Competing alliances in a four-species cyclic ecosystem

Z. Wang*, A. Bayliss, V. A. Volpert

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We consider a four-species cyclic community with species ui where i=1,…,4. For each i, species ui exhibit a crowding effect, competing with members of its own species (intraspecies competition). Furthermore, there is interspecies competition as ui competes with ui+1 (mod 4). We also allow for species mobility via Fickian diffusion. A feature of this cyclic competition scheme is that there are two alliances of non-competing species, d13, consisting only of species u1 and u3, and d24, consisting only of species u2 and u4. We focus on the parameter regime where these two alliances are the only stable states. We consider the invasion problem, where these two states are initially adjacent, and derive conditions where one state displaces the other. We focus primarily on the standstill problem, where both alliances are evenly matched, dividing the parameter space into two regions - one where d13 displaces d24 and one where d24 displaces d13. We concentrate on three specific parameter regimes: (i) moderate competition - the interspecies and intraspecies competitions are roughly of equal strength, (ii) strong competition - interspecies competition is significantly larger than intraspecies competition, and (iii) slow competitors - the species comprising one alliance are significantly less mobile than the species comprising the other alliance. We employ asymptotic and perturbation methods to determine the outcome of the invasion (winning alliance) and compare our analytic results (generally very positively) with both numerical computations and, when parameters are such that the problem can be reduced to the two-species problem, with previously obtained results.

Original languageEnglish (US)
Article number128396
JournalApplied Mathematics and Computation
StatePublished - Mar 1 2024


  • Asymptotics
  • Cyclic ecosystem
  • Perturbative method
  • Traveling waves
  • Wave speed

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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