Methods of measuring and describing competition

The general principles outlined in the previous section will be translated into explicit formulas and illustrated for some simple models in this section. I begin by giving for­mulas for very small press perturbations in a neutral parameter.

and

These will be referred to as measures 1 and 2 below. The negative sign in expression (3.1) is present because increased mortality of species j has a negative effect on its

Measuring competition: a consumer-resource framework • 37 own population growth, and interaction signs are usually defined by the impact of positive effects on (or addition of) the initiator. Both measures above represent the response of population size of species i to a very small press perturbation (Bender et al. 1984) in the per capita mortality of species j. The standardization involved in expression (3.2) makes it equivalent to the competition coefficient in the LV mod­el given in Chapter 1. If increased immigration (rather than mortality) is used as the perturbation, the sign of expression (3.1) is reversed, as this represents a positive per­turbation to species j.

For characterizing the strength of interspecific competition, the second measure above seems preferable to the first in at least two respects. Firstly, as noted above, it is independent of the neutral parameter used, so it produces the same measure if immigration (or another neutral consumer parameter), rather than per capita mortal­ity were used. That is not true of measure (3.1). Secondly, measure (3.2) is consistent with the common method of measuring competition in experimental studies (i.e., measuring the change in a recipient population for a given change in the initiator’s population). In a two-competitor system, measure 2 corresponds to the competi­tion coefficient α_ij in the traditional parameterization of the Lotka-Volterra model in which intraspecific effects of a consumer on its own per capita growth rate have been scaled to one (as in Chapter 1). Thus, measure 2 is a ratio of inter- to intraspecific effects.

On the other hand, because it is a ratio, measure 2 does not always identify cases where the change in the receiver species’ population (measure 1) is large or small in magnitude.

a larger number of other theoretical examples. This possibility was not considered in Schoeners earlier (1993) review of indirect effects, where he stated (p. 366) that, ‘In the most precise existing theoretical formulation, an effect is defined as a change in equilibrium population size caused by a change in the input or abundance of anoth­er species.' The italics have been added because of the problematic fact that increased input may cause decreased equilibrium abundance of the perturbed species. As noted above, hydra effects (where increased input decreases abundance) occur in a signifi­cant fraction of simple models of small groups of interacting species, and such effects can be large in magnitude (Abrams 2002, 2009a; Abrams and Matsuda 2005; Abrams and Cortez 2015a, Cortez and Abrams 2016). In systems with shared resources, hydra effects will often produce positive effects according to measure 2. Very small effects of the parameter on the abundance of the perturbed species can imply that measure 2 is large in magnitude even when measure 1 is relatively small.

Neither expression (3.1) nor (3.2), which are both defined for a particular equi­librium point, provides a full description of an interaction because both change depending on the initial mortality rates. In theoretical studies, this is not a problem, because the values can be determined for any set of mortality rates. Unfortunately, the dependence on the mortality rate makes it difficult to impossible to characterize the ‘strength' of competition by a single number. As I argue below (see also Abrams 2001c), a full quantitative description of the consumer-resource dynamics is essential for understanding the relative change in abundances produced by any given pertur­bation to one consumer. The dependence of sign and magnitude of relative effects on both initial population sizes and the size of the perturbation makes single mea­sures of competition insufficient for understanding the dynamics of most competitive systems. However, these are not problems with the Lotka-Volterra model, where competitive effects are simply assumed to be independent of the abundances of the competitors, and independent of any neutral parameter.

Even in a two-species system, the outcome of a press perturbation to the per capi­ta mortality rate is usually significantly altered if there are sustained non-equilibrium dynamics in the system. In this case, expressions (3.1) and (3.2) should be reformulat­ed as effects on long-term mean densities, i.e., equilibrium population sizes should be replaced by long-term averages of population size. Effects on long-term mean den­sities may differ in sign from those defined by the change in the equilibrium point in systems having endogenously driven cycles due to a saturating (type II) functional response in one or both consumers (Armstrong and McGehee 1980; see Box 2.1). This is also the case when sustained environmental fluctuations drive population cycles (e.g., Abrams 2004b). These issues have largely been ignored in the competition lit­erature based on LV equations.

Changes in non-neutral parameters are likely to be involved in most responses of competitive communities due to climate change (Gilbert et al. 2014; Amarasekare 2015), as well as many of the changes produced by the addition of higher-level preda­tors (Abrams 1995; Suraci et al. 2019). The presence of adaptive phenotypic plasticity (including behaviour) in one or more species means that changes in neutral and non­neutral parameters are often coupled (Abrams 2019). This implies a wider range of potential signs of interactions when changes in non-neutral parameters occur either as a direct response to the environment or as an indirect response to changes in neu­tral parameters or resource abundances. The absence of resources from many formal definitions of competition (and from the LV equations) has delayed the development of theory for understanding these non-neutral parameters. Thus, it is important that the impact of non-neutral parameters be more fully incorporated into theory of inter­actions transmitted by resources, i.e., that expressions (3.1) or (3.2) should only be two of many measures of potential interactions transmitted or influenced by shared resources.

Having reviewed a number of complications involved with measures (3.1) and (3.2) in consumer-resource models, it is useful to return to the Lotka-Volterra model, both to illustrate the use of these expressions, and to show their exceptionally simple properties in the case of this particular model.

Here Ti (which is still the maximum per capita growth rate) can be understood as the maximum per capita birth minus death rate of species i, and it is a neutral param­eter. The α_ii terms are strengths of intraspecific competition, which were given by 1∕K_i in the traditional parameterization of Chapter 1. The α_ij terms are strengths of interspecific competition, which were given by α_ij /K_i in Chapter 1. The general for­mulas for effects of greater per capita mortality given by measures 1 and 2 above can be represented by examining the effect of a reduced r_i; alternatively, one can simply examine a positive change in the r_i and change the sign of expression (3.1). The inter­specific effect of a small increase in r₂ on the equilibrium N₁ is given by α₁₂∕(α₁₂α₂₁ - α₁₁α₂₂); this is always negative, as the denominator must be negative for the equilibri­um to be stable. The two species will not coexist if this condition on the denominator is not met.

Measure 2 above is given here by the ratio of the effect of r₂ on the equilibrium N₁ relative to its effect on the equilibrium N₂; this quantity is -α₁₂∕α₁₁. This is the same ratio of effects on the two population sizes if N₂ were maintained at a slightly higher value by some external interventions (and so was removed as a dynamic variable). In this case, eq. (3.3a) alone determines the abundances. The α_ij terms do not change with the addition or loss of another competing species in the LV model, although the formulas for the equilibrium abundances do change. In the two-competitor case, the increase in r₂ required to bring about extinction of competitor 1 can be linearly extrapolated from the effect of a very small change in r₂. There are no abrupt jumps in equilibrium population size with any small perturbation. Thus, the standard the­oretical approach for investigating indirect interactions—a press change in a neutral parameter—yields the same measures of interaction strength and conditions for coex­istence as would be obtained by manipulating population size directly and holding it at the new density. In addition, the magnitude of the perturbation does not affect the relative population changes of the two species. These claims are true provided that the LV model accurately describes the system. As shown below, this simplifying feature does not characterize most consumer-resource models.

3.2