Peter A. Abrams

Effect of increased productivity on the abundances of trophic levels

Increasing productivity can have a variety of effects on the abundances of higher trophic levels. Previous work, based on models with one type per trophic level, has suggested that increasing nutrient input to the lowest trophic level (1) always increases the abundance of the highest trophic level and (2) increases the abundances of levels that are an even number of levels below the top, while leaving others unchanged. This article investigates how these predictions are altered by the presence of different species or types (i.e., heterogeneity) within trophic levels. The models investigated are Lotka-Volterra-type models of food webs with two or three trophic levels and one or two types per level. Less complete results are derived for models with more levels and more types per level. Several different food web structures may cause the highest level to be independent of nutrient input or actually decrease with nutrient input. However, the majority of possible food web configurations with two species on some or all levels can produce productivity-abundance relationships similar to those obtained when trophic levels are homogeneous. Factors other than heterogeneity within levels can affect the productivity abundance relationships; these factors and available information on relationships in nature are discussed.

PREDATORS THAT BENEFIT PREY AND PREY THAT HARM PREDATORS: UNUSUAL EFFECTS OF INTERACTING FORAGING ADAPTATIONS

This article explores models of four-species food chains in which two species adaptively balance food intake and risk o e predation. The purpose is to determine how the interaction of adaptive foraging behaviors in two or more species influences the interactions in the entire food chain. It is shown first that such interacting behaviors will often reach a stable equilibrium. Differential equations are then derived to describe population dynamics when the behaviors remain close to their optima. These equations are very different from Lotka-Volterra-type food web models. The per capita growth rate of each species is affected by the density of every other species in the food chain, and the largest effects may be caused by the species that are most distant in the food chain. Increased predator density may increase the fitness of their prey, while increased prey density may decrease the fitness of their predator. Such unusual interactions are most likely when the adaptive species increase foraging effort in response to increased food availability. The nature of the indirect effects can often be deduced from knowing the sign of the second derivative of foraging cost-and-benefit functions.

Character displacement and niche shift analyzed using consumer-resource models of competition

This paper analyzes the adaptive responses to competition (both character displacement and niche shift) in a two consumer-two resource model. The model includes density dependence that is unrelated to the resources that are explicit in the model. This could be due to another resource dimension, parasites, or interference competition. Competitors adapt by changing their relative consumption rate constants on the two resource types. This model can result in mutually divergent, parallel, or mutually convergent displacement of competitors. Parallel displacement may entail net divergence, net convergence, or no net change. Parallel change with net convergence is most likely when the competitors have similar constraints on the possible values of consumption rate constants, unequal allopatric abundances, and significant intraspecific density dependence. Numerical calculations of displacements are presented for several models and the effect of a number of different possible alterations of the model are discussed. The evolution of resource handling and processing efficiency, and displacement in the presence of additional selective pressures on the character are considered in detail. The results have implications for questions about maximization of population size, the relationship of character displacement and the competition coefficient, and null models in the study of competition. Differences between this and previous theoretical works are discussed. It is argued that conditions allowing parallel or convergent displacement are not biologically unlikely, and possible examples are discussed. Data on resource partitioning seem to be more consistent with the results reached here than with previous theory.

Limiting similarity and the form of the competition coefficient

The question of whether there is a limit to the similarity of competing species has previously been investigated by a number of authors. These studies have all used the Lotka-Volterra model of competition, and have assumed that the competition coefficient olij may be calculated using the expression, oiij = J” U,(R) U,(R) W/s (U,(R))* dR. In this paper, the generality of this formula is questioned and two alternative expressions for olij are proposed. When these expressions are used in an analysis of limiting similarity, qualitatively different conclusions emerge regarding the existence and nature of this limit, using either deterministic or stochastic models. The relevance of these findings to theories of character convergence and similarity barriers is discussed. The available field evidence does not strongly support the validity of the formula for aij used in previous studies. Since a given method of calculating ajj must be derived from a higher level model, it is suggested that the Lotka-Volterra model is not sufficient in an investigation of limiting similarity. Beginning with Lotka, Volterra, and Gause, most ecologists have held the view that extremely similar organisms cannot coexist for long periods of time. However, it has not been until fairly recently that there has been a more quantitative attempt to determine how similar competing organisms may be, and still coexist in an equilibrium community. The motivation for this line of study stems

High Competition with Low Similarity and Low Competition with High Similarity: Exploitative and Apparent Competition in Consumer‐Resource Systems

This article investigates the relationship between the similarity of resource capture abilities and the amount of competition between two consumer species that exploit common resources. Most of the analysis is based on a consumer‐resource model introduced by Robert MacArthur. Contrary to many statements in the literature and in textbooks, measures of competition may decrease as similarity increases and may be greatest when similarity of the two species sets of resource capture rates is very low. High competition with low similarity may occur whether competition is measured by a competition coefficient near equilibrium or is measured by the proportional increase in a species population density when its competitor is removed. However, these two measures may differ considerably and may change in opposite directions with a given change in similarity. The general conditions required for such counterintuitive relationships between similarity and competition are that the consumer species have relatively low resource requirements for successful reproduction and that the resources be self‐reproducing. These same conditions also frequently lead to exclusion of one or more resources via apparent competition, and this is always true of MacArthurs model. A variety of other models of competition are analyzed, and circumstances most likely to produce large competitive effects with little overlap are identified.

