Selection and stability of wave trains behind predator invasions in a model with non-local prey competition

Merchant, Sandra M.; Nagata, Wayne

doi:10.1093/imamat/hxu048

Cited by 29 publications

(26 citation statements)

References 35 publications

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“…see Murray 2002), or (ii) derived from an individual-based model for movement, under certain scaling arguments (e.g. see Hillen and Painter 2013).…”

Section: Homogeneous Tissue: Single Population Modelmentioning

confidence: 99%

A Nonlocal Model for Contact Attraction and Repulsion in Heterogeneous Cell Populations

et al. 2015

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Instructing others to move is fundamental for many populations, whether animal or cellular. In many instances, these commands are transmitted by contact, such that an instruction is relayed directly (e.g. by touch) from signaller to receiver: for cells, this can occur via receptor-ligand mediated interactions at their membranes, potentially at a distance if a cell extends long filopodia. Given that commands ranging from attractive to repelling can be transmitted over variable distances and between cells of the same (homotypic) or different (heterotypic) type, these mechanisms can clearly have a significant impact on the organisation of a tissue. In this paper, we extend a system of nonlocal partial differential equations (integrodifferential equations) to provide a general modelling framework to explore these processes, performing linear stability and numerical analyses to reveal its capacity to trigger the self-organisation of tissues. We demonstrate the potential of the framework via two illustrative applications: the contact-mediated dispersal of neural crest populations and the self-organisation of pigmentation patterns in zebrafish.

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“…see Murray 2002), or (ii) derived from an individual-based model for movement, under certain scaling arguments (e.g. see Hillen and Painter 2013).…”

Section: Homogeneous Tissue: Single Population Modelmentioning

confidence: 99%

A Nonlocal Model for Contact Attraction and Repulsion in Heterogeneous Cell Populations

et al. 2015

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“…Merchant and Nagata showed the destabilizing effect of nonlocal prey competition and reported various complex spatiotemporal patterns such as stationary periodic in space patterns, wave trains, and irregular spatiotemporal oscillations in [41]. In a subsequent study [44], they studied the correlation between the properties of selected wave trains and the standard deviation of nonlocality, and suggested that highly nonlocal systems are more likely to produce spatiotemporal chaos. Segal et al [42] investigated pattern formations in a nonlocal model of competing populations and showed that the population distribution can eventually evolve into localized island structure.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of a Diffusive Two-Prey-One-Predator Model with Nonlocal Intra-Specific Competition for Both the Prey Species

2020

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Investigation of interacting populations is an active area of research, and various modeling approaches have been adopted to describe their dynamics. Mathematical models of such interactions using differential equations are capable to mimic the stationary and oscillating (regular or irregular) population distributions. Recently, some researchers have paid their attention to explain the consequences of transient dynamics of population density (especially the long transients) and able to capture such behaviors with simple models. Existence of multiple stationary patches and settlement to a stable distribution after a long quasi-stable transient dynamics can be explained by spatiotemporal models with nonlocal interaction terms. However, the studies of such interesting phenomena for three interacting species are not abundant in literature. Motivated by these facts here we have considered a three species prey–predator model where the predator is generalist in nature as it survives on two prey species. Nonlocalities are introduced in the intra-specific competition terms for the two prey species in order to model the accessibility of nearby resources. Using linear analysis, we have derived the Turing instability conditions for both the spatiotemporal models with and without nonlocal interactions. Validation of such conditions indicates the possibility of existence of stationary spatially heterogeneous distributions for all the three species. Existence of long transient dynamics has been presented under certain parametric domain. Exhaustive numerical simulations reveal various scenarios of stabilization of population distribution due to the presence of nonlocal intra-specific competition for the two prey species. Chaotic oscillation exhibited by the temporal model is significantly suppressed when the populations are allowed to move over their habitat and prey species can access the nearby resources.

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“…Returning to (22), we let x → ∞, implying that r → 0, φ → φ 0 , ψ → ψ 0 , and the speed of the wave front approaches c, so we calculate via (26) that f (t) = β; it is not necessary to calculate f (t) in order to proceed. The steady states (r, φ, ψ) = (r s , φ s , ψ s ) must satisfy r s φ s = 0 from (22a).…”

Section: Without Loss Of Generality Choosementioning

confidence: 99%

“…Ecologists, in general, would be interested in largeamplitude PTWs that have a more significant impact on the surrounding ecosystem. This is the natural direction for future work on PTW selectionlittle has been done in this area, with the exception of recent work by Merchant and Nagata [21,22], who developed a new method of PTW prediction that retains accuracy further from the Hopf bifurcation by assuming the existence of a front between the spatially homogeneous steady state and the selected PTW. • Nonlocal dispersal: For many natural populations, diffusion is considered to be a crude representation of movement, failing to capture instances of rare long-distance dispersal events.…”

Section: Examplementioning

confidence: 99%

Periodic Traveling Waves Generated by Invasion in Cyclic Predator--Prey Systems: The Effect of Unequal Dispersal

Bennett¹,

Sherratt²

2017

SIAM J. Appl. Math.

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Periodic traveling waves (wavetrains) have been an invaluable tool in the understanding of spatiotemporal oscillations observed in ecological data sets. Various mechanisms are known to trigger this behavior, but here we focus on invasion, resulting in a predator-prey-type interaction. Previous work has focused on the normal form reduction of PDE models to the well-understood λ-ω equations near a Hopf bifurcation, though this is valid only when assuming an equal rate of dispersion for both predators and prey-an unrealistic assumption for many ecosystems. By relaxing this constraint, we obtain the complex Ginzburg-Landau normal form equation, which has a one-parameter family of periodic traveling wave solutions, parametrized by the amplitude. We derive a formula for the wave amplitude selected by invasion before investigating the stability of the solutions. This gives us a complete description of small-amplitude periodic traveling waves in the governing model ecosystem.

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Selection and stability of wave trains behind predator invasions in a model with non-local prey competition

Cited by 29 publications

References 35 publications

A Nonlocal Model for Contact Attraction and Repulsion in Heterogeneous Cell Populations

A Nonlocal Model for Contact Attraction and Repulsion in Heterogeneous Cell Populations

Dynamics of a Diffusive Two-Prey-One-Predator Model with Nonlocal Intra-Specific Competition for Both the Prey Species

Periodic Traveling Waves Generated by Invasion in Cyclic Predator--Prey Systems: The Effect of Unequal Dispersal

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