Cycle Chains and the LPA Model

Abstract

The LPA model is a three dimensional system of nonlinear difference equations that has found many applications in population dynamics and ecology. In this paper, we consider a special case of the model (a case that approximates that used in experimental bifurcation and chaos studies) and prove several theorems concerning the existence and stability of periodic cycles, invariant loops, and chaos. Key is the notion of synchronous orbits (i.e. orbits lying on the coordinate planes). The main result concerns the existence of an invariant loop of synchronous orbits that bifurcates, in a nongeneric way, from the trivial equilibrium. The geometry and dynamics of this invariant loop are characterized. Specifically, it is shown that the loop is a cycle chain consisting of synchronous heteroclinic orbits connecting the three temporal phases of a synchronous 3-cycle. We also show that a period doubling route to chaos occurs within the class of synchronous orbits.

A discrete model of avian influenza with seasonal reproduction and transmission

“…The assessment enables us to better understand this disease and to inform policymakers developing effective prevention and control policies to minimize the impacts of the disease. Discrete models have been applied in ecology and epidemiology to illustrate population dynamics and disease transmission [1,3] and periodic functions or periodic coefficients have been used in models to study the impact of seasonality or periodicity [6,16]. In this paper, we have proposed and analysed the discrete model with the seasonality because both poultry's reproduction and avian influenza's transmission are seasonal.…”

“…3 The poultry size infected with avian influenza from first year to 50th year for β = 45.8 in system (19). Other parameters are fixed as Figure 5.…”

Some Discrete Competition Models and the Principle of Competitive Exclusion

“…An interesting open question is whether the phenomena (a) and (b) that arise in the competition LPA model, and in simpler competition models of the form (15), are in contradiction to the principle of competitive exclusion or whether they might, in some way, be reconciled with that principle.…”

“…In this cycle the juvenile and adult classes are temporally separated and, for this reason, we refer to the cycle as a synchronous 2-cycle [15], [21]. This synchronous 2-cycle is GAS within the invariant coordinate axes.…”

On a boom and bust year class cycle

“…After reading [2] we realized that for a special case one can prove rather easily that a heteroclinic boundary cycle exists. The next section is devoted to (an analysis of) that special case.…”