Pattern Formation In Predator Research Articles

Spatial patterns through diffusion-driven instability in modified predator–prey models with chaotic behaviors

Understanding the connection between spatial patterns in population densities and ecological heterogeneity is significant to the understanding of population dynamics and the governance of species in a given domain. In this paper, the spatiotemporal complexity of prey–predator dynamics with fractional Laplacian derivative and migration is investigated. To provide good guidelines on the choice of parameters, the linear stability analysis of the models was investigated. The fractional Laplacian operator was defined in terms of the left- and right-handed Riemann–Liouville derivative which in turn were approximated by using the fourth-order compact difference scheme, and the resulting system of ODEs was advanced in time using the fourth-order exponential time-differencing Runge–Kutta method. In the simulation experiments, different Turing dynamics such as spots, stripes, and other chaotic patterns are observed. Overall, pattern formation in predator–prey models is useful for understanding the dynamics of ecological systems, predicting the long-term behavior of the system, and studying the impact of environmental factors on the dynamics of the system.

Cross-diffusion induced Turing patterns on multiplex networks of a predator–prey model

Predator–prey models have generated growing interest across disciplines ranging from mathematics to ecology. The theory of pattern formation in predator–prey systems organized in monolayer networks has often been investigated, due to its significance in both theoretical advances and practical applications. Here we broaden the theory to the case of multiplex networks, which are easily found in diverse areas, such as neuroscience, social networks, and transportation systems. Moreover, we incorporate the model with cross-diffusion by considering that each specie usually has a specific movement tendency. By carrying on the linear analysis, we get the theoretical Turing instability region and find that the homogeneous fixed point can become unstable due to either the topology of multiplex networks or the cross-diffusion, resulting in various Turing patterns. Furthermore, experimental simulation results are in great agreement with theoretical findings, verifying the theoretical analysis’ validity.

The effect of landscape fragmentation on Turing-pattern formation.

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 between patch types, and individuals may have a preference for one of the two habitat types. We apply homogenization theory to derive an appropriately averaged model, to which we apply stability analysis for Turing patterns. We then study three scenarios in detail and find mechanisms by which diffusion-driven instabilities may arise even if the local interaction and movement rates do not indicate it.

On a cannibalistic predator–prey model with prey defense and diffusion

Protective instincts of prey can strongly influence the development of species and also reduces the risk of predation. Cannibalism is frequent in predators and it has strong influence in the evolution of predator species. The consequences of cannibalism is largely neglected when studying the evolution of phenotype prey defenses in predator–prey systems. Motivated by this biological fact, we propose a predator–prey model that involves (a) prey defense; (b) predator cannibalism; and (c) diffusion. We discuss conditions for existence of all biological feasible equilibrium points for non-spatial system. Kolmogorov analysis is performed to examine appearance of limit cycle and chaos in non-spatial system. We obtain local and global stability conditions around all biological feasible equilibrium points. The direction and stability of the Hopf-bifurcation is determined at the vicinity of the concomitant population state of the non-spatial system. In order to study spatio-temporal dynamics, we derive local stability conditions of diffusive system and obtain conditions for the occurrence of Turing instability. Numerical simulations are performed to explore the cases which are beyond the range of analytical methods. Bifurcation diagrams show that non-spatial system has rich dynamics. Various spatial structures such as stripe, spot, rhombus, irregular and target wave patterns of interacting species through Turing and Hopf-Turing instability in two dimensional spatial domain are portrayed and analysed at finite length in order to substantiate the applicability of the present model. Numerical results reveal the impact of drivers of pattern formation in predator–prey systems which provides a better understanding of complex predator–prey interaction.

Laplacian matrices and Turing bifurcations: revisiting Levin 1974 and the consequences of spatial structure and movement for ecological dynamics

We revisit a seminal paper by Levin (Am Nat 108:207–228, 1974), where spatially mediated coexistence and spatial pattern formation were described. We do so by reviewing and explaining the mathematical tools used to evaluate the dynamics of ecological systems in space, from the perspective of recent developments in spatial population dynamics. We stress the importance of space-mediated stability for the coexistence of competing species and explore the ecological consequences of space-induced instabilities (Turing instabilities) for spatial pattern formation in predator–prey systems. Throughout, we link existing theory to recent developments in discrete spatially structured metapopulations, such as our understanding of how ecological dynamics occurring on a network can be analyzed using the Laplacian matrix and its associated eigenvalue spectrum. We underline the validity of Levin’s message, over 40 years later, and suggest it has ever-growing implications in a changing and increasingly fragmented world.

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

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 predator–prey system, the Hopf bifurcation is indeed essential for the onset of chaos whilst the Turing instability is not.