Multiple existence of indefinite nonlinear diffusion problem in population genetics

Monotone and nonmonotone clines with partial panmixia across a geographical barrier

“…Remark 10. If f satisfies (5), (6), and (24), then by the definition of β 1 and β 2 in Theorem 4.1(ii), we see that for every β ∈ (β 1 , β 2 ), the line βp and the graph of f (p) have two points of intersection besides the origin. The situation is exactly the same as in Fig.…”

“…Remark 5. Since a concave up/down function can have at most two points of intersection with a straight line, if f satisfies (5), (6), and (24)…”

“…Here we summarize our results on clines in Theorems 3.7, 3.8, 3.11, and 3.12; and Remarks 6 and 7. Suppose that (4)- (6) and (24) hold. Assume α > 1, β > 0, p ± is a nontrivial solution pair of (7) and determinesp as in (1e).…”

“…We focus on unimodal selection functions f described by (6) that satisfy either (23) or (24) as in Section 3. We first consider nonmonotone clines of (1) with p − = p + = 0.…”

“…In PDE models on the evolution of gene frequencies, clines are nonconstant equilibrium solutions. The general cline theory of regular migration-selection model can be found in a series of systematic studies by Lou and Nagylaki [8,9,10]; in the review papers by Nagylaki and Lou [17], Lou et al [11], and Bürger [1]; and in recent works by Hofbauer and Su [5,6], Nakashima [22,23,24], Sovrano [26], and Feltrin and Sovrano [2,3].…”