Dynamics of a intraguild predation model with generalist or specialist ...

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Only IGP model with specialist predator can have both boundary attractor ... Keywords Intraguild predation · Generalist Predator · Specialist Predator · Extinction ...
Journal of Mathematical Biology manuscript No. (will be inserted by the editor)

Dynamics of a intraguild predation model with generalist or specialist predator

Yun Kang · Lauren Wedekin

Received: date / Accepted: date

Abstract Intraguild predation (IGP) is a combination of competition and predation which is the most basic system in food webs that contains three species where two species that are involved in a predator/prey relationship are also competing for a shared resource or prey. We formulate two intraguild predation (IGP: resource, IG prey and IG predator) models: one has generalist predator while the other one has specialist predator. Both models have Holling-Type I functional response between resource-IG prey and resource-IG predator; Holling-Type III functional response between IG prey and IG predator. We provide sufficient conditions of the persistence and extinction of all possible scenarios for these two models, which give us a complete picture on their global dynamics. In addition, we show that both IGP models can have multiple interior equilibria under certain parameters range. These analytical results indicate that IGP model with generalist predator has ”top down” regulation by comparing to IGP model with specialist predator. Our analysis and numerical simulations suggest that: 1. Both IGP models can have multiple attractors with complicated dynamical patterns; 2. Only IGP model with specialist predator can have both boundary attractor and interior attractor, i.e., whether the system has the extinction of one species or the coexistence of three species depending on initial conditions; 3. IGP model with generalist predator is prone to have coexistence of three species.

Yun Kang Applied Sciences and Mathematics, Arizona State University, Mesa, AZ 85212, USA. E-mail: [email protected] Lauren Wedekin Applied Sciences and Mathematics, Arizona State University, Mesa, AZ 85212, USA. E-mail: [email protected]

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Keywords Intraguild predation · Generalist Predator · Specialist Predator · Extinction · Persistence · Multiple Attractors

1 Introduction Competition and predation have commonly been recognized as important factors in community ecology. Ecologists more recently began to acknowledge an interaction between the two where potentially competing species are also involved in a predator-prey relationship (Polis and Holt 1992). Holt and Polis (1997) thus define intraguild predation (IGP) to be this mixture of competition and predation, i.e., intraguild predation (IGP) describes an interaction in which two species that compete for shared resources also eat each other. IGP has been documented extensively in both terrestrial and aquatic communities (Hall 2011), and within and between a wide variety of taxa including parasitoids and pathogens (Brodeur and Rosenheim 2000).

Fig. 1: The schematic diagram of Intraguild predation model.

IGP is a specific case of omnivory (McCann et al. 1998) with the addition of competition for a shared resource whose simplest form involves three species: IG predator, IG prey, and shared prey/resource (see Figure 1). The IG prey feeds on only the shared prey while the IG predator feeds on both the IG prey and the shared prey. IGP is extremely prevalent (Arim and Marquet 2004) and it is a recurring theme in community ecology which can serve as the building blocks to study even more complex systems and lead to breakthroughs in conservation biology (Brodeur and Rosenheim 2000). Theoretical models and empirical evidence suggest that IGP can lead to spatial and temporal exclusion of intraguild predators, competitive coexistence, alternative stable states or hydra effects (Polis and Holt 1992; Moran et al. 1996; Holt and Polis 1997; Ruggieri and Schreiber 2005; Amarasekare 2007 & 2008; Hall 2011; Sieber and Hilker 2011; Yeakel et al 2011; Sieber and Hilker 2012). IGP not only may have indirect effects at other trophic levels but also may either enhance or impede biological control (Sunderland et al. 1997; Rosenheim 1998; Brodeur and Rosenheim 2000). For instance, in terrestrial arthropod communities, the effects of one predator on another may release

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extra-guild herbivores from intense predation, thereby reducing plant productivity through cascading events (Spiller and Schoener 1990; Diehl 1993; Brodeur and Rosenheim 2000). Most biological systems, including those involving IGP, are comprised of a wide variety of predators, both generalists and specialists. Generalist predators typically survive on an assortment of prey, giving themselves a sort of environmental buffer when resources begin to run short while specialist predators have a much narrower diet and require more specific environmental conditions. Ecologists have studied the impacts of generalist vs. specialist predators separately along with the potential outcome of interactions between them (Hassell and May 1986; Hanski et al. 1991; Snyder and Ives 2003) in various environments. In this article, we develop two IGP models for the interactions of shared prey/resource and IG prey subject to attacks by a generalist or specialist IG predator, where “specialist” is defined such that the shared prey/resource and IG prey are the only two resources available to the IG predator while “generalist” is defined such that the IG predator can survive in the absence of the shared prey/resource and IG prey. The main purpose of this article has two-fold: 1. How does generalist vs. specialist IG predator affect species persistence and extinction of IGP models? 2. How does different functional response between IG prey and IG predator affect the dynamics of IGP models? The rest of this article is organized as follows: In Section 2, we derive two IGP models where one has generalist predator and the other one has specialist predator. Both models have Holling-Type I functional response between resource-IG prey and resource-IG predator; Holling-Type III functional response between IG prey and IG predator. In Section 3, we show sufficient conditions of the persistence and extinction of all possible scenarios for the two IGP models formulated in Section 2. Our analytical results give us a complete picture on their global dynamics. In Section 4, we focus on the number of interior number of IGP models with generalist or specialist predator and study their possible multiple attractors. In the last section, we summary our results and discuss their biological implications.

2 Model Derivations Mathematic modeling is frequently used to study food web dynamics, and to aid in understand phenomena that occur in nature. Recall that IGP is essentially the combination of competition and predation into a multispecies subsystem, the simplest being a three-species subsystem. Holt and Polis (1997) ignited IGP modeling by exploring the implications of incorporating IGP into pre-existing models for exploitative competition and simple food chains made up of predator-prey relationships. They took the standard Lotka-Volterra predator-

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prey model and incorporated IGP into it. Let P (t), G(t), M (t) be the population biomass of shared prey (e.g., plant), IG prey (e.g., forest pest like gypsy moth) and IG predator (e.g., rodent) respectively at time t, then their model appears as follows: P K)

dP dt

= P (r(1 −

dG dt

= G(bg ag P − αm M − dg )

dM dt

= M (bm am P + βαm G − dm )

− am M − ag G) (1)

where r is the per capita growth rate of the shared prey; K is the carrying capacity of the shared prey; ai is the predation rate of species i to the shared prey; αm is the predation rate of IG predator to IG prey; bi , i = g, m is the conversion of resource consumption into reproduction for species i; β conversion rate of IG predator from IG prey; di , i = g, m is the density-independent death rate of species i. Motivated by the IGP model proposed by Holt and Polis (1997), we derive two IGP models where one has a generalist predator that feeds on both the intraguild prey and shared prey along with outside resources, and the other one has a specialist predator that feeds only on the intraguild prey and shared prey. In addition, we suppose that species P, G, M satisfy the following ecological assumptions: – In the absence of species G and M , species P follows a logistic growth function. – In the absence of both P and G, if M is a generalist predator, then it follows a logistic growth function. While if M is a specialist predator then it relies on only P and G for survival. – Predator M feeds on both P and G, but IG prey G only feeds on resource P . – G feeds on P and M feeds on P following a Holling type I functional response because searching time is the only factor of time consumption that will limit the predation rate where handling time and other more complicated dynamics are not applicable (Holling 1959) – M feeds on G following a Holling type III functional response. This assumption fits the case when G is forest insect and M is small mammal that follows experiments done by Schauber et al. (2004) who found a positive relationship between predation rate and pupae densities, causing an accelerating response. Then based on the assumptions above, a continuous time IGP model with specialist predator can be described as follows: dP dt dG dt dM dt

h i = P rp (1 − KPp ) − ag G − am M h i G = G eg ag P − GaM 2 +b2 − dg h i 2 = M em am P + eGm2aG − d 2 m +b

(2)

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while a continuous time IGP model with generalist predator can be described as follows:

dP dt dG dt dM dt

i h = P rp (1 − KPp ) − ag G − am M h i G = G eg ag P − GaM 2 +b2 − dg h i aG2 = M rm (1 − KMm ) + em am P + eGm2 +b 2

(3)

Parameters values are assigned according to biological meanings in Table (1). Ki and ri assign a carrying capacity and growth to species i when considering a logistic growth function; ai is used as a predation rate of species i for the resource, and ei is the conversion efficiency of biomass between two trophic levels. The searching rate decreases the amount of attainable biomass for consumption. Furthermore, the amount of biomass transferred from one trophic level to another decreases drastically between species. Parameters a and b describe the type III functional response between M and G. Notes: Our IGP models (2) and (3) differ from the IGP model (1) proposed by Holt and Polis (1997) in the functional response between IG prey and IG predator, i.e., (1) has Holling-Type I functional response while our models have Holling-Type III functional response. In addition, our IGP model with generalist predator (3) follows logistic growth function

dM dt

= rm M (1 −

M Km )

in the absence of the shared resource P and IG

prey G. Note that we assume that both the IG-prey and the IG-predator have a linear functional response to the basal (or shared) resource, and the trophic complexity comes about because of the relationship (i.e., Holling-Type III functional response) between the IG-predator and IG-prey. The per capita mortality rate of the latter initially rises with its own density, then falls. More generally, one might have complex trophic interactions of this sort, for all three of the trophic interactions contained in the model. However, it is quite useful to have models like (2) and (3), with a modicum of such complexity. Biologically, we could argue that the Holling-Type III functional response might emerge from interacting behavioral reactions by both the predator and the prey, for instance, prey may start aggregating at higher densities, making it harder for individual prey to be caught, whereas at low density, the main factor governing attacks is how much attention the IG-predator gives to the IG-prey. The basal (or shared) resource by contrast might be assumed to be rather passive, thus, the interactions of the basal resource v.s. IG-prey and the basal resource v.s. IG-predator, can take the form of Holling-Type I functional response. In the following section, we investigate sufficient conditions of the extinction and persistence for all species for both (2) and (3).

