The affiliation for Evgeni Burovski was given as Higher School of Economics; the correct affiliation is National Research University, Higher School of Economics. In Box 1, "SciPy is an open-source package that builds on the strengths of Python and Numeric, providing a wide range of fast scientific and numeric functionality" was used as the box title; this has been moved to the beginning of the box text and a new title has been provided: "Excerpt from the SciPy 0.1 release announcement (typos corrected), posted 20 August 2001 on the Python-list mailing list. " From the original first sentence of this box, "(text following the % symbol indicates that a typo in the original text has been corrected in the version reproduced here)" has been deleted, and "% hanker to Hankel" and "% Netwon to Newton" have been deleted from the ends of the special functions row and the optimization row, respectively. In the first sentence of the ndimage section of Box 2, "nonlinear filter" has been changed to plural. At the end of the first paragraph of the section "SciPy matures, " "The library was expanded carefully, with the patience affordable in open-source projects and via best practices common in industry" has been changed to "The library was expanded carefully, with the patience affordable in open-source projects and via best practices, which are increasingly common in the scientific Python ecosystem and industry. " In Table 2, "Inequality constraint" has been changed to plural. In the "Nonlinear optimization: global minimization" section, "scipy.optimize.differentialevolution" had been changed to "scipy.optimize.differential_evolution. " In the first sentence of the section "Maintainers and contributors, " "SciPy developer guide" has been changed to "SciPy contributor guide" and the URL has been changed from

This paper concerns pattern formation in a class of reaction-advection-diffusion systems modeling the population dynamics of two predators and one prey. We consider the biological situation that both predators forage along the population density gradient of the preys which can defend themselves as a group. We prove the global existence and uniform boundedness of positive classical solutions for the fully parabolic system over a bounded domain with space dimension N = 1, 2 and for the parabolic-parabolic-elliptic system over higher space dimensions. Linearized stability analysis shows that prey-taxis stabilizes the positive constant equilibrium if there is no group defense while it destabilizes the equilibrium otherwise. Then we obtain stationary and time-periodic nontrivial solutions of the system that bifurcate from the positive constant equilibrium. Moreover, the stability of these solutions is also analyzed in detail which provides a wave mode selection mechanism of nontrivial patterns for this strongly coupled system. Finally, we perform numerical simulations to illustrate and support our theoretical results.

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We consider the following fully parabolic Keller-Segel system    ut = ∇ • (D(u)∇u − S(u)∇v) + u(1 − u γ), x ∈ Ω, t > 0, vt = ∆v − v + u, x ∈ Ω, t > 0, ∂u ∂ν = ∂v ∂ν = 0, x ∈ ∂Ω, t > 0 over a multi-dimensional bounded domain Ω ⊂ R N , N ≥ 2. Here D(u) and S(u) are smooth functions satisfying: D(0) > 0, D(u) ≥ K 1 u m 1 and S(u) ≤ K 2 u m 2 , ∀u ≥ 0, for some constants K i ∈ R + , m i ∈ R, i = 1, 2. It is proved that, when the parameter pair (m 1 , m 2) lies in some specific regions, the system admits global classical solutions and they are uniformly bounded in time. We cover and extend [22, 28], in particular when N ≥ 3 and γ ≥ 1, and [3, 29] when m 1 > γ − 2 N if γ ∈ (0, 1) or m 1 > γ − 4 N +2 if γ ∈ [1, ∞). Moreover, according to our results, the index 2 N is, in contrast to the model without cellular growth, no longer critical to the global existence or collapse of this system.

No abstract