1
|
Henley L, Finch D, Mathews F, Jones O, Woolley TE. A simple and fast method for estimating bat roost locations. ROYAL SOCIETY OPEN SCIENCE 2024; 11:231999. [PMID: 38660598 PMCID: PMC11040240 DOI: 10.1098/rsos.231999] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/03/2024] [Revised: 03/04/2024] [Accepted: 03/05/2024] [Indexed: 04/26/2024]
Abstract
Bats play a pivotal role in pest control, pollination and seed dispersal. Despite their ecological significance, locating bat roosts remains a challenging task for ecologists. Traditional field surveys are time-consuming, expensive and may disturb sensitive bat populations. In this article, we combine data from static audio detectors with a bat movement model to facilitate the detection of bat roosts. Crucially, our technique not only provides a point prediction for the most likely location of a bat roost, but because of the algorithm's speed, it can be applied over an entire landscape, resulting in a likelihood map, which provides optimal searching regions. To illustrate the success of the algorithm and highlight limitations, we apply our technique to greater horseshoe bat (Rhinolophus ferrumequinum) acoustic data acquired from six surveys from four different UK locations and over six different times in the year. Furthermore, we investigate what happens to the accuracy of our predictions in the case that the roost is not contained within the area spanned by the detectors. This innovative approach to searching rural environments holds the potential to greatly reduce the labour required for roost finding, and, hence, enhance the conservation efforts of bat populations and their habitats.
Collapse
Affiliation(s)
- Lucy Henley
- Cardiff School of Mathematics, Cardiff University, CardiffCF24 4AG, UK
| | - Domhnall Finch
- University of Sussex, John Maynard Smith Building, BrightonBN1 9RH, UK
- National Parks and Wildlife Service, North DublinD07 N7CV, Ireland
| | - Fiona Mathews
- University of Sussex, John Maynard Smith Building, BrightonBN1 9RH, UK
| | - Owen Jones
- Cardiff School of Mathematics, Cardiff University, CardiffCF24 4AG, UK
| | - Thomas E. Woolley
- Cardiff School of Mathematics, Cardiff University, CardiffCF24 4AG, UK
| |
Collapse
|
2
|
Henley L, Jones O, Mathews F, Woolley TE. Bat Motion can be Described by Leap Frogging. Bull Math Biol 2024; 86:16. [PMID: 38197980 PMCID: PMC10781826 DOI: 10.1007/s11538-023-01233-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/17/2023] [Accepted: 11/01/2023] [Indexed: 01/11/2024]
Abstract
We present models of bat motion derived from radio-tracking data collected over 14 nights. The data presents an initial dispersal period and a return to roost period. Although a simple diffusion model fits the initial dispersal motion we show that simple convection cannot provide a description of the bats returning to their roost. By extending our model to include non-autonomous parameters, or a leap frogging form of motion, where bats on the exterior move back first, we find we are able to accurately capture the bat's motion. We discuss ways of distinguishing between the two movement descriptions and, finally, consider how the different motion descriptions would impact a bat's hunting strategy.
Collapse
Affiliation(s)
- Lucy Henley
- Cardiff School of Mathematics Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK
| | - Owen Jones
- Cardiff School of Mathematics Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK
| | - Fiona Mathews
- University of Sussex, John Maynard Smith Building, Falmer, Brighton, BN1 9RH, UK
| | - Thomas E Woolley
- Cardiff School of Mathematics Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK.
| |
Collapse
|
3
|
Boundary Conditions Cause Different Generic Bifurcation Structures in Turing Systems. Bull Math Biol 2022; 84:101. [PMID: 35953624 PMCID: PMC9372019 DOI: 10.1007/s11538-022-01055-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/13/2022] [Accepted: 07/15/2022] [Indexed: 11/13/2022]
Abstract
Turing’s theory of morphogenesis is a generic mechanism to produce spatial patterning from near homogeneity. Although widely studied, we are still able to generate new results by returning to common dogmas. One such widely reported belief is that the Turing bifurcation occurs through a pitchfork bifurcation, which is true under zero-flux boundary conditions. However, under fixed boundary conditions, the Turing bifurcation becomes generically transcritical. We derive these algebraic results through weakly nonlinear analysis and apply them to the Schnakenberg kinetics. We observe that the combination of kinetics and boundary conditions produce their own uncommon boundary complexities that we explore numerically. Overall, this work demonstrates that it is not enough to only consider parameter perturbations in a sensitivity analysis of a specific application. Variations in boundary conditions should also be considered.
