Prove that the solution to a delayed differential equation is positive for all $t>0$.

Question

Prove that for the IVP \begin{cases} x'(t)=cx(t)[1-x(t-r)] \\ x(\mu)=\phi(\mu) & \mu \in [-r,0] \end{cases} for every $\phi\in C([-r,0],\mathbb{R})$ with $\phi(0)>0$, has a unique solution which exists for all $t>0$ and remains positive for all $t>0$.

Notes: The problem I have in solving this question is to prove that the solution $x(t)$, remains positive for all t>0.

My attempt: First we establish uniqueness and existence of the solution $x(t)$. $cx(t)[1-x(t-r)]$ and it's derivative with respect to $x$ is a continuous function so it's uniqueness and existence is guaranteed on the interval $[-r,\sigma]$ (where $\sigma>0$).

Next to establish that $x(t)\geq0$. Because if $x(t)=0$, then $x'(t)=0$. So any solution $x(t)$ with history function $\phi(0)>0$ cannot be negative, as the it cannot pass through $x(t)=0$ (a function with derivative $0$ at value $a$ cannot pass through $a$).

My problem is proving $x(t)>0$ for all $t>0$ and proving global existence.

For example suppose $c>0$ and $r$ is large enough such that $x(t-r)>1$ when $0<x(t)<1$ then $x'(t)<0$ and $|x'(t)|>0$. So $x(t)=0$ for some time $t>0$ in this case.

Answer 1

Some hints:

• The solution of your problem is obviously continuous.
• The initial condition is positive.
• What does $x'$ look like for $x→0$?
• The answer has nothing to do with the delay.