An application of Wazewski's method to a non-linear boundary value problem which arises in population genetics

Summary

A non-linear boundary value problem is treated using the principle of T. Wazewski. The equation is d ²/d, x ² p (x)+s (x) p (1−p)=0 where s (x) is non zero near ±∞. The boundary condition on p at ±∞ is 0 and 1 according as sgn s (±∞) is −1 or +1. Two essentially different cases are treated, namely sgn s (+ ∞)= ±sgn s (-∞). A radially symmetric problem with x ε R ² is also discussed. The Wazewski principle allows one to describe the sets of initial data which satisfy the boundary conditions at + ∞ and at -∞ and to show how they intersect. The problem arises in the study of clines in population genetics theory.