Richard H. Elderkin

On the Steady State of an Age-Dependent Model for Malaria

Publisher Summary This chapter discusses some basic analysis of the steady state equations of an age-dependent model of malaria, suggested by Klaus Dietz in cooperation with others at the World Health Organization. The model considers population density of the sporozoite (asexual) malarial parasite in a host; the death rate of the sporozoites; and the population density of the gametocyte (sexual) malarial parasite in a host. The sporozoite is involved directly in the effect of the disease on the host but not in the transfer of disease, while the gametocyte has little symptomatic effect but is the key to the spread of the disease via the mosquito. The chapter describes different assumptions that are incorporated in the model.

Predator-prey Interactions with Delays Due to Juvenile Maturation

This paper focuses on predator-prey models with juvenile/mature class structure for each of the predator and prey populations in turn, further classified by whether juvenile or mature individuals are active with respect to the predation process. These models include quite general prey recruitment at every stage of analysis, with mass action predation, linear predator mortality as well as delays in the dynamics due to maturation. As a base for comparison we briefly establish that the similar model without delays cannot support sustained oscillation, but it does have predator extinction or global approach to predator-prey coexistence depending on whether the ratio

Seed dispersal in a patchy environment with global age dependence

\alpha

Solitary Invariant Sets

of per predator predation at prey carrying capacity to the predator death rate is less than or greater than one. Our first model shows the effect of introducing an invulnerable juvenile prey class, appropriate, e.g., for some host-parasite interactions. In contrast our second model shows the effect of limiting predation to a prey juvenil...

Contour interpolation of random data

A model of seed population dynamics proposed by S. A. Levin, A. Hastings, and D. Cohen is presented and analyzed. With the environment considered as a mosaic of patches, patch age is used along with time as an independent variable. Local dynamics depend not only on the local state, but also on the global environment via dispersal modelled by an integral over all patch ages. Basic technical properties of the time varying solutions are examined; necessary and sufficient conditions for nontrivial steady states are given; and general sufficient conditions for global asymptotic stability of these steady states are established. Primary tools of analysis include a hybrid Picard iteration, fixed point methods, monotonicity of solution structure, and upper and lower solutions for differential equations.

Population models with globally age-dependent dynamics: on computing the steady state

Publisher Summary This chapter elaborates the solitary invariant sets. Any isolated invariant set is solitary and any isolating neighborhood is a neighborhood of solitude. In this case, the elliptic set is always empty. It is not known what the simplest structure for a neighborhood of solitude is in general; however, certain simplifications, which are always possible, have been discovered. In the study of isolated invariant sets, it is desirable to limit ones attention to certain specially structured isolating neighborhoods, the isolating blocks. The first simplification of the structure of a neighborhood of solitude involves reducing the set along which δR is tangent to the flow. For this, the notion of generic contact, which was introduced by Pugh in his study of the Poincare index formula, is used. The chapter reviews the example of a Cr isolating block. Here, the tangency set is either empty or a submanifold that separates the ingress region from the egress region in δR. At each point of the tangency set, the orbit is directed transverse to the tangency set and into the egress region. In general, it is not possible to choose the isolating block so that the tangency set is empty. For general neighborhoods of solitude, it is not only necessary to have a nonempty tangency set but also may even be necessary that some orbits be tangent to this tangency set. Generic contact means that such tangencies of all orders have been minimized locally.

Separatrices and solitary periodic solutions

Abstract This paper addresses the problem of predicting the region of safe passage for a ship given widely spaced and random depth soundings. The problem is recognized as one of interpolation and surface generation. Shepards method of inverse distance weighting (IDW) is presented as an appropriate modelling technique. The history and workings of this technique are described. The parameters of IDW are discussed, and their influence on the outcome is explained. This is followed by a discription of the computer programs developed to implement this technique and apply it to the data provided. The resultant safe region is presented in graphical form. A possible testing scheme is presented, followed by a discussion of possible predictors of error. An evaluation system designed to compare one interpolation technique to another is also presented. Finally the advantages and disadvantages of IDW are discussed.

Separatrix Structure for Regions Attracted to Solitary Periodic Solutions

Abstract We consider the problem of computing the steady state for a class of differential equations where either the dynamic, (∂/∂ a + ∂/∂ t ) u ( a, t ) or the boundary state u (0, t ) depends on the global state u (·, t ). A fixed point technique for establishing the existence of steady state solutions can be adapted naturally for use as an iterative method of computing these solutions. In some cases of current interest, the method is efficient and practical. In some other cases, where better methods are available, the conditions which are sufficient to guarantee convergence of our method have interesting interpretation which provides insight into the nature of stability in the differential equation.

Nonlinear, globally age-dependent population models: Some basic theory

A separatrix trajectory of a general solution to an ordinary differential equation is one which differs topologically from near by trajectories. Maximal regions of parallel flow are separated by a union of separatrices. The structure of these solutions has been a useful tool in the qualitative theory, especially in the plane (see, for example, [‘2, 6, 71). We will consider the separatrix structure of a flow near a solitary periodic solution in 3-space (cf. [Sj). A periodic orbit y is solitary if it has a compact neighborhood (neighborhood of solitude) G such that any negative (respectively, positive) semitrajectory contained in i? has its 4imit (resp., w-limit) at y. Within a neighborhood of solitude, trajectories are distinguished by their eventual behavior in time. For those sets of trajectories which are elhptic, that is, are contained in L’ and hence have CLand w-limit at y, an analysis of separatrix structure in a slightly different situation has already been set forth [a]. Our interest here will be primarily in the set &I+ of positively attracted trajectories, i.e.,. those which have w-limit at y, but leave U in the negative time direction. In contrast with the situation in two-dimensional settings, where each separatrix trajectory is thought of as separating two canonical regions, our analysis of separatrix structure must be concerned with connected components of the union of all separatrices. Thus, whereas a study of a planar flow is concerned with the geometry of individual trajectories, we must concern ourselves with the geometry of “surfaces” of separatrices. In general, these surfaces may be quite different from manifolds, even for C” flows. Our approach here is to restrict attention to those flows whose separatrix sets satisfy some kind of “manifold hypothesis.” We will also demand that no positive semitrajectory with initial state in the boundary of A !be internally tangent to the boundary of the neighborhood of solitude. Under a strict version of these hypotheses, a classification of regions of 9, is given by boundary type.

HARVESTING PROCEDURES WITH MANAGEMENT POLICY IN ITERATIVE DENSITY-DEPENDENT POPULATION MODELS

This chapter discusses the separatrix structure for regions attracted to solitary periodic solutions. Under strict hypotheses, regions attracted to the periodic solution are identified by boundary type. It is found that of four possible types, three have no internal separatrices. The chapter reviews the separatrix structure in the remaining type. It also presents a weaker hypothesis for comparison and further presents examples to point out differences between the weaker and strict hypotheses. Each circle separates a region of egress from a region of ingress of ƒ relative to U , with the component of ƒ normal to δU having nonzero derivative in a direction in δU transverse to τ , everywhere on τ . The subset of τ where ƒ is tangent to τ is either empty or a finite set of points collectively denoted χ . Each point of χ separates an open subarc of τ where ƒ points toward a region of ingress from an open subarc of τ where ƒ points toward a region of egress. The component of ƒ tangent to τ has nonzero derivative in the direction of τ, everywhere on χ .