IS PREDATOR‐MEDIATED COEXISTENCE POSSIBLE INUNSTABLE SYSTEMS?

Simple mathematical models are used to study the interaction between two prey species that share common resources and a common predator. In contrast to several previous analyses, this one concentrates on systems that may exhibit cyclic or chaotic dynamics due to the predator’s saturating functional response. Under very general conditions, sustained fluctuations make coexistence of prey species more difficult than in comparable systems that are stable. In the model analyzed in greatest detail, there is a limiting similarity in prey vulnerabilities in cycling systems, although such a limit does not exist in similar stable systems. The responses of mean population densities to model parameters often differ qualitatively between stable and unstable systems. In stable systems with two prey species, enrichment always increases the density of the prey that is poorer at resource exploitation and better at predator avoidance, while decreasing the density of the other prey. In contrast, in some unstable systems, en...

[1]  A. Hastings,et al.  Chaos in a Three-Species Food Chain , 1991 .

[2]  R. May,et al.  Interspecific competition, predation and species diversity: a comment. , 1972, Journal of theoretical biology.

[3]  M. Kot,et al.  8. Differential systems in ecology and epidemiology , 1986 .

[4]  Joel E. Cohen,et al.  Community Food Webs: Data and Theory , 1990 .

[5]  S. L. Lima,et al.  Behavioral decisions made under the risk of predation: a review and prospectus , 1990 .

[6]  E. Berlow,et al.  The Keystone Species Concept: Variation in Interaction Strength in a Rocky Intertidal Habitat , 1994 .

[7]  R. Holt,et al.  APPARENT COMPETITION OR APPARENT MUTUALISM? SHARED PREDATION WHEN POPULATIONS CYCLE , 1998 .

[8]  Richard Levins,et al.  Coexistence in a Variable Environment , 1979, The American Naturalist.

[9]  P. Yodzis,et al.  Introduction to Theoretical Ecology , 1989 .

[10]  P. Yodzis,et al.  Body Size and Consumer-Resource Dynamics , 1992, The American Naturalist.

[11]  Sergio Rinaldi,et al.  Slow-fast limit cycles in predator-prey models , 1992 .

[12]  Josef Hofbauer,et al.  The theory of evolution and dynamical systems , 1988 .

[13]  Peter A. Abrams,et al.  The theory of limiting similarity , 1983 .

[14]  Hal Caswell,et al.  Predator-Mediated Coexistence: A Nonequilibrium Model , 1978, The American Naturalist.

[15]  Contributions to the global analysis of 3-D Lotka-Volterra equations: Dynamic boundedness and indirect interactions in the case of one predator and two prey , 1991 .

[16]  Peter A. Abrams,et al.  The nonlinearity of competitive effects in models of competition for essential resources , 1987 .

[17]  J. Lubchenco.  Plant Species Diversity in a Marine Intertidal Community: Importance of Herbivore Food Preference and Algal Competitive Abilities , 1978, The American Naturalist.

[18]  Paul H. Harvey,et al.  The Analysis of Biological Populations , 1972 .

[19]  T. Vincent,et al.  Trade-Offs and Coexistence in Consumer-Resource Models: It all Depends on what and where you Eat , 1996, The American Naturalist.

[20]  James P. Grover,et al.  Simple Rules for Interspecific Dominance in Systems with Exploitative and Apparent Competition , 1994, The American Naturalist.

[21]  Stephen P. Ellner,et al.  Chaos in a Noisy World: New Methods and Evidence from Time-Series Analysis , 1995, The American Naturalist.

[22]  P. Morin,et al.  Effects of Food Chain Length and Omnivory on Population Dynamics in Experimental Food Webs , 1996 .

[23]  S Rinaldi,et al.  Remarks on food chain dynamics. , 1996, Mathematical biosciences.

[24]  Peter J. Morin,et al.  Food Web Architecture and Population Dynamics in Laboratory Microcosms of Protists , 1993, The American Naturalist.

[25]  James P. Grover,et al.  Assembly Rules for Communities of Nutrient-Limited Plants and Specialist Herbivores , 1994, The American Naturalist.

