Self-similar Spreading in a Merging-Splitting Model of Animal Group Size

  • Jian-Guo Liu
  • B. NiethammerEmail author
  • Robert L. Pego


In a recent study of certain merging-splitting models of animal-group size (Degond et al. in J Nonlinear Sci 27(2):379–424, 2017), it was shown that an initial size distribution with infinite first moment leads to convergence to zero in weak sense, corresponding to unbounded growth of group size. In the present paper we show that for any such initial distribution with a power-law tail, the solution approaches a self-similar spreading form. A one-parameter family of such self-similar solutions exists, with densities that are completely monotone, having power-law behavior in both small and large size regimes, with different exponents.


Fish schools Bernstein functions Complete monotonicity Heavy tails Convergence to equilibrium 

Mathematics Subject Classification

45J05 70F45 92D50 37L15 44A10 35Q99 



BN and RLP acknowledge support from the Hausdorff Center for Mathematics and the CRC 1060 on Mathematics of emergent effects, Universität Bonn. This material is based upon work supported by the National Science Foundation under Grants DMS 1514826 and 1812573 (JGL) and DMS 1515400 and 1812609 (RLP), partially supported by the Simons Foundation under Grant 395796, by the Center for Nonlinear Analysis (CNA) under National Science Foundation PIRE Grant No. OISE-0967140, and by the NSF Research Network Grant No. RNMS11-07444 (KI-Net). JGL and RLP acknowledge support from the Institut de Mathématiques, Université Paul Sabatier, Toulouse and the Department of Mathematics, Imperial College London under Nelder Fellowship awards.


  1. 1.
    Bertoin, J.: Eternal solutions to Smoluchowski’s coagulation equation with additive kernel and their probabilistic interpretations. Ann. Appl. Probab. 12(2), 547–564 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bonacini, M., Niethammer, B., Velázquez, J.J.L.: Self-similar solutions to coagulation equations with time-dependent tails: the case of homogeneity smaller than one. Commun. Partial Differ. Equ. 43(1), 82–117 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Degond, P., Liu, J.-G., Pego, R.L.: Coagulation-fragmentation model for animal group-size statistics. J. Nonlinear Sci. 27(2), 379–424 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Feller, W.: An Introduction to Probability Theory and its Applications, vol. II, 2nd edn. John Wiley & Sons Inc., New York (1971)zbMATHGoogle Scholar
  5. 5.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 editionGoogle Scholar
  6. 6.
    Iyer, G., Leger, N., Pego, R.L.: Limit theorems for Smoluchowski dynamics associated with critical continuous-state branching processes. Ann. Appl. Probab. 25(2), 675–713 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Giri, A Kumar, Kumar, J., Warnecke, G.: The continuous coagulation equation with multiple fragmentation. J. Math. Anal. Appl. 374(1), 71–87 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Liu, J.-G., Pego, R.L.: On generating functions of Hausdorff moment sequences. Trans. Am. Math. Soc. 368(12), 8499–8518 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ma, Q., Johansson, A., Sumpter, D.J.T.: A first principles derivation of animal group size distributions. J. Theor. Biol. 283(1), 35–43 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    McLaughlin, D.J., Lamb, W., McBride, A.C.: An existence and uniqueness result for a coagulation and multiple-fragmentation equation. SIAM J. Math. Anal. 28(5), 1173–1190 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Melzak, Z.A.: A scalar transport equation. Trans. Am. Math. Soc. 85, 547–560 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Menon, G., Pego, R.L.: Approach to self-similarity in Smoluchowski’s coagulation equations. Commun. Pure Appl. Math. 57(9), 1197–1232 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Menon, G., Pego, R.L.: The scaling attractor and ultimate dynamics for Smoluchowski’s coagulation equations. J. Nonlinear Sci. 18(2), 143–190 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Niethammer, B., Throm, S., Velázquez, J.J.L.: Self-similar solutions with fat tails for Smoluchowski’s coagulation equation with singular kernels. Ann. Inst. H. Poincaré Anal. Non Linéaire 33(5), 1223–1257 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Niethammer, B., Velázquez, J.J.L.: Self-similar solutions with fat tails for Smoluchowski’s coagulation equation with locally bounded kernels. Commun. Math. Phys. 318, 505–532 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Niwa, H.S.: Power-law versus exponential distributions of animal group sizes. J. Theor. Biol. 224(4), 451–457 (2003)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Riordan, J.: Combinatorial Identities. John Wiley & Sons Inc., New York (1968)zbMATHGoogle Scholar
  18. 18.
    Schilling, R.L., Song, R., Vondraček, Z.: Bernstein Functions. de Gruyter Studies in Mathematics, vol. 37. Walter de Gruyter & Co., Berlin (2010)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Physics and Department of MathematicsDuke UniversityDurhamUSA
  2. 2.Institut für Angewandte MathematikUniversität BonnBonnGermany
  3. 3.Department of Mathematics and Center for Nonlinear AnalysisCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations