Self-organised spatial patterns and chaos in a ratio-dependent predator–prey system

Abstract

Mechanisms and scenarios of pattern formation in predator-prey systems have been a focus of many studies recently as they are thought to mimic the processes of ecological patterning in real-world ecosystems. Considerable work has been done with regards to both Turing and non-Turing patterns where the latter often appears to be chaotic. In particular, spatiotemporal chaos remains a controversial issue as it can have important implications for population dynamics. Most of the results, however, were obtained in terms of 'traditional' predator-prey models where the per capita predation rate depends on the prey density only. A relatively new family of ratio-dependent predator-prey models remains less studied and still poorly understood, especially when space is taken into account explicitly, in spite of their apparent ecological relevance. In this paper, we consider spatiotemporal pattern formation in a ratio-dependent predator-prey system. We show that the system can develop patterns both inside and outside of the Turing parameter domain. Contrary to widespread opinion, we show that the interaction between two different type of instability, such as the TuringHopf bifurcation, does not necessarily lead to the onset of chaos; on the contrary, the emerging patterns remain stationary and almost regular. Spatiotemporal chaos can only be observed for parameters well inside the Turing-Hopf domain. We then investigate the relative importance of these two instability types on the onset of chaos and show that, in a ratio-dependent predatorprey system, the Hopf bifurcation is indeed essential for the onset of chaos whilst the Turing instability is not.

Spatio-temporal organization in a morphochemical electrodeposition model: Hopf and Turing instabilities and their interplay

“…This feature of our model can be in part explained by observing that it displays interesting dynamics in the spatially uniform case since the physically relevant steady state can lose its stability via a transcritical bifurcation as well as via a supercritical Hopf bifurcation. This latter circumstance seems to be noteworthy because recent studies [2,3] have shown that the emergence of spatio-temporal patterns in reaction-diffusion systems can be related to the presence of unstable homogeneous states, like those found beyond a supercritical Hopf bifurcation.…”

“…The coupling between these two kinds of instabilities was first theoretically investigated in the 90's in the framework of the Lengyel-Epstein model of the CIMA reaction [56]. Since then it has become a key topic for pattern formation in the reaction-diffusion framework and has been extensively investigated in chemistry, physics and biology [2,3,24,33,48,49,55].…”

“…In this respect, within the ecological context, a fascinating debate is currently going on: see e.g. [2,3]. In [3] the authors use the concept of generalized models developed in [31] to show that in a modified diffusive predator-prey Rosenzweig-McArthur model, spatially irregular self-sustained non-stationary patterns or even spatio-temporal chaos can appear for parameter values in the neighbourhood of the TH bifurcation point.…”

“…In [3] the authors use the concept of generalized models developed in [31] to show that in a modified diffusive predator-prey Rosenzweig-McArthur model, spatially irregular self-sustained non-stationary patterns or even spatio-temporal chaos can appear for parameter values in the neighbourhood of the TH bifurcation point. Instead, in [2], where a two species reaction-diffusion predator-prey system with a ratio-dependent functional response is considered, spatio-temporal chaos is not found in the close vicinity of the TH bifurcation, but rather for parameter values far from the bifurcation point. These two examples show how the occurrence of chaotic spatio-temporal dynamics near a TH point is still a controversial issue.…”

Pattern Formation in Tri-Trophic Ratio-Dependent Food Chain Model

“…Pattern formation for two species model [11][12][13][14][16][17][18][19]] based on coupled reaction diffusion equations has been intensively investigated. The necessary and sufficient condition for Turing instability, which leads to the formation of spatial patterns, has been derived [9,16,17,20] and very interesting patterns are also obtained from the numerical simulation results [9,21,22].…”