This book is a research level case study of mathematical modeling, as opposed to computer modeling, of the dynamics of two competing plant species or strains, or genotypes. The process modeled in fact is a binary mixture of two strains of the annual legume subterranean clover, and it is common to many annual plants. The models were developed in response to four major questions on long term outcomes of binary mixtures.
The book begins with an introductory chapter which sets the scene with a nonmathematical outline of the growth dynamics of clover. Formulation of the models from these dynamics is quite a long task and Chapter 2 is devoted to this. There are two main models: Model 1 and 2, having the same mathematical structure, which is in the form of a pair of nonlinear convolution equations. A special case of Model 1 yields a simpler model, called Model 1A, which has the form of a pair of first order nonlinear difference equations. Chapter 3 introduces the analysis of Model 1A by embedding it in a more general family of models, called Model G. The chapter ends with a fairly long review of recent theoretical work on modeling the growth of annual plants having seed banks.
Model G can have any number of equilibria, but the authors assume there is at most one interior equilibrium, because this is what happens for Model 1A. Chapter 4 presents a detailed analysis of the case where there is no interior equilibrium. The last section of this chapter treats a degenerate form of Model G which cannot correspond to any possible configuration of Model 1A. Chapter 5 covers similar topics when there is a single interior equilibrium and in the case of Model 1A, a Lyapunov function is used to give an alternative proof that the interior equilibrium is stable and attracting.
Models 1 and 2 are treated in Chapter 6, using a pair of difference equations of order equal to the current time variable. Chapter 7 opens with a recapitulation of the most important theoretical findings from the previous chapters and then discusses their significance for data obtained from field experiments for several binary mixtures of clover. Chapter 8 points to a possible direction of future research while restricting the discussion to a single strain of subclover.
This book is written in a mathematical style, that is, theorem-proof- discussion. The results and their consequences can be gleaned without having to plough through the proofs and the latter are always easily identified. The book is useful either for mathematicians or mathematically literate biologists. The mathematical models given in this monograph are an attempt to formalize knowledge gained over many years of observation and experimentation with competing strains of subterranean clover in the south-west of Western Australia.