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 ba-

sic 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 com-

plicated 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: yun.kang@asu.edu

Lauren Wedekin

Applied Sciences and Mathematics, Arizona State University, Mesa, AZ 85212, USA.

E-mail: lwedekin@asu.edu

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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 multi-

species 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:

dPdt = P (r(1− P

K )− amM − agG)

dGdt = G(bgagP − αmM − dg)dMdt = M(bmamP + βαmG− dm)

(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:

dPdt = P

[rp(1− P

Kp)− agG− amM

]dGdt = G

[egagP − aMG

G2+b2 − dg]

dMdt = M

[emamP + emaG

2

G2+b2 − dm] (2)

5

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

dPdt = P

[rp(1− P

Kp)− agG− amM

]dGdt = G

[egagP − aMG

G2+b2 − dg]

dMdt = M

[rm(1− M

Km) + emamP + em aG2

G2+b2

] (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 dMdt = rmM(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).

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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

Kp, y =

agG

rp, z =

amM

rp, τ = rpt, γ1 =

egagKp

rp, a1 =

ama

egKp,

and

β =rpb

ag, d1 =

dgegagKp

, γ2 =emamKp

rp, a2 =

a

amKp, d2 =

dmemamKp

.

x′ = x(1− x− y − z)

y′ = γ1y(x− a1yz

y2+β2 − d1)

z′ = γ2z(x+ a2y

2

y2+β2 − d2).

(4)

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

x =P

Kp, y =

agG

rp, z =

amM

rp, τ = rpt, γ1 =

egagKp

rp, a1 =

ama

egKp, β =

rpb

ag,

and

d1 =dg

egagKp, γ2 =

emamKp

rp, a2 =

a

amKp, a3 =

rmemamKp

, a4 =rprm

Kma2memKp, d2 =

dmemamKp

.

7

x′ = x(1− x− y − z)

y′ = γ1y(x− a1yz

y2+β2 − d1)

z′ = γ2z(a3 − a4z + x+ a2y

2

y2+β2

).

(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 supτ→∞

x(τ) ≤ 1 anda3a4≤ lim inf

τ→∞z(τ) ≤ lim sup

τ→∞z(τ) ≤ 1 + a2 + a3

a4.

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 x′ = x(1− x− y − z) we have x′ ≤ x(1− x), thus we can conclude that

lim supτ→∞

x(τ) ≤ 1.

Similarly, the positive invariant property indicates that for any initial condition taken in R3+, we have the

following inequality from (5)

z′ = γ2z

(a3 − a4z + x+

a2y2

y2 + β2

)≥ γ2z(a3 − a4z)⇒ lim inf

τ→∞z(τ) ≥ a3

a4.

On the other hand, we have

z′ = γ2z

(a3 − a4z + x+

a2y2

y2 + β2

)≤ γ2z(a3 − a4z + 1 + a2)⇒ lim sup

τ→∞z(τ) ≤ a3 + 1 + a2

a4.

Define v = x+ θ1y + θ2z where (1− γiθi) > 0, i = 1, 2 and a1γ1θ1 > a2γ2θ2, then we have

v′ = x′ + θ1y′ + θ2z

′

= x(1− x)− (1− γ1θ1)xy − (1− γ2θ2)xz − (a1γ1θ1 − a2γ2θ2) y2zy2+β2 − γ1θ1d1y − γ2θ2d2z

≤ x(1− x)− γ1θ1d1y − γ2θ2d2z ≤ (min{γ1d1, γ2d2}+ 1)x−min{γ1d1, γ2d2}v.

Since for any ε > 0, there exists a T > 0 such that x(τ) < 1 + ε for all τ > T . Therefore, for τ > T , we have

v′ < (min{γ1d1, γ2d2}+ 1)(1 + ε)−min{γ1d1, γ2d2}v.

Thus, we can conclude that

lim supτ→∞

v(τ) ≤ min{γ1d1, γ2d2}+ 1

min{γ1d1, γ2d2}.

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 = γ1x+ y. Then we have

v′ = γ1x′ + y′ = γ1x(1− x− z) + γ1y(− a1yz

y2 + β2− d1) ≤ γ1(x− d1y) ≤ γ1(1 + γ1d1 − d1v).

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This implies that

lim supτ→∞

v(τ) ≤ 1 + d1γ1d1

⇒ lim supτ→∞

y(τ) ≤ 1 + d1γ1d1

.

