English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We prove the existence of novel, shock-fronted travelling wave solutions to amodel of wound healing angiogenesis studied in Pettet et al., IMA J. Math. App.Med., 17, 2000. In this work, the authors showed that for certain parametervalues, a heteroclinic orbit in the phase plane representing a smoothtravelling wave solution exists. However, upon varying one of the parameters,the heteroclinic orbit was destroyed, or rather cut-off, by a wall ofsingularities in the phase plane. As a result, they concluded that under thisparameter regime no travelling wave solutions existed. Using techniques fromgeometric singular perturbation theory and canard theory, we show that atravelling wave solution actually still exists for this parameter regime: weconstruct a heteroclinic orbit passing through the wall of singularities via afolded saddle canard point onto a repelling slow manifold. The orbit leavesthis manifold via the fast dynamics and lands on the attracting slow manifold,finally connecting to its end state. This new travelling wave is no longersmooth but exhibits a sharp front or shock. Finally, we identify regions inparameter space where we expect that similar solutions exist. Moreover, wediscuss the possibility of more exotic solutions.

Novel solutions for a model of wound healing angiogenesis