Resource-Based Models of Mutualism

Abstract

Mutualist interactions are thought to be ubiquitous, spanning all levels of biological organisation, and involving most species on Earth. However, in contrast to population interactions such as competition and predation, a comprehensive and succinct theoretical explanation of mutualism has proved elusive. We use a new modelling framework that represents obligation, mutualist benefits and mutualist costs in an extended consumer resource approach to develop simple, consistent models of mutualism. We show how populations may stably transition between facultative and obligate mutualism and demonstrate that our solutions do not depend on saturating functions. We show facultative and obligate mutualisms between autotrophs and heterotrophs.

Appendix

1.1 Conservative Normal (CN) Systems

The CN framework captures fundamental ecological properties of food web-based living systems with mathematical rules. These rules formalise basic ecological concepts, principally that all organisms have to consume resources to survive, and that these resources are finite. The constancy of the total mass of a limiting nutrient is fundamental to modelling ecosystems [31] and to the CN framework. Rather than debate the merits of this view, although it is recognised that ecosystems generally recycle over 90% of their limiting nutrient [34], we draw analogy with the common assumption of linearity. Although it may be argued that no real ecological system has exactly linear processes, the simplifying assumption of linearity in some ecosystem models has led to substantial advances in theoretical ecology, as it has in other fields.

Detailed descriptions of the CN framework have been published elsewhere [9, 10], so, here we just present the essence of the framework. We preface this brief summary with the observation that obligate mutualism is not a ‘normal’ interaction, where we use ‘normal’ in the context of the CN framework. While the vast majority of published applied ecological models satisfy the CN framework, and the CN rules can be used to winnow out models that do not make ecological sense, there are ecologically realistic models that do not simply satisfy all the CN rules. We label these exceptions ‘exotic’ systems as they are sensible, but require modification of what constitutes a resource to include ‘catalysts’. There are a number of such exotic systems: populations where too much resource ‘poisons’ the environment, populations that co-operatively hunt, for which there is an ‘optimum’ population size; omnivorous predators that benefit from consuming certain resources that they cannot alone survive on; and as we shall explain below, obligate mutualists.

We consider a general n population system of the form:

$$ {\dot{x}}_i={x}_i{f}_i\left({x}_1,{x}_2,\cdots, {x}_n\right),i=1,2,\cdots, n. $$

(A1)

The ‘life functions’ f_i may include any form (such as Holling Type I, II or III) for the functions that represent the interactions between populations, and include parameters that define the magnitude of the interactions.

1.2 CN Rule 0: Measuring the System

We assume each interacting population is sufficiently large in number that we can ignore the typical individual and instead define a measure of the population mass in the isolated physical volume that the ecosystem occupies. At time zero (t = 0), we measure the amount of the limiting nutrient in each living population ${\widehat{x}}_i$ present in the ecosystem, together with the amount of nonliving nutrient (i.e. dead organic matter such as detritus and dissolved organic matter and re-mineralised inorganic nutrient) $\widehat{N}$ available to those n interacting populations:

$$ {\widehat{x}}_1+{\widehat{x}}_2+\dots +{\widehat{x}}_n+\widehat{N}={\widehat{N}}_T. $$

(A2)

We then scale the measurements ${\widehat{x}}_i,\widehat{N}$ by the total measure of nutrient pool ${\widehat{N}}_T$ that is cycling in the system, so that the scaled measurements x_i, N are fractions of the total recycling nutrient in the system:

$$ {x}_1(0)+{x}_2(0)+\dots +{x}_n(0)+N(0)=1, $$

(A3)

with 0 < x_i(0), N(0) < 1, each living population x_i(t) is now measured in terms of the fraction of the total amount of cycling nutrient (in all forms) that is bound into the living tissues of the individuals of that population. The nonliving nutrient fraction N(t) represents all other forms of nutrient, that is, all forms of inorganic nutrient and organic nutrient in various stages of remineralization. For convenience, we refer to this as just inorganic nutrient as we only make the distinction here between autotrophs and heterotrophs.

1.3 CN Rule 1: Describing Changes in Living Populations

The CN framework requires that the per capita population growth rates are independent of the way in which we measure the living populations, and satisfy:

$$ \frac{1}{x_i}\frac{d{x}_i}{dt}={\widehat{f}}_i\left({x}_1,{x}_2,\cdots, {x}_n;N\right). $$

(A4)

The life functions ${\widehat{f}}_i$ describe how each population grows (or dies) dependent on interactions with the other populations in the system and with inorganic nutrient. Recall that the ${\widehat{f}}_i$ implicitly include parameters that quantify the rates of environmental interactions.

