lo que nos referimos usando la voz inglesa swarming. Estos modelos se basan en EDOs (ecuaciones diferenciales ordinarias) y muestran un comportamiento asintótico complejo y rico en patrones, que mostramos numéricamente. Además, comentamos cómo se conectan estos modelos de partículas con las ecuaciones en derivadas parciales para describir la evolución de densidades de individuos de forma continua. Las cuestiones matemáticas relacionadas con la estabilidad de de estos modelos de EDP's (ecuaciones en derivadas parciales) despiertan gran interés en la investigación en biología matemáticaIn this short note we review some of the individual based models of the collective motion of agents, called swarming. These models based on ODEs (ordinary differential equations) exhibit a complex rich asymptotic behavior in terms of patterns, that we show numerically. Moreover, we comment on how these particle models are connected to partial differential equations to describe the evolution of densities of individuals in a continuum manner. The mathematical questions behind the stability issues of these PDE (partial differential equations) models are questions of actual interest in mathematical biology researc

Effects of fish movement and environmental variability in the design and success of a marine protected area

Marine Protected Areas (MPA) are broadly used to protect marine ecosystems, restore biomass, and increasingly as tools in fisheries management for benthic stocks. Nevertheless, MPAs are seldom used to target pelagic species due the challenges of designing an effective MPA in a highly dynamic environment. It is believed that highly mobile organisms will get few benefits, since they leave the protected area too frequently. One possible solution is to compensate for such movement with larger MPAs. Nevertheless, uncertainty about the benefits in the face of vagaries about fish movement make it unlikely that such efforts would be successfully pursued.Although it is a generally accepted that MPAs provide multiple benefits if well designed, empirical demonstrations of benefits from MPA are hard to obtain. They require long term evaluations, and as a consequence, comparisons between alternative MPA designs are almost nonexistent. Simulation models provide an alternative to empirical approaches that allow tests of designs and forecasts of potential outcomes. To date, most of the simulation models of MPAs have been developed for benthic systems, where simplified assumptions about fish and fisherman movement are reasonable. Fortunately, with the advent of more realistic fish movement models, new approaches are now possible that can combine complex individual-based models of movement, population dynamics and virtual MPA systems. The use of these new complex simulation models can guide the optimization of MPA design to increase both stock sizes and fisheries yields.The goal of the research presented in this dissertation is to study the potential of a large MPA to protect a pelagic stock and determine how fish movement characteristics and complex environmental dynamics influence the optimal design criteria for a successful pelagic MPA. The findings are timely given increasing interest in developing large no fishing zones to protect overfished pelagic stocks, especially for those taxa whose distribution spans more than one exclusive economic zone or resides in international waters. For this purpose I implemented a simulation model that incorporates detailed fish movement and their responses to complex environmental forcing to study the effect of fish movement on the efficacy of MPAs of different size and location.In Chapter 1, I examined the effects of movement assumptions on the effectiveness of different MPA sizes on fish stocks and fisheries yields. I compared the results to prior modelling studies of MPA design that assumed simple random movement. I explored four movement assumptions with increasing behavioural complexity: a) random, diffusive movement, b) aggregations, c) aggregations that respond to environmental forcing (sea surface temperature), and d) aggregations that respond to environmental forcing and are transported by currents. I found that MPAs can protect pelagic stocks and increase fisheries yields. The optimal MPA size to maximize fisheries benefits increased with fish movement complexity from an area of ~10% when diffusive movement was assumed to ~30% when the movement included aggregations, responses to the sea surface temperature (SST) and transport by currents.In Chapter 2, I studied the effect of spatial environmental variability in the design of the MPA. For this purpose, I specifically explored the same four movement assumptions with increasing behavioural complexity: a) random, diffusive movement, b) aggregations, c) aggregations that respond to environmental forcing (sea surface temperature), and d) aggregations that respond to environmental forcing and are transported by currents, to compare the outcomes from several MPA designs in terms of placement and size. The goal was to analyse how complexities of fish movement and environmental dynamics theoretically affect the predicted outcomes from different MPA designs. In this chapter I found that increasing the size of an MPA reduces the risk of selecting poor or suboptimal placements and protects a larger fraction of the stock while maintaining similar fisheries yields. Simplified movement assumptions (e.g. diffusive movement) underestimated the relative importance of MPA placement and overestimated both the expected stock size and the mean time to recovery. These findings highlight the value of understanding fish movement dynamics and their responses to environmental forcing, providing further evidence that MPAs have the potential to protect, rebuild and manage pelagic stocks.In Chapter 3, I included the effects of temporal variability in environmental forcing. I used the MPA model to simulate a complex pelagic environment with a spatially and temporally heterogeneous sea surface temperature that drove the organism’s movement dynamics. The goal was to include the effects of a highly variable environment on MPA design and successfully increase stock size and fisheries yields. The results suggested that when temporal environmental variability occurs, the expected fisheries benefits are significantly reduced relative to simulations with constant environments. The reduced fishery benefits occur even when the stock density levels were similar, suggesting that in highly variable pelagic environments MPAs may require more of a tradeoff between protect the stocks and increasing fisheries yields.The results of this research suggest that MPAs can effectively be used to protect and rebuild pelagic stocks, and have the potential to increase fisheries yields. Nevertheless, to effectively design MPAs with conservation and management goals, it is necessary to understand the organism’s movement dynamics and how they respond to environmental forcing. Models like the one used in this dissertation have the potential to inform future management decisions and help to design future pelagic MPAs that produce multiple benefits in a variety of environmental settings

Parallel Modeling of Fish Interaction

This paper summarizes our work on a parallel algorithm for an interacting particle model, derived from the model by Czirok, Vicsek, et. al. [3, 4, 5, 13, 14]. Our model is particularly geared toward simulating the behavior of fish in large shoals. In this paper, the background and motivation for the problem are given, as well as an introduction to the mathematical model. A discussion of implementing this model in MATLAB and C++ follows. The parallel implementation is discussed with challenges particular to this mathematical model and how the authors addressed these challenges. Load balancing was performed and is discussed. Finally, a performance analysis follows, using a performance metric to compare the MATLAB, C++, and parallelized code.