John M. Bownds

On the behavior of solutions of predator-prey equations with hereditary terms

Abstract Some global properties of solutions of the classical integrodifferential systems, introduced by Volterra in his study of two species predator-prey populations, are studied. It is shown for large logistic loads that the predator goes to extinction and the prey tends to its carrying capacity. By use of a nonlinear approximation it is shown that for smaller logistic loads a “critical point” is asymptotically stable, while for sufficiently small logistic loads this point is unstable. These cases are demonstrated numerically for the original integrodifferential system using parameters which were computed on the basis of experimental data of S. Utida for bean-weevil vs. braconid-wasp interactions. Moreover, numerical solutions suggest further varied behavior of solutions of this system.

On Numerically Solving Nonlinear Volterra Integral Equations with Fewer Computations

A method is described whereby certain nonlinear Volterra integral equations may be numerically solved by computing the solution of a system of ordinary differential equations. In case the kernel is not finitely decomposable, certain two-dimensional approximating techniques are employed. In such a case, there may be a trade-off between computational effort and accuracy. Several explicit error estimates are given, and numerical examples illustrate the applicability of the method as it compares with other methods.

Some stability theorems for systems of volterra integral equations

Many types of stability have been studied in the theory of ordinary differential equations, Our present purpose is to study perturbed Volterra integral equations (which are a generalization of the initial value problem for ordinary differential equations). As a natural generalization of these concepts for ordinary differential equations, we define stability, uniform stability, and asymptotic stability for integral equations and prove various theorems for linear and perturbed integral equations

Theory and performance of a subroutine for solving Volterra Integral Equations

AbstractThe author considers Volterra Integral Equations of either of the two forms

A representation formula for linear Volterra integral equations

u(x) = f(x) + \int\limits_a^x {k(x - t)g(u(t))dt, a \leqslant } x \leqslant b,

A note on solving Volterra integral equations with convolution kernels

wheref, k, andg are continuous andg satisfies a local Lipschitz condition, or

Uniqueness for algebraically regular solutions to pathological differential equations

u(x) = f(x) + \int\limits_a^x {\sum\limits_{j = 1}^m {c_j (x)g_j (t,u(t))dt} ,}