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Delayed logistic equation as a limit of long memory Markov chains

Eldon Barros, Dirk Erhard, Tertuliano Franco, Milton Jara

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

Delayed logistic equation as a limit of long memory Markov chains

Abstract

We introduce and analyze a long-memory continuous-time Markov chain on whose jump mechanism depends explicitly on a state in the past. From the present state , the process jumps to or , each at rate , where denotes the state located jumps backward in time. Here the delay is fixed and is the scaling parameter. The initial condition is prescribed by a vector of length , all of whose entries are equal to . Using a genuine space-time replacement lemma, we prove that, as , the rescaled process converges to a deterministic limit governed by the Delayed Logistic Equation (also known as the Hutchinson equation) with delay and initial condition for .

Paper Structure

This paper contains 8 sections, 11 theorems, 78 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Model and Statement of Results
  3. Proofs
  4. Proof of the Theorem 2.1
  5. Uniform moment bounds
  6. Replacement lemma
  7. Convergence at negative times and tightness
  8. Computational Simulations

Key Result

Theorem 2.1

Fix $\tau > 0$ and $\mu \in (0,1)$. For each $N \in \mathbb N$, let $X^N$ be the continuous-time Markov chain defined in chain. Define the scaled process Then, for every $T > 0$, the sequence of processes $\{Y^N(t):t\in[-\tau,T]\}$ converge in probability in the Skorohod topology of $D([-\tau,T], {\mathbb R})$ to the unique continuous solution $u:[-\tau,T]\to {\mathbb R}$ to the Delayed Logistic

Figures (6)

  • Figure 1: Jump rates for the long memory Markov chain considered here. Note that the origin is an absorbent state. Above, the real value $x_0$ denotes the present state while $x_{-\lfloor \tau N\rfloor}$ denotes the state $\lfloor \tau N\rfloor$ jumps backward in time.
  • Figure 2: This simulation starts from the invariant profile $u=1$. Therefore, one expects that output of the simulation of the Markov chain to be approximately constant. This precisely what we see in the picture noting the drawing's scale: the oscillation in the $y$-axis is less than $0.05$ around the invariant value $1$, hence less than $0.5\%$ with respect to the invariant value $u=1$.
  • Figure 3: This simulation has delay $\tau = 1$ and starts at $u=2$. Since $\tau < \pi$, which is the critical value of the Hopf's bifurcation of the delayed Logistic Equation, the invariant profile $u\equiv 1$ is attractive. Moreover, contrarily to the classical logistic differential equation, which does not go below one once starting from $u>1$, the delayed logistic does, as we can see in the picture.
  • Figure 4: This simulation has delay $\tau = 2.5<\pi$, so it lies in the subcritical regime and starts at $u=1.1$. The proximity of the starting point with the invariant profile (constant equal to one) creates some noise in the simulation.
  • Figure 5: Another picture in the subcritical regime with delay $\tau = 2.5< \pi$, starting at $u=2$. The simulation is smoother in comparison with the previous picture. We believe is due to a bigger distance of the starting point to the invariant profile equal to one.
  • ...and 1 more figures

Theorems & Definitions (22)

  • Theorem 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 12 more