Collapse
|
4
|
Woolley TE, Hill W, Hogan C. Accounting for dimensional differences in stochastic domain invasion with applications to precancerous cell removal. J Theor Biol 2022; 541:111024. [PMID: 35108550 DOI: 10.1016/j.jtbi.2022.111024] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/07/2021] [Revised: 01/06/2022] [Accepted: 01/10/2022] [Indexed: 11/26/2022]
Abstract
We consider a specific form of domain invasion that is an abstraction of pancreatic tissue eliminating precancerous mutant cells through juxtacrine signalling. The model is explored discretely, continuously, stochastically and deterministically, highlighting unforeseen nonlinear dependencies on the dimension of the solution domain. Specifically, stochastically simulated populations invade with a dimension dependent wave speed that can be over twice as fast as their deterministic analogues. Although the wave speed can be analytically derived in the cases of small domains, the probabilistic state space grows exponentially and, thus, we use numeric simulation and curve fitting to predict limiting dynamics.
Collapse
Affiliation(s)
- Thomas E Woolley
- Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK.
| | - William Hill
- Cancer Evolution and Genome Instability Laboratory, The Francis Crick Institute, London, NW1 1AT, UK; European Cancer Stem Cell Research Institute, School of Biosciences, Cardiff University, Hadyn Ellis Building, Maindy Road, Cardiff, CF24 4HQ, UK
| | - Catherine Hogan
- European Cancer Stem Cell Research Institute, School of Biosciences, Cardiff University, Hadyn Ellis Building, Maindy Road, Cardiff, CF24 4HQ, UK
| |
Collapse
|
5
|
Vittadello ST, Leyshon T, Schnoerr D, Stumpf MPH. Turing pattern design principles and their robustness. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2021; 379:20200272. [PMID: 34743598 PMCID: PMC8580431 DOI: 10.1098/rsta.2020.0272] [Citation(s) in RCA: 18] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 06/24/2021] [Indexed: 05/05/2023]
Abstract
Turing patterns have morphed from mathematical curiosities into highly desirable targets for synthetic biology. For a long time, their biological significance was sometimes disputed but there is now ample evidence for their involvement in processes ranging from skin pigmentation to digit and limb formation. While their role in developmental biology is now firmly established, their synthetic design has so far proved challenging. Here, we review recent large-scale mathematical analyses that have attempted to narrow down potential design principles. We consider different aspects of robustness of these models and outline why this perspective will be helpful in the search for synthetic Turing-patterning systems. We conclude by considering robustness in the context of developmental modelling more generally. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.
Collapse
Affiliation(s)
- Sean T. Vittadello
- School of BioSciences, University of Melbourne, Melbourne, Victoria 3010, Australia
| | - Thomas Leyshon
- Department of Life Sciences, Imperial College London, London, UK
| | - David Schnoerr
- Department of Life Sciences, Imperial College London, London, UK
| | - Michael P. H. Stumpf
- School of BioSciences, University of Melbourne, Melbourne, Victoria 3010, Australia
- School of Mathematics and Statistics, University of Melbourne, Melbourne, Victoria 3010, Australia
| |
Collapse
|
6
|
Leyshon T, Tonello E, Schnoerr D, Siebert H, Stumpf MPH. The design principles of discrete turing patterning systems. J Theor Biol 2021; 531:110901. [PMID: 34530030 DOI: 10.1016/j.jtbi.2021.110901] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/16/2021] [Revised: 08/15/2021] [Accepted: 09/06/2021] [Indexed: 10/20/2022]
Abstract
The formation of spatial structures lies at the heart of developmental processes. However, many of the underlying gene regulatory and biochemical processes remain poorly understood. Turing patterns constitute a main candidate to explain such processes, but they appear sensitive to fluctuations and variations in kinetic parameters, raising the question of how they may be adopted and realised in naturally evolved systems. The vast majority of mathematical studies of Turing patterns have used continuous models specified in terms of partial differential equations. Here, we complement this work by studying Turing patterns using discrete cellular automata models. We perform a large-scale study on all possible two-species networks and find the same Turing pattern producing networks as in the continuous framework. In contrast to continuous models, however, we find these Turing pattern topologies to be substantially more robust to changes in the parameters of the model. We also find that diffusion-driven instabilities are substantially weaker predictors for Turing patterns in our discrete modelling framework in comparison to the continuous case, in the sense that the presence of an instability does not guarantee a pattern emerging in simulations. We show that a more refined criterion constitutes a stronger predictor. The similarity of the results for the two modelling frameworks suggests a deeper underlying principle of Turing mechanisms in nature. Together with the larger robustness in the discrete case this suggests that Turing patterns may be more robust than previously thought.