6 Parameters

Biological Meanings

rp

maximum growth rate of resource

rm

maximum growth rate of IG predator

Kp

carrying capacity of resource

Km

carrying capacity of IG predator

ag

predation rate of IG prey for resource

am

predation rate of IG predator for resource

a

maximum population of IG prey killed by IG predator

b

IG prey density at which the population killed by IG predator reached half of its maximum

eg

bio-mass conversion rate from resource to IG prey

em

bio-mass conversation rate from IG prey to IG predator

dg

the death rate of IG prey

dm

the death rate of IG predator

Table 1: Parameters Table of System (2) and (3)

3 Mathematical Analysis Assume that all parameters are strictly positive, then for the convenience of mathematical analysis, we can simplify System (2) as (4) by letting x=

P ag G am M eg ag Kp am a ,y = ,z = , τ = rp t, γ1 = , a1 = , Kp rp rp rp eg Kp

and β=

rp b dg em am Kp a dm , d1 = , γ2 = , a2 = , d2 = . ag eg ag Kp rp am Kp em am Kp x0 = x(1 − x − y − z)   yz y 0 = γ1 y x − ya21+β 2 − d1   y2 z 0 = γ2 z x + ya22+β . 2 − d2

Similarly, System (3) can be rewritten as (5) by letting x=

P ag G am M eg ag Kp am a rp b ,y = ,z = , τ = rp t, γ1 = , a1 = ,β = , Kp rp rp rp eg K p ag

and d1 =

dg em am Kp a rm rp rm dm , γ2 = , a2 = , a3 = , a4 = , d2 = . 2 eg ag Kp rp am Kp em a m K p Km am em Kp em am Kp

(4)

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x0 = x(1 − x − y − z)   yz y 0 = γ1 y x − ya21+β 2 − d1   y2 z 0 = γ2 z a3 − a4 z + x + ya22+β . 2

(5)

Then we can show the following lemma holds for System (4) and (5): Lemma 1 (Positively invariant and bounded) Both System (4) and (5) are positively invariant and bounded in R3+ . Moreover, we have lim sup x(τ ) ≤ 1 and τ →∞

a3 1 + a2 + a3 ≤ lim inf z(τ ) ≤ lim sup z(τ ) ≤ . τ →∞ a4 a4 τ →∞

Proof By the continuity argument, we can easily prove that both System (4) and (5) are positively invariant in R3+ . Then from the equation of x0 = x(1 − x − y − z) we have x0 ≤ x(1 − x), thus we can conclude that lim sup x(τ ) ≤ 1. τ →∞

Similarly, the positive invariant property indicates that for any initial condition taken in R3+ , we have the following inequality from (5)   a3 a2 y 2 ≥ γ2 z(a3 − a4 z) ⇒ lim inf z(τ ) ≥ . z 0 = γ2 z a3 − a4 z + x + 2 2 τ →∞ y +β a4 On the other hand, we have   a2 y 2 a3 + 1 + a2 ≤ γ2 z(a3 − a4 z + 1 + a2 ) ⇒ lim sup z(τ ) ≤ z 0 = γ2 z a3 − a4 z + x + 2 . 2 y +β a4 τ →∞ Define v = x + θ1 y + θ2 z where (1 − γi θi ) > 0, i = 1, 2 and a1 γ1 θ1 > a2 γ2 θ2 , then we have v 0 = x0 + θ 1 y 0 + θ 2 z 0 2

z = x(1 − x) − (1 − γ1 θ1 )xy − (1 − γ2 θ2 )xz − (a1 γ1 θ1 − a2 γ2 θ2 ) y2y+β 2 − γ1 θ1 d1 y − γ2 θ2 d2 z

≤ x(1 − x) − γ1 θ1 d1 y − γ2 θ2 d2 z ≤ (min{γ1 d1 , γ2 d2 } + 1)x − min{γ1 d1 , γ2 d2 }v. Since for any  > 0, there exists a T > 0 such that x(τ ) < 1 +  for all τ > T . Therefore, for τ > T , we have v 0 < (min{γ1 d1 , γ2 d2 } + 1)(1 + ) − min{γ1 d1 , γ2 d2 }v. Thus, we can conclude that lim sup v(τ ) ≤ τ →∞

min{γ1 d1 , γ2 d2 } + 1 . min{γ1 d1 , γ2 d2 }

Therefore, System (4) is bounded in R3+ . Since we have shown that both species x and z are bounded for System (5), we only need to show that species y is also bounded. This can be done by defining v = γ1 x + y. Then we have v 0 = γ1 x0 + y 0 = γ1 x(1 − x − z) + γ1 y(−

a1 yz − d1 ) ≤ γ1 (x − d1 y) ≤ γ1 (1 + γ1 d1 − d1 v). y2 + β 2

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This implies that lim sup v(τ ) ≤ τ →∞

1 + d1 γ1 1 + d1 γ1 ⇒ lim sup y(τ ) ≤ . d1 d1 τ →∞

Therefore, System (5) is also bounded in R3+ . Notes: Lemma 1 indicates that our IGP systems are biological meaningful. In addition, species z is persistent in System (5). The positive invariant and bounded properties showed in this lemma allow us to obtain theoretical results on sufficient conditions of species’ persistence and extinction in the following subsections.

3.1 Boundary equilibria and their stability It is easy to check that System (4) has the following boundary equilibria if di < 1, i = 1, 2: (0, 0, 0), (1, 0, 0), (d1 , 1 − d1 , 0), and (d2 , 0, 1 − d2 ) while System (5) has the following boundary equilibria if d1 < 1 and  (0, 0, 0), (1, 0, 0), (d1 , 1 − d1 , 0),

a4 − a3 a3 + 1 , 0, a4 + 1 a4 + 1

a3 a4

< 1:

 and

  a3 0, 0, . a4

By evaluating the eigenvalues of their associated Jacobian matrices of System (4) and (5) at these equilibria, we are able to perform local stability analysis and obtain the sufficient conditions of the local stability of these boundary equilibria for System (4) and (5) are stated in the following lemma: Lemma 2 (Boundary equilibria) Both System (4) and (5) always have the extinction equilibrium (0, 0, 0) which are always unstable. Sufficient conditions of the local stability of the boundary equilibria are listed in Table 2-3. BE point

Conditions for stability

Conditions for instability

(0, 0, 0)

Never

Always

(1, 0, 0)

d1 > 1 and d2 > 1

d1 < 1; or d2 < 1

(d2 , 0, 1 − d2 )

d2 < d1 and d2 < 1

d2 > d1 ; or d2 > 1

(d1 , 1 − d1 , 0)

d1 +

a2 (d1 −1)2 (−1+d1 )2 +β 2

< d2 and d1 < 1

d1 +

a2 (d1 −1)2 (−1+d1 )2 +β 2

> d2 or d1 > 1

Table 2: Specialist Predator (4): Local stability conditions for boundary equilibria (BE)

Notes: Lemma 2 indicates follows:

9 BE point

Conditions for stability

Conditions for instability

(0, 0, 0)   3 0, 0, a a

Never

Always

a3 a4

a3 a4

4

(1, 0, 0) 

a4 −a3 1+a3 , 0, a a4 +1 4 +1

>1

Never 

(d1 , 1 − d1 , 0)

0
0 such that d1 > 1 + , then we have the following inequality if time τ is large enough, y 0 = γ1 y (x − d1 ) < γ1 y (1 +  − d1 )

