Existence for semi-infinite cline with neutral belt

In this paper we study the semi-infinite Neumann problem, which models the allele frequency variation in a semi-infinite habitat with a neutral belt. We prove that there exists an unbounded solution branch emanating from ( k 1 , 0 ) , where k 1 is the smallest root of k = arctan ⁡ [ α / ( 1 + α β k ) ] . The system admits only the zero solution when k ≤ k 1 , while it possesses a positive solution when k > k 1 ; this solution does not exceed 1 in C [ 0 , + ∞ ) and decays exponentially to 0 . These results confirm the necessity part of Nagylaki's conjecture.