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 a3a4< 1:

(0, 0, 0), (1, 0, 0), (d1, 1− d1, 0),

(a4 − a3a4 + 1

, 0,a3 + 1

a4 + 1

)and

(0, 0,

a3a4

).

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+β2 < d2 and d1 < 1 d1 +a2(d1−1)2

(−1+d1)2+β2 > d2 or d1 > 1

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

Notes: Lemma 2 indicates follows:

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BE point Conditions for stability Conditions for instability

(0, 0, 0) Never Always(0, 0, a3

a4

)a3a4

> 1 a3a4

< 1

(1, 0, 0) Never Always(a4−a3a4+1

, 0, 1+a3a4+1

)0 < a4−a3

1+a4< d1

a4−a31+a4

> d1; or a4 < a3

(d1, 1− d1, 0) Never Always

Table 3: Generalist Predator (5): Local stability conditions for boundary equilibria (BE)

1. For System (4), if (1, 0, 0) is locally asymptotically stable, then (4) has only two equilibrium: (0,0,0) and

(1, 0, 0).

2. For System (4), the equilibria (d1, 1 − d1, 0) and (d2, 0, 1 − d2) can not be both locally asymptotically

stable at the same time.

3. For System (5), if (0, 0, a3a4 ) is locally asymptotically stable, then (5) does not have the equilibrium(a4−a3a4+1 , 0,

1+a3a4+1

).

4. The big difference between System (4) and System (5) in the local stability of the boundary equilibria is

that both (1, 0, 0) and (d1, 1− d1, 0) are always unstable for (5) while they can be locally asymptotically

stable for (4) under certain conditions. This implies that species z can invade species x and y for (5).

3.2 Global dynamics: extinction and persistence

Notice that both System (4) and (5) have the same following subsystem in the case that species z is absent,

i.e., z = 0

x′ = x(1− x− y)

y′ = γ1y (x− d1)(6)

where its dynamics can be summarized as the following proposition:

Proposition 1 (Subsystem of species x and y) The subsystem (6) is globally stable at (1, 0) if and only

if d1 ≥ 1 while it is globally stable at (d1, 1− d1) if and only if 0 < d1 < 1.

Proof From Lemma 1, we have

lim supτ→∞

x(τ) ≤ 1.

Thus, for any ε > 0 such that d1 > 1 + ε, then we have the following inequality if time τ is large enough,

y′ = γ1y (x− d1) < γ1y (1 + ε− 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 x′ ≥ x(1 − x − δ) ⇒ lim infτ→∞ 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 Poincare-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) = 1xy ,

then we have

∂ (φ(x, y)x′)

∂x+∂ (φ(x, y)y′)

∂y= −1

y< 0 for all (x, y) ∈ R2

+.

Therefore, according to Dulac’s criterion (Guckenheimer and Holmes 1983), we can conclude that the tra-

jectory 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

Poincare-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

x′ = x(1− x− z)

z′ = γ2z (x− d2) .(7)

and

x′ = x(1− x− z)

z′ = γ2z (a3 − a4z + x) .(8)

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

stable at(

0, a3a4

)if and only if a4 ≤ a3 while it is globally stable at

(a4−a31+a4

, a3+11+a4

)if and only if a4 > a3.

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(

0, a3a4

)and two

unstable equilibria (0, 0) and (1, 0). Since we have lim infτ→∞ z(τ) ≥ a3a4

from Lemma 1, therefore we have

the following inequality if a4 < a3 when time τ large enough:

x′ = x(1− x− z) ≤ x(1− a3a4

)⇒ lim supτ→∞

x(τ) = 0.

Therefore, we have limτ→∞ z(τ) = a3a4

if a3 > a4.

If a4 > a3, then (8) has an interior equilibrium(a4−a31+a4

, a3+11+a4

)which is locally asymptotically stable while

(0, 0), (1, 0) and(

0, a3a4

)are saddle nodes. By Poincare-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) = 1xz , then we have

∂ (φ(x, y)x′)

∂x+∂ (φ(x, y)y′)

∂y= −1

z− a4

x< 0 for all (x, z) ∈ R2

+.

Therefore, according to Dulac’s criterion (Guckenheimer and Holmes 1983), we can conclude that the tra-

jectory of any initial condition taken in R2+ converges to an equilibrium point. Since the only locally stable

equilibrium of (8) when a4 > a3 is the interior equilibrium(a4−a31+a4

, a3+11+a4

), therefore, it must be global stable.

