ORCID Profile
0000-0002-9906-6768
Current Organisation
Macquarie University
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Lie Groups, Harmonic and Fourier Analysis | Pure Mathematics | Partial Differential Equations | Algebraic and Differential Geometry
Publisher: American Institute of Mathematical Sciences (AIMS)
Date: 2019
Publisher: EDP Sciences
Date: 09-06-2010
Publisher: Cambridge University Press (CUP)
Date: 23-06-2015
DOI: 10.1017/S0956792515000261
Abstract: When faced with slowly depleting resources (such as decrease in precipitation due to climate change), complex ecological systems are prone to sudden irreversible changes (such as desertification) as the resource level dips below a tipping point of the system. A possible coping mechanism is the formation of spatial patterns, which allows for concentration of sparse resources and the survival of the species within “ecological niches” even below the tipping point of the homogeneous vegetation state. However, if the change in resource availability is too sudden, the system may not have time to transition to the patterned state and will pass through the tipping point instead, leading to extinction. We argue that the deciding factors are the speed of resource depletion and the amount of the background noise (seasonal climate changes) in the system. We illustrate this phenomenon on a model of patterned vegetation. Our analysis underscores the importance of, and the interplay between, the speed of climate change, heterogeneity of the environment, and the amount of seasonal variability.
Publisher: American Physical Society (APS)
Date: 24-10-2016
Publisher: American Physical Society (APS)
Date: 06-03-2015
Publisher: Cambridge University Press (CUP)
Date: 10-04-2013
DOI: 10.1017/S0956792513000089
Abstract: In a one-dimensional domain, the stability of localized spike patterns is analysed for two closely related singularly perturbed reaction–diffusion (RD) systems with Brusselator kinetics. For the first system, where there is no influx of the inhibitor on the domain boundary, asymptotic analysis is used to derive a non-local eigenvalue problem (NLEP), whose spectrum determines the linear stability of a multi-spike steady-state solution. Similar to previous NLEP stability analyses of spike patterns for other RD systems, such as the Gierer–Meinhardt and Gray–Scott models, a multi-spike steady-state solution can become unstable to either a competition or an oscillatory instability depending on the parameter regime. An explicit result for the threshold value for the initiation of a competition instability, which triggers the annihilation of spikes in a multi-spike pattern, is derived. Alternatively, in the parameter regime when a Hopf bifurcation occurs, it is shown from a numerical study of the NLEP that an asynchronous , rather than synchronous, oscillatory instability of the spike litudes can be the dominant instability. The existence of robust asynchronous temporal oscillations of the spike litudes has not been predicted from NLEP stability studies of other RD systems. For the second system, where there is an influx of inhibitor from the domain boundaries, an NLEP stability analysis of a quasi-steady-state two-spike pattern reveals the possibility of dynamic bifurcations leading to either a competition or an oscillatory instability of the spike litudes depending on the parameter regime. It is shown that the novel asynchronous oscillatory instability mode can again be the dominant instability. For both Brusselator systems, the detailed stability results from NLEP theory are confirmed by rather extensive numerical computations of the full partial differential equations system.
Publisher: American Physical Society (APS)
Date: 14-02-2013
Publisher: Springer Science and Business Media LLC
Date: 17-12-2014
DOI: 10.1007/S11538-014-0053-5
Abstract: A hybrid asymptotic-numerical method is formulated and implemented to accurately calculate the mean first passage time (MFPT) for the expected time needed for a predator to locate small patches of prey in a 2-D landscape. In our analysis, the movement of the predator can have both a random and a directed component, where the diffusivity of the predator is isotropic but possibly spatially heterogeneous. Our singular perturbation methodology, which is based on the assumption that the ratio [Formula: see text] of the radius of a typical prey patch to that of the overall landscape is asymptotically small, leads to the derivation of an algebraic system that determines the MFPT in terms of parameters characterizing the shapes of the small prey patches together with a certain Green's function, which in general must be computed numerically. The expected error in approximating the MFPT by our semi-analytical procedure is smaller than any power of [Formula: see text], so that our approximation of the MFPT is still rather accurate at only moderately small prey patch radii. Overall, our hybrid approach has the advantage of eliminating the difficulty with resolving small spatial scales in a full numerical treatment of the partial differential equation (PDE). Similar semi-analytical methods are also developed and implemented to accurately calculate related quantities such as the variance of the mean first passage time (VMFPT) and the splitting probability. Results for the MFPT, the VMFPT, and splitting probability obtained from our hybrid methodology are validated with corresponding results computed from full numerical simulations of the underlying PDEs.
Publisher: Elsevier BV
Date: 06-2021
Publisher: The Royal Society
Date: 12-11-2018
Abstract: For a large class of reaction–diffusion systems with large diffusivity ratio, it is well known that a two-dimensional stripe (whose cross-section is a one-dimensional homoclinic spike) is unstable and breaks up into spots. Here, we study two effects that can stabilize such a homoclinic stripe. First, we consider the addition of anisotropy to the model. For the Schnakenberg model, we show that (an infinite) stripe can be stabilized if the fast-diffusing variable (substrate) is sufficiently anisotropic. Two types of instability thresholds are derived: zigzag (or bending) and break-up instabilities. The instability boundaries sub ide parameter space into three distinct zones: stable stripe, unstable stripe due to bending and unstable due to break-up instability. Numerical experiments indicate that the break-up instability is supercritical leading to a ‘spotted-stripe’ solution. Finally, we perform a similar analysis for the Klausmeier model of vegetation patterns on a steep hill, and examine transition from spots to stripes. This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)’.
