The Dynamics of the Schoener-polis-holt Model of Intra-guild Predation

Intraguild predation occurs when one species (the intraguild predator) pre-dates on and competes with another species (the intraguild prey). A classic model of this interaction was introduced by Gary Polis and Robert Holt building on a model of competing species by Thomas Schoener. A global analysis reveals that this model exhibits generically six dynamics: extinction of one or both species; coexistence about a globally stable equilibrium; contingent exclusion in which the first established species prevents the establishment of the other species; contingent coexistence in which coexistence or displacement of the intraguild prey depend on initial conditions; exclusion of the intraguild prey; and exclusion of the intraguild predator. Implications for biological control and community ecology are discussed. 1. Introduction. Intraguild predation occurs when one species (the intraguild predator) preys on and competes for limiting resources with another species (the intraguild prey). This mixture of predation and competition is observed in a diversity of freshwater, marine, and terrestrial food webs [2, 3]. For example, in freshwater streams, bluegills and insects compete for plankton, and bluegills prey on insects. In deserts, spiders and scorpions compete for insects and prey on one another. In [3, 2], which coined the phrase " intraguild predation, " Gary Polis and Robert Holt modeled intraguild predation by modifying a model of competition by Thomas Schoener [5]. If N 1 is the density of the intraguild prey and N 2 is the density of the intraguild predator, then this model is given by dN 1 dt

1. Introduction.Intraguild predation occurs when one species (the intraguild predator) preys on and competes for limiting resources with another species (the intraguild prey).This mixture of predation and competition is observed in a diversity of freshwater, marine, and terrestrial food webs [2,3].For example, in freshwater streams, bluegills and insects compete for plankton, and bluegills prey on insects.In deserts, spiders and scorpions compete for insects and prey on one another.In [3,2], which coined the phrase "intraguild predation," Gary Polis and Robert Holt modeled intraguild predation by modifying a model of competition by Thomas Schoener [5].If N 1 is the density of the intraguild prey and N 2 is the density of the intraguild predator, then this model is given by where a i > 0 is the rate at which a species finds the resource, b 1 > 0 (respectively b 2 ) is the efficiency at which the intraguild prey (resp.intraguild predator) convert the basal resource into intraguild prey (resp.intraguild predators), c i > 0 is the per-capita death rate, α > 0 is the rate at which the intraguild predator encounters the intraguild prey, β > 0 is the efficiency at which the consumed intraguild prey is converted into intraguild predators, and 280 E. RUGGIERI AND S. J. SCHREIBER S > 0 is the rate at which the resource enters the system.In the absence of intraguild predation (i.e., α = β = 0), Schoener [5] showed that (generically) one species competitively excluded the other.Holt and Polis [1,3] illustrated that the inclusion of intraguild predation (i.e., α > 0 and β > 0) could mediate coexistence between the two species, reverse competitive dominance, and generate alternative stable states.In this article, we provide a complete mathematical analysis of a generalization of this model.
An unfortunate feature of the original Schoener model is that the "resource function" R(N 1 , N 2 ) = S a 1 N 1 +a 2 N 2 is not defined at the origin.A modified version of this resource function was presented by Schoener in 1978 and is given by where γ is the natural "decay rate" of the resource [6].This resource function has a simple mechanistic interpretation, as it corresponds to the quasi steady state of the equation In this article, we study the dynamics of under the assumption that γ > 0. The remainder of the article is structured as follows.In section 2, we state and illustrate our main results.In sections 3 and 4, we prove our main results.In section 5, we discuss the implications of these results for biological control and the structure of ecological communities.
2. Main results.To ensure that (2) has solutions defined for all forward time, we begin with a proposition that ensures that all solutions to (2) enter a compact subset of the nonnegative quadrant.PROPOSITION 2.1.Let (N 1 (t), N 2 (t)) be a solution to (2) we get Equation ( 2) always has the equilibrium (0, 0).The per-capita growth rate of species i at (0, 0) is a i b i S γ − c i .When a i b i S γ − c i > 0, there is a positive equilibrium on the N i axis given by N i = b i S c i − γ a i at which the resource level is R = c i a i b i .The missing species can invade at these boundary equilibria if its per-capita growth rate at this equilibrium is positive.The per-capita growth rate of the intraguild prey at the intraguild predator equilibrium (0, b 2 S c 2 − γ a 2 ) is given by The per-capita growth rate of the intraguild predator at the intraguild prey equilibrium The following theorem characterizes the behaviors of (2) when both species can invade the origin.The conclusions of this theorem are illustrated in Fig. 1.The proof is given in section 3.

