Segregation effects and gap formation in cross-diffusion models

Burger, Martin; Carrillo, José A.; Pietschmann, Jan‐Frederik; Schmidtchen, Markus

doi:10.4171/ifb/438

Cited by 18 publications

(45 citation statements)

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“…The entropy dissipation techniques used there are not only essential to define the solutions, but also to determine their long-time asymptotics. Similar results have only been achieved for systems with drift terms in a handful of specific cases [6,5].…”

Section: Introductionsupporting

confidence: 74%

Bound-Preserving Finite-Volume Schemes for Systems of Continuity Equations with Saturation

Bailo,

Carrillo,

2021

Preprint

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We propose finite-volume schemes for general continuity equations which preserve positivity and global bounds that arise from saturation effects in the mobility function. In the particular case of gradient flows, the schemes dissipate the free energy at the fully discrete level. Moreover, these schemes are generalised to coupled systems of non-linear continuity equations, such as multispecies models in mathematical physics or biology, preserving the bounds and the dissipation of the energy whenever applicable. These results are illustrated through extensive numerical simulations which explore known behaviours in biology and showcase new phenomena not yet described by the literature.

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Section: Introductionsupporting

confidence: 74%

Bound-Preserving Finite-Volume Schemes for Systems of Continuity Equations with Saturation

Bailo,

Carrillo,

2021

Preprint

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“…Originating in spatial ecology [42][43][44], cross-diffusion is widely recognized as a mechanism for pattern formation [45]. Recent interest in cross-diffusion has led to advances in analytical understanding of these systems [46][47][48][49]. Since this paper offers three variations on a novel cross-diffusion system, new avenues are opened for further numerical and analytical study to better understand the properties and behavior of these systems, such as the analytical work done on the two-gang system [33].…”

Section: Discussionmentioning

confidence: 99%

“…Following the same steps used to derive the continuum equations in Section 3, but now substituting (3) with (46), it can easily be shown that the resulting system of equations for j = 1, 2, . .…”

Section: Threat Level Model (Variation 2)mentioning

confidence: 99%

A Multispecies Cross-Diffusion Model for Territorial Development

Alsenafi

Barbaro

2021

Mathematics

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We develop an agent-based model on a lattice to investigate territorial development motivated by markings such as graffiti, generalizing a previously-published model to account for K groups instead of two groups. We then analyze this model and present two novel variations. Our model assumes that agents’ movement is a biased random walk away from rival groups’ markings. All interactions between agents are indirect, mediated through the markings. We numerically demonstrate that in a system of three groups, the groups segregate in certain parameter regimes. Starting from the discrete model, we formally derive the continuum system of 2K convection–diffusion equations for our model. These equations exhibit cross-diffusion due to the avoidance of the rival groups’ markings. Both through numerical simulations and through a linear stability analysis of the continuum system, we find that many of the same properties hold for the K-group model as for the two-group model. We then introduce two novel variations of the agent-based model, one corresponding to some groups being more timid than others, and the other corresponding to some groups being more threatening than others. These variations present different territorial patterns than those found in the original model. We derive corresponding systems of convection–diffusion equations for each of these variations, finding both numerically and through linear stability analysis that each variation exhibits a phase transition.

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“…Originating in spatial ecology [35,36,37], cross-diffusion is widely recognized as a mechanism for pattern formation [38]. Recent interest in cross-diffusion has led to advances in analytical understanding of these systems [39,40,41,42]. Since this paper offers three variations on a novel cross-diffusion system, new avenues are opened for further numerical and analytical study to better understand the properties and behavior of these systems, such as the analytical work done on the two-gang system [26].…”

Section: Discussionmentioning

confidence: 99%

“…If one follows the derivation of the continuum in Section 4 but replacing (3) with (39), it can be easily shown that the resulting system of equations for j = 1, 2, . .…”

Section: Timidity Model (Variation 1)mentioning

confidence: 99%

A Multispecies Cross-Diffusion Model for Territorial Development

Alsenafi¹,

Barbaro²

2021

Preprint

View full text Add to dashboard Cite

We develop an agent-based model on a lattice to investigate territorial development motivated by markings such as graffiti, generalizing a previously-published model to account for K groups instead of two groups. We then analyze this model and present two novel variations. Our model assumes that agents' movement is a biased random walk and that the agents' movement is away from rival groups' markings. All interactions between agents is indirect, mediated through the markings. We numerically demonstrate that in a system of three groups, the groups segregate in certain parameter regimes. Starting from the discrete model, we also formally derive the continuum system of 2K convectiondiffusion equations for our model. These equations exhibit cross-diffusion due to the avoidance of the rival groups' markings. Linear stability analysis is performed to study the phase transition that the system undergoes as parameters are varied. Two new variations of the model are introduced and numerically studied. Both exhibit a phase transition, with segregation dynamics that are distinct both from each other and from the original model.

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Segregation effects and gap formation in cross-diffusion models

Cited by 18 publications

References 0 publications

Bound-Preserving Finite-Volume Schemes for Systems of Continuity Equations with Saturation

Bound-Preserving Finite-Volume Schemes for Systems of Continuity Equations with Saturation

A Multispecies Cross-Diffusion Model for Territorial Development

A Multispecies Cross-Diffusion Model for Territorial Development

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