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SIAM J. on Applied Dynamical Systems

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2012

Volume 11

partial Issue 2 | 2012 | pp. 567-683

Issue 1 | 2012 | pp. 1-566

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Volume 11, Issue 2 (partial)

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Select this Article		$R_0$ Analysis of a Spatiotemporal Model for a Stream Population H. W. Mckenzie, Y. Jin, J. Jacobsen, and M. A. Lewis SIAM J. Appl. Dyn. Syst. 11, pp. 567-596 (30 pages) Online Publication Date: April 12, 2012 Full Text: \| Download PDF
	+ -	Show Abstract Water resources worldwide require management to meet industrial, agricultural, and urban consumption needs. Management actions change the natural flow regime, which impacts the river ecosystem. Water managers are tasked with meeting water needs while mitigating ecosystem impacts. We develop process-oriented advection-diffusion-reaction equations that couple hydraulic flow to population growth, and we analyze them to assess the effect of water flow on population persistence. We present a new mathematical framework, based on the net reproductive rate $R_0$ for advection-diffusion-reaction equations and on related measures. We apply the measures to population persistence in rivers under various flow regimes. This work lays the groundwork for connecting $R_0$ to more complex models of spatially structured and interacting populations, as well as more detailed habitat and hydrological data.

Select this Article		Reliable Computation of Robust Response Tori on the Verge of Breakdown Jordi-Lluís Figueras and Àlex Haro SIAM J. Appl. Dyn. Syst. 11, pp. 597-628 (32 pages) Online Publication Date: April 12, 2012 Full Text: \| Download PDF
	+ -	Show Abstract We prove the existence and local uniqueness of invariant tori on the verge of breakdown for two systems: the quasi-periodically driven logistic map and the quasi-periodically forced standard map. These systems exemplify two scenarios: the Heagy–Hammel route for the creation of strange nonchaotic attractors and the nonsmooth bifurcation of saddle invariant tori. Our proofs are computer-assisted and are based on a tailored version of the Newton–Kantorovich theorem. The proofs cannot be performed using classical perturbation theory because the two scenarios are very far from the perturbative regime, and fundamental hypotheses such as reducibility or hyperbolicity either do not hold or are very close to failing. Our proofs are based on a reliable computation of the invariant tori and a careful study of their dynamical properties, leading to the rigorous validation of the numerical results with our novel computational techniques.

Select this Article		Pattern Formation in a Model of Acute Inflammation Kevin Penner, Bard Ermentrout, and David Swigon SIAM J. Appl. Dyn. Syst. 11, pp. 629-660 (32 pages) Online Publication Date: April 17, 2012 Full Text: \| Download PDF
	+ -	Show Abstract We seek to understand patterns that form due to acute inflammation in the skin in the absence of specific pathogenic stimuli. By incorporating inhibition (represented by an anti-inflammatory cytokine) into a classical Keller–Segel chemotaxis model, we create a novel model that produces a variety of spatial patterns. We find that the dynamical instability in both homogeneous and nonhomogeneous steady states arises only when the inhibitor dynamics are sufficiently slow. We present simulation results that motivate the nonlinear analysis of the model and illustrate a variety of interesting dynamic two-dimensional spatial patterns that form, including isolated traveling pulses, rotating waves, and patterns that do not settle to a regular behavior.

Select this Article		The Persistence of a Slow Manifold with Bifurcation K. Uldall Kristiansen, P. Palmer, and R. M. Roberts SIAM J. Appl. Dyn. Syst. 11, pp. 661-683 (23 pages) Online Publication Date: May 01, 2012 Full Text: \| Download PDF
	+ -	Show Abstract This paper considers the persistence of a slow manifold with bifurcation in a slow-fast two degree of freedom Hamiltonian system. In particular, we consider a system with a supercritical pitchfork bifurcation in the fast space which is unfolded by the slow coordinate. The model system is motivated by tethered satellites. It is shown that an almost full measure subset of a neighborhood of the slow manifold's normally elliptic branches persists in an adiabatic sense. We prove this using averaging and a blow-up near the bifurcation.