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Therefore, we have lim supτ →∞ y(τ ) = 0 if d1 > 1. This indicates that for any δ > 0, there exists a T large enough such that we have x0 ≥ x(1 − x − δ) ⇒ lim inf τ →∞ x(τ ) = 1. Since lim supτ →∞ x(τ ) ≤ 1, thus, limτ →∞ x(τ ) = 1. Therefore, (1, 0) is globally stable when d1 > 1. If d1 < 1, then (6) has an interior equilibrium (d1 , 1−d1 ) which is locally asymptotically stable while both (1, 0) and (0, 0) are saddle nodes. By Poincar´e-Bendixson Theorem (Guckenheimer and Holmes 1983), the omega limit set of System (6) is either a fixed point or a limit cycle. Now define a scalar function φ(x, y) =

1 xy ,

then we have ∂ (φ(x, y)x0 ) ∂ (φ(x, y)y 0 ) 1 + = − < 0 for all (x, y) ∈ R2+ . ∂x ∂y y Therefore, according to Dulac’s criterion (Guckenheimer and Holmes 1983), we can conclude that the trajectory of any initial condition taken in R2+ converges to an equilibrium point. Since the only locally stable equilibrium of (6) when d1 < 1 is the interior equilibrium (d1 , 1 − d1 ), therefore, it must be global stable. In the case that d1 = 1, (6) has two equilibria (0, 0) and (1, 0) where (0, 0) is unstable. Then according to Poincar´e-Bendixson Theorem (Guckenheimer and Holmes 1983) again, we can conclude that all trajectories should converge to (1, 0), thus, (1, 0) is global stable. The necessary conditions are easily derived from the local stability.



The subsystems of System (4) and (5) that includes only species x and z can be represented in (7) and (8), respectively x0 = x(1 − x − z)

(7)

z 0 = γ2 z (x − d2 ) . and x0 = x(1 − x − z)

(8)

z 0 = γ2 z (a3 − a4 z + x) . where their dynamics can be summarized as the following proposition: Proposition 2 (Subsystem of species x and z) The subsystem (7) is globally stable at (1, 0) if and only if d2 ≥ 1 while it is globally stable at (d2 , 1 − d2 ) if and only if 0 < d2 < 1. The subsystem (8) is globally     4 −a3 a3 +1 , if and only if a4 > a3 . stable at 0, aa43 if and only if a4 ≤ a3 while it is globally stable at a1+a 1+a4 4 Proof The proof of the first part of Proposition 2 is similar to the proof shown in Proposition 1, thus, we only show the second part of Proposition 2 in details.

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If a4 < a3 , then (8) has only one locally asymptotically stable boundary equilibria unstable equilibria (0, 0) and (1, 0). Since we have lim inf τ →∞ z(τ ) ≥

a3 a4



0, aa34



and two

from Lemma 1, therefore we have

the following inequality if a4 < a3 when time τ large enough: x0 = x(1 − x − z) ≤ x(1 − Therefore, we have limτ →∞ z(τ ) =

a3 a4

a3 ) ⇒ lim sup x(τ ) = 0. a4 τ →∞

if a3 > a4 .

  4 −a3 a3 +1 If a4 > a3 , then (8) has an interior equilibrium a1+a , which is locally asymptotically stable while 1+a 4 4   a3 (0, 0), (1, 0) and 0, a4 are saddle nodes. By Poincar´e-Bendixson Theorem (Guckenheimer and Holmes 1983), the omega limit set of System (8) is either a fixed point or a limit cycle. Now define a scalar function φ(x, y) =

1 xz ,

then we have ∂ (φ(x, y)x0 ) ∂ (φ(x, y)y 0 ) 1 a4 + =− − < 0 for all (x, z) ∈ R2+ . ∂x ∂y z x

Therefore, according to Dulac’s criterion (Guckenheimer and Holmes 1983), we can conclude that the trajectory of any initial condition taken in R2+ converges to an equilibrium point. Since the only locally stable   4 −a3 a3 +1 equilibrium of (8) when a4 > a3 is the interior equilibrium a1+a , 1+a4 , therefore, it must be global stable. 4   a3 In the case that a3 = a4 , (8) has 0, a4 and the other two unstable equilibria (0, 0) and (1, 0). Then according to Poincar´e-Bendixson Theorem (Guckenheimer and Holmes 1983) again, we can conclude that     all trajectories should converge to 0, aa34 , thus, 0, aa34 is global stable. The necessary conditions are easily derived from the local stability.



Notes: Proposition 1 and 2 indicate that the subsystem of both System (4) and (5) have relative simple dynamics by applying Poincar´e-Bendixson Theorem and Dulac’s criterion (Guckenheimer and Holmes 1983). To proceed the statement and proof of our main results of this section, we provide the definition of persistence and permanence as follows: Definition 1 (Persistence of single species) We say species x is persistent in R3+ for either System (4) or (5) if there exists constants 0 < b < B, such that for any initial condition with x(0) > 0, the following inequality holds b ≤ lim inf x(τ ) ≤ lim sup x(τ ) ≤ B. τ →∞

τ →∞

Similar definitions hold for species y and z. Definition 2 (Permanence of a system) We say System (4) is permanent in R3+ if there exists constants 0 < b < B, such that for any initial condition taken in R3+ with x(0)y(0)z(0) > 0, the following inequality holds b ≤ lim inf min{x(τ ), y(τ ), z(τ )} ≤ lim sup max{x(τ ), y(τ ), z(τ )} ≤ B. τ →∞

τ →∞

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From Lemma 1, we know that species z in System (5) is persistent for any initial value taken in the interior of R3+ . Similarly, we should expect that species x of System (4) is persistent in R3+ (see Theorem 1). The following theorem 1 implies that the local stability of (1, 0, 0) of System (4) and the local stability of (0, 0, aa43 ) of System (5) indicate their global stability. Theorem 1 (Persistence of single species) Species x is persistent in R3+ for System (4) which is global stable at (1, 0, 0) if di > 1, i = 1, 2. Similarly, species z is persistent in R3+ for System (5) which is global stable at (0, 0, aa34 ) if a3 > a4 . Proof Notice that the omega limit set of the y-z subsystem of System (4) is (0, 0, 0), thus according to Theorem 2.5 of Hutson (1984), we can conclude species is persistent in R3+ since dx = (1 − x − y − z) = 1 > 0. xdt (0,0,0) (0,0,0) Define V = γ2 a2 y + γ1 a1 z, then for any  > 0 such that 1 +  < min{d1 , d2 }, there exists a time T large enough such that for any τ > T we have V 0 = γ2 a2 y 0 + γ1 a1 z 0 = γ1 γ2 a2 y(x − d1 ) + γ1 γ2 a1 z(x − d2 ) ≤ γ1 γ2 a2 y(1 +  − d1 ) + γ1 γ2 a1 z(1 +  − d2 ) ≤ max{1 +  − d1 , 1 +  − d2 }(γ2 a2 y + γ1 a1 z) = max{1 +  − d1 , 1 +  − d2 }V. Since max{1 +  − d1 , 1 +  − d2 } < 0, thus, we have limτ →∞ V (τ ) = 0. This implies that the limiting system of (4) is x0 = x(1 − x), therefore, limτ →∞ x(τ ) = 1. This indicates that (1, 0, 0) is global stable for System (4) if di > 1, i = 1, 2. Now assume that a3 > a4 for System (5), then from Lemma 1, we can conclude that for any  > 0 such that 1 + 
T we have     a a3 1− a3 + t 0 4 x = x(1 − x − y − z) ≤ x(1 − z) ≤ x 1 − +  ⇒ x(t) < x(0)e → 0 as t → ∞. a4

Since species y feeds on species x, thus, we have limτ →∞ y(τ ) = 0. This implies that the limiting system of (5) is z 0 = γ2 z(a3 − a4 z), therefore, limτ →∞ z(τ ) = System (5) if a3 > a4 .

a3 a4 .

This indicates that (0, 0, aa34 ) is global stable for



Theorem 2 (Persistence of two species) For System (4), species x and y are persistent in R3+ if d2 > min{1, d1 }. In addition, System (4) has global stability at (d1 , 1 − d1 , 0) if d2 > 1 > d1 and d2 > 1 + a2 . 2

a2 (d1 −1) Similarly, species x and z are persistent in R3+ if d1 + (−1+d 2 2 > d2 & d1 < 1 or d2 < 1 < d1 . In addition, 1 ) +β

System (4) has global stability at (d2 , 0, 1 − d2 ) if d1 > 1 > d2 . For System (5), species x and z are persistent

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in R3+ if a3 < a4 . If, in addition, d1 > 1, then System (5) has global stability at species y and z are persistent in R3+ if a3 < a4 and

a4 −a3 1+a4



a4 −a3 a3 +1 1+a4 , 0, 1+a4



. Similarly,

> d1 .