1.4 CN Rule 2: Constancy of Total Nutrient Mass

We make the assumption that there is no population migration or nutrient flow in to or out of the model domain and require that the total mass of recycling nutrient in the model domain remains constant for all time (i.e. ${\widehat{N}}_T$ is constant). The living population fractions x_i(t) and the nonliving nutrient fraction N(t) then satisfy a constancy of total nutrient mass constraint for all time t > 0:

$$ {x}_1(t)+{x}_2(t)+\dots +{x}_n(t)+N(t)=1. $$

(A5)

This fundamental constraint allows us to eliminate N(t) from the living population equations ${\widehat{f}}_i\left({x}_1,\cdots, {x}_n:1-\sum {x}_i\right)\equiv {f}_i\left({x}_1,\cdots, {x}_n\right)$ so that (A4) may be written in the form (A1), for x_i(t) > 0. The elimination of an explicit differential equation for the nonliving nutrient N(t) means that all the remaining differential equations in the model have the same (Kolmogorov) form (A1), which simplifies the analysis of CN models.

Equation (A5) allows us to define a lid {x₁ + x₂ + … + x_n = 1} (i.e. N = 0), on the model’s state space x_i > 0 for all i. The lid completes the closure of the state space in which reasonable model solutions exist, and defines the ecospace E:

$$ E\equiv \left\{0<{x}_i,0<{x}_1+{x}_2+\dots +{x}_n<1\right\}. $$

(A6)

1.5 CN Rule 3: Normal Ecosystems

All living populations x_j require food to survive and grow. This food may be inorganic nutrient in the cases of autotrophs, or prey (i.e. other living organisms) in the cases of heterotrophs. These resources R_j are finite and limit the growth of population x_j when they become depleted. We define two basic criteria that a living population (measured by x_j) must comply with:

• when its resources are maximal (R_j = 1, a feast), the populationx_j must be able to grow; and

• when there is no resource available (R_j = 0, a famine), the population x_j must die.

This means that each life function f_j must satisfy the natural resource constraints:

$$ {f}_j{\left|{}_{R_j=1}>0>{f}_j\right|}_{R_j=0}. $$

(A7)

Evaluation of this rule on the boundaries of the ecospace may place constraints on allowable parameter values. The CN framework places further constraints on the life functions, that in a normal ecology the f_j change monotonically along resource rays within E. As we follow a life function from any point of minimum resource on the boundary of E to any point of maximum resource along such a ray, where the direction of the resource ray is given by the unit vector defined by the usual direction cosines (γ_kis the angle between this direction vector and the kth co-ordinate axis (cosγ₁, cosγ₂, ⋯, cosγ_n))the life function f_j must monotonically increase:

$$ \cos {\gamma}_1\frac{\partial {f}_j}{\partial {x}_1}+\cos {\gamma}_2\frac{\partial {f}_j}{\partial {x}_2}+\dots +\cos {\gamma}_n\frac{\partial {f}_j}{\partial {x}_n}>0. $$

(A8)

Obligate mutualist populations require the presence of another population in addition to their resource(s) in order to grow but the interaction is ‘catalytic’ in that there is (usually) no direct exchange of mass between the populations. Obligate mutualist systems are therefore not ‘normal’ CN systems, similarly to populations that have optimal population sizes (for example cooperative hunters) or omnivores that benefit from multiple resources but consume some resources that they cannot survive on independently. We address such ‘exotic’ obligate systems after completing the description of the CN framework with the Consistency Condition below.

1.6 A Consistency Condition

When resources are explicitly represented using local mass balance (as in a predator consuming prey) then we easily find CN systems, but if population benefits are more implicitly defined as in mutualism models, then we need to check that resource is available for such benefits. The mass consistency condition provides such a check. The consistency condition guarantees that a CN model cannot predict a negative mass, that is, that solutions.

