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 $x_0(1+\frac{1}{N})$ or $x_0(1-\frac{x_{-\lfloor \tau N\rfloor }}{N^2})$, each at rate $\frac{1}{2}$, 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 $N\to \mathrm{\infty }$, 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 $t\in [-\tau ,0]$.