There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Statistics for Spatial Data.

N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

http://people.physik.hu-berlin.de/~lindner/PDF/physreport.pdf

Effects of noise in excitable systems

Mathematical modelling of the movement of animals, micro-organisms and cells is of great relevance in the fields of biology, ecology and medicine. Movement models can take many different forms, but the most widely used are based on the extensions of simple random walk processes. In this review paper, our aim is twofold: to introduce the mathematics behind random walks in a straightforward manner and to explain how such models can be used to aid our understanding of biological processes. We introduce the mathematical theory behind the simple random walk and explain how this relates to Brownian motion and diffusive processes in general. We demonstrate how these simple models can be extended to include drift and waiting times or be used to calculate first passage times. We discuss biased random walks and show how hyperbolic models can be used to generate correlated random walks. We cover two main applications of the random walk model. Firstly, we review models and results relating to the movement, dispersal and population redistribution of animals and micro-organisms. This includes direct calculation of mean squared displacement, mean dispersal distance, tortuosity measures, as well as possible limitations of these model approaches. Secondly, oriented movement and chemotaxis models are reviewed. General hyperbolic models based on the linear transport equation are introduced and we show how a reinforced random walk can be used to model movement where the individual changes its environment. We discuss the applications of these models in the context of cell migration leading to blood vessel growth (angiogenesis). Finally, we discuss how the various random walk models and approaches are related and the connections that underpin many of the key processes involved.

https://royalsocietypublishing.org/doi/pdf/10.1098/rsif.2008.0014

Random walk models in biology.

‘Bioconvection’ is the name given to pattern-forming convective motions set up in suspensions of swimming micro-organisms. ‘Gyrotaxis’ describes the way the swimming is guided through a balance between the physical torques generated by viscous drag and by gravity operating on an asymmetric distribution of mass within the organism. When the organisms are heavier towards the rear, gyrotaxis turns them so that they swim towards regions of most rapid downflow. The presence of gyrotaxis means that bioconvective instability can develop from an initially uniform suspension, without an unstable density stratification. In this paper a continuum model for suspensions of gyrotactic micro-organisms is proposed and discussed; in particular, account is taken of the fact that the organisms of interest are non-spherical, so that their orientation is influenced by the strain rate in the ambient flow as well as the vorticity. This model is used to analyse the linear instability of a uniform suspension. It is shown that the suspension is unstable if the disturbance wavenumber is less than a critical value which, together with the wavenumber of the most rapidly growing disturbance, is calculated explicitly. The subsequent convection pattern is predicted to be three-dimensional (i.e. with variation in the vertical as well as the horizontal direction) if the cells are sufficiently elongated. Numerical results are given for suspensions of a particular algal species (Chlamydomonas nivalis); the predicted wavelength of the most rapidly growing disturbance is 5–6 times larger than the wavelength of steady-state patterns observed in experiments. The main reasons for the difference are probably that the analysis describes the onset of convection, not the final, nonlinear steady state, and that the experimental fluid layer has finite depth.

/pdf/the-growth-of-bioconvection-patterns-in-a-uniform-suspension-g28mbyqlqk.pdf

The growth of bioconvection patterns in a uniform suspension of gyrotactic micro-organisms

We present the first mathematical model to account for the evolution of the abdominal aortic aneu- rysm The artery is modelled as a two-layered, cylindrical membrane using nonlinear elasticity and a physiologically realistic constitutive model It is subject to a constant systolic pressure and a physiological axial prestretch The development of the aneurysm is assumed to be a conse- quence of the remodelling of its material constituents Microstructural 'recruitment' and fibre density variables for the collagen are introduced into the strain energy density functions This enables the remodelling of colla- gen to be addressed as the aneurysm enlarges An axi- symmetric aneurysm, with axisymmetric degradation of elastin and linear differential equations for the remodel- ling of the fibre variables, is simulated numerically Using physiologically determined parameters to model the abdominal aorta and realistic remodelling rates for its constituents, the predicted dilations of the aneurysm are consistent with those observed in vivo An asymmetric aneurysm with spinal contact is also modelled, and the stress distributions are consistent with previous studies

https://www.maths.gla.ac.uk/~nah/reprints/WHH04journal.pdf

A mathematical model for the growth of the abdominal aortic aneurysm

A biased random walk model for the trajectories of swimming micro-organisms.

The effect of gyrotaxis on the linear stability of a suspension of swimming, negatively buoyant micro-organisms is examined for a layer of finite depth. In the steady basic state there is no bulk fluid motion, and the upwards swimming of the cells is balanced by diffusion resulting from randomness in their shape, orientation and swimming behaviour. This leads to a bulk density stratification with denser fluid on top. The theory is based on the continuum model of Pedley, Hill & Kessler (1988), and employs both asymptotic and numerical analysis. The suspension is characterized by five dimensionless parameters: a Rayleigh number, a Schmidt number, a layer-depth parameter, a gyrotaxis number G, and a geometrical parameter measuring the ellipticity of the micro-organisms. For small values of G, the most unstable mode has a vanishing wavenumber, but for sufficiently large values of G, the predicted initial wavelength is finite, in agreement with experiments. The suspension becomes less stable as the layer depth is increased. Indeed, if the layer is sufficiently deep an initially homogeneous suspension is unstable, and the equilibrium state does not form. The theory of Pedley, Hill & Kessler (1988) for infinite depth is shown to be appropriate in that case. An unusual feature of the model is the existence of overstable or oscillatory modes which are driven by the gyrotactic response of the micro-organisms to the shear at the rigid boundaries of the layer. These modes occur at parameter values which could be realized in experiments.

/pdf/growth-of-bioconvection-patterns-in-a-suspension-of-3f54fg4l4h.pdf

Growth of bioconvection patterns in a suspension of gyrotactic micro-organisms in a layer of finite depth

The novel three-dimensional (3D) mathematical model for the development of abdominal aortic aneurysm (AAA) of Watton et al. Biomech Model Mechanobiol 3(2): 98-113, (2004) describes how changes in the micro-structure of the arterial wall lead to the development of AAA, during which collagen remodels to compensate for loss of elastin. In this paper, we examine the influence of several of the model's material and remodelling parameters on growth rates of the AAA and compare with clinical data. Furthermore, we calculate the dynamic properties of the AAA at different stages in its development and examine the evolution of clinically measurable mechanical properties. The model predicts that the maximum diameter of the aneurysm increases exponentially and that the ratio of systolic to diastolic diameter decreases from 1.13 to 1.02 as the aneurysm develops; these predictions are consistent with physiological observations of Vardulaki et al. Br J Surg 85:1674-1680 (1998) and Lanne et al. Eur J Vasc Surg 6:178-184 (1992), respectively. We conclude that mathematical models of aneurysm growth have the potential to be useful, noninvasive diagnostic tools and thus merit further development.

https://www.maths.gla.ac.uk/~nah/reprints/WattonHill-BMMB2007.pdf

Nicholas A. Hill

Papers

The growth of bioconvection patterns in a uniform suspension of gyrotactic micro-organisms

A mathematical model for the growth of the abdominal aortic aneurysm

A biased random walk model for the trajectories of swimming micro-organisms.

Growth of bioconvection patterns in a suspension of gyrotactic micro-organisms in a layer of finite depth

Evolving mechanical properties of a model of abdominal aortic aneurysm.