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

Abstract

It is demonstrated that a Kolmogorov-type competition model featuring species allocation and gain functions can possess multiple coexistence states. Two examples are constructed: one in which the two competing species possess rectangular allocation functions but distinct gain functions, and the other in which one species has a rectangular allocation function, the second species has a bi-rectangular allocation function, and the two species share a common gain function. In both examples, it is shown that the species nullclines may intersect multiple times within the interior of the first quadrant, thus creating both locally stable and unstable equilibrium points. These results have important applications in the study of plant competition for sunlight, in which the allocation functions describe the vertical placement of leaves for two competing species, and the gain functions represent rates of photosynthesis performed by leaves at different heights when shaded by overlying leaves belonging to either species.

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

“…It can therefore capture in real time subtle (and usually nonlinear) changes in leaf photosynthesis rates at different heights, which result when plant populations change in leaf area over time. Although initial applications of the model focused on species possessing VLPs that are constant functions over a height interval (rectangular profiles) or a pair of disjoint height intervals (bi-rectangular profiles) [19,21,22], it can equally well accommodate species with big-leaves positioned at multiple discrete heights in a canopy. The purpose of this study is to examine this latter situation.…”

“…In [19] it was shown that there exist pairs of species with rectangular VLPs and distinct gain functions for which any finite number of stable and unstable equilibrium points are possible. It was also shown that at least two equilibrium points are possible even if the species share the same gain function, provided that one of their VLPs is bi-rectangular and surrounds the other.…”

“…Toward this end, it is known that (i) if two species have allocation profiles which are supported on non-overlapping intervals then either competitive exclusion or coexistence at a single globally attracting equilibrium point occurs [21]; (ii) if two species have rectangular (i.e. uniform) allocation profiles supported on overlapping but non-nested intervals, and the species share a common gain function, then either competitive exclusion or coexistence at a single globally attracting equilibrium point occurs [21]; (iii) there exist pairs of species possessing rectangular allocation profiles and distinct gain functions such that multiple non-trivial equilibrium points occur [19]; (iv) there exist pairs of species, one with a rectangular allocation profile surrounded by a birectangular allocation profile belonging to the other, such that multiple non-trivial equilibrium points occur, even when the two species share the same gain function [19]; (v) if two species possess rectangular allocation profiles, they share a common gain function, and there exists some τ > 0 such that s 2 (z) = s 1 (z − τ ) and C 2 = C 1 + bτ , then the competitive ability of species 2 diminishes from dominant to compatible to inferior as the allocation difference τ increases [22].…”