2003/07/01 by J. M. Cushing, J.M. Cushing · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Medicine · #Evolution and Genetic Dynamics #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Dynamics and Pattern Formation
paper · doi:10.1080/1023619021000042216
openalex publication_date 2003/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
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.