Spatial pattern formation in reaction–diffusion models: a computational approach

Wenrui Hao, Chuan Xue

Research output: Contribution to journalArticlepeer-review

21 Scopus citations


Reaction–diffusion equations have been widely used to describe biological pattern formation. Nonuniform steady states of reaction–diffusion models correspond to stationary spatial patterns supported by these models. Frequently these steady states are not unique and correspond to various spatial patterns observed in biology. Traditionally, time-marching methods or steady state solvers based on Newton’s method were used to compute such solutions. However, the solutions that these methods converge to highly depend on the initial conditions or guesses. In this paper, we present a systematic method to compute multiple nonuniform steady states for reaction–diffusion models and determine their dependence on model parameters. The method is based on homotopy continuation techniques and involves mesh refinement, which significantly reduces computational cost. The method generates one-parameter steady state bifurcation diagrams that may contain multiple unconnected components, as well as two-parameter solution maps that divide the parameter space into different regions according to the number of steady states. We applied the method to two classic reaction–diffusion models and compared our results with available theoretical analysis in the literature. The first is the Schnakenberg model which has been used to describe biological pattern formation due to diffusion-driven instability. The second is the Gray–Scott model which was proposed in the 1980s to describe autocatalytic glycolysis reactions. In each case, the method uncovers many, if not all, nonuniform steady states and their stabilities.

Original languageEnglish (US)
Pages (from-to)521-543
Number of pages23
JournalJournal of Mathematical Biology
Issue number1-2
StatePublished - Jan 1 2020
Externally publishedYes

Bibliographical note

Funding Information:
Funding was provided by US National Science Foundation (NSF CAREER Award 1553637, NSF DMS 1818769) and American Heart Association (Grant No. 17SDG33660722).

Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.


  • Homotopy continuation
  • Pattern formation
  • Reaction–diffusion equations


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