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Mean-Field Game for Gene Expression of Beetles

Yiming Jiang, Yuan Lou, Yawei Wei, Fei Zeng, Zelin Zhang

TL;DR

The paper addresses the probability of expression of a size-controlling gene in beetles under competition between two size-based populations, formulated as a multipopulation mean-field game with a coupling function $p(t)$. It introduces a nonstandard Hamiltonian $H_k(x,p,\partial_x u_k)$ and proves the existence and uniqueness of a solution $(u_1,u_2,m_1,m_2,p)$ to the coupled HJB-FP system using a Schauder fixed-point approach. Existence is obtained by decoupling $p(t)$, solving the HJB and FP equations for fixed $p$, and then constructing a self-consistent $p$ via a fixed-point argument; uniqueness follows from an ODE-based contraction/monotonicity analysis of the $p$-coupling. A concrete quadratic-quadratic example validates the model and numerical simulations indicate that the gene-expression probability stabilizes to a fixed value (approximately $p(T)\approx 0.25$) across various initial conditions, highlighting robustness of trait-expression dynamics under population structure.

Abstract

In this paper, we investigate the probability of the expression of genes that control the size of beetles under competitive relationships. We use the mean field game (MFG) theory in multiple populations to characterize the different competitive pressures of large and small beetles in the population, and simulate the probability of gene expression in finite time $[0, T]$. Therefore, we prove the existence and uniqueness of the solution of the equation under some assumptions.

Mean-Field Game for Gene Expression of Beetles

TL;DR

The paper addresses the probability of expression of a size-controlling gene in beetles under competition between two size-based populations, formulated as a multipopulation mean-field game with a coupling function . It introduces a nonstandard Hamiltonian and proves the existence and uniqueness of a solution to the coupled HJB-FP system using a Schauder fixed-point approach. Existence is obtained by decoupling , solving the HJB and FP equations for fixed , and then constructing a self-consistent via a fixed-point argument; uniqueness follows from an ODE-based contraction/monotonicity analysis of the -coupling. A concrete quadratic-quadratic example validates the model and numerical simulations indicate that the gene-expression probability stabilizes to a fixed value (approximately ) across various initial conditions, highlighting robustness of trait-expression dynamics under population structure.

Abstract

In this paper, we investigate the probability of the expression of genes that control the size of beetles under competitive relationships. We use the mean field game (MFG) theory in multiple populations to characterize the different competitive pressures of large and small beetles in the population, and simulate the probability of gene expression in finite time . Therefore, we prove the existence and uniqueness of the solution of the equation under some assumptions.

Paper Structure

This paper contains 6 sections, 8 theorems, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Existence of a Solution
  4. Uniqueness
  5. Example
  6. Conclusions

Key Result

Theorem 1.1

If the function $u_1(x, T), u_2(x, T)\in C^1([0,T], R)$, and $m_1(x, 0)$, $m_2(x, 0)\in C^1([0,T], \mathbb{P}), Q(t) \in C^1([0,T], R)$ and satisfy Assumption 2.1 and 2.2, then there exists a unique probability function $p(t) \in [0,T] \times [0,1]$ such that the quintuple $(u_1, u_2, m_1, m_2, p)$ for $k=1,2$.

Figures (2)

  • Figure 1: $p(0)-p(T)$ with interval 0.1
  • Figure 2: $p(0)-p(T)$ with interval 0.01

Theorems & Definitions (15)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 5 more