[Paper Review] A Mean Field Approximation of the Bolker-Pacala Population Model

Mariya Bessonov, Stanislav Molchanov+1 more|arXiv (Cornell University)|Feb 5, 2013

Stochastic processes and statistical mechanics19 references3 citations

TL;DR

This paper develops a mean field approximation of the Bolker-Pacala spatial population model by modeling it as a logistic Markov chain on a lattice. It establishes a local central limit theorem and large deviations results for the invariant distribution, showing that large deviations decay as $ L^{-1/2} e^{-L} $, and derives asymptotics for first passage times to interval boundaries, revealing exponential growth in expected hitting times with system size $ L $. The analysis relies on asymptotic properties of confluent hypergeometric functions and provides global limit theorems via Kurtz's functional laws.

ABSTRACT

We approximate the Bolker-Pacala model of population dynamics with the logistic Markov chain and analyze the latter. We find the asymptotics of the degenerated hypergeometric function and use these to prove a local CLT and large deviations result. We also state global limit theorems and obtain asymptotics for the first passage time to the boundary of a large interval.

Motivation & Objective

• To develop a tractable mean field approximation of the stochastic spatial Bolker-Pacala population model, which incorporates both spatial dynamics and competition.
• To analyze the resulting logistic Markov chain on $ \mathbb{Z}^d $ to understand its invariant distribution and long-time behavior.
• To establish limit theorems—local CLT and large deviations—for the invariant measure under mean field scaling.
• To derive asymptotic expressions for the first passage time from the equilibrium state to the boundaries of large symmetric intervals.
• To provide global limit theorems (functional LLN and CLT) for the logistic Markov chain using results from Kurtz.

Proposed method

• Model the Bolker-Pacala process as a discrete-time Markov chain on $ \mathbb{Z}^d $ with birth, death, and competition dynamics.
• Use a mean field approximation to reduce the spatially structured process to a one-dimensional logistic birth-death process with state-dependent rates.
• Analyze the invariant distribution using the transition probabilities and solve the associated recurrence relations involving ratios of birth and death rates.
• Derive asymptotics for the confluent hypergeometric function arising in the invariant measure via integral approximations and the Euler-Maclaurin formula.
• Apply the local central limit theorem and large deviations principle to the invariant measure, with the large deviations rate dominated by $ L^{-1/2} e^{-L} $.
• Use potential theory and first passage time equations to derive asymptotics for $ \mathbb{E}\tau_{n^* \to \{n_1,n_2\}} $, showing exponential dependence on $ L $.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the invariant distribution of the logistic Markov chain under mean field scaling?
• RQ2 How do large deviations from the equilibrium state scale with system size $ L $ in the logistic Markov chain?
• RQ3 What is the asymptotic expected first passage time from the equilibrium state to the boundaries of a large symmetric interval?
• RQ4 Can global limit theorems (functional LLN and CLT) be established for the logistic Markov chain using known stochastic process theory?
• RQ5 How does the symmetry of the process around the equilibrium point $ n^* $ influence the first passage time distribution?

Key findings

• The large deviations probability for the invariant distribution decays as $ L^{-1/2} $ times a constant times $ e^{-L} $, indicating strong exponential decay with system size $ L $.
• The local central limit theorem holds for the invariant distribution of the logistic Markov chain, with convergence rates derived from asymptotic analysis of confluent hypergeometric functions.
• The expected first passage time from the equilibrium point $ n^* $ to the interval boundaries $ n_1 $ and $ n_2 $ scales as $ \sqrt{L}^{-1} \exp\left( \frac{b}{\gamma}L \ln \rho_1 + \delta_1(1 - \ln \rho_1)n^* \right) $, showing exponential growth with $ L $.
• The asymptotic analysis of $ \psi_2(x) $, the solution to the recurrence for the invariant measure, shows that $ \psi_2(n_2) \asymp \exp\left( \frac{1}{\omega} \int_0^{\delta_2(1 - \mu/b)} \ln(1+x) \, dx \right) $, with $ \omega = \gamma/(bL) $.
• Under symmetry assumptions, the first passage time from $ n^* $ to $ \{n_1,n_2\} $ is asymptotically half the expected time to hit $ n_1 $ from $ n_2 $, leading to the key estimate in equation (4.23).
• The expected recurrence time to the equilibrium state $ n^* $ is $ O(\sqrt{L}) $, while to a state $ n^* + \delta L $ it is $ \sqrt{L} e^{O(L)} $, indicating extremely long return times for large deviations.