Obligate Mutualism in a Resource-Based Framework

Abstract

Obligate mutualist interactions appear to be ubiquitous in nature but cannot be described by the simple models that have been so effective for developing the theory of other population interactions including competition, predation, mixotrophy, and facultative mutualism. We present a teaching framework that extends the standard Lotka-Volterra analysis of these interactions to the more complicated obligate mutualism. This provides a useful addition to applications of dynamical systems theory for mathematics students and an advanced course in population dynamics for ecology students. The theoretical framework used in this work is based on explicitly accounted resources and per capita rates of change for populations that are negative when they have no resources and positive when they have maximal resources. We extend the Lotka-Volterra models by including terms that capture the "catalytic" effect of obligation, reflecting that while one population may be necessary for survival of another, the obligated population does not necessarily consume it. A key attribute of our consumer-resource approach is that the catalytic services provided by obligate mutualists are treated as pseudoresources for the purposes of determining these rates of change. This framework allows all major ecosystem population interactions to be modeled within a single, simple consumer-resource framework, and it reveals how populations can smoothly change their trophic status through a continuum of stable coexistence states.

Carrying capacity – A capricious construct

“…, is equivalent to the classic fail diagram of Lotka-Volterra models of obligate mutualism (Figure 1(a)). We emphasise this similarity in Figure 4 The reasons for this are explained in detail in [30].…”

“…Our results show that it essentially the abstraction of a resource as a carrying capacity, that constrains a three-dimensional problem to two dimensions, that is the cause of the problem. In previous work, we showed that particular mutualist processes such as plant-pollinator interactions and mutualist autotrophs can be successfully modelled in this framework [30] . Loreau [31] suggests that Lotka-Volterra models are unable to produce useful answers for obligate mutualism because they ignore fundamental mass-balance constraints.…”

The eco-evolutionary modelling of populations and their traits using a measure of trait differentiation

“…Evolutionary bounds may stem from physical considerations such as maximum or minimum body size, as Kempes et al [25] suggest, environmental tolerances, or population traits that, for example, may limit mutualism benefits as Cropp and Norbury [11,13] have shown theoretically. The use of a bounded trait distribution such as a beta distribution has the advantage of allowing phenotypes to explore the bounds of a trait, and for eco-evolutionary equilibrium points at the bounds to potentially provide useful information on the eco-evolutionary system dynamics.…”