Geoffrey Butler

Uniformly persistent systems

Conditions are given under which weak persistence of a dynamical system with respect to the boundary of a given set implies uniform persistence.

A Mathematical Model of the Chemostat with Periodic Washout Rate

In its simplest form, the chemostat consists of several populations of microorganisms competing for a single limiting nutrient. If the input concentration of nutrient and the washout rate are constant, theory predicts and experiment confirms that at most one of the populations will survive. In nature, however, one may expect the input concentration and washout rate to vary with time. In this paper we consider a model for the chemostat with periodic washout rate. Conditions are found for competitive exclusion to hold, and bifurcation techniques are employed to show that under suitable circumstances there will be coexistence of the competing populations in the form of positive periodic solutions.

Coexistence of competing predators in a chemostat

An analysis is given of a mathematical model of two predators feeding on a single prey growing in the chemostat. In the case that one of the predators goes extinct, a global stability result is obtained. Under appropriate circumstances, a bifurcation theorem can be used to show that coexistence of the predators occurs in the form of a limit cycle.

Bifurcation from a limit cycle in a two predator-one prey ecosystem modeled on a chemostat

A three dimensional system of ordinary differential equations modeling two predators competing for a renewable resource is analyzed and a periodic solution established in the open octant by a bifurcation from a two dimensional limit cycle. The basis equations are similar to those of the chemostat. This result confirms numerical evidence from [7], partially answering a question raised there.

Rapid oscillation, nonextendability, and the existence of periodic solutions to second order nonlinear ordinary differential equations

Abstract Consider the second order scalar ordinary differential equation x″(t) + ƒ(t, x(t)) = 0 (′ = d dt ) , where ƒ(t, x) is ω-periodic in t and ƒ(t, 0) = 0 for all t. The usual existence results for periodic solutions employing degree theory or other fixed point arguments are generally unhelpful in this case, since the periodic solution that they predict may well be the trivial solution. Jacobowitz has recently succeeded in applying the Poincare-Birkhoff “twist” theorem to demonstrate that this equation has infinitely many (nontrivial) periodic solutions when ƒ satisfies a suitable “strong nonlinearity” condition with respect to x. Essential to his method of proof, however, is the condition xƒ(t, x) > 0 for all t (x = ≠ 0) . In this note we show how this hypothesis may be relaxed, by modifying a technique used by the author when considering the problem of the global existence of solutions which occurs with the removal of the sign condition.

A generalization of a lemma of Bihari and applications to pointwise estimates for integral equations

1. The most commonly used generalization of the Bellman-Gronwall inequality is due to I. Bihari [ 11. See also [2] where related integral inequalities are discussed and more references given. The integral equation associated with Bihari’s inequality is nonlinear in the function of interest (see Section 2), but linear as a function of the integral involved. Recent special uses of integral inequalities in which the associated integral equation is nonlinear in the integral have been considered in papers by D. Willett [3] and H.E. Gollwitzer [4]. In Section 2 of this paper, Bihari’s inequality is generalized. Some results of Muldowney and Wong in [5] are improved, while it is shown that in some situations results of Willett and Gollwitzer are considerable sharpened. Results in terms of maximal solutions and uniqueness theorems are anticipated in a later paper. Fred Brauer considered such extensions of Bihari’s inequality in [6]. S C. Chu and F. T. Metcalf in [7] produced an explicit maximal solution for an integral inequality involving a general Volterra integral.

The oscillatory behavior of a second order nonlinear differential equation with damping

Our intention is to find conditions on the coefficients p and q under which all continuable solutions of (1.1) oscillate. By continuable, we mean a solution which is defined on a half-line [T, 00); such a solution is said to oscillate if it has arbitrarily large zeros. Throughout we shall be concerned with a superlinear restoring force f; that is f(u) is continuously differentiable and monotone increasing for all u, f(0) = 0 and s:T (&/j(u)) < co. This class of functions includes f(u) = / u joL sgn u (0~ > 1). Occasionally our results apply to a larger class of functions f. In addition, p and q will always be assumed to be continuous (locally integrable would suffice) and g will be assumed to be locally Lipschitz continuous. Our procedure is to consider various classes of damping function g for each of which we seek the appropriate conditions on p and q which will serve as oscillation criteria. Sometimes we shall be able to express our criteria in the form of necessary and sufficient conditions for oscillation. In the absence of damping, there is a very large body of literature devoted to the corresponding equation

Oscillation results for selfadjoint differential systems

Abstract We consider the second-order differential system, (1) ( R ( t ) Y ′)′ + Q ( t ) Y = 0, where R , Q , Y are n × n matrices with R ( t ), Q ( t ) symmetric and R ( t ) positive definite for t ϵ [ a , + ∞) ( R ( t ) > 0, t ⩾ a ). We establish sufficient conditions in order that all prepared solutions Y ( t ) of (1) are oscillatory; that is, det Y ( t ) vanishes infinitely often on [ a , + ∞). The conditions involve the smallest and largest eigen-values λ n ( R −1 ( t )) and λ 1 (∝ a t Q ( s ) ds ), respectively. The results obtained can be regarded as generalizing well-known results of Leighton and others in the scalar case.

Global dynamics of a selection model for the growth of a population with genotypic fertility differences

A population growth model is considered for a one locus two allele problem with selection based entirely on fertility differences. A local stability analysis is carried out for the critical points — which include possible polymorphic states — of the resulting nonlinear differential equations. The methods of dynamical systems theory are applied to obtain limiting genotypic proportions for every initial state. Thus the results are global and there are no periodic solutions.

Limiting the complexity of limit sets in self-regulating systems☆☆☆

In the analysis of models of ecosystems one seeks to discover conditions that limit the complexity of the possible behavior of solutions since “intuitively” one feels that the full range of possible complex behaviors in systems of order three or more ought not to occur for simple ecological interactions. It has been shown by Hirsch [6,7] that the solutions of competitive and cooperative systems have limit sets which cannot be more complicated than invariant sets of systems of one lower dimension. In particular, autonomous 2-dimensional systems of these types have only “trivial” dynamics in the sense that all bounded solutions approach equilibrium asymptotically. In planar systems, for example, the absence of limit cycles makes the dynamics trivial in the sense that bounded solutions can have only critical points or orbits connecting critical points in their omega limit sets [3]. In this note we prove a theorem which appears to be useful in limiting the complexity of limit sets for a class of biologically important equations.