Ray Redheffer

Solution of the stability problem for a class of generalized volterra prey-predator systems☆

where ei,pij, and qj are real constants, ui are real-valued continuous functions of t for t > 0, and zii = dui/dt. Both equations are required for i = 1, 2,..., m. Condition (lb) is here considered to be a restriction on the vector e = (ei) rather than on the matrix p = (pii) as is more usual. The hypothesis ~~(0) > 0, qi > 0 is appropriate in applications but as far as the mathematical development is concerned, it could be replaced by qiui(0) > 0. A discussion of historical background can be found in the papers [4,5], to which the present paper is a sequel. Suffice it to say here that the chief landmarks are: Volterra’s book [6], La Salle’s theorem [ 1, 21, and Krikorian’s analysis of the case n = 3 [3]. References [4,5] provide useful

Nonautonomous SEIRS and Thron models for epidemiology and cell biology

Abstract Remarks on positivity of certain vector-valued functions are applied to the SEIRS equations for infectious disease and to a nonlinear system recently introduced in connection with cell biology by Dr. Dennis Thron. In both cases the formerly constant coefficients are replaced by functions of time, and in both cases we give conditions under which the solutions exist globally and are positive. For the nonautonomous SEIRS equations conditions are also given which ensure that the limiting population as t →∞ contains susceptible individuals only, hence is disease-free.

A comparison theorem for difference inequalities

In the study of growth and decay properties of nonlinear evolution equations (in particular hyperbolic and parabolic partial differential equations) as t → ∞, the following difference inequality (which is satisfied by an energy expression) arises:

A theorem of La Salle-Lyapunov type for parabolic systems

(*)\mathop {\sup }\limits_{t \leq s \leq t + 1} u{(s)^{1 + \alpha }} \leq c{(1 + t)^r}(u(t) - u(t + 1)) + g(t)

Mean values and the nonautonomous May--Leonard equations

(*) .

Quenching in Time-Delay Systems: A Summary and a Counterexample

Throughout this paper X is a normed linear space over the real field and the norm of an element x e X is Ixl. Extension to the case of complex scalars involves the use of Re(x, y) instead of (x, y), and the use of Re cy instead of cy in the definition of (x, y). The set of n onnegative reals is R +, and u = u(t) is a function R ÷ ~ X. Differential inequalities are given for t > 0, and the functions in them are continuous for t >-0. Real numbers or real valued functions are generally denoted by Greek letters, as 0t, 5, e, ~ (an exception is t). We set

The spectral radius and Liapunov's theorem

This paper deals with the boundary value problem for a nonlinear system of parabolic differential equations for

Positive linear functionals and the order cone

u = u(t,x)