Periodic orbits near heteroclinic cycles in a cyclic replicator system

Abstract

A species is semelparous if every individual reproduces only once in its life and dies immediately after the reproduction. While the reproduction opportunity is unique per year and the individual's period from birth to reproduction is just n years, the individuals that reproduce in the ith year (modulo n) are called the ith year class, i = 1, 2, . . . , n. The dynamics of the n year-class system can be described by a differential equation system of Lotka-Volterra type. For the case n = 4, there is a heteroclinic cycle on the boundary as shown in previous works. In this paper, we focus on the case n = 4 and show the existence, growth and disappearance of periodic orbits near the heteroclinic cycle, which is a part of the conjecture by Diekmann and van Gils (SIAM J Appl Dyn Syst 8:1160-1189, 2009). By analyzing the Poincaré map near the heteroclinic cycle and introducing a metric to measure the size of the periodic orbit, we show that (i) when the average competitive degree among subpopulations (year classes) in the system is weak, there exists an asymptotically stable periodic orbit near the heteroclinic cycle which is repelling; (ii) the periodic orbit grows in size when some competitive degree increases, and converges to the heteroclinic cycle when the average competitive degree tends to be strong; (iii) when the average competitive degree is strong, there is no periodic orbit near the heteroclinic cycle which becomes asymptotically stable. Our results provide explanations why periodic solutions expand and disappear and why all but one subpopulation go extinct.

Asymptotic Poincaré maps along the edges of polytopes

“…We will now present two examples, both replicators, illustrating the procedure detailed in the previous section. The second example belongs to a class of systems studied by Wang et al [24]. By Theorem 7.8 the dynamics of these two systems are chaotic, i.e., their flows contain horse-shoes, in sufficiently large levels.…”

“…This would help to understand the bifurcations taking place in the polytope's interior as the parameters cross the boundary between adjacent regions. For example, higher dimensional cases of the systems studied at [24] could be investigated. In each parametric region, the mentioned tools can be used to detect and characterize some of its invariant dynamical structures such as heteroclinic cycles, periodic points, hyperbolic invariant sets, invariant manifolds and invariant foliations, which are essential to understand the model's dynamics in the polytope's interior.…”

“…The second example is a cyclic replicator system that models a semelparous species (each individual reproduces once in its life time and dies thereafter). These models were studied by Diekmann and van Gils [9] and by Wang et al [31] in lower dimensions.…”

Complex dynamic of the system of nonlinear difference equations in the Hilbert space

“…It is very important in mathematical biology as a model which describes dynamics of populations. There exist many papers which are dedicated to investigation of such problems in continuous and discrete cases (see for example recently works [12] - [50]). As a rule such problems are regular.…”