[26]  P. Abrams.  Resource Productivity-Consumer Species Diversity: Simple Models of Competition in Spatially Heterogeneous Environments , 1988 .

[27]  Mathew A. Leibold,et al.  A Graphical Model of Keystone Predators in Food Webs: Trophic Regulation of Abundance, Incidence, and Diversity Patterns in Communities , 1996, The American Naturalist.

[28]  Richard E. Lenski,et al.  Effect of resource enrichment on a chemostat community of bacteria and bacteriophage , 1997 .

[29]  D. Doak,et al.  The Keystone-Species Concept in Ecology and ConservationManagement and policy must explicitly consider the complexity of interactions in natural systems , 1993 .

[30]  Kevin S. McCann,et al.  Biological Conditions for Chaos in a Three‐Species Food Chain , 1994 .

[31]  Peter A. Abrams,et al.  Predators that Benefit Prey and Prey that Harm Predators: Unusual Effects of Interacting Foraging Adaptation , 1992, The American Naturalist.

[32]  O. Phillips,et al.  THE EQUILIBRIUM AND STABILITY OF SIMPLE MARINE BIOLOGICAL SYSTEMS. II. HERBIVORES , 1974 .

[33]  M. Hassell.  The dynamics of arthropod predator-prey systems. , 1979, Monographs in population biology.

[34]  D. Tilman.  Resource competition and community structure. , 1983, Monographs in population biology.

[35]  C. S. Holling,et al.  The functional response of predators to prey density and its role in mimicry and population regulation. , 1965 .

[36]  Marcus W. Feldman,et al.  Species Packing and Predation Pressure , 1975 .

[37]  Peter A. Abrams,et al.  The Effects of Enrichment of Three‐Species Food Chains with Nonlinear Functional Responses , 1994 .

[38]  S. Hsu.  Predator-mediated coexistence and extinction☆ , 1981 .

[39]  Norihiko Adachi,et al.  Existence and bifurcation of stable equilibrium in two-prey, one-predator communities , 1983 .

[40]  R. Paine.  Food webs : linkage, interaction strength and community infrastructure , 1980 .

[41]  M. A. Leibold,et al.  Resource Edibility and the Effects of Predators and Productivity on the Outcome of Trophic Interactions , 1989, The American Naturalist.

[42]  Peter Turchin,et al.  Complex Dynamics in Ecological Time Series , 1992 .

[43]  A Hastings,et al.  Chaos in one-predator, two-prey models: general results from bifurcation theory. , 1994, Mathematical biosciences.

[44]  P. Abrams.  Implications of Dynamically Variable Traits for Identifying, Classifying, and Measuring Direct and Indirect Effects in Ecological Communities , 1995, The American Naturalist.

[45]  Michael E. Gilpin,et al.  Spiral Chaos in a Predator-Prey Model , 1979, The American Naturalist.

[46]  M. Rosenzweig.  Paradox of Enrichment: Destabilization of Exploitation Ecosystems in Ecological Time , 1971, Science.

[47]  Pa Abrams,et al.  Prey evolution as a cause of predator-prey cycles , 1997 .

[48]  Robert A. Armstrong,et al.  Prey Species Replacement along a Gradient of Nutrient Enrichment: A Graphical Approach , 1979 .

[49]  R. McGehee,et al.  Coexistence of species competing for shared resources. , 1976, Theoretical population biology.

[50]  K. Fujii,et al.  Complexity-stability relationship of two-prey-one-predator species system model: local and global stability. , 1977, Journal of theoretical biology.

[51]  Peter Chesson,et al.  Predator-Prey Theory and Variability , 1978 .

[52]  Peter A. Abrams,et al.  Effect of Increased Productivity on the Abundances of Trophic Levels , 1993, The American Naturalist.

[53]  R. Paine.  Food Web Complexity and Species Diversity , 1966, The American Naturalist.

[54]  W. Murdoch,et al.  Predation and Population Stability , 1975 .

[55]  P. Abrams.  Variability and Adaptive Behavior: Implications for Interactions between Stream Organisms , 1997, Journal of the North American Benthological Society.