Delayed logistic equation as a limit of long memory Markov chains

Abstract

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 $x_0\left(1-\frac{x_{-\lfloor \ddot{I}„N\rfloor }}{N^2}\right)$, each at rate $\frac{1}{2}$, where $x_{-\lfloor \ddot{I}„N\rfloor }$ denotes the state located $\lfloor \ddot{I}„N\rfloor$ jumps backward in time. Here the delay $\ddot{I}„>0$ is fixed and $N$ is the scaling parameter. The initial condition is prescribed by a vector of length $\lfloor \ddot{I}„N\rfloor +1$, all of whose entries are equal to $\hat{I}¼N$. 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 $\ddot{I}„$ and initial condition $\ddot{I}(t)\equiv \hat{I}¼$ for $t\in [-\ddot{I}„,0]$.

16 pages, 6 figures