Then (2) exhibits five types of dynamics:
1. Global coexistence.If Γ 1 > 0 and Γ 2 > 0, then there exists a unique equilibrium ( N 1 , N 2 ) in the positive quadrant, and lim t→∞ (N 1 (t), N 2 (t)) = ( N 1 , N 2 ) for every solution satisfying N 1 (0) > 0 and N 2 (0) > 0. 2. Contingent exclusion.If Γ 1 < 0 and Γ 2 < 0 then there exists a unique equilibrium ( N 1 , N 2 ) in the positive quadrant with a one dimensional stable manifold S that separates the positive orthant into two connected components.For every solution with (N 1 (0), N 2 (0)) lying below S, we have that lim t→∞ (N . For every solution with (N 1 (0), N 2 (0)) lying above S, we have that lim t→∞ (N and A 2 1 > B 1 , then there exist two positive equilibria ) is a saddle with a one-dimensional stable manifold S that separates the positive orthant into an upper and lower region.Moreover, for every solution with (N 1 (0), N 2 (0)) > (0, 0) lying below S, and The following proposition characterizes the behavior of (2) when one of the species can not invade the origin.The proof is given in section 4. PROPOSITION 2.2.Define Γ 2 as above.
3. Proof of Theorem 2.1.Assume that a i b i S γ > c i for i = 1, 2. We begin by proving that there are no periodic solutions or heteroclinic loops in the positive quadrant for (2).Recall that the Dulac criterion states that if there exists a positive continuously differentiable realvalued function Ψ(N 1 , N 2 ) on the positive quadrant such that ∂ΨF 1 ∂N 1 < 0 in the positive quadrant, then there are no closed orbits or heteroclinic loops in the positive quadrant.Choosing Ψ(N 1 , N 2 ) = 1 N 1 N 2 yields the desired result.Since (as we shall shortly prove) equation ( 2) generically has only a finite number of equilibria, Poincaré-Bendixson theory implies that generically every solution (N 1 (t), N 2 (t)) of ( 2) with N i (0) ≥ 0 for i = 1, 2 converges to an equilibrium.To understand the equilibria of (2), we examine the nullclines.The equations, dN i dt = 0, have a pair of solutions.For each species, one of its nullclines exists on the opposing axis.For example, one nullcline for the prey is the N 2 axis .The nontrivial nullcline for the intraguild predator is the curve This intraguild predator nullcline is a hyperbola with a vertical asymptote at N 2 = c 2 αβ and an oblique asymptote along the line of this nullcline is positive.Consequently, the graph of (3) does not intersect the negative quadrant (see Figure 2), and no equilibria can reside in this quadrant.Similarly, the nontrivial intraguild prey nullcline is the curve This intraguild prey nullcline has the same oblique asymptote N 2 = − γ a 2 − a 1 a 2 N 1 as the oblique asymptote of (3).It also has a horizonal asymptote at N 2 = − c 1 α .Moreover, both components of this intraguild prey nullcline are strictly decreasing as functions of N 1 (see Figure 2).
In addition to the three equilibria on the axes, we can solve the nontrivial nullcline equations (3) and (4) for N 1 and N 2 .If b 2 = b 1 β, then this yields the two additional possibilities for equilibria where We will use the fact that for i, j = 1, 2 with i = j.Alternatively, when b 2 = b 1 β and A i = 0 for i = 1, 2, solving (3) and ( 4) yields Finally if b 2 = b 1 β and A i = 0 for i = 1 or 2, there is no solution to (3) and (4).Let ( N 1 , N 2 ) be an equilibrium with N 1 > 0 and N 2 > 0. This can occur only if α > 0 and β > 0. Linearizing the right-hand side of (2) about ( N 1 , N 2 ) yields The trace of J is negative.Moreover, we claim that det J = 0 if and only if Substituting the preceding equation into det J yields Hence, det J = 0 if and only if ε = 0, equivalently, if and only if A 2 1 = B 1 .These observations imply that so long as A 2 i = B i or b 2 = βb 1 , any equilibrium in the positive quadrant is either a sink (i.e., eigenvalues of J have negative real parts) or a saddle (i.e., J has a positive and negative eigenvalue).
