How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses

Abstract

In Rosenzweig–MacArthur models of predator–prey dynamics, Allee effects in prey usually destabilize interior equilibria and can suppress or enhance limit cycles typical of the paradox of enrichment. We re-evaluate these conclusions through a complete classification of a wide range of Allee effects in prey and predator's functional response shapes. We show that abrupt and deterministic system collapses not preceded by fluctuating predator–prey dynamics occur for sufficiently steep type III functional responses and strong Allee effects (with unstable lower equilibrium in prey dynamics). This phenomenon arises as type III functional responses greatly reduce cyclic dynamics and strong Allee effects promote deterministic collapses. These collapses occur with decreasing predator mortality and/or increasing susceptibility of the prey to fall below the threshold Allee density (e.g. due to increased carrying capacity or the Allee threshold itself). On the other hand, weak Allee effects (without unstable equilibrium in prey dynamics) enlarge the range of carrying capacities for which the cycles occur if predators exhibit decelerating functional responses. We discuss the results in the light of conservation strategies, eradication of alien species, and successful introduction of biocontrol agents.

Introduction

Predator–prey models of the Rosenzweig–MacArthur type (i.e. including logistic growth of the prey and a type II functional and numerical response of the predator) predict destabilization of interior equilibria and the emergence of stable limit cycles with increasing carrying capacity of the environment (Rosenzweig, 1971, Gilpin, 1972). This phenomenon, known as the paradox of enrichment, did not show up in empirical tests with aquatic predator–prey systems (Murdoch et al., 1998, McCauley et al., 1999). Several explanations of this apparent contradiction have passed theoretical and empirical scrutiny: sigmoidal or other similar non-linear functional responses of the predators (Murdoch, 1969, Oaten and Murdoch, 1975, Hassell and Comins, 1978, Nunney, 1980, Abrams and Roth, 1994, Sugie et al., 1996, Collings, 1997, Oksanen et al., 2001, Gross et al., 2004), mutual interference between predators (Ruxton et al., 1992), hiding of prey (Ruxton, 1995), induction of defenses (Vos et al., 2004, Verschoor et al., 2004), limiting nutrients and more generally the prey quality (Sommer, 1992, Loladze et al., 2000, Andersen et al., 2004), heterogeneity in the prey population with respect to edibility (Abrams and Walters, 1996, Genkai-Kato and Yamamura, 1999, Bohannan and Lenski, 1999, Persson et al., 2001), and patterned distribution in space (Scheffer and De Boer, 1995, Nisbet et al., 1998, Holyoak, 2000, Jansen and de Roos, 2000).

Another mechanism that can prevent predator–prey systems from exhibiting sustained cycles is a (demographic) Allee effect, i.e. positive density dependence in prey population growth at low prey densities (Stephens et al., 1999). Strong Allee effects—with negative population growth at low densities—lead to extinction if the population falls below a threshold size or density. If troughs in otherwise plausible prey cycles extend below the Allee threshold density, both prey and predators go extinct. This mechanism has already been shown to lead to a reduced potential for cycles and an increased propensity for system collapse (Courchamp et al., 2000, Kent et al., 2003, Webb, 2003, Zhou et al., 2005).

The Allee effect in prey may be caused by predation or by processes inherent to the prey life history (Dennis, 1989, Sinclair et al., 1998, Stephens et al., 1999, Boukal and Berec, 2002, Liermann and Hilborn, 2001, Gascoigne and Lipcius, 2004). Empirical evidence for Allee effects in single-species animal and plant populations is widespread (Lamont et al., 1993, Hopper and Roush, 1993, Groom, 1998, Kuussaari et al., 1998, Hackney and McGraw, 2001, Morris, 2002, Liebhold and Bascompte, 2003). Some of these observations (e.g. Hackney and McGraw, 2001) in fact point to a weak rather than a strong Allee effect, such that the prey per-capita population growth rate is reduced at low densities but remains positive.

Considerable attention has been given to the role of human exploitation in promoting collapse of animal populations subject to strong Allee effects (e.g. Rowe et al., 2004, Hutchings and Reynolds, 2004) as well as the role of predators (Sinclair et al., 1998, Courchamp and Macdonald, 2001, Gascoigne and Lipcius, 2004, Mooring et al., 2004, Sarnelle and Knapp, 2004). However, theoretical investigations of predator–prey dynamics have only dealt with a limited set of predator functional responses and strong Allee effects (references above). Given that predation is a general ecological mechanism, a more thorough analysis is needed, especially because insights in the role of predation and Allee effects for ecosystem persistence may well be important for population management and species conservation (Sinclair et al., 1998, Courchamp et al., 1999, Gascoigne and Lipcius, 2004).

In this article, we examine how system collapse and limit cycles, characteristic of the paradox of enrichment, are influenced by Allee effects inherent to prey life history on the one hand and the (constant, decelerating or sigmoidal) shape of the predator's functional response on the other. In particular, we ask to what extent these mechanisms reduce the propensity to exhibit sustained cycles with increasing prey carrying capacity (or decreased predator mortality) in the framework of a Rosenzweig–MacArthur predator–prey model. This is done by bifurcation analysis of this model, extended to include a continuum of functional response shapes and a wider variety of Allee effects than usually considered in other model exercises. This type of analysis has not been carried out before. As we will show, it is critically important to distinguish between weak and strong Allee effects and different types of predator functional responses.