The Influence of Double Delays in a Diffusive Predator-Prey System: A Study Using the Stability Switching Curve Method

Academic Background

The predator-prey model is one of the fundamental models in ecology for studying population interactions. Although these models may seem simple, they can generate complex dynamic structures and even lead to chaotic trajectories in certain cases. The rate at which predators consume prey, known as the functional response, plays a crucial role in these models. Functional responses can be categorized into those that depend solely on prey and those that depend on both prey and predators, such as the Holling I-IV types and the Beddington-DeAngelis and Crowley-Martin types.

In recent years, researchers have begun to focus on the impact of delays on predator-prey systems. Delays are prevalent in ecological systems, such as reproduction delays in predators and harvest delays caused by human activities. The presence of delays makes the stability analysis of systems more complex, especially when multiple delays are involved. The study of double delays has not yet received sufficient attention in ecology and mathematical modeling, but their impact on system dynamics holds significant theoretical and practical importance.

This paper aims to study a diffusive predator-prey system with a Crowley-Martin functional response and harvest delay, exploring the synergistic effects of harvest delay and reproduction delay on the system’s stability. Using the stability switching curve method, the authors analyze the stability of coexisting equilibrium points and investigate how delay parameters influence system dynamics through Hopf bifurcation and Turing bifurcation.

Source of the Paper

This paper was co-authored by Lakpa Thendup Bhutia, Samir Biswas, Tapan Kumar Kar, and Bidhan Bhunia, all from the Department of Mathematics at the Indian Institute of Engineering Science and Technology, Shibpur. The paper was published on February 15, 2025, in the journal Nonlinear Dynamics, with the DOI 10.1007/s11071-025-11015-4.

Research Process

1. Analysis of Spatial Instability in the Non-Delayed System

In the absence of delays (τ₁ = τ₂ = 0), the authors first investigated the stability of the diffusive predator-prey system. By linearizing the system and analyzing its characteristic equation, the authors derived the stability conditions for the coexisting equilibrium point and explored the impact of the diffusion coefficient on Turing instability. The study found that when the diffusion coefficient δ₂ exceeds a critical value δ₂*, the system undergoes Turing bifurcation, leading to the formation of spatial patterns.

Experimental Results: Through numerical simulations, the authors validated the theoretical analysis. When δ₂ < δ₂, the system remains stable; when δ₂ > δ₂, the system exhibits spatial instability, forming Turing patterns. This indicates that the diffusion coefficient plays a key role in the system’s stability.

2. Analysis of Stability Switching Curves in the Delayed System

After introducing double delays, the authors employed the stability switching curve method to study the impact of delays on the system’s stability. By analyzing the roots of the characteristic equation, the authors determined the crossing set for each wave number and derived the Hopf bifurcation curves. The study found that as the delay parameters pass through these curves, the system’s stability undergoes switching.

Experimental Results: Numerical simulations showed that when the harvest delay τ₁ and reproduction delay τ₂ cross the stability switching curves, the coexisting equilibrium point of the system undergoes Hopf bifurcation, leading to the emergence of periodic solutions. Specifically, when the reproduction delay is low, the harvest delay has a minor impact on the system’s dynamics; however, when the reproduction delay is moderate, the harvest delay induces stability switching phenomena.

3. Analysis of the Direction and Stability of Hopf Bifurcation

To further understand the impact of delays on system dynamics, the authors used normal form theory and the center manifold theorem to analyze the properties of Hopf bifurcation. By calculating the bifurcation direction and the stability of solutions, the authors determined whether the periodic solutions were supercritical or subcritical.

Experimental Results: The study found that when the delay parameters cross the Hopf bifurcation curves, the system generates stable periodic solutions. This indicates that delays not only affect the system’s stability but may also lead to complex dynamic behaviors.

Research Findings

Through the stability switching curve method, this paper systematically studied the impact of double delays on the stability of the diffusive predator-prey system. The research found:

1. Diffusion Coefficient: The diffusion coefficient plays a crucial role in the system’s stability, especially when it exceeds a critical value, leading to Turing bifurcation and the formation of spatial patterns.
2. Delay Parameters: Delay parameters significantly impact the system’s stability. When delay parameters cross the stability switching curves, the coexisting equilibrium point undergoes Hopf bifurcation, resulting in periodic solutions.
3. Synergistic Effects of Harvest and Reproduction Delays: The interaction between harvest delay and reproduction delay induces stability switching phenomena, particularly when the reproduction delay is moderate, where the harvest delay has a more pronounced impact on system dynamics.

Research Highlights

1. Study of Double Delays: This paper is the first to combine harvest delay and reproduction delay, exploring their synergistic effects on the stability of the diffusive predator-prey system, filling a research gap in the field.
2. Stability Switching Curve Method: The authors employed the stability switching curve method to systematically analyze the impact of delay parameters on system stability, providing a new tool for studying systems with multiple delays.
3. Numerical Simulation Validation: Extensive numerical simulations validated the theoretical analysis, enhancing the credibility of the research findings.

Practical Applications

This research not only holds significant theoretical importance but also provides references for the management of actual ecological systems. For example, fishermen can regulate harvest delays (e.g., by using fishing nets of different sizes) to control fishing activities, thereby maintaining the sustainability of fishery resources. Additionally, the findings offer new insights into understanding the impact of multiple delays on ecosystem dynamics, aiding in the prediction and control of ecosystem stability.

Through theoretical analysis and numerical simulations, this paper provides an in-depth exploration of the impact of double delays on the stability of the diffusive predator-prey system, offering important theoretical and practical references for related fields.