Kolmogorov-type competition model with finitely supported allocation profiles and its applications to plant competition for sunlight

Abstract

A Kolmogorov-type competition model featuring allocation profiles, gain functions, and cost parameters is examined. For plant species that compete for sunlight according to the canopy partitioning model [R.R. Vance and A.L. Nevai, Plant population growth and competition in a light gradient: a mathematical model of canopy partitioning, J. Theor. Biol. 245 (2007), pp. 210-219] the allocation profiles describe vertical leaf placement, the gain functions represent rates of leaf photosynthesis at different heights, and the cost parameters signify the energetic expense of maintaining tall stems necessary for gaining a competitive advantage in the light gradient. The allocation profiles studied here, being supported on three alternating intervals, determine "interior" and "exterior" species. When the allocation profile of the interior species is a delta function (a big leaf) then either competitive exclusion or coexistence at a single globally attracting equilibrium point occurs. However, if the allocation profile of the interior species is piecewise continuous or a weighted sum of delta functions (multiple big leaves) then multiple coexistence states may also occur.

A Kolmogorov-type competition model with multiple coexistence states and its applications to plant competition for sunlight

“…In particular, if the common gain function has the particular form (1.2), what is the maximum number of equilibrium points for species with two given types of allocation functions? Answers for these types of questions were recently obtained in [9] for finitely supported allocation functions, but we do not know the answers for the type of allocation functions considered here.…”

“…When can we find infinitely many, or even uncountably many equilibrium points? Although this question remains open for allocation functions which are rectangular, or indeed piecewise continuous, we have recently answered it in the affirmative for allocation functions with finite support [9]. Notice that it is not in general clear that our construction in Section 2 can be iterated infinitely often, since the number Q may increase at each step of the construction, and this increase needs to be bounded by the sum of a convergent series if we are to iterate our construction infinitely often.…”