Thomas G. Hallam

Structural sensitivity of grazing formulations in nutrient controlled plankton models

SummaryStability and persistence properties of a family of non-spatial plankton models, each differentiated by its herbivore grazing term, are analytically compared. The dynamic persistence function in the model is shown to operate uniformly even though stability configuration characteristics of the model may be topologically distinct. The persistence threshold for each model indicates that total nutrient is a fundamental biological control. In the parameter space, all of the models studied are structurally unstable; however, an important bifurcation mechanism associated with this instability governs persistence. While, topologically, model transfigurement through parameter modulation is non-continuous, the biological populations evolve in a continuous or a lower semicontinuous manner. A basic conclusion of the paper is that fundamental problems for these marine ecological models remain unresolved since each of the models is a structurally unstable system for a fixed dynamically persistent ecology.

On the nonexistence of L p solutions of certain nonlinear differential equations

The asymptotic behavior of the solutions of ordinary nonlinear differential equations will be considered here. The growth of the solutions of a differential equation will be discussed by establishing criteria to determine when the differential equation does not possess a solution that is an element of the space L p (0, ∞)( p ≧ 1).

BOUNDS FOR SOLUTIONS OF REACTION-DIFFUSION EQUATIONS

Publisher Summary This chapter describes bounds for the solutions of reaction–diffusion equations. Reaction–diffusion equations have been employed as models for problems in chemical morphogenesis. Reaction processes are usually nonlinear in character; hence, approximate solutions to the dynamical equations representing these phenomena are important for understanding behavior and can often be profitably exploited in the quantitative and qualitative study of such processes. The chapter describes a general method for constructing upper and lower bounds for the solutions of initial-boundary value problems associated with nonlinear reaction–diffusion equations. The nonlinear analysis approximation technique employs Lagranges theory of first-order characteristics, a transformation that is required to be monotone in the solution argument, and differential inequalities.

On nonlinear functional perturbation problems for ordinary differential equations

Abstract The linear homogeneous system of differential equations is a good basic model for perturbation problems because many useful properties of the solutions of linear equations are readily established. In this article, conditions are imposed upon a linear system of differential equations which imply that the linear system is either conditionally uniformly stable, conditionally asymptotically stable, or conditionally uniformly asymptotically stable. Then, a class of perturbation, terms is found which asymptotically preserves the solutions of the linear system that satisfy a prescribed growth condition. The class of perturbation terms is sufficiently general to include functions which possess retarded, advanced, or a combination of retarded and advanced arguments.

On convergence of the solutions of a functional differential equation

Abstract A sufficient condition is given for the solutions of a functionally perturbed linear system of ordinary differential equations to have limits at ± ∞.

On the intrinsic structure of differential equation models of ecosystems

Abstract Some intrinsic properties of differential equation models of ecosystems are formulated. The properties, which are long-termed, are classified into fundamental, stability, and sensitivity substructures. An aspect of the ecological term “resilient” — asymptotic stability uniformly for all small parameter variations — is introduced in a mathematical setting. In the sensitivity substructure, bounded sensitivity and continuity of sensitivity with respect to solutions are recognized as intrinsic properties. Illustrations of these two properties are given.

On representation and boundedness of solutions of a functional differential equation

The validity and uniqueness of the representation of solutions of (2) by a singular integral equation is discussed. Unique representations of the solutions that are bounded on the real line R, on R+ = [0, ~) , or on R_ = ( ~ , 0] are given. Unbounded solutions of (2) that are essentially generated by the unbounded solutions of (1) are found. In equations (1) and (2), x and y are elements of an n-dimensional vector space X, and A(t) is a continuous, n × n matrix defined on R. Let Y(t) denote the fundamental matrix of (1) that satisfies the initial condition Y(0) = I,, where I, is the n x n identity matrix. Let B denote a compact subset of R and suppose that ~ e C[R x B, R]. For each t e R and each x e C[R, X], define the composition transformation T(t, x) of (2) by