Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition

Merchant, Sandra M.; Nagata, Wayne

doi:10.1016/j.tpb.2011.10.001

Cited by 63 publications

(60 citation statements)

References 23 publications

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“…Some of this work concerns the differential adhesion hypothesis, which is the direct antecedent of this paper. Another active research area concerns spatiotemporal patterns in oscillatory systems with nonlocal terms (Merchant and Nagata 2011;Nagata and Merchant 2014;Sherratt 2014). But by far, the largest part of the recent literature concerns swarming in animal populations.…”

Section: Discussionmentioning

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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Section: Discussionmentioning

confidence: 99%

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

et al. 2015

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“…The results presented in Lemma 3.8 can be determined directly by studying the positivity of k T ( ) defined by (14) . Lemma 3.9.…”

Section: Lemma 36 the Region R 1 ∩ R 4 Is Exactly Turing Instabilitmentioning

confidence: 99%

“…4,5 This functional response has been followed by numerous other interaction functionals by including many other factors such as searching for the prey pack by predators, saturation of predator consumption, herd shape, so on; see, for instance, other works. [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] The impact of self-(resp. cross-)diffusion on the dynamics of predator-prey models has been always scrutinized since the pioneering work of Turing.…”

Section: Introductionmentioning

confidence: 99%

“…It is well known that the competition of the prey at a locative location can not only depend on the locative prey density but also depend on the average of the thereabout prey. The effect of the nonlocal competition has been studied widely in different cases, as the single species, 26 predator-prey interaction, 14,[27][28][29][30][31][32] competing populations, 33 and so on. For instance, the papers 14,32,34 and many other papers deal with the presence of the nonlocal competition in a prey population.…”

Section: Introductionmentioning

confidence: 99%

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Pattern formation of a diffusive predator‐prey model with herd behavior and nonlocal prey competition

Djilali

2020

Math Methods in App Sciences

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In this paper, we study the influence of the nonlocal interspecific competition of the prey population on the dynamics of the diffusive predator‐prey model with prey social behavior. Using the linear stability analysis, the conditions for the positive constant steady state at which undergoes Hopf bifurcation, T‐H bifurcation (Turing‐Hopf bifurcation) are investigated. The Turing patterns occur in the presence of the nonlocal competition and cannot be found in the original system. For determining the dynamical behavior near T‐H bifurcation point, the normal form of the T‐H bifurcation has been used. Some graphical representations are provided to illustrate the theoretical results.

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“…Other examples include PTWs and patterns of regional synchrony in geometrid moths in Northern Fennoscandia [5][6][7][8][9], PTWs and intermittent synchrony in red grouse [10,11], and PTWs in larch budmoth in the European Alps [12][13][14]; see Sherratt & Smith [15] for further examples. Modelling studies have demonstrated a number of potential causes for this spatial asynchrony, including invasions [16][17][18][19][20][21], heterogeneous habitats [13,[22][23][24], migration between subpopulations [25], migration driven by pursuit and evasion [26] and hostile habitat boundaries [27][28][29][30]. This paper concerns the last of these.…”

Section: Introductionmentioning

confidence: 99%

Generation of periodic travelling waves in cyclic populations by hostile boundaries

Sherratt

2013

Proc. R. Soc. A.

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Many recent datasets on cyclic populations reveal spatial patterns with the form of periodic travelling waves (wavetrains). Mathematical modelling has identified a number of potential causes of this spatial organization, one of which is a hostile habitat boundary. In this paper, the author investigates the member of the periodic travelling wave family selected by such a boundary in models of reaction–diffusion type. Using a predator–prey model as a case study, the author presents numerical evidence that the wave generated by a hostile (zero-Dirichlet) boundary condition is the same as that generated by fixing the population densities at their coexistence steady-state levels. The author then presents analysis showing that the two waves are the same, in general, for oscillatory reaction–diffusion models with scalar diffusion close to Hopf bifurcation. This calculation yields a general formula for the amplitude, speed and wavelength of these waves. By combining this formula with established results on periodic travelling wave stability, the author presents a division of parameter space into regions in which a hostile boundary will generate periodic travelling waves, spatio-temporal disorder or a mixture of the two.

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Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition

Cited by 63 publications

References 23 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

Pattern formation of a diffusive predator‐prey model with herd behavior and nonlocal prey competition

Generation of periodic travelling waves in cyclic populations by hostile boundaries

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