Proof Since species x is persistent for System (4), thus, sufficient conditions for the persistence of species x and y or the persistence of species x and z are equivalent to sufficient conditions for the persistence of y, z, respectively. From Lemma 1 and Theorem 1, we can restrict the dynamics of System (4) on a compact set C = [b, B] × [0, B] × [0, B]. The omega limit set of the x-z subsystem restricted on C has two situations according to Proposition 2: If d2 > 1, then the omega limit set of the x-z subsystem restricted on C for System (4) is (1, 0, 0) while if d2 < 1, the omega limit set is (d2 , 0, 1 − d2 ). Therefore, according to Theorem 2.5 of Hutson (1984), we can conclude species y is persistent in R3+ if   a1 yz dy = γ1 x − 2 = 1 − d1 > 0 ⇒ d2 > 1 > d1 − d1 d2 > 1 and ydt (1,0,0) y + β2 (1,0,0) or   dy a1 yz d2 < 1 and = γ1 x − 2 − d1 = d2 − d1 > 0 ⇒ 1 > d2 > d1 . ydt (d2 ,0,1−d2 ) y + β2 (d2 ,0,1−d2 ) Similarly, we can conclude that species z is persistent in R3+ for System (4) if   dz a2 y 2 d1 > 1 and = γ2 x + 2 − d = 1 − d2 > 0 ⇒ d1 > 1 > d2 2 zdt (1,0,0) y + β2 (1,0,0) or d1 < 1 and

  a2 (1 − d − 1)2 dz a2 y 2 = γ2 x + 2 − d = d1 + − d2 > 0. 2 2 zdt (d1 ,1−d1 ,0) y +β (1 − d1 )2 + β 2 (d1 ,1−d1 ,0)

If d2 > 1 > d1 , then species x and y are persistent R3+ for System (4). If, in addition, species z goes to extinction, then the limiting system of (4) is the x-y subsystem. Since d1 < 1, thus, according to Proposition 1,   y2 we can conclude that (d1 , 1−d1 , 0) is global stable. Assume that d2 > 1+a2 . From z 0 = γ2 z x + ya22+β 2 − d2 and Lemma 1, then we have follows if time is large enough:   a2 y 2 0 z = γ2 z x + 2 − d2 < γ2 z (1 + a2 − d2 ) ⇒ z(τ ) → 0 as τ → ∞. y + β2 Therefore, System (4) has global stability at (d1 , 1 − d1 , 0) if d2 > 1 > d1 and d2 > 1 + a2 . If d1 > 1 > d2 , then species x and z are persistent R3+ for System (4). If, in addition, species y goes to extinction, then the limiting system of (4) is the x-z subsystem. Since d2 < 1, thus, according to Proposition   yz 1, we can conclude that (d2 , 0, d2 ) is global stable. From y 0 = γ1 y x − ya21+β and Lemma 1, then we 2 − d1 have follows if time is large enough:   a1 yz y 0 = γ1 y x − 2 − d < y 0 = γ1 y (1 − d1 ) ⇒ y(τ ) → 0 as τ → ∞. 1 y + β2

14

Therefore, System (4) has global stability at (d2 , 0, 1 − d2 ) if d1 > 1 > d2 . According to Lemma 1, we can also restrict the dynamics of System (5) on the compact set C = [b, B] × [0, B] × [0, B]. Since species z is persistent for System (5), thus, sufficient conditions for the persistence of species x and z or the persistence of species y and z are equivalent to sufficient conditions for the persistence of x, y, respectively. The omega limit set of the y-z subsystem restricted on C is (0, 0, aa34 ). Therefore, according to Theorem 2.5 of Hutson (1984), we can conclude species x is persistent in R3+ if dx a3 = (1 − x − y − z) =1− > 0 ⇒ a3 < a4 . a3 a xdt (0,0, a4 ) a (0,0, a3 ) 4 4   yz Since y 0 = γ1 y x − ya21+β ≤ γ1 y (x − d1 ), thus, according to Lemma 1, species y of System (5) goes to 2 − d1 extinct if d1 > 1. Therefore, the limiting system is the x-z subsystem if a3 < a4 and d1 > 1. Then according   a3 +1 4 −a3 , 0, to Proposition 2, System (5) has global stability at a1+a 1+a4 . 4 Since species y feeds on species x, thus its persistence requires the persistence of species x, i.e., a3 < a4 .   a3 +1 4 −a3 This implies that the omega limit set of the x-z subsystem restricted on C for System (5) is a1+a , 0, . 1+a 4 4 Therefore, according to Theorem 2.5 of Hutson (1984), we can conclude species y is persistent in R3+ if   dy a4 − a3 a1 yz − d1  a4 −a3 a3 +1  = − d1 > 0.  a4 −a3 a3 +1  = γ1 x − 2 2 ydt 1+a4 ,0, 1+a4 y +β 1 + a4 1+a4 ,0, 1+a4 Therefore, sufficient condition for the persistence of species y is a3 < a4 and

a4 −a3 1+a4

> d1 .



Notes: Sufficient conditions for the persistence of two species for both System (4) and (5) are stated in Table 4. For both (4) and (5), the persistence of species y indicates that both species x and y are persistent. From Theorem 2, we can obtain sufficient conditions for the persistence of three species by looking at sufficient conditions for the persistence of two species: For System (4), if d1 +

a2 (d1 −1)2 (−1+d1 )2 +β 2

> d2 > d1 & d2 < 1, then

species x and y are persistent and species x and z are also persistent, thus, species x, y, z are also persistent. For System (5), the persistence of species y indicates that the persistence of three species since species z is always persistent and species y feeds on species x. Thus, we have the following corollary for sufficient conditions of the persistence of three species for both System (4) and (5). Corollary 1 (Persistence of three species) System (4) is permanent in R3+ , i.e., three species x, y, z are all persistent in R3+ , if the following inequalities hold d1 +

a2 (d1 − 1)2 > d2 > d1 & d2 < 1. (−1 + d1 )2 + β 2

Similarly, System (5) is permanent in R3+ if the following inequalities hold a3 a4 − a3 < 1& > d1 . a4 1 + a4

15 Persistent Species

Sufficient Conditions for (4)

Sufficient Conditions for (5)

species x

Always

a3 a4

d2 & d1 < 1 or d2
d2 & d1 < 1 or d2
1 species x, y species x, z

d1 < min{1, d2 } d1 +

a2 (d1 −1)2 (−1+d1 )2 +β 2

a4 −a3 1+a4

> d1

a4 −a3 1+a4

> d1

d 2 > d 1 & d2 < 1

Table 4: Persistence results of (4) and (5)

Notes: Sufficient conditions for the persistence of three species for both System (4) and (5) are also stated in Table 4. From Theorem 1, we know that species x, z are always persistent for System (4), (5) respectively. Since species y feeds on species x for both systems, thus we have the following cases: 1. For System (4): From Theorem 1 and 2, the extinction of species y happens when System (4) has global stability either at (d2 , 0, 1 − d2 ) or at (1, 0, 0), i.e., d2 < min{1, d1 } or d1 > 1. Since species y feeds on species x, thus, according to Theorem 1, the extinction of species x happens when System (4) has global stability at (1, 0, 0), i.e., di > 1, i = 1, 2. 2. Similarly, for System (5), species y goes to extinction if d1 > 1 or

a3 a4

> 1 and species x goes to extinction

if a3 > a4 . The summary of sufficient conditions for the extinction of single species are stated in the following Theorem 3 (see Table 5). Theorem 3 (Extinction of single species) Sufficient conditions for the extinction of species y, z of System (4) and sufficient conditions for the extinction of species x, y of System (5) are stated in Table 5.

Extinct Species

Sufficient Conditions for (4)

Sufficient Conditions for (5)

species x

Never

a3 a4

species y

d2 < min{1, d1 } or d1 > 1

d1 > 1 or

species z

d2 > 1 + a2

Never

species x, y

Never

a3 a4

species y, z

di > 1, i = 1, 2

Never

>1 a3 a4

>1

>1

Table 5: Extinction results of (4) and (5)

16

Proof Here we only show that if d2 < min{1, d1 } or d1 > 1, then species y goes to extinction for System (4) γ1

while other results listed in Table 5 can be obtained from Theorem 1-2. Define a scalar function V = yz − γ2 , then according to System (4), we have dV dτ

γ1

γ1

= y 0 z − γ2 − γγ12 yz − γ2 −1 z 0    γ1 γ1 yz = yz − γ2 γ1 x − ya21+β − yz − γ2 γ1 z x + 2 − d1   yz a2 y 2 = γ1 V x − ya21+β 2 − d1 − x − y 2 +β 2 + d2

a2 y 2 y 2 +β 2

− d2



≤ γ1 V (−d1 + d2 ) . Therefore, if d1 > d2 , then we have γ1

V (τ ) = y(τ )z − γ2 (τ ) → 0 as τ → ∞. According to Theorem 2, species z is persistent when d2 < 1 < d1 for System (4). This implies that y(τ ) → 0 as τ → ∞. If d1 > 1, then according to the proof of Theorem 2, we also have y(τ ) → 0 as τ → ∞. Therefore, species y goes to extinction for System (4) if d2 < min{1, d1 } or d1 > 1.



