SCHWEITZER 1979 Problem 10

Prove that if $a_i(i=1,2,3,4)$ are positive constants, $a_2-a_4 > 2$ , and $a_1a_3-a_2 > 2$ , then the solution $(x(t),y(t))$ of the system of differential equations $$\.{x}=a_1-a_2x+a_3xy,$$ $$\.{y}=a_4x-y-a_3xy \;\;\;(x,y \in \mathbb{R})$$ with the initial conditions $x(0)=0, y(0) \geq a_1$ is such that the function $x(t)$ has exactly one strict local maximum on the interval $[0, \infty)$ .

L. Pinter, L. Hatvani