Delayed Logistic Equation as a Limit of Long Memory Markov Chains

Abstract

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