Mollusc seashells grow through the local deposition and calcification of material at the shell opening by a soft and thin organ called the mantle. Through this process, a huge variety of shell structures are formed. Previous models have shown that these structural patterns can largely be understood by examining the mechanical interaction between the deformable mantle and the rigid shell aperture to which it adheres. In this paper we extend this modelling framework in two distinct directions. For one, we incorporate a mechanical feedback in the growth of the mollusc. Second, we develop an initial framework to couple the two primary and orthogonal modes of pattern formation in shells, which are termed antimarginal and commarginal ornamentation. In both cases we examine the change in shell morphology that occurs due to the different mechanical influences and evaluate the hypotheses in light of the fossil record.

John Blake (1947–2016) was a leader in fluid mechanics, his two principal areas of expertise being biological fluid mechanics on microscopic scales and bubble dynamics. He produced leading research and mentored others in both Australia, his home country, and the UK, his adopted home. This article reviews John Blake’s contributions in biological fluid mechanics, as well as gives the author’s personal viewpoint as one of the many graduate students and researchers who benefitted from his supervision, guidance and inspiration. The key topics from biological mechanics discussed are: “squirmer” models of protozoa, the method of images in Stokes flow and the “blakelet” solution, discrete cilia modelling via slender body theory, physiological flows in respiration and reproduction, blinking stokeslets in microorganism feeding, human sperm motility and embryonic nodal cilia.

We have proposed a three-species hybrid food chain model with multiple time delays. The interaction between the prey and the middle predator follows Holling type (HT) II functional response, while the interaction between the top predator and its only food, the middle predator, is taken as a general functional response with the mutual interference schemes, such as Crowley–Martin (CM), Beddington–DeAngelis (BD) and Hassell–Varley (HV) functional responses. We analyse the model system which employs HT II and CM functional responses, and discuss the local and global stability analyses of the coexisting equilibrium solution. The effect of gestation delay on both the middle and top predator has been studied. The dynamics of model systems are affected by both factors: gestation delay and the form of functional responses considered. The theoretical results are supported by appropriate numerical simulations, and bifurcation diagrams are obtained for biologically feasible parameter values. It is interesting from the application point of view to show how an individual delay changes the dynamics of the model system depending on the form of functional response.

We study the numerical Adomian decomposition method for the pricing of European options under the well-known Black–Scholes model. However, because of the nondifferentiability of the pay-off function for such options, applying the Adomian decomposition method to the Black–Scholes model is not straightforward. Previous works on this assume that the pay-off function is differentiable or is approximated by a continuous estimation. Upon showing that these approximations lead to incorrect results, we provide a proper approach, in which the singular point is relocated to infinity through a coordinate transformation. Further, we show that our technique can be extended to pricing digital options and European options under the Vasicek interest rate model, in both of which the pay-off functions are singular. Numerical results show that our approach overcomes the difficulty of directly dealing with the singularity within the Adomian decomposition method and gives very accurate results.

In the theory of spontaneous combustion, identifying the critical value of the Frank-Kamenetskii parameter corresponds to solving a bifurcation point problem. There are two different numerical methods used to solve this problem—the direct and indirect numerical methods. The latter finds the bifurcation point by solving a partial differential equation (PDE) problem. This is a better method to find the bifurcation point for complex geometries. This paper improves the indirect numerical method by combining the grid-domain extension method with the matrix equation computation method. We calculate the critical parameters of the Frank-Kamenetskii equation for some complex geometries using the indirect numerical method. Our results show that both the curve of the outer boundary and the height of the geometries have an effect on the values of the critical Frank-Kamenetskii parameter, however, they have little effect on the critical dimensionless temperature.

We consider an initial–boundary value problem that involves a partial differential equation with a functional term. The problem is motivated by a cell division model for size structured cell cohorts in which growth and division occur. Although much is known about the large time asymptotic behaviour of solutions to these problems for constant growth rates, general solution techniques are rare. We analyse the case where the growth rate is linear and the division rate is a monomial, and we develop a method to determine the general solution for a general class of initial data. The large time dynamics of solutions for this case are significantly different from the constant growth rate case. We show that solutions approach a time-dependent attracting solution that is periodic in the time variable.