Biological swarms of insects, fish, birds, and other organisms influence ecological and evolutionary dynamics, serve as prototypes for algorithm development of autonomous vehicles, and are rich sources of patterns. Swarms are shaped by an interaction of endogenous social forces with exogenous environmental ones. A kernel describing the social forces is a key element of swarm models. To improve the mathematical understanding of these models, to clarify how exogenous and endogenous forces interact to influence aggregation pattern and morphology, and to provide biological insight into aphids and locusts, we carry out three projects. (1) We analyze and simulate a broadly applicable differential equation based model describing organisms whose movement is due to endogenous and exogenous forces. We connect properties of swarm patterns to properties of the two forces. (2) In experiment, we measure trajectories of pea aphids moving within a group. Using the model from (1), we quantify the social kernel. (3) We generalize the study from (1) by incorporating a dynamically evolving kernel that models social and antisocial behavioral phases in desert locusts. This work will suggest swarm control strategies to minimize aggregation and hence ameliorate some of the negative economic and humanitarian impacts of locust swarms.
Turing patterns arise from a competition between reaction and diffusion in diverse chemical, biological, and physical systems. To contribute to the mathematical and chemical understanding of forced patterns, we perform experiments on the chlorine dioxide - iodine - malonic acid (CDIMA) Turing system. Specifically, we force Turing patterns with a time-periodic light source and measure how the resulting pattern suppression scales with forcing parameters.
These projects incorporate education through research involvement of twelve undergraduate student researchers, including women, other underrepresented minorities, and post-baccalaureate students bound for graduate study. To forge links between experiment and theory, some students will perform experimental work and all will train within the PI's experimental/theoretical lab. Students will interact with theorists and experimentalists at other universities and undergraduate colleges. Lab and curricular enhancements will provide benefits beyond the term of the grant.
Project 1: Biological aggregations are groups such as bird flocks and fish schools. The PI and collaborators built a continuum model for biological aggregations. The model describes large populations that interact socially and are influenced by the environment. Exact solutions exist to the model, and these have unusual features such as sharp edges and high concentrations of organisms. The model also helps explain the importance of dimensionality on flying insect swarms. Project 2: Locusts are destructive insects whose swarms cause billions of dollars in economic losses. Locusts are known to have two behavioral phases. Solitarious locusts are repelled from others, while gregarious locusts form social aggregations such as hopper bands. Phase change can occur when a solitarious locust is in a crowded region or a gregarious locust is in a sparse region. The PI abd collaborators built a model of locust phase change and spatial dynamics to better understand the conditions under which dangerous hopper bands might form. Studying the model revealed conditions under which dispersed locusts might instead form gregarious clumps. The transition to gregarious aggregations turns out to be hysteretic (not the same forwards and backwards), suggesting why swarm prevention is more important than swarm control. Project 3: A challenge of aggregation modeling is to construct individual-level rules that are well-tied to experimental data. The PI gathered experimental data on pea aphid groups moving in a featureless circular arena. The data is described by a model in which each aphid transitions randomly between a moving and a stationary state, and moving aphids follow a particular type of movement called a random walk. The motion state transition and random walk parameters are well-modeled as depending on distance d to an aphid's nearest neighbor, providing evidence of social interactions. For large d, aphids move faster, turn narrowly, and are less likely to stop. For small d, aphids move more slowly, turn widely, and are more likely to stop; this constitutes a passive aggregation mechanism (as opposed to directed motion towards neighbors). Project 4: Reaction-diffusion systems in diffusively coupled layers arise in diverse biological subfields and chemical experiments. The PI investigated a broad class of two-layer reaction-diffusion models. For two layers with identical reactions, there exist eight scenarios for transitions from the trivial (uniform) state. One scenario involves the formation of patterns with two length scales whose ration may be tuned by altering the strength with which layers are coupled together. Numerical simulations show the formation of rare square patterns. Results of this award were disseminated in 5 peer reviewed journal publications, 24 research talks, and 14 talks related to teaching (curriculum and/or pedagogy). This project provided 16 undergraduate summer research experiences to 14 students (including two women). Seven undergraduates (including two women) collaborated on two published manuscripts. Four undergraduates (including two women) presented work at the Joint Mathematics Meetings. The PI incorporated examples of swarming dynamics into upper-division undergraduate courses on differential equations/dynamical systems and mathematical modeling (via seminar-style discussions of literature). He also developed flipped instruction methods for calculus and differential equations and disseminated them at his own institution and others. The PI also collaborated with faculty and/or students from institutions including Wofford College, Harvey Mudd College, Cal State Northridge, and Univ. of British Columbia. The PI hosted a visiting research (from Harvey Mudd) for four summers. The PI co-organized the week-long workshop Insect Self-Organization and Swarming at the Mathematical Biosciences Institute (March 2011) which brought together mathematicians, engineers, and theoretical and experimental biologists. He spoke on biological swarming models at the multidisciplinary Chinese-American Frontiers of Science meeting, part of the Kavli Frontiers program of the National Academy of Sciences. The meeting brought together a broad-based scientific audience of mathematicians, biologists, physicists, geoscientists, engineers, and more. Similarly, he then co-organized the Indo-American Frontiers of Science meeting in 2013. The PI’s work provided insight into strategies for coping with destructive locust swarms. Results suggest that preventing swarm formation is likely a more effective strategy than attempting to control an existing swarm with pesticides.
