Existence and non-existence of spatial patterns in a ratio-dependent predator–prey model (original) (raw)
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Self-organised spatial patterns and chaos in a ratio-dependent predator–prey system
Theoretical Ecology, 2011
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 Turing-Hopf 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.
Spatiotemporal pattern formation of Beddington-DeAngelis-type predator-prey model
In this paper, we investigate the emergence of a predator-prey model with Beddington-DeAngelistype functional response and reaction-diffusion. We derive the conditions for Hopf and Turing bifurcation on the spatial domain. Based on the stability and bifurcation analysis, we give the spatial pattern formation via numerical simulation, i.e., the evolution process of the model near the coexistence equilibrium point. We find that for the model we consider, pure Turing instability gives birth to the spotted pattern, pure Hopf instability gives birth to the spiral wave pattern, and both Hopf and Turing instability give birth to stripe-like pattern. Our results show that reaction-diffusion model is an appropriate tool for investigating fundamental mechanism of complex spatiotemporal dynamics. It will be useful for studying the dynamic complexity of ecosystems.
Turing Patterns in a Predator-Prey System with Self-Diffusion
Abstract and Applied Analysis, 2013
For a predator-prey system, cross-diffusion has been confirmed to emerge Turing patterns. However, in the real world, the tendency for prey and predators moving along the direction of lower density of their own species, called self-diffusion, should be considered. For this, we investigate Turing instability for a predator-prey system with nonlinear diffusion terms including the normal diffusion, cross-diffusion, and self-diffusion. A sufficient condition of Turing instability for this system is obtained by analyzing the linear stability of spatial homogeneous equilibrium state of this model. A series of numerical simulations reveal Turing parameter regions of the interaction of diffusion parameters. According to these regions, we further demonstrate dispersion relations and spatial patterns. Our results indicate that self-diffusion plays an important role in the spatial patterns. *
Mutual Interference Between Predators Can Give Rise to Turing Spatial Patterns
Ecology, 2002
The study of spatial patterns in the distribution of organisms is a central issue in ecology. Here we address the question of whether predator-prey interactions can induce nonuniform distributions. We study how diffusion affects the stability of predatorprey coexistence equilibria and show a new difference between ratio-and prey-dependent models. Recently, Peter Abrams and Lev Ginzburg reviewed the controversial issue of what kind of function better describes the rate of prey consumption by an average predator, the so-called ''predator functional response.'' Here, we focus on reaction-diffusion predatorprey models with and without predator dependence in the functional response. We show that classical prey-dependent models cannot give rise to spatial structures through diffusiondriven instabilities; however, predator-dependent models with the same degree of complexity can. The origin of predator dependence in the rate of prey consumption is the mutual interference between predators. Therefore, we show that this mechanism can generate patchiness in a homogeneous environment under certain conditions of trophic interaction and predator-prey relative diffusion.
The effect of landscape fragmentation on Turing-pattern formation
Mathematical Biosciences and Engineering, 2022
Diffusion-driven instability and Turing pattern formation are a well-known mechanism by which the local interaction of species, combined with random spatial movement, can generate stable patterns of population densities in the absence of spatial heterogeneity of the underlying medium. Some examples of such patterns exist in ecological interactions between predator and prey, but the conditions required for these patterns are not easily satisfied in ecological systems. At the same time, most ecological systems exist in heterogeneous landscapes, and landscape heterogeneity can affect species interactions and individual movement behavior. In this work, we explore whether and how landscape heterogeneity might facilitate Turing pattern formation in predator–prey interactions. We formulate reaction-diffusion equations for two interacting species on an infinite patchy landscape, consisting of two types of periodically alternating patches. Population dynamics and movement behavior differ bet...
Spatiotemporal pattern formation in a prey-predator model under environmental driving forces
Journal of Physics: Conference Series, 2015
Many existing studies on pattern formation in the reaction-diffusion systems rely on deterministic models. However, environmental noise is often a major factor which leads to significant changes in the spatiotemporal dynamics. In this paper, we focus on the spatiotemporal patterns produced by the predator-prey model with ratio-dependent functional response and density dependent death rate of predator. We get the reaction-diffusion equations incorporating the self-diffusion terms, corresponding to random movement of the individuals within two dimensional habitats, into the growth equations for the prey and predator population. In order to have to have the noise added model, small amplitude heterogeneous perturbations to the linear intrinsic growth rates are introduced using uncorrelated Gaussian white noise terms. For the noise added system, we then observe spatial patterns for the parameter values lying outside the Turing instability region. With thorough numerical simulations we characterize the patterns corresponding to Turing and Turing-Hopf domain and study their dependence on different system parameters like noise-intensity, etc.
Pattern formation and spatiotemporal chaos in the presence of boundaries
Physical review, 1997
In this paper, we consider a diffusive predator-prey system with modified Holling-Tanner functional response under homogeneous Neumann boundary condition. The qualitative analysis and Hopf bifurcation of the original ODE system are discussed, the conditions of the Turing instability for the reactiondiffusion system are derived, and the Turing space in the parameters space is achieved. We present the results of numerical simulations in order to validate the obtained analytical findings. We found some interesting spatiotemporal patterns when parameter values are taken in Turing-Hopf domain, in which the dynamics shows spatiotemporal behavior that is influenced by temporal oscillations as well as by Turing instabilities. With the help of numerical simulations, we identified the different types of spatial patterns in this diffusive predatorprey system, including stationary spatial pattern, periodic competing dynamics, and spatiotemporal chaos.
Ecological Complexity, 2018
Spatio-temporal chaos is an intriguing part of the spatio-temporal pattern formation, observed in many interacting population models when their heterogeneous distributions within their habitats and movement from one location to the other are taken care of within the modeling approach. When the homogeneous steady-states become unstable, the solutions of the corresponding reaction-diffusion systems never approach a stationary state rather exhibit an irregular nature with respect to both space and/or time. Cross-diffusion terms are incorporated in a system of reaction-diffusion equations to model the situation where presence, absence, abundance of one species influence the movement of another species and vice-versa. In this work, cross-diffusion is considered in a prey-predator model with ratiodependent functional response along with the self-diffusion terms. After deriving the Turing instability conditions in terms of cross-diffusion parameters, extensive numerical simulations are carried out to study the effect of cross-diffusion on the chaotic dynamics and stationary Turing patterns generated in the system containing self-diffusion terms only. Appropriate numerical tools are used to characterize the spatio-temporal chaos. Route to spatio-temporal chaos and its disappearance are discussed in detail. The chaotic dynamics of the self-diffusion model may be suppressed leading to a stationary state or preserved depending on the cross-diffusion coefficients. The stationary patches of both the species generated in the Turing domain remain stationary but their configurations may change due to the effect of densitydependent cross-diffusion. On the other hand, stationary patches generated in the Turing-Hopf domain change to spatio-temporal chaos for higher dispersal rates of predator avoidance by prey.
Formation and Persistence of Spatiotemporal Turing Patterns
2007
This article is concerned with the stability and long-time dynamics of structures arising from a structureless state. The paradigm is suggested by developmental biology, where morphogenesis is thought to result from a competition between chemical reactions and spatial diffusion. A system of two reaction-diffusion equations for the concentrations of two morphogens is reduced to a finite system of ordinary differential equations. The stability of bifurcated solutions of this system is analyzed, and the long-time asymptotic behavior of the bifurcated solutions is established rigorously. The Schnakenberg and Gierer-Meinhardt equations are discussed as examples.