Global versus local coupling models and theoretical stability analysis of size-structure dynamics in plant populations

A local coupling model for growth dynamics of plant populations is proposed, in which each individual occupies a square lattice point and follows logistic growth with potential maximum relative growth rate c₀ being reduced by the competitive effects from eight interacting neighbors. The competitive effects are given by the competition function W, which describes the degree of competitive asymmetry between individuals, β and the distance-dependent intensity of competition between individual i and j, α(ij), as a coupling constant. Then a global coupling model corresponding to the local coupling model is proposed, in which every individual interacts with all the other individuals in the field. In the global coupling model, the coupling constant α as the same parameter for all the individuals was employed instead of α(ij)'s in the local coupling model, α was given as an average of α(ij)'s of eight interacting neighbors in the local coupling model. Both the coupling constants were normalized by the number of interacting individuals to make simulation results comparable between the models. Numerical simulations revealed that the local coupling model can be simulated by the corresponding global coupling model fairly well if population growth dynamics are continuous without deaths or new recruits. In both the models, multi-layered structure of size distribution was more likely to emerge under asymmetric and/or intense competition than under symmetric and/or weak competition. This conforms to the widely known phenomenon of size-structure dynamics in plant populations. Since theoretical analysis is impossible for the local coupling model, linear stability analysis of size-structure dynamics was made for the global coupling model. It was theoretically shown that if α < c₀/(1 + β), mono-layered size structure is stable; if α > c₀/(1 + β), multi-layered size structure is stable. As α and/or β increases (decreases), multi-layered (mono-layered) size structure gets stable. As c₀ (i.e. seedlings' relative growth rate) increases, the domain of stable mono-layered (multi-layered) size structure becomes larger (smaller). Therefore, the above simulation results were supported by linear stability analysis of the dynamical systems. Ecological implications of these theoretical results are discussed concerning the relationship between the stability of stand size structure and the degree of competitive asymmetry (multi-layered versus mono-layered).