Maureen P. Edwards

Application of Lie group analysis to a core group model for sexually transmitted diseases

Abstract Lie group analysis is applied to a core group model for sexually transmitted disease formulated by Hadeler and Castillo-Chavez [Hadeler K P and Castillo-Chavez C, A core group model for disease transmission, Math. Biosci. 128 (1995), 41–55]. Several instances of integrability even linearity are found which lead to the general solution of the model. A discussion of such solutions is presented and it is shown how they complement Hadeler and Castillo-Chavez’s qualitative analysis.

Systematic construction of hidden nonlocal symmetries for the inhomogeneous nonlinear diffusion equation

We consider a class of inhomogeneous nonlinear diffusion equations (INDE) that arise in solute transport theory. Hidden nonlocal symmetries that seem not to be recorded in the literature are systematically determined by considering an integrated equation, obtained using the general integral variable, rather than a system of first-order partial differential equations (PDEs) associated with the concentration and flux of a conservation law. Reductions for the INDE to ordinary differential equations (ODEs) are performed and some invariant solutions are constructed.

Exact transient solutions to nonlinear diffusion-convection equations in higher dimensions

The complete Lie algebra of classical infinitesimal symmetries of the nonlinear diffusion-convection equation in two and three dimensions is presented. Except for some cases involving constant diffusivity, a complete reduction to an ordinary differential equation is not possible. However, closed-form solutions are obtained for special forms of both the 2D and 3D nonlinear diffusion-convection equations, using a symmetry reduction and an additional physical constraint. This extends the small list of closed-form transient solutions already known.

Closed-form solutions for unsaturated flow under variable flux boundary conditions

It is shown that from any solution of the linear diffusion equation, we may construct a solution of a realistic form of the Richards equation for unsaturated flow. Compared to the usual direct linearization method, our inverse approach involves a quite different sequence of transformations. This opens the possibility of exact solutions with a wider variety of continuously varying flux boundary conditions. Closed-form solutions are presented for two examples. In these, the varying water flux boundary conditions resemble (i) the passage of a peaking storm and (ii) the continuous opening of a valve preceding a steady water supply. Unlike earlier more systematic approaches to this problem, our method does not require the numerical solution of an integral equation.

Analytical Solutions for Two-Dimensional Solute Transport with Velocity-Dependent Dispersion

A form of the solute transport equation is transformed from Cartesian to streamline coordinates. Symmetry analysis of this equation with a point water source reveals a 5-parameter symmetry group. Exploitation of the rich symmetry properties of the equation leads to a number of associated reduced partial differential equations - that is, partial differential equations where the number of independent variables has been reduced by one. Using further symmetry reductions and other transformation techniques, the construction of new solutions for non-radial solute transport on a background of radial water flow is possible.

The diffusive Lotka-Volterra predator-prey system with delay

Semi-analytical solutions for the diffusive Lotka-Volterra predator-prey system with delay are considered in one and two-dimensional domains. The Galerkin method is applied, which approximates the spatial structure of both the predator and prey populations. This approach is used to obtain a lower-order, ordinary differential delay equation model for the system of governing delay partial differential equations. Steady-state and transient solutions and the region of parameter space, in which Hopf bifurcations occur, are all found. In some cases simple linear expressions are found as approximations, to describe steady-state solutions and the Hopf parameter regions. An asymptotic analysis for the periodic solution near the Hopf bifurcation point is performed for the one-dimensional domain. An excellent agreement is shown in comparisons between semi-analytical and numerical solutions of the governing equations.

Quadratic autocatalysis with non-linear decay

We provide a detailed, and thorough, investigation into the concentration multiplicity and dynamic stability of a prototype non-linear chemical mechanism: quadratic autocatalysis subject to non-linear decay in a continuously stirred tank reactor. This model was previously investigated in the literature using numerical path-following techniques. The contribution of this study is the application of singularity theory and degenerate Hopf-bifurcation theory to obtain analytical representations of many of the features of interest in this system. In particular, we use these presentations to identify critical values of an unfolding parameter below which specified phenomenon are no longer exhibited.

Compactly Supported Solutions of Reaction–Diffusion Models of Biological Spread

Lie group analysis is one of the most useful techniques for analyzing the analytic structure of the solutions of differential equations. Here, reaction–diffusion (RD) modelling of biological invasion is used to illustrate this fact in terms of identifying the conditions that the diffusion and reaction terms must satisfy for their solutions to have compact support. Biological invasion, such as the spread of viruses on the leaves of plants and the invasive spread of animals and weeds into new environments, has a well-defined progressing compactly supported spatial \(\mathbb {R}^2\) structure. There are two distinct ways in which such progressing compact structure can be modelled mathematically; namely, cellular automata modelling and reaction–diffusion (RD) equation modelling. The goal in this paper is to review the extensive literature on RD equations to investigate the extent to which RD equations are known to have compactly supported solutions. Though the existence of compactly supported solutions of nonlinear diffusion equations, without reaction, is well documented, the conditions that the reaction terms should satisfy in conjunction with such nonlinear diffusion equations, for the compact support to be retained, has not been examined in specific detail. A possible partial connection relates to the results of Arrigo, Hill, Goard and Broadbridge, who examined, under various symmetry analysis assumptions, situations where the diffusion and reaction terms are connected by explicit relationships. However, it was not investigated whether the reaction terms generated by these relationships are such that the compact support of the solutions is maintained. Here, results from a computational analysis for the addition of different reaction terms to power law diffusion are presented and discussed. It appears that whether or not the reaction term is zero, as a function of its argument at zero, is an important consideration. In addition, it is confirmed algebraically and graphically that the shapes of compactly supported solutions are strongly controlled by the choice of the reaction term.

Modeling the positioning of trichomes on the leaves of plants

A continuing and future challenge in plant science is the “genetics of geometry” [3]: the recovery of information about the dynamics of the genetic mechanisms by which plants control the development of various features of their geometry. Some representative publications dealing with such issues include: (i) the modeling of plant architecture using L-systems and rewriting [18], (ii) the genetic control of floral development [4,10], and (iii) the positioning of the trichomes (hairs) on the leaves of plants such as Arabidopsis thaliana [23, 25]. It is the positioning of trichomes which is examined in this chapter. The use of reaction-diffusion models is compared with cellular signaling and switching models. It is concluded that, in performing simulations to understand the dynamics of the mechanisms that control pattern formation in plants, it is necessary to work with a cellular model of the plant organ being studied in order to improve on current understanding about how the genetics controls the signaling and switching between cells to produce the observed patterns.