Aggregation-diffusion Equations for Collective Behavior in the Sciences
A surprisingly large number of complex phenomena can be modeled as systems of point particles that interact under short- and long-range forces. While researchers typically use this modeling framework to study many-body systems (such as gravitational collapse, electron transport in semiconductors, and kinetic equations in statistical physics), it has recently found applications in other areas of physics as well as biology, social science, machine learning, and even optimization. These applications are diverse and span both microscopic and macroscopic scales; examples include ion channel transport, chemotaxis, bacterial orientation, cellular adhesion, angiogenesis, animal herding, and human crowd motion. One can approximate the interaction effects between the “individuals” in each case with long-range attractive forces (e.g., ligand binding, electrical interactions, or social preferences) and short-range repulsions (due to volume constraints or crowding).
Several agent-based (microscopic) modeling approaches—such as cellular automata and driven Brownian particles—can describe these phenomena and encode complex behavior. However, the best way to understand and rigorously analyze such behavior is to study the mean-field limit (or its variants) — i.e., the limit of the microscopic model as the number of agents becomes large. For instance, consider two types of cells, both of which are able to express different surface proteins and ligands (such as cadherins or nectins). Assume that the nuclei of the first cell type are located at positions
where
![<strong>Figure 1.</strong> Interaction forces of an agent-based model for cell adhesion. Figure courtesy of [3].](/media/xyanbzwb/figure1.jpg)
Under suitable assumptions, the model’s associated empirical measures (weighted sums of Dirac deltas that are supported at each particle’s position) can approximate the macroscopic normalized cell densities in the many-particle limit:
Given the biological context, it is reasonable to suppose that repulsion is purely localized and the attraction forces only act within a cutoff radius
Each equation in this system is a particular case of the aggregation-diffusion equation
which is a generic continuum equation for the kinematic evolution of a population density
Gradient Flows
We can understand the aggregation-diffusion equation in
in a suitable metric space, i.e., the 2-Wasserstein space of probability densities over
![<strong>Figure 2.</strong> Numerical approximation of a geodesic curve via the 2-Wasserstein distance between the characteristic sets of Pac-Man and the Ghost (suitably normalized). An animation of this interpolation is <a href="https://figshare.com/articles/media/Wasserstein_Geodesic_between_PacMan_and_Ghost/7665377?file=14240948" target="_blank">available online.</a> Figure courtesy of [4].](/media/m05dsrtv/figure2.jpg)
Due to the lack of linear structure in the 2-Wasserstein space, it is difficult to precisely construct solutions to the aggregation-diffusion equation in
for
If the functional
Mathematical Biology Applications
Mathematical biology is a prominent application area for aggregation-diffusion systems. Cell population models are particularly relevant and often use the aggregation-diffusion equation (in either scalar or system form) to describe the motion of cells as they concentrate or separate from a target. The aforementioned diffusion terms are consistent with population pressure effects, whereby groups of cells naturally gravitate away from areas of high concentration. In addition, the aggregation term has classically served as a model for chemotaxis, as cells often direct their movement along the increasing gradient of a chemical agent that is produced by either external sources or the cells themselves. The Keller-Segel model for chemotaxis [12] is an archetypal example of the aggregation-diffusion equation in mathematical biology.
![<strong>Figure 3.</strong> <em>In vitro</em> experiments from [11] compared to <em>in silico</em> experiments from [7]. The numerical schemes are based on finite volumes, as in [2]. A corresponding animation is <a href="https://figshare.com/articles/media/Front_propagation_and_intermingling_of_cell_types_experiments_versus_mathematical_model_simulations_/7707890?file=14343011" target="_blank">available online.</a> Figure adapted from [7].](/media/025i00jo/figure3.jpg)
