July  2015, 20(5): 1461-1480. doi: 10.3934/dcdsb.2015.20.1461

A free boundary problem for aggregation by short range sensing and differentiated diffusion

1. 

King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia

2. 

Courant Institute of Mathematical Sciences, New York University (NYU), 251 Mercer Street, New York, N.Y. 10012-1185, United States

Received  January 2014 Revised  November 2014 Published  May 2015

On the $d$-dimensional torus we consider the drift-diffusion equation, where the diffusion coefficient may take one of two possible values depending on whether the locally sensed density is below or above a given threshold. This can be interpreted as an aggregation model for particles like insect populations or freely diffusing proteins which slow down their dynamics within dense aggregates. This leads to a free boundary model where the free boundary separates densely packed aggregates from areas with a loose particle concentration.
    The paper has a rigorous part and a formal part. In the rigorous part we prove existence of solutions to the distributional formulation of the model. In the second, formal, part we derive the strong formulation of the model including the free boundary conditions and characterize stationary solutions giving necessary conditions for the emergence of stationary plateaus. We conclude that stationary aggregation plateaus in this situation are either spherical, complements of sphericals or stripes, which has implications for biological applications.
    Finally, numerical simulations in one and two dimensions are used to give evidence for the long time convergence to stationary states which feature aggregations.
Citation: Jan Haškovec, Dietmar Oelz. A free boundary problem for aggregation by short range sensing and differentiated diffusion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1461-1480. doi: 10.3934/dcdsb.2015.20.1461
References:
[1]

G. Alberti, S. Bianchini and G. Crippa, Structure of level sets and sard-type properties of lipschitz maps, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 12 (2013), 863-902.

[2]

F. Bolley, J. A. Cañizo and J. A. Carrillo, Stochastic mean-field limit: non-Lipschitz forces and swarming, Math. Models Methods Appl. Sci., 21 (2011), 2179-2210. doi: 10.1142/S0218202511005702.

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C. Brangwynne, C. Eckmann, D. Courson, A. Rybarska, C. Hoege, J. Gharakhani, F. Jülicher and A. Hyman, Germline p granules are liquid droplets that localize by controlled dissolution/condensation, Science, 324 (2009), 1729-1732. doi: 10.1126/science.1172046.

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M. Burger, J. Haškovec and M.-T. Wolfram, Individual based and mean-field modeling of direct aggregation,, Physica D: Nonlinear Phenomena., ().  doi: 10.1016/j.physd.2012.11.003.

[5]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2010.

[6]

R. Jeanson, C. Rivault, J.-L. Deneubourg, S. Blanco, R. Fournier, C. Jost and G. Theraulaz, Self-organized aggregation in cockroaches, Animal Behaviour, 69 (2005), 169-180. doi: 10.1016/j.anbehav.2004.02.009.

[7]

P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of mathematics: Stochastic modelling and applied probability, Springer, 1992. doi: 10.1007/978-3-662-12616-5.

[8]

H. P. McKean, Propagation of chaos for a class of non-linear parabolic equations, in Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), Air Force Office Sci. Res., Arlington, Va., (1967), 41-57.

[9]

A.-S. Sznitman, Topics in propagation of chaos, in Ecole d'Eté de Probabilités de Saint-Flour XIX - 1989 (ed. P.-L. Hennequin), vol. 1464 of Lecture Notes in Mathematics, Springer Berlin Heidelberg, (1991), 165-251. doi: 10.1007/BFb0085169.

[10]

L. Trefethen, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, Cornell University, [Department of Computer Science and Center for Applied Mathematics], 1996, http://www.books.google.com/books?id=KFDnJAAACAAJ.

show all references

References:
[1]

G. Alberti, S. Bianchini and G. Crippa, Structure of level sets and sard-type properties of lipschitz maps, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 12 (2013), 863-902.

[2]

F. Bolley, J. A. Cañizo and J. A. Carrillo, Stochastic mean-field limit: non-Lipschitz forces and swarming, Math. Models Methods Appl. Sci., 21 (2011), 2179-2210. doi: 10.1142/S0218202511005702.

[3]

C. Brangwynne, C. Eckmann, D. Courson, A. Rybarska, C. Hoege, J. Gharakhani, F. Jülicher and A. Hyman, Germline p granules are liquid droplets that localize by controlled dissolution/condensation, Science, 324 (2009), 1729-1732. doi: 10.1126/science.1172046.

[4]

M. Burger, J. Haškovec and M.-T. Wolfram, Individual based and mean-field modeling of direct aggregation,, Physica D: Nonlinear Phenomena., ().  doi: 10.1016/j.physd.2012.11.003.

[5]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2010.

[6]

R. Jeanson, C. Rivault, J.-L. Deneubourg, S. Blanco, R. Fournier, C. Jost and G. Theraulaz, Self-organized aggregation in cockroaches, Animal Behaviour, 69 (2005), 169-180. doi: 10.1016/j.anbehav.2004.02.009.

[7]

P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of mathematics: Stochastic modelling and applied probability, Springer, 1992. doi: 10.1007/978-3-662-12616-5.

[8]

H. P. McKean, Propagation of chaos for a class of non-linear parabolic equations, in Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), Air Force Office Sci. Res., Arlington, Va., (1967), 41-57.

[9]

A.-S. Sznitman, Topics in propagation of chaos, in Ecole d'Eté de Probabilités de Saint-Flour XIX - 1989 (ed. P.-L. Hennequin), vol. 1464 of Lecture Notes in Mathematics, Springer Berlin Heidelberg, (1991), 165-251. doi: 10.1007/BFb0085169.

[10]

L. Trefethen, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, Cornell University, [Department of Computer Science and Center for Applied Mathematics], 1996, http://www.books.google.com/books?id=KFDnJAAACAAJ.

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