Prove that if ai(i=1,2,3,4) are positive constants, a2−a4>2, and a1a3−a2>2, then the solution (x(t),y(t)) of the system of differential equations x˙=a1−a2x+a3xy, y˙=a4x−y−a3xy(x,y∈R) with the initial conditions x(0)=0,y(0)≥a1 is such that the function x(t) has exactly one strict local maximum on the interval [0,∞).
L. Pinter, L. Hatvani functionreal analysisreal analysis unsolved