A new paradigm in cell adhesion models suggests that short-range interaction potentials can capture ligand binding through filopodia (as in Figure 1). One example of such a model [7] considers two species of cells, each of which is attracted to both itself and the other species in the form of
José A. Carrillo delivered an invited lecture about this research at the 10th International Congress on Industrial and Applied Mathematics, which took place in Tokyo, Japan, last year.
Acknowledgments: The authors were supported by the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for Complex Particle Dynamics: Phase Transitions, Patterns and Synchronization) of the European Research Council Executive Agency under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 883363). In addition, David Gómez-Castro was partially supported by grants RYC2022-037317-I and PID2023-151120NA-I00 from the Spanish government.
References
[1] Ambrosio, L., Gigli, N., & Savaré, G. (2008). Gradient flows in metric spaces and in the space of probability measures. In Lectures in mathematics ETH Zürich. Basel, Switzerland: Birkhäuser.
[2] Bailo, R., Carrillo, J.A., & Hu, J. (2020). Fully discrete positivity-preserving and energy-dissipating schemes for aggregation-diffusion equations with a gradient-flow structure. Commun. Math. Sci., 18(5), 1259-1303.
[3] Carrillo, J.A., Colombi, A., & Scianna, M. (2018). Adhesion and volume constraints via nonlocal interactions determine cell organisation and migration profiles. J. Theoret. Biol., 445, 75-91.
[4] Carrillo, J.A., Craig, K., Wang, L., & Wei, C. (2021). Primal dual methods for Wasserstein gradient flows. Found. Comput. Math., 22, 389-443.
[5] Carrillo, J.A., Craig, K., & Yao, Y. (2019). Aggregation-diffusion equations: Dynamics, asymptotics, and singular limits. In N. Bellomo, P. Degond, & E. Tadmor (Eds.), Active particles, Volume 2: Advances in theory, models, and applications (pp. 65-108). Modeling and simulation in science, engineering and technology. Cham, Switzerland: Birkhäuser.
[6] Carrillo, J.A., Hittmeir, S., Volzone, B., & Yao, Y. (2019). Nonlinear aggregation-diffusion equations: Radial symmetry and long time asymptotics. Invent. Math., 218(2), 889-977.
[7] Carrillo, J.A., Murakawa, H., Sato, M., Togashi, H., & Trush, O. (2019). A population dynamics model of cell-cell adhesion incorporating population pressure and density saturation. J. Theoret. Biol., 474, 14-24.
[8] Falcó, C., Baker, R.E., & Carrillo, J.A. (2024). A local continuum model of cell-cell adhesion. SIAM J. Appl. Math., 84(3), S17-S42.
[9] Gómez-Castro, D. (2024). Beginner’s guide to aggregation-diffusion equations. SeMA J.
[10] Jordan, R., Kinderlehrer, D., & Otto, F. (1998). The variational formulation of the Fokker-Planck equation. SIAM J. Math. Anal., 29(1), 1-17.
[11] Katsunuma, S., Honda, H., Shinoda, T., Ishimoto, Y., Miyata, T., Kiyonari, H., … Togashi, H. (2016). Synergistic action of nectins and cadherins generates the mosaic cellular pattern of the olfactory epithelium. J. Cell Biol., 212(5), 561-575.
[12] Keller, E.F., & Segel, L.A. (1970). Initiation of slide mold aggregation viewed as an instability. J. Theoret. Biol., 26(3), 399-415.
[13] Steinberg, M.S. (1962). On the mechanism of tissue reconstruction by dissociated cells. I. Population kinetics, differential adhesiveness, and the absence of directed migration. Proc. Natl. Acad. Sci., 48(9), 1577-1582.
[14] Vázquez, J.L. (2006). The porous medium equation: Mathematical theory. Oxford, U.K.: Oxford University Press.
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About the Authors
Rafael Bailo
Assistant professor, Eindhoven University of Technology
Rafael Bailo is an assistant professor of mathematics at the Eindhoven University of Technology. His work deals with the numerical analysis of kinetic equations and other partial differential equations. He is also interested in collective dynamics, self-organization, and the control of agent-based models.

José A. Carrillo
Professor, University of Oxford
José A. Carrillo is a Professor of the Analysis of Partial Differential Equations at the University of Oxford’s Mathematical Institute and a Tutorial Fellow in Applied Mathematics at The Queen´s College of Oxford.

David Gómez-Castro
Postdoctoral Research Fellow, Universidad Autónoma de Madrid
David Gómez-Castro is a Ramón y Cajal Postdoctoral Research Fellow at the Universidad Autónoma de Madrid.

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