Delayed logistic equation as a limit of long memory Markov chains

We introduce and analyze a long-memory continuous-time Markov chain on $\mathbb{R}_{+}$ whose jump mechanism depends explicitly on a state in the past. From the present state $x_0$, the process jumps to $x_0\left(1+\frac{1}{N}\right)$ or $x_0\left(1-\frac{x_{-\lfloor \tau N \rfloor}}{N^2}\right)$, each at rate $\tfrac{1}{2}$, where $x_{-\lfloor \tau N \rfloor}$ denotes the state located $\lfloor \tau N \rfloor$ jumps backward in time. Here the delay $\tau > 0$ is fixed and $N$ is the scaling parameter. The initial condition is prescribed by a vector of length $\lfloor \tau N \rfloor + 1$, all of whose entries are equal to $\mu N$. Using a genuine space-time replacement lemma, we prove that, as $N \to \infty$, the rescaled process converges to a deterministic limit governed by the Delayed Logistic Equation (also known as the Hutchinson equation) with delay $\tau$ and initial condition $\rho(t) \equiv \mu$ for $t \in [-\tau, 0]$.