Primary-Productivity Gradients and Short-Term Population Dynamics in Open Systems

Abstract

We present three models representing the trophic and behavioral dynamics of a simple food chain (primary producers, grazers, and predators) at temporal scales shorter than the scale of consumer reproduction, and at the spatial scales typically employed in field experiments. These models incorporate flexible behavioral responses of organisms to their predators and resources in spatially heterogeneous environments that are open to immigration and emigration. The basic models include passive immigration at all trophic levels, producer growth rates and losses to grazer consumption, grazer emigration rate as a behavioral response to producer and predator densities, grazer losses to predator consumption, and predator emigration as a function of grazer density. We model this system as: (1) a set of ordinary differential equations (''well-mixed model''); (2) a set of partial differential equations describing a population of discrete grazers foraging on discrete patches of primary producers (''discrete-grazer model''); and (3) a set of simulation rules describing the movement and foraging of individual grazers and the growth of primary producers on discrete patches in explicit space (''individual-based model''). The ordinary differential-equation models produced similar results to individual-based models with wellmixed producers, and the discrete-grazer and individual-based models produced similar results when grazers possessed a long-term memory of patch reward rates. The well-mixed and discrete-grazer models thus represent specific, limiting cases of the general individualbased model.Multiple equilibria and sustained oscillations are possible but are less likely in the discrete-grazer and individual-based models than in the well-mixed model, because localized foraging of discrete grazers leads to the rapid development of spatial heterogeneity in producer biomass and, hence, to a decrease in overall primary production. All models predict that stable equilibrium densities of all trophic levels increase with enrichment, provided grazers increase their emigration rates as predator density increases. If increasing predator density leads to decreasing grazer-emigration rates, predator and grazer densities increase, but producer biomass may increase or decrease with enrichment. These results contrast with predictions from models that assume ideal free distributions of grazers and/or predators with respect to their resources. Our models also predict that densities at all trophic levels will increase with increasing producer immigration, and that producer density will decline with increasing grazer immigration and increase with increasing predator immigration. Our qualitative findings on enrichment are used to interpret an experiment dealing with the short-term dynamics of a stream community open to grazers and predators.

Top-down and bottom-up processes in grassland and forested streams

“…Vertical bars denote 1 SE increase in grazer growth or density in streams (Hill et al 1995), lower productivity of biofilms at forested locations may have contributed to the overall lower invertebrate biomass at those sites. Moreover, models of open systems dominated by mobile consumers (e.g., Deleatidium) that decrease emigration as resource abundance increases predict that abundance will increase at all trophic levels with an increase in resource availability (Nisbet et al 1997). In our case, difference in the productivity of biofilms is unlikely to be the only factor involved in determining consumer abundance because substantial differences in primary production at open sites would be required to overshadow the large differences in standing crop associated with trout.…”

“…Most models of food chains have been developed for closed systems, but many systems are open to immigration and emigration of organisms, which can significantly affect predator impact (e.g., Cooper et al 1990). Few population models have investigated the simultaneous effects of predators and resource supply on the dynamics of open systems (Wootton and Power 1993;Nisbet et al 1997;Diehl et al 2000). In contrast to traditional food-chain theory, some models of open systems predict that densities at all trophic levels will increase with enrichment (Nisbet et al 1997).…”

“…Few population models have investigated the simultaneous effects of predators and resource supply on the dynamics of open systems (Wootton and Power 1993;Nisbet et al 1997;Diehl et al 2000). In contrast to traditional food-chain theory, some models of open systems predict that densities at all trophic levels will increase with enrichment (Nisbet et al 1997). This is because immigration into a system is independent of the systems internal state, whereas emigration and behaviour of organisms in open systems may be influenced by predator presence as well as resource supply (Forrester et al 1999;Diehl et al 2000).…”

“…Inputs of terrestrial plant matter to open systems can have complex effects on the interactions between prey and predators. Nisbet et al (1997) predicted that the abundance at all trophic levels would increase with inputs of detritus because of low consumer emigration rates. Indeed, patterns and responses of stream invertebrate populations in natural streams have supported this prediction (Richardson 1991;Wallace et al 1997).…”

Long-term effects of local disturbance history on mobile stream invertebrates

“…The possibility that patchy disturbances can affect the longterm distribution of mobile stream organisms has recently gained some theoretical support, as well. Nisbet et al (1997) modelled invertebrate grazer dynamics in lotic grazer-algae systems and showed that patches can develop persistent differences in grazer densities depending on the initial conditions in the patch and grazer immigration from adjacent patches.…”