Fordyce A. Davidson

Growth and function of fungal mycelia in heterogeneous environments.

As decomposer organisms, pathogens, plant symbionts and nutrient cyclers, fungi are of fundamental importance in the terrestrial environment. Moreover, in addition to their well-known applications in industry, many species also have great potential in environmental biotechnology. The study of this important class of organisms is difficult through experimental means alone due to the heterogeneity of their natural growth habitat and the microscopic scale of growth. In this work we present a mathematical model for colony expansion that is derived through consideration of the growth characteristics on the microscale. The model equations are of mixed hyperbolic-parabolic type and are treated with a numerical scheme that preserves positivity and conserves mass. The numerical solutions are compared against experimental results in a variety of environments. Thus the effect of different translocation mechanisms on fungal growth and function are identified. The derivation and analysis of an approximation to the full model yields further results concerning basic properties of mycelial growth. Finally, the acidification of the growth habitat is considered and the model thus provides important predictions on the functional consequences of the redistribution of internally-located material.

Solubilization of calcium phosphate as a consequence of carbon translocation by Rhizoctonia solani

A model system based on arrays of three concentric rings of discrete agar droplets is described which allowed study of fungal growth in vitro in nutritionally-heterogeneous conditions. Droplets containing different combinations of glucose and calcium phosphate were used to study the consequences of spatially separating these components in relation to metal phosphate solubilization by Rhizoctonia solani. A pH indicator, bromocresol purple, was added to the agar to visualise the localised production of acidity by the fungus. In the presence of the fungus, solubilization of calcium phosphate on homogeneous agar plates only occurred when glucose was present in the underlying medium. However, solubilization occurred in droplets containing calcium phosphate, but no glucose, when glucose was present in other droplets within the tessellation and where fungal hyphae spanned the droplets. This demonstrates that substrate was transported via mycelia from glucose-containing domains, with the functional consequence of metal phosphate solubilization. In another design, where the inner ring of droplets contained glucose and the outer ring contained only calcium phosphate, acidification of all droplets in the outer ring was observed when the inner droplets contained glucose. However, solubilization of calcium phosphate only occurred when the concentration of glucose in the inner droplets was greater than 2% (w/v). This indicated that a threshold concentration of carbon source may be required before such mechanisms of solubilization are invoked. There was also evidence for reverse translocation of substrate from newly colonised glucose-containing droplets in the outer ring to the central droplets, where fungal growth had originated.

Periodic solutions for a delayed predator-prey model of prey dispersal in two-patch environments

Abstract A delayed periodic Lotka–Volterra type predator-prey model with prey dispersal in two-patch environments is investigated. By using Gaines and Mawhins continuation theorem of coincidence degree theory and by means of a suitable Lyapunov functional, a set of easily verifiable sufficient conditions are obtained to guarantee the existence, uniqueness and global stability of positive periodic solutions of the system. Numerical simulations are given to illustrate the feasibility of our main results.

Context-dependent macroscopic patterns in growing and interacting mycelial networks

Fungal mycelia epitomize, at the cellular level of organization, the growth and pattern-generating properties of a wide variety of indeterminate (indefinitely expandable) living systems. Some of the more important of these properties arise from the capacity of an initially dendritic system of protoplasm filled, apically extending hyphal tubes to anastomose. This integrational process partly restores the symmetry lost during the proliferation of hyphal branches from a germinating spore and so increases the scope for communication and transfer of resources across the system. Growth and pattern generation then depend critically on processes that affect the degree to which resistances to energy transfer within the system are sustained, bypassed or broken down. We use a system of reaction diffusion equations augmented with appropriate initial data to model the processes of expansion and pattern formation within growing mycelia. Such an approach is a test of the feasibility of the hypothesis that radical, adaptive shifts in mycelial pattern can be explained by purely contextual, rather than genetic, changes. Thus we demonstrate that phenotype does not necessarily equate solely to genotype—environment interactions, but may include the physical role in self-organization played by the boundary between the two.