Notes: From numerical simulations, it suggests that for System (4), species z goes to extinction if d2 > ac where ac ∈ [0, 1 + a2 ]. In the case that d2 > 1 + a2 , species z definitely goes extinct. The main results of this section can be summarized in the following two-dimensional bifurcation diagrams (see Figure 2-3) where G.S. indicates the global stability and L.S. indicates the local stability. By comparing Figure 2 and Figure 3, we have the following conclusions that can partially answer the first question proposed in the introduction: 1. The relative growth rates γi , i = 1, 2 do not affect species persistence for both (4) and (5). The persistence of our IGP model with specialist predator (4) are determined by the relative death rate of IG prey and predator di , while the persistence of our IGP model with generalist predator (5) are determined by the availability of outside resource, i.e., a3 and a4 . This suggests that Model (5) has “top down” regulation. 2. Notice that, for (4), two nontrivial boundary equilibria are Exy = (d1 , 1 − d1 , 0) and Exz = (d2 , 0, 1 − d2 ), the persistence condition for species y is 1 − d1 > 1 − d2 ⇒ d1 < min{1, d2 }; While for (5), two nontrivial   1+a3 3 , 0, boundary equilibria are Exy = (d1 , 1 − d1 , 0) and Exz = aa44−a +1 a4 +1 , the persistence condition for species y is 1 − d1 >

1+a3 a4 +1

⇒ d1
1

II: species x persistent

0.8

a4

0.6 a4=(a3+d1)/(1−d1)

I: (0,0,a3/a4) G.S. 0.4 a4=a3

0.2

0 0

0.2

0.4

a3

0.6

0.8

1

Fig. 3: Two dimensional bifurcation diagram of a3 v.s. a4 for System (5).

Since the two cases mentioned above are the cases when System (4)-(5) have multiple attractors, to continue our study, we first explore the possible number of interior equilibria for both System (4) -(5). Assume that (x∗ , y ∗ , z ∗ ) is an interior equilibrium of System (4), then we have 1 − x∗ − y ∗ − z ∗ = 0 x∗ − x∗ +

(9) ∗ ∗

∗ ∗



y z x − d1 a1 y z − d1 = 0 ⇒ ∗ 2 = (y ∗ )2 + β 2 (y ) + β 2 a1

(10)

a2 (y ∗ )2 (y ∗ )2 d2 − x∗ − d = 0 ⇒ = ⇒ y∗ = 2 (y ∗ )2 + β 2 (y ∗ )2 + β 2 a2

s

β 2 (d2 − x∗ ) . x∗ + a2 − d2

(11)

Thus, from (10)-(11), we can conclude that max{d1 , d2 − a2 } < x∗ < d2 < 1 and a2 (x∗ − d1 ) a2 (x∗ − d1 ) ∗ y = z = ∗ a1 (d2 − x ) a1 (d2 − x∗ )

s



Then from (9), we have 1 − x∗ − β 1−x∗ β



=

q

(d2 −x∗ ) x∗ +a2 −d2



a2 β(x∗ − d1 ) β 2 (d2 − x∗ ) = p . ∗ x + a2 − d2 a1 (d2 − x∗ )(x∗ + a2 − d2 )

a1



a2 β(x∗ −d1 ) (d2 −x∗ )(x∗ +a2 −d2 )

a1 (d2 −x∗ )+a2 (x∗ −d1 )



a1 (d2 −x∗ )(x∗ +a2 −d2 ) √ a1 (1−x∗ ) (d2 −x∗ )(x∗ +a2 −d2 ) = β

=

= 0 which implies that

a1 d2 −a2 d1 −(a1 −a2 )x∗ a1



(d2 −x∗ )(x∗ +a2 −d2 )

a1 d2 − a2 d1 − (a1 − a2 )x∗ > 0 √

2 d1 subject to max{d1 , d2 − a2 } < x∗ < min{d2 , a1 ad21 −a −a2 }. Let f1 =

a1 (1−x)

(d2 −x)(x+a2 −d2 ) β

and f2 = a1 d2 −

a2 d1 − (a1 − a2 )x, then we can see that the interior equilibria of System (4) is determined by the intercepts

19

of f1 and f2 in the first quadrant. Notice that f1 is convex and f2 is a straight line with restriction that a1 d2 − a2 d1 a1 d2 − a2 d1 > 0 and max{d1 , d2 − a2 } < x < min{d2 , }, a1 − a2 a1 − a2 then we can classify the possible scenarios in the following categories: 1. If a1 > a2 > d2 , then we have

a1 d2 −a2 d1 a1 −a2

> d2 . Notice that even if f1 and f2 have an intercept in the first

quadrant, it is possible for System (4) has no interior equilibrium due to the restriction that max{d1 , d2 − a2 } < x < d2 . For example Graph (d) of Figure 4 has no interior equilibrium since the x-coordinator of the intercept of f1 and f2 is less than d1 . In summary, System (4) can have zero, one or two interior equilibria (see Figure 4). 2. If min{a1 , d2 } > a2 , then we have

a1 d2 −a2 d1 a1 −a2

> d2 . Notice that even if f1 and f2 have an intercept in

the first quadrant, it is possible for System (4) has no interior equilibrium due to the restriction that max{d1 , d2 − a2 } < x < d2 . For example Graph (d) of Figure 5 has no interior equilibrium since the x-coordinator of the intercept of f1 and f2 is less than d1 . It is also possible for Graph (b) of Figure 5 has no interior equilibrium if all the x-coordinators of the intercepts of f1 and f2 are less than d1 . In summary, System (4) can also have zero, one or two interior equilibria (see Figure 5). 3. If a2 > max{a1 , d2 }, then we have

a1 d2 −a2 d1 a1 −a2

=

a2 d1 −a1 d2 a2 −a1

< d1 since d2 > d1 . Notice that even if f1 and

f2 have an intercept in the first quadrant, it is possible for System (4) has no interior equilibrium due to the restriction that max{d1 , d2 − a2 } < x < d2 . For example Graph (c) and (e) of Figure 6 have no interior equilibrium since the x-coordinator of the intercept of f1 and f2 is less than d1 . System (4) can have zero or one interior equilibrium (see Figure 6). 4. If d2 > a2 > a1 , then we have

a1 d2 −a2 d1 a1 −a2

=

a2 d1 −a1 d2 a2 −a1

< d1 since d2 > d1 . Notice that even if f1 and f2

have an intercept in the first quadrant, it is possible for System (4) has no interior equilibrium due to the restriction that max{d1 , d2 − a2 } < x < d2 . For example Graph (c) and (f) of Figure 7 have no interior equilibrium since the x-coordinator of the intercept of f1 and f2 is less than d1 . System (4) can also have zero, one or two interior equilibria (see Figure 7). From the discussions above and the schematic graphical representations (see Figure 4, 5, 6 and 7), we can conclude that System (4) may have zero, one or two interior equilibria. If (x∗ , y ∗ , z ∗ ) is an interior equilibrium of System (5), then we have 1 − x∗ − y ∗ − z ∗ = 0 a1 y ∗ z ∗ a1 y ∗ z ∗ x∗ − ∗ 2 − d1 = 0 ⇒ 1 − d1 − y ∗ − z ∗ − ∗ 2 =0 2 (y ) + β (y ) + β 2 a2 (y ∗ )2 a2 (y ∗ )2 = 0 ⇒ 1 + a3 − y ∗ − (1 + a4 )z ∗ + ∗ 2 =0 a3 − a4 z ∗ + x∗ + ∗ 2 2 (y ) + β (y ) + β 2

(12) (13) (14)

20

y

y a1 > a2 > d2

a1 > a2 > d2

d2 − a2

0

d2 − a2

a1 d2 −a2 d1 a1 −a2

d2

0

(a)

a1 d2 −a2 d1 a1 −a2

x

(b)

y

y a1 > a2 > d2

a1 > a2 > d2

x = d1

d2 − a2

d2

x

x = d1 a1 d2 −a2 d1 a1 −a2

0

d2

x

d2 − a2

a1 d2 −a2 d1 a1 −a2

0

(c)

(d)

Fig. 4: For System (4) when a1 > a2 > d2 subject to max{d1 , d2 − a2 } < x < d2 }. The solid curve is f1 ; the dashed line is f2 while the dotted line is x = d1 . Graph (a) and (d) have no interior equilibrium; Graph (c) has one interior equilibrium and Graph (b) has two interior equilibria.

