Periodic orbits near heteroclinic cycles in a cyclic replicator system

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.