Case 1 -Global coexistence.Assume Γ i > 0 for i = 1, 2. We claim that there is a unique equilibrium in the positive quadrant.To prove this claim and identify the correct expression of this positive equilibrium, we need to consider three cases.For the first case, assume that b

)), N −
1 and N + 1 must be real and have the same sign.Moreover, as there are no equilibria in the negative quadrant, N − 1 and N + 1 are positive.Hence, is the unique equilibrium in the positive quadrant.For the second case, assume that b 1 β < b 2 .Then, C 1 < 0 and C 2 < 0.Moreover, Γ i > 0 and b 1 β < b 2 imply that B 1 < 0, and B 2 > 0. Hence, 5)), N − 2 and N + 1 are real and have the same sign.Moreover, as there are no equilibria in the negative quadrant, N − 2 and N + 2 are positive.Hence, is the unique equilibrium in the positive quadrant.For the third case, assume that b 1 β = b 2 .In this case the only equilibrium is given by (N * 1 , N * 2 ).Since Γ i > 0 for i = 1, 2 and there are no equilibria in the negative quadrant, equation (6) implies that (N * 1 , N * 2 ) must lie in the positive quadrant.Let ( N 1 , N 2 ) denote the unique equilibrium in the positive quadrant.Since we have either b 2 = b 1 β or A 2  1 − B 1 = 0 at this equilibrium, the equilibrium is either a sink or a saddle.Since the equilibria on the axes are a source (the origin) and two saddles and all solutions starting in the positive quadrant must converge to an equilibrium, the Poincaré-Bendixson theory implies that all solutions starting in the positive quadrant must converge to ( N 1 , N 2 ).
Case 2: Contingent exclusion.Assume that Γ 1 < 0 and Γ 2 < 0. We claim that there is a unique equilibrium in the positive quadrant.To prove this claim and identify the correct expression of this positive equilibrium, we need to consider three cases.For the first case, assume that b As N + 2 and N − 2 have the same sign and there are no equilibria in the negative quadrant, is the unique equilibrium in the positive quadrant.For the second case, assume that b As N + 1 and N − 1 have the same sign and there are no equilibria in the negative quadrant, is the unique equilibrium in the positive quadrant.For the third case, assume that b 1 β = b 2 .In this case the only equilibrium is given by (N * 1 , N * 2 ).Since Γ i < 0 for i = 1, 2 and there are no equilibria in the negative quadrant, equation (6) implies that (N * 1 , N * 2 ) must lie in the positive quadrant.Let ( N 1 , N 2 ) denote the unique equilibrium in the positive quadrant.Since we have either b 2 = b 1 β or A 2  1 − B 1 = 0 at this equilibrium, this equilibrium is either a sink or a saddle.However, since ) are sinks and the origin is a source, ( N 1 , N 2 ) must be a saddle.Moreover, the stable manifold of ( N 1 , N 2 ) separates the basins of attraction of ( b 1 S c 1 − γ a 1 , 0) and (0, b 2 S c 2 − γ a 2 ).Case 3: Intraguild predator displacement.Assume that Γ 1 > 0 and Γ 2 < 0. Since Γ 2 < 0, the intraguild predator nullcline given by (3) must lie below the N 1 axis at Since the prey nullcline given by ( 4) is decreasing and passes through the point ( b 1 S c 1 − γ a 1 , 0) and the predator nullcline has a vertical asymptote at N 1 > b 1 S c 1 − γ a 1 , these nullclines must intersect at an equilibrium ( N 1 , N 2 ) with N 1 > 0 and N 2 < 0. Using this observation, we claim there are no equilibria in the positive quadrant.To prove this claim, we consider three cases.First, assume that b 1 β > b 2 .Then, B 1 > 0 and B 2 > 0. Hence, when A 2 i − B i ≥ 0 (i.e., the equilibria are real), N ± 1 have the same sign and N ± 2 have the same sign.Our observation about ( N 1 , N 2 ) implies that N ± 1 > 0 and N ± 2 < 0. Hence, there are no equilibria in the positive quadrant.Second, assume that b 1 β < b 2 .Since C i < 0 and B i < 0 for i = 1, 2 in this case, we get and there is no equilibrium in the positive quadrant. Since ) is a saddle, and there are no equilibria in the positive quadrant, the Poincaré-Bendixson theory implies that all solutions in the positive quadrant converge to ( b 1 S c 1 − γ a 1 , 0).Cases 4 and 5: Intraguild prey displacement and contingent coexistence.Assume that then there are no equilibria in the positive quadrant.Second, consider the case that b 1 β > b 2 .Then C i > 0 and B i < 0 for i = 1, 2. Hence, Therefore, there are no equilibria in the positive quadrant.Third, consider the case that b 1 β = b 2 .Since Γ 1 Γ 2 < 0, equation (6) implies that N * 1 and N * 2 are of opposite sign.Therefore, there are no equilibria in the positive quadrant.