Thus, from (13)-(14), we have z∗ =

(1 − d1 − y ∗ )((y ∗ )2 + β 2 ) (1 + a2 + a3 − y ∗ )((y ∗ )2 + β 2 ) − a2 β 2 ∗ and z = (y ∗ )2 + β 2 + a1 y ∗ ((y ∗ )2 + β 2 )(1 + a4 )

subject to x∗ > d1 , z ∗ >

a3 a4



and y ∗ < 1 − d1 . This implies that

∗ ∗ 2 2 2 (1−d1 −y ∗ )((y ∗ )2 +β 2 ) 3 −y )((y ) +β )−a2 β = (1+a2 +a (y ∗ )2 +β 2 +a1 y ∗ ((y ∗ )2 +β 2 )(1+a4 ) [(1+a2 +a3 −y∗ )((y∗ )2 +β 2 )−a2 β 2 ]((y∗ )2 +β 2 +a1 y∗ ) (1 − d1 − y ∗ ) = (1+a4 )((y ∗ )2 +β 2 )2

d2 x

21

y

d2 − a2

0

y

a1 > a2 d2 > a2

a1 d2 −a2 d1 a1 −a2

d2

0

d2 − a2

a1 > a2 d2 > a2

d2 − a2

a1 d2 −a2 d1 a1 −a2

x

(b)

a1 > a2 d2 > a2

y

x = d1

x = d1 0

d2

x

(a)

y

a1 > a2 d2 > a2

a1 d2 −a2 d1 a1 −a2

d2

x

(c)

0

d2 − a2

a1 d2 −a2 d1 a1 −a2

d2

(d)

Fig. 5: For System (4) when min{a1 , d2 } > a2 subject to max{d1 , d2 − a2 } < x < d2 }. The solid curve is f1 ; the dashed line is f2 while the dotted line is x = d1 . Graph (a) and (d) have no interior equilibrium; Graph (c) has one interior equilibrium and Graph (b) has two interior equilibria.

Let g1 = (1 − d1 − y) and g2 =

[(1+a2 +a3 −y)(y2 +β 2 )−a2 β 2 ](y2 +β 2 +a1 y) (1+a4 )(y 2 +β 2 )2

, then the intercepts of the functions g1

and g2 can provide us the information on the number of the interior equilibrium of System (5). In addition, we have the follows:

1. g2 (y) has exactly one x-intercept (y o , 0) (i.e., g2 (y o ) = 0) due to the fact that (1 + a2 + a3 − y)(y 2 + β 2 ) = a2 β 2 has exactly one positive real root.

x

22

2. g2 (1) =

[(1+a2 +a3 −1)(1+β 2 )−a2 β 2 ](1+β 2 +a1 ) (1+a4 )(1+β 2 )2

> 0 and g2 (1 + a2 + a3 ) < 0, thus, we have

1 − d1 < 1 < y o < 1 + a2 + a3 . 3. g1 (0) = 1 − d1 and g2 (0) =

1+a3 1+a4

=1−

a4 −a3 1+a4 .

Thus, we can classify System (5) into the following two cases depending on g1 (0) > g2 (0) ⇐⇒ d1
1 + a4 1 + a4

1. If the inequality g1 (0) > g2 (0) ⇐⇒ d1
g1 (1 − d1 ) = 0, then the possible number of interior equilibrium is one (see Graph (b) in Figure 8), two (see Graph (c) in Figure 8) and three (see Graph (d) in Figure 8). 2. If the inequality g1 (0) < g2 (0) ⇐⇒ d1 >

a4 −a3 1+a4

holds, then according to Lemma 2, we can conclude   1+a3 3 that System (5) is locally asymptotically stable at the boundary equilibrium aa44−a +1 , 0, a4 +1 . Since g2 (1 − d1 ) > g1 (1 − d1 ) = 0, then the possible number of interior equilibrium is zero (see Graph (a) in Figure 9) or two (see Graph (c) in Figure 9).

From the discussions above and the schematic graphical representations (see Figure 8-9, 6), we can conclude that System (5) may have zero, one, two or three interior equilibria where one or three interior equilibria happens when System (5) is permanent (see Figure 8). For convenience, let f1 (xc ) = maxd2 −a2 ≤x≤d2 {f1 (x)} and g2 (y2 ) = max0≤y≤1+a2 +a3 {g2 (y)}. Then according to all discussions above and sufficient conditions on the global stability at the boundary equilibrium, we can summarize the results on the possible number of interior equilibrium for System 4 and System 5 in the following theorem: Theorem 4 (Number of interior equilibrium) Sufficient conditions for none, one, two or three interior equilibria of System 4 and System 5 can be partially summarized as the following table 6 Notes: Theorem 4 suggests that both System (4) can have multiple interior equilibria when it is or has locally asymptotically stable boundary equilibrium while System (5) can have multiple interior equilibria when it is permanent. Now we are investigating the possible multiple attractors for both System (4) and (5). Multiple attractors of System (4): Here we consider two cases, i.e., when System (4) is permanent and is locally asymptotically stable at (d1 , 1 − d1 , 0): 1. Let r1 = 25; r2 = 1; β = .1; a1 = 1; a2 = .6; d1 = .15; d2 = .54, then according to Corollary 1, we know that System (4) is permanent. From numerical simulations (see Figure 10), we can see that (4) has two attractors where is a locally stable interior equilibrium and the other is a limit cycle.

23 Number

of

interior

Sufficient Conditions for (4)

Sufficient Conditions for (5)

equilibrium None

1. d1 < d2 ; or 2. di > 1, i = 1 or 2; or 3. a2 > a1 , d2 > d1 &d2
a4 ; or 2. No easy expression (see Graph (a) of Figure 9).

(see Graph (a) of Figure 6-7); or 4. a2 < a1 , d2 > d1 &d2 − a2 > One

1. a2


f2 (d1 )&b2 − a2 < a1 d2 −a2 d1 a1 −a2

< d2 (see Graph (b), (d) of

Figure 6 and Graph (b) of Figure 7) Two

1. a2




d1 , f1 (xc )

f2 (xc )&d1 + a2 < d2 < (see

Graph

2. a2

>

(b) a1 , d2

2 d1 f2 (xc )& a1 ad2 −a 1 −a2

Three

>

a1 d2 −a2 d1 a1 −a2

d1

>

a4 −a3 &g2 (yc ) 1+a4

> ∈

g1 (yc )& there some y

5);

or

(yc , y o ) such that g2 (y)

d1 , f1 (xc )

>

(see Graph (c) of Figure 9). However,

< d1 < d2 − a2 (see

numerical simulations do not find this

of >

Figure

Graph (d) of Figure 7).

case.

Never

d1
1 − d2 ⇒ d1 < min{1, d2 }; While for (5), two nontrivial   1+a3 3 boundary equilibria are Exy = (d1 , 1 − d1 , 0) and Exz = aa44−a +1 , 0, a4 +1 , the persistence condition for species y is 1 − d1 >

1+a3 a4 +1

⇒ d1
d2 , then the population of species z goes to infinity which drives both species x and y go extinct. – If d1 > 1 and d2 > 1 + a2 , then Model (16) has global stability at (1, 0, 0). – If d1 < 1 and d2 > a2 + d1 , then Model (16) has local stability at (1, 0, 0). The nontrivial equilibrium (1, 0, 0) is global stable if d1 < 1 and d2 > a2 + 1. – If 1 + a2 > d2 > a2 + d1 , then Model (16) has a unique interior equilibrium which is always unstable while it has local stability at (1, 0, 0). This implies that Model (16) does not have alternative stable states (i.e., multiple attractors). In fact, the coexistence of three species seems impossible from numerical simulations. This suggests that the

27

increasing return to scale of foraging by the IG predator with increasing of density IG prey destablizes the system while the Holling-Type III functional response generates rich dynamics with the possibility of the coexistence. These results presented above may help to explain dynamics in terms of specific parameters which enables biologists to pinpoint parameters that will be most beneficial to maintain or alter persistence and extinction of species. Our study may have important implications in conservation biology and agriculture which may provide useful insights that can help aid policy makers in making decisions regarding conservation of specific species.

Acknowledgement The research of Y.K. is partially supported by Simons Collaboration Grants for Mathematicians (208902).