Publisher: IOP Publishing
Date: 30-03-2023
Abstract: For a bounded 2D planar domain Ω, we investigate the impact of domain geometry on oscillatory translational instabilities of N -spot equilibrium solutions for a singularly perturbed Schnakenberg reaction-diffusion system with activator-inhibitor diffusivity ratio. An N -spot equilibrium is characterized by an activator concentration that is exponentially small everywhere in Ω except in N well-separated localized regions of extent. We use the method of matched asymptotic analysis to analyze Hopf bifurcation thresholds above which the equilibrium becomes unstable to translational perturbations, which result in -frequency oscillations in the locations of the spots. We find that stability to these perturbations is governed by a nonlinear matrix-eigenvalue problem, the eigenvector of which is a 2 N -vector that characterizes the possible modes (directions) of oscillation. The 2 N × 2 N matrix contains terms associated with a certain Green’s function on Ω, which encodes geometric effects. For the special case of a perturbed disk with radius in polar coordinates r = 1 + σ f ( θ ) with 0 ε ≪ σ ≪ 1 , θ ∈ [ 0 , 2 π ) , and f ( θ ) 2 π -periodic, we show that only the mode-2 coefficients of the Fourier series of f impact the bifurcation threshold at leading order in σ . We further show that when f ( θ ) = cos 2 θ , the dominant mode of oscillation is in the direction parallel to the longer axis of the perturbed disk. Numerical investigations on the full Schnakenberg PDE are performed for various domains Ω and N -spot equilibria to confirm asymptotic results and also to demonstrate how domain geometry impacts thresholds and dominant oscillation modes.
Publisher: Cambridge University Press (CUP)
Date: 30-07-2018
DOI: 10.1017/S0956792518000426
Abstract: In the singular perturbation limit ε → 0, we analyse the linear stability of multi-spot patterns on a bounded 2-D domain, with Neumann boundary conditions, as well as periodic patterns of spots centred at the lattice points of a Bravais lattice in $\\mathbb{R}^2$ , for the Brusselator reaction–diffusion model $$ \\begin{equation*} v_t = \\epsilon^2 \\Delta v + \\epsilon^2 - v + fu v^2 \\,, \\qquad \\tau u_t = D \\Delta u + \\frac{1}{\\epsilon^2}\\left(v - u v^2\\right) \\,, \\end{equation*} $$ where the parameters satisfy 0 & f & 1, τ & 0 and D & 0. A previous leading-order linear stability theory characterizing the onset of spot litude instabilities for the parameter regime D = ${\\mathcal O}$ (ν −1 ), where ν = −1/log ϵ , based on a rigorous analysis of a non-local eigenvalue problem (NLEP), predicts that zero-eigenvalue crossings are degenerate. To unfold this degeneracy, the conventional leading-order-in-ν NLEP linear stability theory for spot litude instabilities is extended to one higher order in the logarithmic gauge ν. For a multi-spot pattern on a finite domain under a certain symmetry condition on the spot configuration, or for a periodic pattern of spots centred at the lattice points of a Bravais lattice in $\\mathbb{R}^2$ , our extended NLEP theory provides explicit and improved analytical predictions for the critical value of the inhibitor diffusivity D at which a competition instability, due to a zero-eigenvalue crossing, will occur. Finally, when D is below the competition stability threshold, a different extension of conventional NLEP theory is used to determine an explicit scaling law, with anomalous dependence on ϵ , for the Hopf bifurcation threshold value of τ that characterizes temporal oscillations in the spot litudes.
Publisher: Cambridge University Press (CUP)
Date: 07-03-2017
DOI: 10.1017/S0956792517000043
Abstract: For certain singularly perturbed two-component reaction–diffusion systems, the bifurcation diagram of steady-state spike solutions is characterized by a saddle-node behaviour in terms of some parameter in the system. For some such systems, such as the Gray–Scott model, a spike self-replication behaviour is observed as the parameter varies across the saddle-node point. We demonstrate and analyse a qualitatively new type of transition as a parameter is slowly decreased below the saddle node value, which is characterized by a finite-time blow-up of the spike solution. More specifically, we use a blend of asymptotic analysis, linear stability theory, and full numerical computations to analyse a wide variety of dynamical instabilities, and ultimately finite-time blow-up behaviour, for localized spike solutions that occur as a parameter β is slowly r ed in time below various linear stability and existence thresholds associated with steady-state spike solutions. The transition or route to an ultimate finite-time blow-up can include spike nucleation, spike annihilation, or spike litude oscillation, depending on the specific parameter regime. Our detailed analysis of the existence and linear stability of multi-spike patterns, through the analysis of an explicitly solvable non-local eigenvalue problem, provides a theoretical guide for predicting which transition will be realized. Finally, we analyse the blow-up profile for a shadow limit of the reaction–diffusion system. For the resulting non-local scalar parabolic problem, we derive an explicit expression for the blow-up rate near the parameter range where blow-up is predicted. This blow-up rate is confirmed with full numerical simulations of the full PDE. Moreover, we analyse the linear stability of this solution that blows up in finite time.