In the case that a3 = a4, (8) has(

0, a3a4

)and the other two unstable equilibria (0, 0) and (1, 0). Then

according to Poincare-Bendixson Theorem (Guckenheimer and Holmes 1983) again, we can conclude that

all trajectories should converge to(

0, a3a4

), thus,

(0, a3a4

)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 Poincare-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 infτ→∞

x(τ) ≤ lim supτ→∞

x(τ) ≤ B.

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 infτ→∞

min{x(τ), y(τ), z(τ)} ≤ lim supτ→∞

max{x(τ), y(τ), z(τ)} ≤ B.

12

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, a3a4 )

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, a3a4 ) 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

xdt

∣∣∣(0,0,0)

= (1− x− y − z)∣∣∣(0,0,0)

= 1 > 0.

Define V = γ2a2y + γ1a1z, then for any ε > 0 such that 1 + ε < min{d1, d2}, there exists a time T large

enough such that for any τ > T we have

V ′ = γ2a2y′ + γ1a1z

′ = γ1γ2a2y(x− d1) + γ1γ2a1z(x− d2)

≤ γ1γ2a2y(1 + ε− d1) + γ1γ2a1z(1 + ε− d2)

≤ max{1 + ε− d1, 1 + ε− d2}(γ2a2y + γ1a1z)

= max{1 + ε− d1, 1 + ε− d2}V.

Since max{1 + ε− d1, 1 + ε− d2} < 0, thus, we have limτ→∞ V (τ) = 0. This implies that the limiting system

of (4) is x′ = 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 ε > 0 such

that 1 + ε < a3a4

, there exists a time T large enough such that for any τ > T we have

x′ = x(1− x− y − z) ≤ x(1− z) ≤ x(

1− a3a4

+ ε

)⇒ x(t) < x(0)e

(1− a3a4 +ε

)t → 0 as t→∞.

Since species y feeds on species x, thus, we have limτ→∞ y(τ) = 0. This implies that the limiting system

of (5) is z′ = γ2z(a3 − a4z), therefore, limτ→∞ z(τ) = a3a4. This indicates that (0, 0, a3a4 ) is global stable for

System (5) if a3 > a4. �

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.

Similarly, species x and z are persistent in R3+ if d1 + a2(d1−1)2

(−1+d1)2+β2 > d2 & d1 < 1 or d2 < 1 < d1. In addition,

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

13

in R3+ if a3 < a4. If, in addition, d1 > 1, then System (5) has global stability at

(a4−a31+a4

, 0, a3+11+a4

). Similarly,

species y and z are persistent in R3+ if a3 < a4 and a4−a3

1+a4> 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

d2 > 1 anddy

ydt

∣∣∣(1,0,0)

= γ1

(x− a1yz

y2 + β2− d1

) ∣∣∣(1,0,0)

= 1− d1 > 0⇒ d2 > 1 > d1

or

d2 < 1 anddy

ydt

∣∣∣(d2,0,1−d2)

= γ1

(x− a1yz

y2 + β2− d1

) ∣∣∣(d2,0,1−d2)

= d2 − d1 > 0⇒ 1 > d2 > d1.

Similarly, we can conclude that species z is persistent in R3+ for System (4) if

d1 > 1 anddz

zdt

∣∣∣(1,0,0)

= γ2

(x+

a2y2

y2 + β2− d2

) ∣∣∣(1,0,0)

= 1− d2 > 0⇒ d1 > 1 > d2

or

d1 < 1 anddz

zdt

∣∣∣(d1,1−d1,0)

= γ2

(x+

a2y2

y2 + β2− d2

) ∣∣∣(d1,1−d1,0)

= d1 +a2(1− d− 1)2

(1− d1)2 + β2− d2 > 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,

we can conclude that (d1, 1−d1, 0) is global stable. Assume that d2 > 1+a2. From z′ = γ2z(x+ a2y

2

y2+β2 − d2)

and Lemma 1, then we have follows if time is large enough:

z′ = γ2z

(x+

a2y2

y2 + β2− d2

)< γ2z (1 + a2 − d2)⇒ z(τ)→ 0 as τ →∞.