References 1. S. Allesina and M. Pascual, 2008. Network structure, predator-prey modules, and stability in large food webs, Theoretical Ecology, 1, 55-64. 2. P. Amarasekare, 2007. Spatial dynamics of communities with intraguild predation: the role of dispersal strategies, American Naturalist, 170, 819-831. 3. P. Amarasekare, 2008. Coexistence of intraguild predators and prey in resource-rich environments, Ecology, 89, 2786-2797. 4. M. Arim and P. A. Marquet , 2004. Intraguild predation: a widespread interaction related to species biology, Ecology Letter, 7, 557-564. 5. J. Brodeur and J. A. Rosenheim, 2000. Intraguild interactions in aphid parasitoids, Entomologia Experimentalis et Applicata, 97, 93-108. 6. S. Diehl, 1993. Relative consumer sizes and the strengths of direct and indirect interactions in omnivorous feeding relationships, Oikos, 68, 151-157. 7. J.S. Elkinton, W. M. Healy, J. P. Buonaccorsi, G. H. Boettner, A. Hazzard, H. Smith, and A. M. Liebhold,1996. Interactions among gypsy moths, white-footed mice, and acorns. Ecology, 77, 2332-2342. 8. J. S. Elkinton, W. M. Healy, J.P. Buonaccorsi, G.H. Boettner, A. Hazzard, H. Smith and A.M. Liebhold,1998. Gypsy moths, mice and acorns. USDA Forest Service. 9. D. L. Finke, and R. F. Denno, 2005. Predator diversity and the functioning of ecosystems: the role of intraguild predation in dampening trophic cascades, Ecology Letter, 8,1299-1306. 10. J. Guckenheimer and P. Holmes, 1983. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag. 11. R. Hall, 2011. Intraguild predation in the presence of a shared natural enemy, Ecology, 92, 352-361. 12. I. Hanski, L. Hansson and H. Henttonen, 1991. Specialist Predators, Generalist Predators, and the Microtine Rodent Cycle, Journal of Animal Ecology, 60, 353-367.

28 13. M. P. Hassell and R. M. May, 1986. Generalist and Specialist Natural Enemies in Insect Predator-Prey Interactions, British Ecological Society, 55,923-940. 14. A. Hastings, 1988. Food Web Theory and Stability, Ecology, 69,1665-1668. 15. F. L. Hastings, F. P. Hain, H. R. Smith, S. P. Cook and J. F. Monahan, 2002. Predation of Gypsy Moth (Lepidoptera: Lymantriidae) Pupae in Three Ecosystems Along the Southern Edge of Infestation, Environ Entomol, 31, 668-675. 16. R. D. Holt and G. A. Polis, 1997. A Theoretical Framework for Intraguild Predation, Am Nat, 149,745-764. 17. V. Hutson, 1984. A theorem on average Liapunov functions. Monatshefte f¨ ur Mathematik,98, 267-275. 18. V. Hutson and K. Schimtt, 1992. Permanence and the dynamics of biological systems, Mathematical Biosciences, 111, 1-71. 19. C. G. Jones, R. S. Ostfield, M. P. Richard, E. M. Schauber and J. O. Wolff, 1998. Chain Reactions Linking Acorns to Gypsy Moth Outbreaks and Lyme Disease Risk, Science, 279,1023-1026. 20. K. McCann, A. Hastings and G. R. Huxel, 1998. Weak trophic interactions and the balance of nature, Nature, 395, 794-798. 21. G. A. Polis, C. A. Myers and R. D. Holt, 1989. The ecology and evolution of intraguild predation: Potential competitors that eat each other, Annual Review of Ecology and Systematics, 20 297-330. 22. G. A. Polis and R. D. Holt, 1992. Intraguild Predation: The Dynamics of Complex Trophic Interactions, Trends Ecol, 7, 151-154. 23. N. Rooney, K. McCann, G. Gellner and J. C. Moore, 2006. Structural asymmetry and the stability of diverse food webs, Nature, 442, 265-269. 24. J. A. Rosenheim, 1998. Higher-order predators and the regulation of insect herbivore populations,

Annual Review of

Entomology, 43, 421-447. 25. E. Ruggieri and S. J. Schreiber, 2005. The dynamics of the Schoener-Polis-Holt model of intraguild predation, Mathematical Biosciences and Engineering, 2, 279-288. 26. E. M. Schauber, R. S. Ostfeld and C. G. Jones, 2004. Type 3 functional response of mice to gypsy moth pupae: is it stabilizing? OIKOS, 107, 592-602. 27. M. Sieber and F. M. Hilker, 2011. Prey, predators, parasites: intraguild predation or simpler community modules in disguise?, Journal of Animal Ecology, 80, 414-421. 28. M. Sieber and F. M. Hilker, 2012. The hydra effect in predatorprey models, Journal of Mathematical Biology, 64, 341-360. 29. A. Singh and R. M. Nisbet, 2007. Semi-discrete host-parasitoid models, Journal of Theoretical Biology, 247, 733-742. 30. W. E. Snyder and A. R. Ives, 2003. Interactions Between Specialist and Generalist Natural Enemies: Parasitoids, Predators, and Pea Aphid Biocontrol. Ecological Society of America, 84, 91-107. 31. D. A. Spiller and T. W. Schoener, 1990. A terrestrial field experiment showing the impact of eliminating top predators on foliage damage, Nature, 347, 469-472. 32. K. D. Sunderland, J. A. Axelsen, K. Dromph, B. Freier, J.-L. Hemptinne, N. H. Holst, P. J. Mols, M. K. Petersen, W. Powell, P. Ruggle, H. Triltsch and L. Winder, 1997. Pest control by a community of natural enemies, Acta Jutlandica, 72, 271-326. 33. J. D. Yeakel, D. Stiefs, M. Novak and T. Gross, 2011. Generalized modeling of ecological population dynamics, Theoretical Ecology, 4, 179-194/

29

Appendix .1 The persistence and extinction results in terms of the original parameters In this Appendix, we convert the persistence and extinction results of scaled models (4)- (5) to the results of their original forms (2)- (3).

BE point (0, 0, 0) (1, 0, 0) 

Conditions for stability

Conditions for instability

Never

Always

dg > eg ag Kg and dm > em am Kp

dm , 0, 1 em am Kp

dm em am Kp





dg ,0 eg ag Kp



dm em am


 em am r b 2 d + ap −1+ e a gK

>

+

g g

dg < eg ag Kg

p

or

g

dg > eg ag Kg

Table 7: Specialist Predator (2): Local stability conditions for boundary equilibria (BE)

BE point (0, 0, 0)   0, 0, amrKm (1, 0, 0) rm (rp −Km am ) K a (rm +am em Kp ) , 0, rmr m+K 2 rp rm +Km Kp a2 p m m Kp am em m em   dg dg ,1 − e a K ,0 eg ag Kp g g p

Conditions for instability

Never

Always

am Km rp

p



Conditions for stability

am Km rp

>1

Never 

0
1

< min{1,

+

dg eg ag

p

1

Table 9: Persistence results of (2) and (3) Extinct Species

Sufficient Conditions for (2)

Sufficient Conditions for (3)

species x

Never

species y

dm em am Kp dg > eg ag Kp dg > eg ag Kp

am Km >1 rp dg >1 eg ag Kp

species z species x, y

Never

species y, z

di ei ai Kp


1, i = m, g

>1

Never

Table 10: Extinction results of (2) and (3)

am Km rp

>1

31

y

a2 > a1 d2 < a 2

y

a2 > a1 d2 < a 2

x = d1

d2 − a2

a1 d2 −a2 d1 a1 −a2

d2

0

d2 − a2

0

x

a1 d2 −a2 d1 a1 −a2

(a)

y

d2 x

(b)

y

a2 > a1 d2 < a2

a2 > a1 d2 < a2

x = d1

x = d1 d2 − a2

0

a1 d2 −a2 d1 a1 −a2

d2

x

d2 − a2

a1 d2 −a2 d1 a1 −a2

0

d2 x

(c) y

(d) y

a2 > a1 d2 < a2

a2 > a1 d2 < a2

x = d1 d2 − a2

a1 d2 −a2 d1 a1 −a2

0

d2 x

a1 d2 −a2 d1 a1 −a2

(e)

b2 − a2

b2

0

x (f)

Fig. 6: For System (4) when max{a1 , d2 } < a2 subject to max{d1 , d2 − a2 } < x < d2 . The solid curve is f1 ; the dashed line is f2 while the dotted line is x = d1 . Graph (a), (c), (e) and (f) have no interior equilibrium while Graph (b) and (d) have one interior equilibrium.

32

y

y

d2 > a2 > a1

d2 > a 2 > a 1

x = d1 d2 − a2

0

a d −a d d2 1 a12 −a22 1

0 d2 − a2 x

a1 d2 −a2 d1 a1 −a2

(a)

d2 > a2 > a1

d2 > a 2 > a 1

x = d1

x = d1 d2 − a2

0

a1 d2 −a2 d1 a1 −a2

d2

0

x

a1 d2 −a2 d1 a1 −a2

d2 − a2

(d) y

d2 > a2 > a1

d2 > a2 > a1

x = d1 0

a1 d2 −a2 d1 a1 −a2

d2 − a2

x = d1 d2

0 x

(e)

d2 x

(c) y

x

(b) y

y

d2

a1 d2 −a2 d1 a1 −a2

d2 − a2

d2 x

(f)

Fig. 7: For System (4) when d2 > a2 > a1 subject to max{d1 , d2 − a2 } < x < d2 . The solid curve is f1 ; the dashed line is f2 while the dotted line is x = d1 . Graph (a), (c) and (f) have no interior equilibrium; Graph (b) and (e) have one interior equilibrium and Graph (d) has two interior equilibria.