Assume that there are no equilibria in the positive quadrant (i.e., b ) is a sink, all solutions in the positive quadrant converge to (0, b 2 S c 2 − γ a 2 ).Alternatively assume that there are two positive equilibria 1 > B 1 , these equilibria are either sinks or saddles.By the Poincaré-Hopf Index theorem, one of the positive equilibria is a sink and the other is a saddle.The stable manifold of this saddle must separate the basins of attraction of the boundary sink and the sink in the positive quadrant.Then, dN 1 dt 1 Case 2: Intraguild predator failure.Suppose that a 2 b 2 S γ < c 2 and a 1 b 1 S γ < c 1 .Let (N 1 (t), N 2 (t)) be a solution to (2) with N 1 (0) ≥ 0 and N 2 (0) > 0.Then, lim t→∞ N 1 (t) = 0. Since a 2 b 2 S γ < c 2 , there exists a T > 0 such that for all t ≥ T .Hence, lim t→∞ N 2 (t) = 0. Suppose that a 2 b 2 S γ < c 2 , a 1 b 1 S γ > c 1 , and Γ 2 < 0.Then, the intraguild predator nullcline (3) lies below the N 1 axis for 0 Since the intraguild prey nullcline (4) lies below the N 1 axis for N 1 > b 1 S c 1 − γ a 1 , there is no equilibrium in the positive quadrant.Since the origin is a saddle and and Γ 2 > 0.Then, the intraguild predator nullcline (3) has a negative N 2 intercept; has an ; and has a vertical asymptote at N 1 = c 2 αβ .Since the intraguild prey nullcline ( 4) is decreasing and positive on the interval 0 , there is a unique intersection ( N 1 , N 2 ) of these nullclines in the positive quadrant.Since there are no periodic orbits in the positive quadrant (cf. the first paragraph of the proof of Theorem 2.1) and the origin and 5. Discussion.Competition and predation are two fundamental processes in ecological communities that have been studied extensively.Intraguild predation combines these two processes in a unique way: species that compete for common resources also predate on one another.In this article, we analyzed a variation of an intraguild predation model introduced by Polis and Holt [1,3], which builds on Schoener's model of competing species [5,6].Our analysis reveals that there are generically six types of dynamics: extinction of one or both species; coexistence about a globally stable equilibrium; competitive exclusion of the intraguild predator; competitive exclusion of the intraguild prey; contingent exclusion in which the first established species prevents the establishment of the other species; and contingent coexistence in which the species coexist or the intraguild prey is displaced, depending on initial conditions.
As observed by Polis and Holt [2], this model of intraguild predation has important implications for classical biological control where a pest species (in our case, the "resource") is regulated by natural enemies (in our case, the intraguild prey or predator).For example, many agricultural pests, such as lepidoptera or hymenoptera ("the resource species"), are attacked by parasitic wasps and predatory insects.Since these predatory insects also attack pests that contain wasp larvae [4], the predators are intraguild predators, and the wasps are their intraguild prey.The relevance of our analysis to biocontrol stems from the observation that the equilibrium pest abundance equals at any equilibrium ( N 1 , N 2 ) supporting the intraguild prey and predator.Hence, the pest equilibrium determined by the intraguild prey is smaller than the pest abundance R( N 1 , N 2 ) determined by an equilibrium supporting both species.Thus, intraguild predators coexisting with intraguild prey may disrupt biological control by raising the equilibrium abundance of the pest.However, this prediction needs to be viewed with caution, as this model does not account for the potentially stabilizing effects of intraguild predation with a dynamic pest species [7].
Our analysis also has various implications for how intraguild predation impacts community structure.Most interestingly, intraguild predation can result in alternative stable states in which community composition depends on the order of species arrivals.In the case of contingent exclusion, the species that arrives first exclusively determines the community composition.Alternatively, in the case of contingent coexistence, there exist alternative states with different levels of species richness.If the intraguild predator arrives first, it prevents the establishment of the intraguild prey.However, if the intraguild prey arrives first, the community can be augmented with the intraguild predator.Our analysis also makes predictions about the effect of environmental productivity on community structure.Assume the intraguild prey can suppress the resource to a lower equilibrium abundance (i.e., c 1 a 1 b 1 < c 2 a 2 b 2 ).Then at low productivity (i.e., S just large enough to support both species), the intraguild prey displaces the intraguild prey (i.e., Γ 1 > 0 and Γ 2 < 0).Alternatively, at high levels of productivity (i.e. S sufficiently large), a reversal occurs in which the intraguild predator exerts sufficient predatory pressure to displace the intraguild prey (i.e., Γ 1 < 0, Γ 2 > 0, and either b 2 ≤ b 1 β or B 1 > A 2 1 ).Thus, only at intermediate levels of productivity are alternative states or coexistence possible.
In conclusion, simple models of intraguild predation exhibit a diversity of behaviors that have noteworthy ecological ramifications.While providing an illuminating starting point, the biological complexities of the real world require us to understand how stage-structure, nonlinear functional responses, spatial heterogeneity, additional species interactions and environmental stochasticity interact with intraguild predation to modulate the effects observed in these simple models.

FIGURE 1 .
FIGURE 1.The phase portraits in the N 1 -N 2 plane corresponding to the conclusions of Theorem 2.1.Solutions (in blue) are plotted along with the nullclines (in red/yellow).In the case of bistability and contingent coexistence, the separatrix of the positive saddle is shown in green.