ARTICLE IN PRESS Theoretical Population Biology 64 (2003) 163–176 http://www.elsevier.com/locate/ytpbi Dynamics and responses to mortality rates ofcompeting predators undergoing predator–prey cycles Peter A. Abrams,a,Ã Chad E. Brassil,a and Robert D. Holtb a Department of Zoology, The University of Toronto, 25 Harbord Street, Toronto, Ont., Canada M5S 3G5 b Department of Zoology, University of Florida, P.O. Box 118525, Gainesville, FL 32611-8525, USA Received 30 April 2002 Abstract Two or more competing predators can coexist using a single homogeneous prey species ifthe system containing all three undergoes internally generated fluctuations in density. However, the dynamics ofspecies that coexist via this mechanism have not been extensively explored. Here, we examine both the nature ofthe dynamics and the responses ofthe mean densities ofeach predator to mortality imposed upon it or its competitor. The analysis ofdynamics uncovers several previously undescribed behaviors for this model, including chaotic fluctuations, and long-term transients that differ significantly from the ultimate patterns offluctuations. The limiting dynamics ofthe system can be loosely classified as synchronous cycles, asynchronous cycles, and chaotic dynamics. Synchronous cycles are simple limit cycles with highly positively correlated densities ofthe two predator species. Asynchronous cycles are limit cycles, frequently of complex form, including a significant period during which prey density is nearly constant while one predator gradually, monotonically replaces the other. Chaotic dynamics are aperiodic and generally have intermediate correlations between predator densities. Continuous changes in density-independent mortality rates often lead to abrupt transitions in mean population sizes, and increases in the mortality rate ofone predator may decrease the population size of the competing predator. Similarly, increases in the immigration rate ofone predator may decrease its own density and increase the density of the other predator. Proportional changes in one predator’s birth and death rate functions can have significant effects on the dynamics and mean densities of both predator species. All of these responses to environmental change differ from those observed when competitors coexist stably as the result ofresource (prey) partitioning. The patterns described here occur in many other competition models in which there are cycles and differences in the linearity of the responses of consumers to their resources. r 2003 Elsevier Science (USA) All rights reserved. 0. Introduction one species decreases its equilibrium density and increases the equilibrium density ofits competitor. Much current thought on competition is based upon Similar qualitative behaviors characterize many models the Lotka–Volterra model oftwo competing species. ofdirect competition in which there is stable coex- Because this model always has a stable equilibrium when istence. Similarly, increasing the immigration rate of two species coexist, the role ofpopulation cycles is that species will increase its density and decrease that of seldom considered in empirical studies when trying to its competitor. Because empirical studies ofcompetitor understand the interactions between coexisting compe- and community structure are guided by theoretical titors. The Lotka–Volterra model also leads to expecta- expectations, it is important to identify situations in tions about the responses ofthe densities ofeach which these expectations are unlikely to be realized. In competitor to parameters affecting the fitness of the this paper, we show that models with explicit resource other species, such as its mortality rate. For instance, dynamics, differences in the linearity of the predators’ continuous increase in the per capita mortality rate of responses, and the potential for sustained oscillations, can exhibit sharply different effects. Because it has long been known that consumer–resource interactions can ÃCorresponding author. Fax: +416-978-8532. E-mail addresses: [email protected] (P.A. Abrams), bras- easily produce cycles (Rosenzweig and MacArthur, [email protected] (C.E. Brassil), [email protected]fl.edu (R.D. 1963), and that cycles can alter conditions for coex- Holt). istence (Armstrong and McGehee, 1980), it is surprising 0040-5809/03/$ - see front matter r 2003 Elsevier Science (USA) All rights reserved. doi:10.1016/S0040-5809(03)00067-4 ARTICLE IN PRESS 164 P.A. Abrams et al. / Theoretical Population Biology 64 (2003) 163–176 that the implications ofcycling forinteractions between (resource), N; are given by competitors have not received more attention. dP1 B1C1N In a previous article (Abrams and Holt, 2002), we ¼ P1 À D1 ; dt 1 þ hC N argued that a mechanism ofcoexistence proposed by 1 dP2 Koch (1974) and by Armstrong and McGehee (1976a, b, ¼ P2ðB2C2N À D2Þ; 1980), McGehee and Armstrong (1977) may play an dt dN N C1NP1 important role in species coexistence in natural commu- ¼ rN 1 À À À C2NP2: ð1a; b; cÞ nities. This mechanism is based on differences in the dt K 1 þ hC1N nonlinearity oftwo competitors’ responses to a single The parameter Bi is the conversion efficiency of food resource combined with consumer–resource cycles gen- into offspring for predator i; h is the handling time per erated by the interaction ofone ofthose consumers with prey item for predator 1, Di is a density independent the resource. The more linear species has a relatively death rate, Ci is a searching predator’s attack rate, and r high resource requirement, and therefore cannot exclude and K are logistic growth parameters. The variables in the species with the more nonlinear response. The latter this equation can be scaled to reduce the number of 0 0 0 generates cycles when alone with the resource; these parameters, as follows: t ¼ rt; N ¼ N=K; and P1 ¼ 0 cycles increase the mean resource density so that linear P1=ðKB1Þ; and P2 ¼ P2=ðKB2Þ: After making substitu- species can invade. Ifthe ‘robustness’ ofcoexistence is tions and dropping the primes on the new variables, the measured by the range ofprey (resource) requirements model becomes for zero population growth (i.e., efficiencies) that allow dP1 a1N coexistence, then Armstrong and McGehee’s model can ¼ P1 À d1 ; produce very robust coexistence. The range ofefficien- dt 1 þ bN cies allowing coexistence via Armstrong and McGehee’s dP2 ¼ P2ða2N À d2Þ; mechanism is often as wide as the range of efficiencies dt allowing coexistence ofcompeting predators which do dN a1NP1 ¼ Nð1 À NÞÀ À a2NP2; ð2a; b; cÞ not cycle, but which have a high degree ofpartitioning dt 1 þ bN ofprey species ( Abrams and Holt, 2002). This observa- where the new parameters are a1 ¼ KB1C1=r; a2 ¼ tion, together with the fact that a significant number of KB2C2=r; b ¼ KC1h; d1 ¼ D1=r; d2 ¼ D2=r: Below we species cycle (Ellner and Turchin, 1995; Kendall et al., refer to predator 1 as the nonlinear predator, and 1998), suggests that cases ofcoexistence via the predator 2 as the linear predator. Armstrong–McGehee mechanism may be reasonably There is no equilibrium point with positive densities common. However, well-supported empirical examples ofall three species. However, the species can coexist for ofcoexistence due to competitor–resource cycles some range ofthe parameters a2 and d2 if P1 and N are absent from the published literature. By describing undergo limit cycles in the absence ofthe second the dynamics ofsuch competitive systems, and how predator (McGehee and Armstrong, 1977; Armstrong they respond to changes in mortality rates, we hope and McGehee, 1980). Limit cycles occur in the to find dynamical signals that may help identify subsystem lacking predator 2 if examples in natural systems. Moreover, Armstrong a1ðb À 1Þ and McGehee’s (1980) model exhibits a wider range d1o : ð3Þ ofdynamics than has been described previously, bðb þ 1Þ including chaos, alternative limit cycles, and long-lasting The right-hand side ofthis inequality will be referred to transients. as the stability threshold value of d1: Here we only consider cases where d1; a1 and b satisfy (3). We consider coexistence to occur ifeach predator species can increase 1. The model when it is rare and the other predator and prey are undergoing their limiting dynamics (stable point or limit We begin by analyzing the model that Armstrong and cycle).ThisdiffersfromArmstrong and McGehee’s McGehee (1980) used to illustrate nonequilibrial coex- (1980) definition ofcoexistence, and there are circum- istence oftwo species on a single biotic resource. This stances when such ‘mutual invasion’ definitions of system is characterized by logistic prey growth, a linear coexistence may be misleading (Armstrong and McGehee, functional response for one predator, and a type-2 1980). The most serious problem is the possibility that functional response for the other predator. We argue initial invasion is eventually followed by exclusion of the below that this model displays many dynamics that are invading species. This outcome was never observed in any common to a much broader set ofmodels characterized numerical solution ofEq. (2), or in any other model by differences in the linearity consumer species and that we consider here. Previous examples ofthis endogenously generated cycles. The dynamics ofthe two phenomenon (the reversal ofinitially successfulinva- predators (consumers), P1 and P2; and the prey sion) have had alternative attractors for the subsystem ARTICLE IN PRESS P.A. Abrams et al. / Theoretical Population Biology 64 (2003) 163–176 165 being invaded (Abrams and Shen, 1989; Case, 1995), a range of a1 values spanning the prey’s intrinsic growth unlike the models considered here. A second potential rate {a1 ¼ 0:2; a1 ¼ 1; a1 ¼ 5}. For each a1; we problem with the mutual invasion definition ofcoex- examined death rates, d1; approximately 20% or 60% istence is that invasion from very low densities may be below the stability threshold set by inequality (3); i.e. impossible, yet an attractor with bounded positive d1 ¼ 0:06545a1; and 0:03273a1: For each ofthese 6 densities ofall species may be attainable, given a parameter combinations, we numerically determined the sufficiently high initial density.