2013
DOI: 10.1007/s10440-013-9823-6
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Abstract: In this paper, we use rigorous numerics to compute several global smooth branches of steady states for a system of three reaction-diffusion PDEs introduced by Iida et al. [J. Math. Biol., 53, 617-641 (2006)] to study the effect of cross-diffusion in competitive interactions. An explicit and mathematically rigorous construction of a global bifurcation diagram is done, except in small neighborhoods of the bifurcations. The proposed method, even though influenced by the work of van den Berg et al. [Math. Comp., … Show more

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Cited by 49 publications
(107 citation statements)
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References 14 publications
(47 reference statements)
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“…see [28] if s > 2 and [29] if s ∈ (1, 2)). This implies that a * b ∈ Ω s , and hence that (Ω s , * ) is an algebra.…”
Section: The Banach Space X S and The Fixed Point Problem T (X) = Xmentioning
confidence: 99%
“…see [28] if s > 2 and [29] if s ∈ (1, 2)). This implies that a * b ∈ Ω s , and hence that (Ω s , * ) is an algebra.…”
Section: The Banach Space X S and The Fixed Point Problem T (X) = Xmentioning
confidence: 99%
“…Third, existing rigorous continuation methods (van den Berg et al 2010;Breden et al 2013) are directly applicable to our novel approach, thus putting the rigorous computation of branches of connecting orbits within direct reach. Of course, from the viewpoint of continuation, all three advantages are crucial.…”
Section: Introductionmentioning
confidence: 99%
“…Once these givens are fixed the roots of the radii polynomials yield not only existence and uniqueness results, but also tight error estimates and isolation bounds for the problem. Radii polynomial methods have been applied successfully to a number of problems in dynamical systems, partial differential equations and delay equations, and we refer the interested reader to Day et al (2007), Breden et al (2013), van den Berg et al (2010), van den Berg et al (2011), Hungria et al (2016) and van den Berg et al (2015) for more discussion and references.…”
Section: Introductionmentioning
confidence: 99%
“…The main distinguishing features of our algorithm are described in Remark 3.1. The verification method based on the uniform contraction principle on each simplex is inspired by the pseudo-arclength continuation methods introduced in [1,11] for smooth one-dimensional manifolds.…”
Section: Introductionmentioning
confidence: 99%
“…In fact, the conditions that allow the rigorous continuation of the solution manifold (i.e. the "Y and Z bounds" (11) and (12) that appear in Lemma 2.2) are akin to the hypothesis of the Newton-Kantorovich Theorem: uniform bounds on, respectively, the inverse and the second derivative of a certain operator. These, in turn, are reminiscent of the "stability and consistency" conditions of the Lax Equivalence Theorem.…”
Section: Introductionmentioning
confidence: 99%