Collapse
Affiliation(s)
- Thomas Leyshon
- Department of Life Sciences, Imperial College London, UK
| | - Elisa Tonello
- FB Mathematik und Informatik, Freine Universität Berlin, Germany
| | - David Schnoerr
- Department of Life Sciences, Imperial College London, UK
| | - Heike Siebert
- FB Mathematik und Informatik, Freine Universität Berlin, Germany
| | - Michael P H Stumpf
- Department of Life Sciences, Imperial College London, UK; Melbourne Integrated Genomics, University of Melbourne, Australia; School of BioScience, University of Melbourne, Australia; School of Mathematics and Statistics, University of Melbourne, Australia.
| |
Collapse
|
7
|
Hill W, Zaragkoulias A, Salvador-Barbero B, Parfitt GJ, Alatsatianos M, Padilha A, Porazinski S, Woolley TE, Morton JP, Sansom OJ, Hogan C. EPHA2-dependent outcompetition of KRASG12D mutant cells by wild-type neighbors in the adult pancreas. Curr Biol 2021; 31:2550-2560.e5. [PMID: 33891893 PMCID: PMC8231095 DOI: 10.1016/j.cub.2021.03.094] [Citation(s) in RCA: 26] [Impact Index Per Article: 8.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/24/2020] [Revised: 02/15/2021] [Accepted: 03/29/2021] [Indexed: 12/22/2022]
Abstract
As we age, our tissues are repeatedly challenged by mutational insult, yet cancer occurrence is a relatively rare event. Cells carrying cancer-causing genetic mutations compete with normal neighbors for space and survival in tissues. However, the mechanisms underlying mutant-normal competition in adult tissues and the relevance of this process to cancer remain incompletely understood. Here, we investigate how the adult pancreas maintains tissue health in vivo following sporadic expression of oncogenic Kras (KrasG12D), the key driver mutation in human pancreatic cancer. We find that when present in tissues in low numbers, KrasG12D mutant cells are outcompeted and cleared from exocrine and endocrine compartments in vivo. Using quantitative 3D tissue imaging, we show that before being cleared, KrasG12D cells lose cell volume, pack into round clusters, and E-cadherin-based cell-cell adhesions decrease at boundaries with normal neighbors. We identify EphA2 receptor as an essential signal in the clearance of KrasG12D cells from exocrine and endocrine tissues in vivo. In the absence of functional EphA2, KrasG12D cells do not alter cell volume or shape, E-cadherin-based cell-cell adhesions increase and KrasG12D cells are retained in tissues. The retention of KRasG12D cells leads to the early appearance of premalignant pancreatic intraepithelial neoplasia (PanINs) in tissues. Our data show that adult pancreas tissues remodel to clear KrasG12D cells and maintain tissue health. This study provides evidence to support a conserved functional role of EphA2 in Ras-driven cell competition in epithelial tissues and suggests that EphA2 is a novel tumor suppressor in pancreatic cancer.
