DOI: 10.1017/s1446181126100388 ISSN: 1446-1811
GLOBAL BIFURCATION AND STABILITY FOR SEMI-INFINITE CLINE
GUOWEI DAI, YINGXIN SUN Abstract
We study the semi-infinite Neumann problem, which models the variation law of a cline in a semi-infinite habitat. Using the bifurcation analysis method, we find that there is a unique solution curve emanating from
$(\arctan \alpha ,0)$
with
$\alpha> 0$
, which is strictly increasing and approaches
$1$
in
$C[0,+\infty )$
. Furthermore, we show that any cline (bifurcation solution) is stable, thereby providing a confirmed answer to a conjecture. Moreover, we also establish the stability of the trivial solution. Our conclusions are consistent with the related numerical results and biological reality.