{x₁(t), x₂(t), ⋯, x_n(t)} of Eq. (A1) which start in E do not leave E through the ‘lid’ of the ecospace defined by x₁ + x₂ + … + x_n = 1. This ensures that N(t) remains positive, and consequently that mass conservation is always physically sensible. Differentiating Eq. (A5) and using (A1) provides the lid consistency condition:

$$ \frac{\mathrm{dN}}{\mathrm{dt}}=-\frac{{\mathrm{dx}}_1}{\mathrm{dt}}-\frac{{\mathrm{dx}}_2}{\mathrm{dt}}\dots -\frac{{\mathrm{dx}}_n}{\mathrm{dt}}=-\sum \limits_{i=1}^n{x}_i{f}_i>0\kern0.36em \mathrm{when}\kern0.49em N=0, $$

(A9)

ensuring N ≥ 0 for all time. Eq. (A9) is an important constraint that should be checked for these systems that include mutualist interactions. It provides generic conditions on the parameters that constrain the magnitude of mutualist benefits that could be obtained without violating the basic principle that mass cannot be ‘borrowed’ from outside the system.

1.7 Obligate Mutualism as an Exotic CN System

The CN framework summarised above is a consumer–resource conceptualisation predicated on explicit accounting of the mass of limiting nutrient. It explicitly accounts for transfers of mass between populations due to resource-limited growth (via inorganic nutrient uptake or predation) and losses due to mortality and predation and includes implicit recycling of limiting nutrient and constancy of total nutrient mass. The CN framework provides simple and intuitive conditions for many ecosystems but requires some modification or extension when applied to ‘exotic’ systems.

The obligation of one population on another, that is where a second population provides a ‘catalytic’ service such that the second population has to be present in order that the first population can grow, is not a normal CN system process as it does not involve a direct explicit transfer of mass. CN Rule 3 requires that a population must be able to grow at any point in the ecospace where a resource is maximal; however, when applying this rule to systems that include the provision of services, we need to treat the service as if it were a resource. This treatment is consistent with the CN framework as it is a zero sum game—the presence of a catalysing population in a closed ecosystem must reduce the quantum of resources available to the other populations.

Consider the simplest example in which an obligate mutualist population x_i has a single resource R_i = x_h and is obligated on the presence of another population x_j (which may be an inorganic resource x₀) to be able to grow. The maximal point of the single resource for the population lies at a vertex of E, where x_h = 1 and all other populations, including the population supplying the service to the obligated population, are zero. The sign constraint of Eq. (A7) cannot be satisfied at such a point. However, if for the purposes of Rule 3, we treat the servicing population as a resource instead of a catalyst, then R_i = x_h + x_j and the point of maximum resource ${R}_i^{\ast }$ lies somewhere on the line of maximum ‘resource’ x_h + x_j = 1. (Note the location of ${R}_i^{\ast }$ depends on the specific nature of the obligate interaction.) CN Rule 3 then stipulates that ${\left.{f}_i\right|}_{R_1^{\ast }}>0$ similarly to any population with multiple resources. This generalisation of what is regarded as a resource captures the effect of the abundance of the obliging population on the utilisation of resources by the obligate population, a key assumption of the Dean [13] and Graves et al. [14] models.

1.8 Obligate Mutualism in Linear CN Systems

It appears that it is not possible to represent obligate mutualism in linear CN systems, in which we include systems with planar zero isosurfaces such as the Lotka-Volterra CN (LVCN) models discussed in Cropp and Norbury [10]. The difficulty in representing obligate mutualism in linear CN systems arises from the requirement for obligation on another population. This eliminates obligate autotroph populations, as autotroph populations in linear CN systems always have a boundary equilibrium point at which they can exist in the absence of any other population. Obligate populations in linear CN systems must then be heterotrophs—in the simplest scenario, these are herbivores that graze on autotrophs, as at least one autotroph must always exist in any CN system. If we ignore the trivial case of obligation, that of a predator on its prey, then on any face of the solution space of the system (the ecospace 1 E) in order that a heterotroph does not exist in the absence of its mutualist benefactor, the zero isoclines on that face cannot intersect. In a heterotroph-autotroph interaction, this means that the autotroph only boundary equilibrium point is stable, as it must lie in the part of the solution space where the heterotroph is not growing, that is, the heterotroph must go extinct. Thus, it appears that it is not possible to represent obligate-obligate mutualism by judiciously arranging planar zero isosurfaces. Here, we use a nonlinear population interaction to build simple heuristic models of obligate mutualism.