Collapse
Affiliation(s)
- William Hill
- European Cancer Stem Cell Research Institute, School of Biosciences, Cardiff University, Hadyn Ellis Building, Maindy Road, Cardiff CF24 4HQ, UK
| | - Andreas Zaragkoulias
- European Cancer Stem Cell Research Institute, School of Biosciences, Cardiff University, Hadyn Ellis Building, Maindy Road, Cardiff CF24 4HQ, UK
| | - Beatriz Salvador-Barbero
- European Cancer Stem Cell Research Institute, School of Biosciences, Cardiff University, Hadyn Ellis Building, Maindy Road, Cardiff CF24 4HQ, UK
| | - Geraint J Parfitt
- European Cancer Stem Cell Research Institute, School of Biosciences, Cardiff University, Hadyn Ellis Building, Maindy Road, Cardiff CF24 4HQ, UK; School of Optometry & Vision Sciences, Cardiff University, Maindy Road, Cardiff CF24 4HQ, UK
| | - Markella Alatsatianos
- European Cancer Stem Cell Research Institute, School of Biosciences, Cardiff University, Hadyn Ellis Building, Maindy Road, Cardiff CF24 4HQ, UK
| | - Ana Padilha
- European Cancer Stem Cell Research Institute, School of Biosciences, Cardiff University, Hadyn Ellis Building, Maindy Road, Cardiff CF24 4HQ, UK
| | - Sean Porazinski
- European Cancer Stem Cell Research Institute, School of Biosciences, Cardiff University, Hadyn Ellis Building, Maindy Road, Cardiff CF24 4HQ, UK; Faculty of Medicine, St Vincent's Clinical School, University of New South Wales, Sydney, Australia
| | - Thomas E Woolley
- School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, UK
| | - Jennifer P Morton
- CRUK Beatson Institute, Glasgow G61 1BD, UK; Institute of Cancer Sciences, University of Glasgow, Glasgow G61 1QH, UK
| | - Owen J Sansom
- CRUK Beatson Institute, Glasgow G61 1BD, UK; Institute of Cancer Sciences, University of Glasgow, Glasgow G61 1QH, UK
| | - Catherine Hogan
- European Cancer Stem Cell Research Institute, School of Biosciences, Cardiff University, Hadyn Ellis Building, Maindy Road, Cardiff CF24 4HQ, UK.
| |
Collapse
|
8
|
Adamer MF, Harrington HA, Gaffney EA, Woolley TE. Coloured Noise from Stochastic Inflows in Reaction-Diffusion Systems. Bull Math Biol 2020; 82:44. [PMID: 32198538 PMCID: PMC7083815 DOI: 10.1007/s11538-020-00719-w] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/13/2019] [Accepted: 03/06/2020] [Indexed: 12/14/2022]
Abstract
In this paper, we present a framework for investigating coloured noise in reaction-diffusion systems. We start by considering a deterministic reaction-diffusion equation and show how external forcing can cause temporally correlated or coloured noise. Here, the main source of external noise is considered to be fluctuations in the parameter values representing the inflow of particles to the system. First, we determine which reaction systems, driven by extrinsic noise, can admit only one steady state, so that effects, such as stochastic switching, are precluded from our analysis. To analyse the steady-state behaviour of reaction systems, even if the parameter values are changing, necessitates a parameter-free approach, which has been central to algebraic analysis in chemical reaction network theory. To identify suitable models, we use tools from real algebraic geometry that link the network structure to its dynamical properties. We then make a connection to internal noise models and show how power spectral methods can be used to predict stochastically driven patterns in systems with coloured noise. In simple cases, we show that the power spectrum of the coloured noise process and the power spectrum of the reaction-diffusion system modelled with white noise multiply to give the power spectrum of the coloured noise reaction-diffusion system.
Collapse
Affiliation(s)
- Michael F Adamer
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK.
| | - Heather A Harrington
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK
| | - Eamonn A Gaffney
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford, UK
| | | |
Collapse
|
9
|
Abstract
Recent experiments on zebrafish pigmentation suggests that their typical black and white striped skin pattern is made up of a number of interacting chromatophore families. Specifically, two of these cell families have been shown to interact through a nonlocal chasing mechanism, which has previously been modeled using integro-differential equations. We extend this framework to include the experimentally observed fact that the cells often exhibit chiral movement, in that the cells chase, and run away, at angles different to the line connecting their centers. This framework is simplified through the use of multiple small limits leading to a coupled set of partial differential equations which are amenable to Fourier analysis. This analysis results in the production of dispersion relations and necessary conditions for a patterning instability to occur. Beyond the theoretical development and the production of new pattern planiforms we are able to corroborate the experimental hypothesis that the global pigmentation patterns can be dependent on the chirality of the chromatophores.
Collapse
Affiliation(s)
- Thomas E Woolley
- Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG Wales, United Kingdom
| |
Collapse
|
10
|
Belmonte-Beitia J, Woolley T, Scott J, Maini P, Gaffney E. Modelling biological invasions: Individual to population scales at interfaces. J Theor Biol 2013; 334:1-12. [DOI: 10.1016/j.jtbi.2013.05.033] [Citation(s) in RCA: 25] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/25/2012] [Revised: 05/24/2013] [Accepted: 05/28/2013] [Indexed: 11/27/2022]
|
11
|
Schumacher LJ, Woolley TE, Baker RE. Noise-induced temporal dynamics in Turing systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:042719. [PMID: 23679461 DOI: 10.1103/physreve.87.042719] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/05/2013] [Indexed: 05/03/2023]
Abstract
We examine the ability of intrinsic noise to produce complex temporal dynamics in Turing pattern formation systems, with particular emphasis on the Schnakenberg kinetics. Using power spectral methods, we characterize the behavior of the system using stochastic simulations at a wide range of points in parameter space and compare with analytical approximations. Specifically, we investigate whether polarity switching of stochastic patterns occurs at a defined frequency. We find that it can do so in individual realizations of a stochastic simulation, but that the frequency is not defined consistently across realizations in our samples of parameter space. Further, we examine the effect of noise on deterministically predicted traveling waves and find them increased in amplitude and decreased in speed.
