Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Determining habitat anomalies in cross-diffusion predator-prey chemotaxis models

Yuhan Li, Hongyu Liu, Catharine W. K. Lo

TL;DR

The paper tackles the inverse problem of uniquely locating a habitat-degradation subdomain $\omega$ and determining the discontinuous ecological interaction rules across $\partial\omega$ from boundary measurements in a multi-species cross-diffusion predator-prey chemotaxis model with prey-taxis. It develops a unified framework that yields uniqueness for both smooth and non-smooth (polyhedral) anomalies, leveraging high-order linearization for the time-dependent case and complex geometric optics (CGO) corner techniques for the stationary case. The main contributions are (i) a rigorous identifiability theory for smooth $\omega$ and $\mathbf{G}$, including a high-order variation method to recover geometry and coefficients, and (ii) a parallel theory for non-smooth polyhedral inclusions granting uniqueness of $\omega$ and $\mathbf{F}$ from boundary data, with CGO-based arguments at corners. These results enable non-invasive ecological sensing, linking boundary observations to internal habitat heterogeneity and altered interaction rules, with potential applications in habitat monitoring, invasive species management, and environmental assessment.

Abstract

This paper addresses an open inverse problem at the interface of mathematical analysis and spatial ecology: the unique identification of unknown spatial anomalies -- interpreted as zones of habitat degradation -- and their associated ecological parameters in multi-species predator-prey systems with multiple chemical signals, using only boundary measurements. We formulate the problem as the simultaneous recovery of an unknown interior subdomain and discontinuous ecological interaction rules across its boundary. A unified theooretical framework is developed that unique determines both the anomaly's geometry and discontinuous coefficients characterizing the altered interactions within the degraded region. Our results cover smooth anomalies in time-dependent systems and are extended to non-smooth polyhedral inclusions in stationary regimes. This work bridges a gap between ecological sensing and the quantitative inference of internal habitat heterogeneity, offering a mathamtical basis for detecting and characterizing habitat degradation from limited external data.

Determining habitat anomalies in cross-diffusion predator-prey chemotaxis models

TL;DR

The paper tackles the inverse problem of uniquely locating a habitat-degradation subdomain and determining the discontinuous ecological interaction rules across from boundary measurements in a multi-species cross-diffusion predator-prey chemotaxis model with prey-taxis. It develops a unified framework that yields uniqueness for both smooth and non-smooth (polyhedral) anomalies, leveraging high-order linearization for the time-dependent case and complex geometric optics (CGO) corner techniques for the stationary case. The main contributions are (i) a rigorous identifiability theory for smooth and , including a high-order variation method to recover geometry and coefficients, and (ii) a parallel theory for non-smooth polyhedral inclusions granting uniqueness of and from boundary data, with CGO-based arguments at corners. These results enable non-invasive ecological sensing, linking boundary observations to internal habitat heterogeneity and altered interaction rules, with potential applications in habitat monitoring, invasive species management, and environmental assessment.

Abstract

This paper addresses an open inverse problem at the interface of mathematical analysis and spatial ecology: the unique identification of unknown spatial anomalies -- interpreted as zones of habitat degradation -- and their associated ecological parameters in multi-species predator-prey systems with multiple chemical signals, using only boundary measurements. We formulate the problem as the simultaneous recovery of an unknown interior subdomain and discontinuous ecological interaction rules across its boundary. A unified theooretical framework is developed that unique determines both the anomaly's geometry and discontinuous coefficients characterizing the altered interactions within the degraded region. Our results cover smooth anomalies in time-dependent systems and are extended to non-smooth polyhedral inclusions in stationary regimes. This work bridges a gap between ecological sensing and the quantitative inference of internal habitat heterogeneity, offering a mathamtical basis for detecting and characterizing habitat degradation from limited external data.
Paper Structure (16 sections, 8 theorems, 107 equations)

This paper contains 16 sections, 8 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Biology background and motivation of our study
  3. Mathematical setup
  4. Technical developments and discussion
  5. Smooth Internal Topological Structures
  6. Admissible classes
  7. Main theorem
  8. High-order variation method
  9. Recovery of the smooth internal domain
  10. Recovery of the coefficient functions
  11. Non-smooth Internal Topological Anomalies for the Stationary Predator-Prey Chemotaxis Model
  12. Geometrical setup
  13. Admissible classes
  14. Main theorem
  15. Recovery of the polyhedral domain
  16. ...and 1 more sections

Key Result

Theorem 1.1

Consider a certain general scenario in which an unknown smooth heterogeneity $\omega$ is present within a larger domain $\Omega$. In this scenario, the domain $\omega$ and its corresponding parameter configurations $\mathbf{G}$ can be uniquely determined by the boundary measurement $\mathcal{M}^{+}_

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Definition 3.3
  • ...and 4 more