Persistence and global stability of a ratio-dependent predator-prey model with stage structure

A ratio-dependent predator-prey model with stage structure for prey is investigated. First, sufficient conditions are derived for the uniform persistence and impermanence of the model. Next, by constructing appropriate Lyapunov functions, a set of easily verifiable sufficient conditions are obtained for the global asymptotic stability of nonnegative equilibria of the model. Numerical simulations are presented to illustrate the validity of our main results.

Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation

We study a scalar reaction-diffusion equation which contains a nonlocal term in the form of an integral convolution in the spatial variable and demonstrate, using asymptotic, analytical and numerical techniques, that this scalar equation is capable of producing spatio-temporal patterns. Fishers equation is a particular case of this equation. An asymptotic expansion is obtained for a travelling wavefront connecting the two uniform steady states and qualitative differences to the corresponding solution of Fishers equation are noted. A stability analysis combined with numerical integration of the equation show that under certain circumstances nonuniform solutions are formed in the wake of this front. Using global bifurcation theory, we prove the existence of such non-uniform steady state solutions for a wide range of parameter values. Numerical bifurcation studies of the behaviour of steady state solutions as a certain parameter is varied, are also presented.

Global stability of a Lotka-Volterra type predator-prey model with stage structure and time delay

A delayed Lotka-Volterra type predator-prey model with stage structure for predator is investigated. It is assumed in the model that the individuals in the predator population may belong to one of two classes: the immatures and the matures, the age to maturity is presented by a time delay, and that the immature predators do not have the ability to prey. By analyzing characteristic equations and using an iterative technique, a set of easily verifiable sufficient conditions are derived for the local and global stability of the nonnegative equilibria of the model. Numerical simulations are carried out to illustrate the validity of our results.

Periodic solutions for a predator-prey model with Holling-type functional response and time delays

A delayed periodic Holling-type predator-prey model without instantaneous negative feedback is investigated. By using the continuation theorem of coincidence degree theory and by constructing suitable Lyapunov functionals, a set of easily verifiable sufficient conditions are derived for the existence, uniqueness and global stability of positive periodic solutions to the model. Numerical simulation is carried out to illustrate the feasibility of our main results.

Periodic solution of a Lotka-Volterra predator-prey model with dispersion and time delays

A periodic Lotka-Volterra predator-prey model with dispersion and time delays is investigated. By using Gaines and Mawhins continuation theorem of coincidence degree theory and by means of a suitable Lyapunov functional, a set of easily verifiable sufficient conditions are derived for the existence, uniqueness and global stability of positive periodic solutions of the system. Sufficient conditions are also established for the uniform persistence of the system. Numerical simulations are presented to illustrate our main results.

A mathematical model for fungal development in heterogeneous environments

Abstract-A mathematical model for the development of fungal mycelia in heterogeneous envi- ronmental conditions is presented. The validity of this model is tested by comparison of numerical simulations with experimental observations. @ 1998 Elsevier Science Ltd. All rights reserved. Keywords-fingal mycelia, Heterogeneous environments. 1. INTRODUCTION Many species of fungi form a mycelium, that is, an indeterminate system of protoplasm-filled, apically extending, branching tubes (hyphae). At the macroscopic level, mycelia produce organi- zational patterns that are found in many other indeterminate systems (e.g., nervous and vascular systems). Therefore, they provide an experimentally and observationally accessible model sys- tem for investigating the dynamic origins of phenotype patterns in such systems. However, their utility in this role has been limited on two counts: first, by the tendency to treat them as purely additive assemblages of effectively discrete, individual lengths of hyphae (hyphal growth units) that duplicate at regular intervals. Second, all previous models have considered fungi grown in perfectly uniform conditions. This is certainly not the situation fungi encounter when growing in any natural environment, and a consideration of the effects of heterogeneity is essential for any true understanding of their form and function. That such environments cause dramatic changes to their growth characteristics has been shown experimentally by Ritz [I], Rayner [2,3], and in the geometric setting discussed below, by Park [4]. In a series of papers, Davidson