Collapse
Affiliation(s)
- Linus J Schumacher
- Centre for Mathematical Biology, Mathematical Institute, University of Oxford, 24-29 St. Giles', Oxford, OX1 3LB, United Kingdom
| | | | | |
Collapse
|
12
|
Maini PK, Woolley TE, Baker RE, Gaffney EA, Lee SS. Turing's model for biological pattern formation and the robustness problem. Interface Focus 2012; 2:487-96. [PMID: 23919129 PMCID: PMC3363041 DOI: 10.1098/rsfs.2011.0113] [Citation(s) in RCA: 105] [Impact Index Per Article: 8.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/11/2011] [Accepted: 01/11/2012] [Indexed: 01/30/2023] Open
Abstract
One of the fundamental questions in developmental biology is how the vast range of pattern and structure we observe in nature emerges from an almost uniformly homogeneous fertilized egg. In particular, the mechanisms by which biological systems maintain robustness, despite being subject to numerous sources of noise, are shrouded in mystery. Postulating plausible theoretical models of biological heterogeneity is not only difficult, but it is also further complicated by the problem of generating robustness, i.e. once we can generate a pattern, how do we ensure that this pattern is consistently reproducible in the face of perturbations to the domain, reaction time scale, boundary conditions and so forth. In this paper, not only do we review the basic properties of Turing's theory, we highlight the successes and pitfalls of using it as a model for biological systems, and discuss emerging developments in the area.
Collapse
Affiliation(s)
- Philip K. Maini
- Centre for Mathematical Biology, Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford OX1 3PN, UK
- Oxford Centre for Integrative Systems Biology, Department of Biochemistry, University of Oxford, South Parks Road OX1 3QU, UK
| | - Thomas E. Woolley
- Centre for Mathematical Biology, Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford OX1 3PN, UK
| | - Ruth E. Baker
- Centre for Mathematical Biology, Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford OX1 3PN, UK
| | - Eamonn A. Gaffney
- Centre for Mathematical Biology, Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford OX1 3PN, UK
| | - S. Seirin Lee
- Center for Developmental Biology, RIKEN, Minatojima-minami 2-2-3, Kobe 650-0047, Japan.
| |
Collapse
|
13
|
Woolley TE, Baker RE, Gaffney EA, Maini PK, Seirin-Lee S. Effects of intrinsic stochasticity on delayed reaction-diffusion patterning systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:051914. [PMID: 23004794 DOI: 10.1103/physreve.85.051914] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/26/2012] [Indexed: 05/03/2023]
Abstract
Cellular gene expression is a complex process involving many steps, including the transcription of DNA and translation of mRNA; hence the synthesis of proteins requires a considerable amount of time, from ten minutes to several hours. Since diffusion-driven instability has been observed to be sensitive to perturbations in kinetic delays, the application of Turing patterning mechanisms to the problem of producing spatially heterogeneous differential gene expression has been questioned. In deterministic systems a small delay in the reactions can cause a large increase in the time it takes a system to pattern. Recently, it has been observed that in undelayed systems intrinsic stochasticity can cause pattern initiation to occur earlier than in the analogous deterministic simulations. Here we are interested in adding both stochasticity and delays to Turing systems in order to assess whether stochasticity can reduce the patterning time scale in delayed Turing systems. As analytical insights to this problem are difficult to attain and often limited in their use, we focus on stochastically simulating delayed systems. We consider four different Turing systems and two different forms of delay. Our results are mixed and lead to the conclusion that, although the sensitivity to delays in the Turing mechanism is not completely removed by the addition of intrinsic noise, the effects of the delays are clearly ameliorated in certain specific cases.
Collapse
Affiliation(s)
- Thomas E Woolley
- Centre for Mathematical Biology, Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford, OX1 3LB, United Kingdom.
| | | | | | | | | |
Collapse
|