[Paper Review] Effects of impulsive harvesting and an evolving domain in a diffusive logistic model

Meng Yue, Zhigui Lin+1 more|arXiv (Cornell University)|Dec 22, 2020

Mathematical and Theoretical Epidemiology and Ecology Models41 references32 citations

TL;DR

This study proposes a diffusive logistic model with impulsive harvesting on a periodically evolving domain, introducing an ecological reproduction index R₀ that determines population persistence or extinction. The key finding is that a large domain evolution rate enhances population survival, while impulsive harvesting consistently reduces population density and can lead to extinction, regardless of harvesting function type (monotone or nonmonotone).

ABSTRACT

In order to understand how the combination of domain evolution and impulsive harvesting affect the dynamics of a population, we propose a diffusive logistic population model with impulsive harvesting on a periodically evolving domain. Initially the ecological reproduction index of the impulsive problem is introduced and given by an explicit formula, which depends on the domain evolution rate and the impulsive function. Then the threshold dynamics of the population under monotone or nonmonotone impulsive harvesting is established based on this index. Finally numerical simulations are carried out to illustrate our theoretical results, and reveal that a large domain evolution rate can improve the population survival, no matter which impulsive harvesting takes place. Contrary, impulsive harvesting has a negative effect on the population survival, and can even lead to the extinction of the population.

Motivation & Objective

• To investigate the combined effects of impulsive harvesting and domain evolution on population persistence.
• To develop a threshold criterion, the ecological reproduction index R₀, for predicting population extinction or survival.
• To establish threshold-type dynamics under both monotone and nonmonotone impulsive harvesting functions.
• To numerically validate theoretical results and explore the interplay between domain evolution rate and harvesting intensity.

Proposed method

• Proposes a reaction-diffusion model on a periodically evolving domain with impulsive harvesting at discrete time intervals T.
• Introduces an explicit ecological reproduction index R₀, derived from the domain evolution rate ρ(t) and the derivative g′(0) of the impulsive function g.
• Analyzes the model using impulsive differential equations with Dirichlet boundary conditions and time-periodic domain evolution.
• Transforms the problem into a moving domain via a change of variables to simplify analysis.
• Applies threshold dynamics theory to classify solutions as either converging to zero (extinction) or to a positive periodic steady state (persistence).
• Performs numerical simulations using Beverton-Holt and Ricker functions for g(u), with varying domain evolution rates ρ(t).

Experimental results

Research questions

• RQ1 How does the domain evolution rate influence population persistence when impulsive harvesting occurs?
• RQ2 What is the role of the impulsive harvesting function’s form (monotone vs. nonmonotone) in determining population survival?
• RQ3 Can a threshold value R₀ be derived to predict extinction or persistence in this hybrid continuous-discrete system?
• RQ4 How do the interplay between harvesting pulses and domain expansion affect long-term population dynamics?
• RQ5 To what extent does impulsive harvesting accelerate population extinction compared to non-harvested scenarios?

Key findings

• A large domain evolution rate (e.g., ρ₂(t) = e⁰.¹⁽¹⁻ᶜᵒˢ πt⁾) leads to R₀ > 1, enabling population persistence even under impulsive harvesting.
• When R₀ < 1 (e.g., ρ₁(t) = e⁻⁰.¹⁽¹⁻ᶜᵒˢ πt⁾), the population always goes extinct regardless of harvesting function type.
• Impulsive harvesting consistently reduces population density and accelerates extinction, as shown in comparisons between simulations with and without impulses.
• For monotone harvesting (Beverton-Holt), the population converges to a positive periodic steady state when R₀ > 1.
• For nonmonotone harvesting (Ricker), the solution converges to an attractive sector when R₀ > 1, indicating stable oscillatory dynamics.
• Numerical results confirm that impulsive harvesting has a negative impact on survival, with extinction occurring even in large domains when harvesting intensity is high.