Publisher: American Physical Society (APS)
Date: 29-12-2014
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2020
DOI: 10.1137/20M1326271
Publisher: Proceedings of the National Academy of Sciences
Date: 20-06-2019
Abstract: One approach to testing biological theories is to determine if they are predictive. We have developed a simple, theoretical treatment of T cell receptor (TCR) triggering that relies on just two physical principles: ( i ) the time TCRs spend in cell–cell contacts depleted of large tyrosine phosphatases and ( ii ) constraints on the size of these contacts imposed by cell topography. The theory not only distinguishes between agonistic and nonagonistic TCR ligands but predicts the relative signaling potencies of agonists with remarkable accuracy. These findings suggest that the theory captures the essential features of receptor triggering.
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2015
DOI: 10.1137/140968604
Publisher: Cambridge University Press (CUP)
Date: 06-05-2011
DOI: 10.1017/S0956792511000179
Abstract: Recent attention has focused on deriving localised pulse solutions to various systems of reaction–diffusion equations. In this paper, we consider the evolution of localised pulses in the Brusselator activator–inhibitor model, long considered a paradigm for the study of non-linear equations, in a finite one-dimensional domain with feed of the inhibitor through the boundary and global feed of the activator. We employ the method of matched asymptotic expansions in the limit of small activator diffusivity and small activator and inhibitor feeds. The disparity of diffusion lengths between the activator and inhibitor leads to pulse-type solutions in which the activator is localised while the inhibitor varies on an O (1) length scale. In the asymptotic limit considered, the pulses become spikes described by Dirac delta functions and evolve slowly in time until equilibrium is reached. Such quasi-equilibrium solutions with N activator pulses are constructed and a differential-algebraic system of equations (DAE) is derived, characterising the slow evolution of the locations and the litudes of the pulses. We find excellent agreement for the pulse evolution between the asymptotic theory and the results of numerical computations. An algebraic system for the equilibrium pulse litudes and locations is derived from the equilibrium points of the DAE system. Both symmetric equilibria, corresponding to a common pulse litude, and asymmetric pulse equilibria, for which the pulse litudes are different, are constructed. We find that for a positive boundary feed rate, pulse spacing of symmetric equilibria is no longer uniform, and that for sufficiently large boundary flux, pulses at the edges of the pattern may collide with and remain fixed at the boundary. Lastly, stability of the equilibrium solutions is analysed through linearisation of the DAE, which, in contrast to previous approaches, provides a quick way to calculate the small eigenvalues governing weak translation-type instabilities of equilibrium pulse patterns.
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2016
DOI: 10.1137/15M1038931
Publisher: Elsevier BV
Date: 2015
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2018
DOI: 10.1137/17M1137759
Publisher: Elsevier BV
Date: 09-2009
Publisher: SAGE Publications
Date: 10-2007
DOI: 10.1016/J.JALA.2007.05.007
Abstract: The use of molecular techniques to inform diagnosis, prognosis, and treatment design will play an important role in the future of medicine. Each new test, however, represents a new cost to the health care system, and significant effort is required to move new techniques to the clinical setting in the most cost-effective and efficient manner. Consequently, there is a compelling need for technological improvements that will facilitate clinical application of novel assays, by reducing cost and complexity in implementation. This is particularly important in cancer pathology. Here, we present a novel applicator technology that enables staining of in idual biopsies in a tissue microarray (TMA) to provide low-cost, multiplexed biomarker testing at the level of intact tissue. The applicator is designed to deliver tens of nanoliters of aqueous reagent to arrayed tissue biopsies kept under a layer of oil-based Liquid Coverslip without contacting the biopsies. A pin consisting of concentric stainless steel electrodes separated by a hydrophobic insulator provides a balance between hydrophobicity and hydrophilicity to hold a reagent droplet on the tip of the pin, whereas a small electrical current passed through the droplet spanning the electrodes is used for drop sensing. This design is more amenable to repeatable manufacturing than a previous prototype, which in initial testing demonstrated successful immunohistochemical and in situ hybridization staining of in idual biopsies in a TMA, but was difficult to produce. This new design was tested to investigate the factors affecting its operation, in terms of the volume of reagent picked up and its ability to successfully deliver reagent to the biopsies.
Publisher: IOP Publishing
Date: 19-12-2019
Publisher: Elsevier BV
Date: 06-2018
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2017
DOI: 10.1137/16M108121X
Publisher: Society for Industrial & Applied Mathematics (SIAM)
Date: 2017
DOI: 10.1137/16M1060169
Start Date: 03-2023
End Date: 02-2025
Amount: $405,000.00
Funder: Australian Research Council
View Funded Activity