The Samardzija–Greller model is an extension of the classical Lotka–Volterra predator–prey system, and this paper investigates the multiscale dynamics in a modified Samardzija–Greller model to take into account the slower timescales in predator reactions rather than prey. We present a comprehensive local stability analysis, pattern formation through diffusive instability and fractional Hopf bifurcations. The analysis of the spatio-temporal model reveals the effects of diffusion coefficients and parameter variations on the dynamical behavior of the slow-fast system. By analyzing the system’s response to changes in the self-diffusion rate of the prey (<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11022_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$d_{\mathbb {X}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>d</mi> <mi mathvariant="double-struck">X</mi> </msub> </math></EquationSource> </InlineEquation>), the intra-species competition rate of the first predator (<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11071_2025_11022_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$d_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>d</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>) and the interaction parameter ‘a’, we observe chaotic patterns for small values of ‘a’, particularly when the prey exhibits strong diffusion. Increases in ‘a’ lead to the emergence of regular, periodic patterns that are homogeneous in space. We discuss in detail how fractional-order models create memory effects that inhibit chaotic transitions, potentially being delayed or avoided in the temporal model. The study shows clear differences in the dynamical regimes between the integer-order and the fractional-order models. The latter model gives more significance to the stabilization effect of the fractional-order derivative on ecological systems and improves our understanding of predator–prey interactions under different parameter settings. Findings clarify the potential to derive ecological stability from emergent patterns and transition into a better understanding of complex ecological processes.

Diffusive instability, patterns and limit cycles in a slow-fast generalized Samardzija–Greller model: a multiscale approach

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发表日期：

2025年4月18日

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Arkaprovo Chakraborty,
Nayana Mukherjee,
P. Veeresha

摘要

The Samardzija–Greller model is an extension of the classical Lotka–Volterra predator–prey system, and this paper investigates the multiscale dynamics in a modified Samardzija–Greller model to take into account the slower timescales in predator reactions rather than prey. We present a comprehensive local stability analysis, pattern formation through diffusive instability and fractional Hopf bifurcations. The analysis of the spatio-temporal model reveals the effects of diffusion coefficients and parameter variations on the dynamical behavior of the slow-fast system. By analyzing the system’s response to changes in the self-diffusion rate of the prey ( $d_{\mathbb {X}}$ $d_{X}$ ), the intra-species competition rate of the first predator ( $d_1$ $d_{1}$ ) and the interaction parameter ‘a’, we observe chaotic patterns for small values of ‘a’, particularly when the prey exhibits strong diffusion. Increases in ‘a’ lead to the emergence of regular, periodic patterns that are homogeneous in space. We discuss in detail how fractional-order models create memory effects that inhibit chaotic transitions, potentially being delayed or avoided in the temporal model. The study shows clear differences in the dynamical regimes between the integer-order and the fractional-order models. The latter model gives more significance to the stabilization effect of the fractional-order derivative on ecological systems and improves our understanding of predator–prey interactions under different parameter settings. Findings clarify the potential to derive ecological stability from emergent patterns and transition into a better understanding of complex ecological processes.