Research & Publications
I am broadly interested in mathematical biology as well as nonlinear dynamical systems with applications to other areas of science. I am particularly interested in understanding and overcoming the limits of current methodologies to capture spatial and temporal complexity in phenomena of scientific interest. I am currently active in the following research areas:
- Reaction-Diffusion Systems: pattern formation, diffusion-driven (Turing) instabilities, spatiotemporal phenomena (including chaos and heterogeneity), RD systems on static and growing manifolds, with applications to developmental biology, population dynamics, and tissue engineering.
- Spatial Population Dynamics: discrete patch and spatially continuous models, random and deterministic dispersal, synchronization of metapopulations, with applications to ecological and epidemiological metapopulation models.
- Nonlinear Dynamical Systems: dissipative dynamics (attractors and absorbing sets), numerical bifurcation analysis, and stochastic differential equations.
In all of these areas I am interested both in the scientific field or application, as well as in the theoretical and computational tools used. I primarily approach problems from a dynamical systems point of view, and I am keen to see applications motivate new theory in this field. However, I am much more motivated by developing approaches to solve problems as they are, rather than trying to force problems into a particular shape in order to use a specific hammer. While I have used several tools in the past, I am more than happy to pick up new methodologies in order to address an interesting problem. I am also happy to pursue questions outside of my main research interests, as long as I am able to provide a useful perspective in addressing them. In short, I am problem-driven rather than technique-constrained.
- R. A. Van Gorder, V. Klika, and A. L. Krause. Turing conditions for pattern forming systems on evolving manifolds. arXiv:1904.09683 [nlin.PS].
- A. L. Krause, V. Klika, T. E. Woolley, and E. A. Gaffney. From one pattern into another: Analysis of Turing patterns in heterogeneous domains via WKBJ. Journal of the Royal Society Interface, in Press arXiv:1908.07219 [nlin.PS].
- M. A. R. Strobl, A. L. Krause, M. Damaghi, R. Gillies, A. R. A. Anderson, and P. K. Maini. Mix & match: Phenotypic coexistence as a key facilitator of solid tumour invasion. Bulletin of Mathematical Biology, in Press. bioRxiv doi.org/10.1101/750810.
- Y. Xu, A. L. Krause, and R. A. Van Gorder. Generalist predator dynamics under Kolmogorov versus non-Kolmogorov models. Journal of Theoretical Biology, 486:110060, 2020.
- A. L. Krause, M. Ellis, and R. A. Van Gorder. Influence of curvature, growth, and anisotropy on the evolution of Turing patterns on growing manifolds. Bulletin of Mathematical Biology, 81(3):1–41, 2019.
- R. A. Van Gorder, H. Kim, and A. L. Krause. Diffusive instabilities and spatial patterning from the coupling of reaction-diffusion processes with Stokes flow in complex domains. Journal of Fluid Mechanics, 877:759–823, 2019.
- E. F. Fussell, A. L. Krause, and R. A. Van Gorder. Hybrid approach to modeling spatial dynamics of systems with generalist predators. Journal of Theoretical Biology, 462:26–47, 2019.
- R. A. Van Gorder, A. L. Krause, and J. A. Kwiecinski. Amplitude death criteria for coupled complex Ginzburg-Landau systems. Nonlinear Dynamics, 97:151–159, 2019.
- F. Sánchez-Garduño, A. L. Krause, J. A. Castillo, and P. Padilla. Turing–Hopf patterns on growing domains: the torus and the sphere. Journal of Theoretical Biology, 481:136–150, 2019.
- A. L. Krause, D. Beliaev, R. A. Van Gorder, and S. L. Waters. Lattice and Continuum Modelling of a Bioactive Porous Tissue Scaffold. IMA Mathematical Medicine and Biology, 36(3):325–360, 2019.
- A. L. Krause, V. Klika, T. E. Woolley, and E. A. Gaffney. Heterogeneity induces spatiotemporal oscillations in reaction-diffusion systems. Physical Review E, 97:052206, 2018.
- A. L. Krause, L. Kurowski, K. Yawar, and R. A. Van Gorder. Stochastic Epidemic Metapopulation Models on Networks: SIS Dynamics and Control Strategies. Journal of Theoretical Biology, 449:35–52, 2018.
- A. L. Krause, A. M. Burton, N. T. Fadai, and R. A. Van Gorder. Emergent structures in reaction-advection-diffusion systems on a sphere. Physical Review E, 97:042215, 2018.
- A. L. Krause, D. Beliaev, R. A. Van Gorder, and S. L. Waters. Bifurcations and dynamics emergent from lattice and continuum models of bioactive porous media. International Journal of Bifurcation and Chaos, 28(11):1830037, 2018.
- V. Klika and A. L. Krause. Beyond Onsager-Casimir relations: Shared dependence of phenomenological coefficients on state variables. The Journal of Physical Chemistry Letters, 9:7021–7025, 2018.
- R. A. Van Gorder, A. L. Krause, F. Brosa Planella, and A. M. Burton. Coupled complex Ginzburg-Landau systems with saturable nonlinearity and asymmetric cross-phase modulation. Annals of Physics, 396:397–428, 2018
- R. M. Eide, A. L. Krause, N. T. Fadai, and R. A. Van Gorder. Predator-prey-subsidy population dynamics on stepping-stone domains with dispersal delays. Journal of Theoretical Biology, 451:19–34, 2018.
- J. A. Kwiecinski, A. Kovacs, A. L. Krause, F. Brosa Planella, and R. A. Van Gorder. Chaotic Dynamics in the Planar Gravitational Many-Body Problem with Rigid Body Rotations. International Journal of Bifurcation and Chaos, 28(05):1830013, 2018.
- J. A. Kwiecinski, A. L. Krause, and R. A. Van Gorder. Effects of tidal torques on 1i/2017 U1 (‘Oumuamua). Icarus, 311:170 – 174, 2018.
- L. Kurowski, A. L. Krause, H. Mizuguchi, P. Grindrod, and R. A. Van Gorder, Two-Species Migration and Clustering in Two-Dimensional Domains. Bulletin of Mathematical Biology, 79(10), pp.2302-2333, 2017
- A. Bassett, A. L. Krause, and R. A. Van Gorder, Continuous dispersal in a model of predator–prey-subsidy population dynamics. Ecological Modelling, 354, pp.115-122, 2017
- A. Krause and B. Wang, Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains. Journal of Mathematical Analysis and Applications, 417(2), pp.1018-1038, 2014
- A. Krause, M. Lewis and B. Wang, Dynamics of the non-autonomous stochastic p-Laplace equation driven by multiplicative noise. Applied Mathematics and Computation, 246, pp.365-376, 2014.