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

1, we can conclude that (d2, 0, d2) is global stable. From y′ = γ1y(x− a1yz

y2+β2 − d1)

and Lemma 1, then we

have follows if time is large enough:

y′ = γ1y

(x− a1yz

y2 + β2− d1

)< y′ = γ1y (1− d1)⇒ y(τ)→ 0 as τ →∞.

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, a3a4 ). Therefore, according

to Theorem 2.5 of Hutson (1984), we can conclude species x is persistent in R3+ if

dx

xdt

∣∣∣(0,0,

a3a4

)= (1− x− y − z)

∣∣∣(0,0,

a3a4

)= 1− a3

a4> 0⇒ a3 < a4.

Since y′ = γ1y(x− a1yz

y2+β2 − d1)≤ γ1y (x− d1), thus, according to Lemma 1, species y of System (5) goes to

extinct if d1 > 1. Therefore, the limiting system is the x-z subsystem if a3 < a4 and d1 > 1. Then according

to Proposition 2, System (5) has global stability at(a4−a31+a4

, 0, a3+11+a4

).

Since species y feeds on species x, thus its persistence requires the persistence of species x, i.e., a3 < a4.

This implies that the omega limit set of the x-z subsystem restricted on C for System (5) is(a4−a31+a4

, 0, a3+11+a4

).

Therefore, according to Theorem 2.5 of Hutson (1984), we can conclude species y is persistent in R3+ if

dy

ydt

∣∣∣(a4−a31+a4

,0,a3+11+a4

) = γ1

(x− a1yz

y2 + β2− d1

) ∣∣∣(a4−a31+a4

,0,a3+11+a4

) =a4 − a31 + a4

− d1 > 0.

Therefore, sufficient condition for the persistence of species y is a3 < a4 and a4−a31+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+β2 > 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

(−1 + d1)2 + β2> d2 > d1 & d2 < 1.

Similarly, System (5) is permanent in R3+ if the following inequalities hold

a3a4

< 1 &a4 − a31 + a4

> d1.

15

Persistent Species Sufficient Conditions for (4) Sufficient Conditions for (5)

species x Always a3a4

< 1

species z d1 +a2(d1−1)2

(−1+d1)2+β2 > d2 & d1 < 1 or d2 <

1 & d1 > 1

Always

species x, y d1 < min{1, d2} a3a4

< 1 & a4−a31+a4

> d1

species x, z d1 +a2(d1−1)2

(−1+d1)2+β2 > d2 & d1 < 1 or d2 <

1 < d1

a3a4

< 1

species x, y, z d1 +a2(d1−1)2

(−1+d1)2+β2 > d2 > d1 & d2 < 1 a3a4

< 1 & a4−a31+a4

> d1

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 a3a4> 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 a3a4

> 1

species y d2 < min{1, d1} or d1 > 1 d1 > 1 or a3a4

> 1

species z d2 > 1 + a2 Never

species x, y Never a3a4

> 1

species y, z di > 1, i = 1, 2 Never

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)

while other results listed in Table 5 can be obtained from Theorem 1-2. Define a scalar function V = yz−γ1γ2 ,

then according to System (4), we have

dVdτ = y′z−

γ1γ2 − γ1

γ2yz−

γ1γ2−1z′

= yz−γ1γ2 γ1

(x− a1yz

y2+β2 − d1)− yz−

γ1γ2 γ1z

(x+ a2y

2

y2+β2 − d2)

= γ1V(x− a1yz

y2+β2 − d1 − x− a2y2

y2+β2 + d2

)≤ γ1V (−d1 + d2) .

Therefore, if d1 > d2, then we have

V (τ) = y(τ)z−γ1γ2 (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

boundary equilibria are Exy = (d1, 1 − d1, 0) and Exz =(a4−a3a4+1 , 0,

1+a3a4+1

), the persistence condition for

species y is 1 − d1 > 1+a3a4+1 ⇒ d1 <

a4−a3a4+1 . Therefore, we can conclude that the persistence of species y

requires it being superior competitor to IG predator.

3. The permanence of (5) holds whenever species y persists. This is not true for IGP model with specialist

predator (4) (see notes in the end of section 4.1). In addition, by comparing Figure 2 to Figure 3, we can

17

see that (5) has much larger region being permanent. This may suggest that IGP model with generalist

predator is prone to have coexistence of three species.

