Non-triangular cross-diffusion systems with predator–prey reaction terms

Desvillettes, Laurent; Soresina, Cinzia

doi:10.1007/s11587-018-0403-y

“…Finally, the same study could be carried out for other quasilinear problems involving crossdiffusion terms. For instance, in the context of predator-prey systems, it is possible to derive by time-scale arguments another type of cross-diffusion terms [8,10]. The linearized analysis suggests that they do not increase the parameter region in which patterns appear, but as in the present work, the global influence cannot be captured only by the linearized analysis.…”

Section: Discussionmentioning

confidence: 56%

On the influence of cross-diffusion in pattern formation

Breden¹,

Kuehn²,

Soresina³

2019

Preprint

1

3

0

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In this paper we consider the Shigesada-Kawasaki-Teramoto (SKT) model to account for stable inhomogeneous steady states exhibiting spatial segregation, which describe a situation of coexistence of two competing species. We provide a deeper understanding on the conditions required on both the cross-diffusion and the reaction coefficients for non-homogeneous steady states to exist, by combining a detailed linearized analysis with advanced numerical bifurcation methods via the continuation software pde2path. We report some numerical experiments suggesting that, when cross-diffusion is taken into account, there exist positive and stable non-homogeneous steady states outside of the range of parameters for which the coexistence homogeneous steady state is positive. Furthermore, we also analyze the case in which self-diffusion terms are considered.

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“…Finally, the same study could be carried out for other quasilinear problems involving cross-diffusion terms. For instance, in the context of predator-prey systems, it is possible to derive by time-scale arguments another type of cross-diffusion terms [10,12]. The linearized analysis suggests that they do not increase the parameter region in which patterns appear, but as in the present work, the global influence cannot be captured only by the linearized analysis.…”

Section: 3mentioning

confidence: 56%

On the influence of cross-diffusion in pattern formation

Breden¹,

Kuehn²,

Soresina³

2021

JCD

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In this paper we consider the Shigesada-Kawasaki-Teramoto (SKT) model to account for stable inhomogeneous steady states exhibiting spatial segregation, which describe a situation of coexistence of two competing species. We provide a deeper understanding on the conditions required on both the cross-diffusion and the reaction coefficients for non-homogeneous steady states to exist, by combining a detailed linearized analysis with advanced numerical bifurcation methods via the continuation software pde2path. We report some numerical experiments suggesting that, when cross-diffusion is taken into account, there exist positive and stable non-homogeneous steady states outside of the range of parameters for which the coexistence homogeneous steady state is positive. Furthermore, we also analyze the case in which self-diffusion terms are considered.

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“…The same study can be performed for other fastreaction systems with multiple time scales and their cross-diffusion limits [60,70].…”

Section: Fast Reaction Limitsmentioning

confidence: 91%

“…In the context of predator-prey interactions, the expression of widely used functional responses can also come out of a systematic process in which one starts with a system of more than two equations with simple reaction terms and performs one [28,113,151,168] or more limits [70,93].…”

Section: Fast Reaction Limitsmentioning

confidence: 99%

A General View on Double Limits in Differential Equations

Kuehn,

Berglund,

Bick

et al. 2021

Preprint

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In this paper, we review several results from singularly perturbed differential equations with multiple small parameters. In addition, we develop a general conceptual framework to compare and contrast the different results by proposing a three-step process. First, one specifies the setting and restrictions of the differential equation problem to be studied and identifies the relevant small parameters. Second, one defines a notion of equivalence via a property/observable for partitioning the parameter space into suitable regions near the singular limit. Third, one studies the possible asymptotic singular limit problems as well as perturbation results to complete the diagrammatic subdivision process. We illustrate this approach for two simple problems from algebra and analysis. Then we proceed to the review of several modern double-limit problems including multiple time scales, stochastic dynamics, spatial patterns, and network coupling. For each example, we illustrate the previously mentioned three-step process and show that already double-limit parametric diagrams provide an excellent unifying theme. After this review, we compare and contrast the common features among the different examples. We conclude with a brief outlook, how our methodology can help to systematize the field better, and how it can be transferred to a wide variety of other classes of differential equations.

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Non-triangular cross-diffusion systems with predator–prey reaction terms

Cited by 12 publications

References 21 publications

On the influence of cross-diffusion in pattern formation

On the influence of cross-diffusion in pattern formation

On the influence of cross-diffusion in pattern formation

A General View on Double Limits in Differential Equations

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