33 x

1+a3 1+a4

< 1 − d1

1+a3 1+a4

x

< 1 − d1

1 − d1

1 − d1

1+a3 1+a4

1+a3 1+a4

y o 1 − d1

0

1 − d1

0

y

1+a3 1+a4

y

(b)

(a) x

yo

x

< 1 − d1

1 − d1

1+a3 1+a4

< 1 − d1

1 − d1

1+a3 1+a4

1+a3 1+a4

0

1 − d1

yo

(c)

y

1 − d1

0

yo

y

(d)

Fig. 8: Graph (b) and (d) are the schematic cases when the inequality g1 (0) > g2 (0) ⇐⇒ d1
1 − d1 .

34

1+a3 1+a4

x

> 1 − d1

1+a3 1+a4

x

1+a3 1+a4

> 1 − d1

1+a3 1+a4

1 − d1

1 − d1

yo

1 − d1

0

0

y

1+a3 1+a4

y

(b)

(a) x

y o 1 − d1

x

> 1 − d1

1+a3 1+a4

> 1 − d1

1+a3 1+a4

1 − d1

1+a3 1+a4

1 − d1

1 − d1

0

yo

(c)

y

yo

0

1 − d1

y

(d)

Fig. 9: Graph (a) and (c) are the schematic cases when the inequality g1 (0) < g2 (0) ⇐⇒ d1 >

a4 −a3 1+a4

holds for System 5. The dashed line is g1 while the solid curve is g2 . Graph (b) and (d) are impossible due to the restriction y o > 1 − d1 .

35

IGP−specialist−time series

IGP−specialist

0.7

6

Black−resource; Blue−Prey; Red−Specialist Predator

r1=25;r2=1;b=0.1;a1=1; a2=0.6;d1=0.15;d2=0.54;

5 Black−resource; Blue−Prey; Red−Specialist Predator

x(0)=.4; y(0)= 0.2; z(0)=0.05

4

3

2

1

0.6

0.5

0.4 r1=25;r2=1;b=0.1;a1=1; a2=0.6;d1=0.15;d2=0.54;

0.3

x(0)=0.4; y(0)= 0.2; z(0)=0.5

0.2

0.1

0

0

20

40

60

80

100 time t

120

140

160

180

0 0

200

(a) Time series with x(0) = .4, y(0) = .2, z(0) = .05.

20

40

60

80

100 time t

120

140

160

180

200

(b) Time series with x(0) = .4, y(0) = .2, z(0) = .5.

IGP−specialist−3D Phase Plane

IGP−specialist−3D Phase Plane

0.06

0.52

0.05

0.5

r1=25;r2=1;b=0.1;a1=1; a2=0.6;d1=0.15;d2=0.54;

0.04

x(0)=.4; y(0)= 0.2; z(0)=0.05

IGP−Predator

IGP−Predator

x(0)=0.4; y(0)= 0.2; z(0)=0.5 r1=25;r2=1;b=0.1;a1=1; a2=0.6;d1=0.15;d2=0.54;

0.03 0.02 0.01

0.48 0.46 0.44 0.42

0 6

0.4 0.2 5

0.5

4

0.4

3 IGP−Prey

0.3 2

0.6

0.15

0.2 1

0.1 0

0.55

0.1

0.5 0.45

0.05 Resource

0

(c) 3D-phase plane with x(0) = .4, y(0) = .2, z(0) = .05.

IGP−Prey

0.4 0

0.35

Resource

(d) 3D-phase plane with x(0) = .4, y(0) = .2, z(0) = .5.

Fig. 10: When r1 = 25; r2 = 1; β = .1; a1 = 1; a2 = .6; d1 = .15; d2 = .54, System (4) is permanent with two interior attractors. In the figures of time series, the black curve is the population of the shared resource; the blue curve is the population of the IG prey and the red curve is the population of the IG predator.

36

IGP−specialist

IGP−specialist−time series

5

0.7 r1=25;r2=1;b=0.1;a1=1; a2=0.01;d1=0.15;d2=0.54;

4

Black−resource; Blue−Prey; Red−Specialist Predator

Black−resource; Blue−Prey; Red−Specialist Predator

4.5

r1=25;r2=1;b=0.1;a1=1; a2=0.01;d1=0.15;d2=0.54;

3.5

x(0)=.4; y(0)= 0.2; z(0)=0.05

3

2.5

2

1.5

1

0.6

x(0)=.4; y(0)= 0.2; z(0)=0.5

0.5

0.4

0.3

0.2

0.1 0.5

0 0

20

40

60

80

100

120

140

160

180

0 0

200

20

40

60

80

time t

(a) Time series with x(0) = 0.4, y(0) = 0.2, z(0) = 0.05.

100 time t

120

140

160

180

200

(b) Time series with x(0) = 0.4, y(0) = 0.2, z(0) = 0.5.

IGP−specialist−3D Phase Plane

IGP−specialist−3D Phase Plane

0.05

r1=25;r2=1;b=0.1;a1=1; a2=0.01;d1=0.15;d2=0.54;

0.5

x(0)=.4; y(0)= 0.2; z(0)=0.5 0.48 r1=25;r2=1;b=0.1;a1=1; a2=0.011;d1=0.15;d2=0.54;

IGP−Predator

IGP−Predator

0.04

x(0)=.4; y(0)= 0.2; z(0)=0.05

0.03

0.02

0.01

0.46 0.44 0.42 0.4

0 5

0.38 0.2 4

0.5 3

0.4 0.3

2 IGP−Prey

0.6

0.15

0.2

1

0.1 0

0

0.55

0.1

0.5 0.45

0.05 Resource

IGP−Prey

0.4 0

0.35

Resource

(c) 3D-phase plane with x(0) = 0.4, y(0) = 0.2, z(0) = (d) 3D-phase plane with x(0) = 0.4, y(0) = 0.2, z(0) = 0.5. 0.05.

Fig. 11: When r1 = 25; r2 = 1; β = .1; a1 = 1; a2 = .01; d1 = .15; d2 = .54, System (4) has two attractors: one is the boundary equilibrium (d1 , 1 − d1 , 0) while the other is an interior equilibrium. In the figures of time series, the black curve is the population of the shared resource; the blue curve is the population of the IG prey and the red curve is the population of the IG predator.

37

IGP−generalist−time series

IGP−generalist−time series

7

0.8

0.7

x(0)=.4; y(0)= 0.4; z(0)=0.05

6

Black−resource; Blue−Prey; Red−Generalist Predator

Black−resource; Blue−Prey; Red−Generalist Predator

r1=25;r2=1;b=0.15;a1=1; a2=0.01;a3=0.1;a4=4.5;d1=0.15;

5

4

3

2

1

r1=25;r2=1;b=0.15;a1=1; a2=0.01;a3=0.1;a4=4.5;d1=0.15;

0.6

x(0)=.4; y(0)= 0.1; z(0)=0.5 0.5

0.4

0.3

0.2

0.1

0 0

20

40

60

80

100 time t

120

140

160

180

0 0

200

(a) Time series with x(0) = .4, y(0) = .4, z(0) = .05.

20

40

60

80

100 time t

120

140

160

180

200

(b) Time series with x(0) = .4, y(0) = .1, z(0) = .5.

IGP−generalist−3D Phase Plane

IGP−generalist−3D Phase Plane

r1=25;r2=1;b=0.15;a1=1; a2=0.01;a3=0.1;a4=4.5;d1=0.15; x(0)=.4; y(0)= 0.4; z(0)=0.05 0.095

0.5

0.09

r1=25;r2=1;b=0.15;a1=1; a2=0.01;a3=0.1;a4=4.5;d1=0.15;

0.45

x(0)=.4; y(0)= 0.1; z(0)=0.5

0.085 IGP−Predator

IGP−Predator

0.4 0.08 0.075 0.07 0.065

0.35 0.3 0.25

0.06 0.2

0.055 IGP−Prey

0.05 6 5

0.7

4

0.6 0.5

3

0.4

2

0.3 0.2

1 0

IGP−Prey

0.12

Resource

0.1 0

(c) 3D-phase plane with x(0) = .4, y(0) = .4, z(0) = .05.

Resource

0.1

0.75

0.08

0.7 0.65

0.06

0.6

0.04

0.55 0.5

0.02 0

0.45 0.4

(d) 3D-phase plane with x(0) = .4, y(0) = .1, z(0) = .5.

Fig. 12: When r1 = 25; r2 = 1; β = .15; a1 = 1; a2 = .01; a3 = 0.1; a4 = 4.5; d1 = .15, System (5) is permanent with two interior attractors. In the figures of time series, the black curve is the population of the shared resource; the blue curve is the population of the IG prey and the red curve is the population of the IG predator.