SHOULD PREY OVERESTIMATE THE RISK OF PREDATION

Mathematical models are used to determine the optimal foraging effort of individuals that face increased risk of predation when they exert greater foraging effort but have imperfect information about the degree of risk. If the fitness cost of underestimating predation risk is less than that of overestimating risk, imperfect information should lead to behavior that is appropriate for a lesser risk than is actually present. Overestimation is favored under the opposite condition. If there is a trade-off between starvation and predation, an animal will usually underestimate (overestimate) risk if the third derivative of the starvation-versus-risk relation is positive (negative), provided uncertainty is not too large. Different, plausible starvation functions can favor either under- or overestimation of risk. If there is a trade-off between reproduction and predation, a more complex condition determines which type of bias is adaptive; this condition involves the reproduction-versus-risk function and its first three derivatives, and again, over- or underestimation of risk can be advantageous. In almost all models, increased accuracy of estimation is favored when costs of increased accuracy are sufficiently small. These results differ from those of previous analyses, and reasons for these differences are discussed.

ALTERNATIVE MODELS OF CHARACTER DISPLACEMENT AND NICHE SHIFT. 2. DISPLACEMENT WHEN THERE IS COMPETITION FOR A SINGLE RESOURCE

Character displacement or nongenetic niche shift, when competitors become sympatric, can result from changes in the absolute abundance of the resources available, as well as from changes in the relative abundance of different types. A model of competition for a single resource by two species of consumers is used to illustrate the range of possible directions of phenotypic change that can occur when the main effect of competition is a change in the absolute resource abundance at equilibrium. Divergence, convergence, and parallel change are all possible responses to competition in sympatry, depending on the type and magnitude of cost involved in increasing the functional response. Parallel change may be the most common mode of displacement for similar species. These adaptive responses to competition may result in an increase or a decrease in the strength of the competitive interaction between the species.

Arguments in Favor of Higher Order Interactions

A higher order interaction occurs whenever one species affects the nature of the interaction between two others. There has been considerable controversy over whether there are higher order interactions in competitive guilds of three or more species. A recent article by Pomerantz (1981) shows that previous experiments widely cited as demonstrating the existence of higher order interactions, are actually consistent with the hypothesis that there are no higher order interactions, but there are nonlinear intraspecific density-dependent effects. I agree that the present experimental evidence for higher order interactions is weak. There are, however, some strong deductive arguments for believing that many competitive systems will have higher order interactions; these arguments are reviewed below. Competitive systems are usually abstractions of consumer-resource systems with several consumers and several resources (Schaffer 1981). If resource population dynamics occur on a sufficiently faster time scale than the consumer population dynamics, it is possible to assume that resource populations are at steadystate values (Schaffer 1981). This result means that competition between consumers can be described without explicit consideration of resource dynamics; resource dynamics will, however, determine the form of the consumer population growth equations. Mechanistic competition models that are derived in this way allow one to relate general features (such as nonlinear density dependence or higher order interactions) to specific assumptions regarding resource utilization. The Lotka-Volterra competition equations may be derived in the manner described above by assuming that consumers with linear functional responses utilize resources with logistic growth (MacArthur 1970, 1972; Schoener 1974; Lawlor 1980; Abrams 1980a). Unfortunately, most resources do not have logistic growth. Pomerantz (1981) notes that nonlinearities in intraspecific density dependence have been observed quite often. Schoener (1973), Harper (1977), and Pomerantz et al. (1980) reviewed studies of density-dependent population growth, and concluded that per capita rates of increase are usually not linear functions of population size. In addition, linear functional responses are relatively uncommon (Hassell 1978; Abrams 1980b). This evidence suggests that exploitation competition generally involves nonlinear density dependence and nonconstant competition coefficients (Abrams 1980a).

DENSITY-INDEPENDENT MORTALITY AND INTERSPECIFIC COMPETITION: A TEST OF PIANKA'S NICHE OVERLAP HYPOTHESIS

A review of the relationship between nonselective density-independent mortality (DIM) and the maximum overlap of competitors in the Lotka-Volterra model fails to support Piankas niche-overlap hypothesis. An analysis of a simple model of exploitative competition reveals that higher DIM may increase the niche separation required for competing species to coexist. This is the opposite of the relationship proposed by Pianka. Confusion regarding this issue may have arisen from a failure to define the intensity of competition. Several possible meanings of this term are presented and discussed in relation to the evolution of interference competition. The nature of correlations between intrinsic rate of increase and competitive ability is especially important in determining whether DIM will ever make coexistence of competitors more likely.

Consumer functional response and competition in consumer-resource systems

Abstract This study investigates the effect of the functional response of resource consumers on the relationship between resource overlap and competition for some two-consumer, two-resource models. Two measures of competition are examined: α, the competition coefficient, and β, an index of the ease of invasion by the second consumer species when the first is at its carrying capacity. A comparison of systems with linear (type-1) and decelerating (type-2) functional responses shows that: (1) Competition coefficients are functions of the population densities of consumers or resources in systems with type-2 responses. (2) Competition coefficients may differ substantially in magnitude between systems with type-1 and type-2 functional responses. (3) The relative handling time of different resources is important in determining the relationship between overlap and competition. Positive correlations between capture rates (per unit resource) and handling times cause the system with type-2 functional responses to exhibit a higher level of competition for a given level of overlap than for the case of negative correlation. (4) If the functional response is type-2 it may be possible to obtain a priority effect in which either consumer species can exclude the other. (5) Invasion may be easier in a system with type-1 functional responses than in a similar system with type-2 functional responses, even when competition coefficients are larger in the former. Accelerating functional responses also affect the relationship between overlap and competition, but realistic models of such responses are likely to be very complex. Several currently accepted ideas in competition theory depend upon the assumption of a linear functional response, and are unlikely to be generally valid.