Richard C. MacCamy

Non-linear age-dependent population dynamics

where P( t ) is the total population at time t and 5 is the growth modulus. This law is clearly inapplicable to situations in which the population competes for resources (e.g., space and food), for in these situations 5 should depend on the size of the population: the larger the population, the slower should be its rate of growth. To overcome this deficiency in the Malthusian law, VERHULST [1845, 1847] assumed that f i= (6o-co0 P ) P (5o, COo=constant). (1.2)

On the diffusion of biological populations

Abstract This paper develops a model for the spatial diffusion of biological populations. Arguments are given in support of a degenerate, non-linear partial differential equation for the population density. Because of this degeneracy, a population which is initially confined to a bounded region spreads out at a finite speed, and may even remain confined for all time. A transformation is given which reduces our equation to an equation which arises in the theory of porous media. Using this transformation we are able to carry over to our theory theorems of existence and uniqueness for the one-dimensional initial-value problem as well as the solution for an initial point source.

Solution of Boundary Value Problems by Integral Equations of the First Kind

This paper discusses an integral equation procedure for the solution of boundary value problems. The method derives from work of Fichera and differs from the more usual one by the use of integral equations of the first kind. The method here extends to equations of higher order than second. Its connection with singular perturbation theory and thin-body theory are indicated by examples. Some numerical experiments are included to indicate how the method operated in specific situations.

Some simple models for nonlinear age-dependent population dynamics

Abstract This paper presents two simple models for nonlinear age-dependent population dynamics. In these models the basic equations of the theory reduce to systems of ordinary differential equations. We discuss certain qualitative aspects of these systems; in particular, we show that for many cases of interest periodic solutions are not possible.

Diffusion models for age-structured populations☆

Abstract This paper discusses some models for age-dependent populations with diffusion in one-dimensional environments. Three simple birth-death mechanisms are used. There are two diffusion processes: one represents random movement and the other movement to avoid crowding. The resulting models are reduced to systems of partial differential equations. These are sufficiently simple that an analysis of the interaction of diffusion and the birth-death processes is possible. An analysis is presented of the large-time behavior of populations in limited environments.

A population model with nonlinear diffusion

Abstract A model is presented for a single species population moving in a limited one-dimensional environment. The birth-death process is specialized by assuming a constant death modulus and a birth modulus which is an exponential in the age. The diffusion mechanism is nonlinear and results in a problem for the space population density which has a degenerate parabolic form and is similarly to the porous media equation. It is shown that the effect of the nonlinearity in the diffusion is to produce an approach to steady state even when the process is birth dominant. The interaction of the birth-death and diffusion processes is studied and is shown to yield a modified birth-death mechanism which is both time and space dependent.

Solution procedures for three-dimensional eddy current problems

Abstract The problem under consideration is that of the scattering of time periodic electromagnetic fields by metallic obstacles. A common approximation here is that in which the metal is assumed to have infinite conductivity. The resulting problem, called the perfect conductor problem, involves solving Maxwells equations in the region exterior to the obstacle with the tangential component of the electric field zero on the obstacle surface. In the interface problem different sets of Maxwell equations must be solved in the obstacle and outside while the tangential components of both electric and magnetic fields are continuous across the obstacle surface. Solution procedures for this problem are given. There is an exact integral equation procedure for the interface problem and an asymptotic procedure for large conductivity. Both are based on a new integral equation procedure for the perfect conductor problem. The asymptotic procedure gives an approximate solution by solving a sequence of problems analogous to the one for perfect conductors.

On absorbing boundary conditions for wave propagation

Abstract This paper is concerned with the development of methods for constructing stable artificial boundary conditions for wavelike equations in a general and automatic way. The one-dimensional problem of a semi-infinite, inhomogeneous, elastic bar is studied here as a prototype situation. For this problem a family of efficient artificial boundary conditions is obtained using geometrical optics in the Laplace transform domain for generating outgoing solutions, together with a stability criterion based on energy integrals to insure that the resulting artificial boundaries are dissipative. Numerical examples illustrate the efficacy of this approach. The paper also includes some remarks about the extension of the proposed method to a more general two-dimensional situation.

Product solutions and asymptotic behavior for age-dependent, dispersing populations

Abstract The paper presents some population models, containing both age structure and diffusion, for which there are stable age distributions. This means distributions which are products of functions of age only with functions of space and time only. For these models a formal calculation is presented showing that the stable distributions will always be the large-time limits. Two kinds of diffusion are studied: random, and directed (which means diffusion to avoid crowding). The principal specialization of the age structure is that the death modulus can be decomposed into the sum of two terms. The first is a function of age only and represents deaths by natural causes, while the second is a function of total population only and represents environmental effects.

Nonlinear Volterra equations on a Hilbert space

Abstract We consider equations of the form, u(t) = − ∝0t A(t − τ)g(u(τ)) dτ + ϑ(t) (I) on a Hubert space H . A(t) is a family of bounded, linear operators on H while g is a transformation on D g ⊂ H which can be nonlinear and unbounded. We give conditions on A and g which yield stability and asymptotic stability of solutions of (I). It is shown, in particular, that linear combinations with positive coefficients of the operators eMt and −eMtsin Mt where M is a bounded, negative self-adjoint operator on H satisfy these conditions. This is shown to yield stability results for differential equations of the form, Q ( d dt ) u = − P ( d dt ) g(u(t)) + χ(t) , on H .