0 0.5 1 1.50

0.5

1

1.5

Two dimensional bifurcation diagram for System (3):d1 v.s. d

2

d1

d2

V: Permanence

d2=d

1+a

2(1−d

1)2/((1−d

1)2+β

2)

d1=d

2

d2=1

d1=1

d2=1+a

2

III: (d1,1−d

1,0) G.S.

I: (1,0,0) G.S.

II: (d2,0,1−d

2) G.S.

IV: (d1,1−d

1,0) L.S.

Fig. 2: Two dimensional bifurcation diagram of d1 v.s. d2 for System (4).

4 Multiple attractors

We have investigated sufficient conditions for species extinction and persistence in the previous section. In

this section, we focus on the possible dynamical patterns, i.e., multiple attractors, for System (4)-(5) in the

following two situations:

1. When both System (4) -(5) are permanent, e.g., the values of parameters are in Region V for System (4)

and Region III for System (5). According to the fixed point theorem (also see Theorem 6.3, Hutson and

Schmitt 1992), the permanence of System (4)-(5) indicates that they have at least one interior equilibrium.

The interesting question is that whether System (4)-(5) can have multiple interior equilibria, therefore,

they may have multiple interior attractors.

2. When System (4) has local stability at (d1, 1− d1, 0) or System (5) has local stability at (a4−a31+a4, 0, 1+a31+a4

),

e.g., the values of parameters are in Region IV for System (4) and Region II for System (5). In this

scenario, we are interested in whether System (4) or (5) can have interior equilibria, therefore, they may

have an interior attractor.

18

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Two dimensional bifurcation diagram for System (4):a3 v.s. a

4

a3

a4

III: Permanence

a4=a

3

a4=(a

3+d

1)/(1−d

1)

II: species x persistent

((a4−a

3)/(1+a

4),0,(1+a

3)/(1+a

4)) G.S. if d

1>1

I: (0,0,a3/a

4) G.S.

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 (9)

x∗ − a1y∗z∗

(y∗)2 + β2− d1 = 0⇒ y∗z∗

(y∗)2 + β2=x∗ − d1a1

(10)

x∗ +a2(y∗)2

(y∗)2 + β2− d2 = 0⇒ (y∗)2

(y∗)2 + β2=d2 − x∗

a2⇒ y∗ =

√β2(d2 − x∗)x∗ + a2 − d2

. (11)

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

z∗ =a2(x∗ − d1)

a1(d2 − x∗)y∗ =

a2(x∗ − d1)

a1(d2 − x∗)

√β2(d2 − x∗)x∗ + a2 − d2

=a2β(x∗ − d1)

a1√

(d2 − x∗)(x∗ + a2 − d2).

Then from (9), we have 1− x∗ − β√

(d2−x∗)x∗+a2−d2 −

a2β(x∗−d1)

a1√

(d2−x∗)(x∗+a2−d2)= 0 which implies that

1−x∗

β = a1(d2−x∗)+a2(x∗−d1)

a1√

(d2−x∗)(x∗+a2−d2)= a1d2−a2d1−(a1−a2)x∗

a1√

(d2−x∗)(x∗+a2−d2)

⇒ a1(1−x∗)√

(d2−x∗)(x∗+a2−d2)β = a1d2 − a2d1 − (a1 − a2)x∗ > 0

subject to max{d1, d2 − a2} < x∗ < min{d2, a1d2−a2d1a1−a2 }. Let f1 =a1(1−x)

√(d2−x)(x+a2−d2)

β and f2 = a1d2 −

a2d1 − (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

a1d2 − a2d1a1 − a2

> 0 and max{d1, d2 − a2} < x < min{d2,a1d2 − a2d1a1 − a2

},

then we can classify the possible scenarios in the following categories:

1. If a1 > a2 > d2, then we have a1d2−a2d1a1−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 a1d2−a2d1a1−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 a1d2−a2d1a1−a2 = a2d1−a1d2

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 a1d2−a2d1a1−a2 = a2d1−a1d2

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 (12)

x∗ − a1y∗z∗

(y∗)2 + β2− d1 = 0⇒ 1− d1 − y∗ − z∗ −

a1y∗z∗

(y∗)2 + β2= 0 (13)

a3 − a4z∗ + x∗ +a2(y∗)2

(y∗)2 + β2= 0⇒ 1 + a3 − y∗ − (1 + a4)z∗ +

a2(y∗)2

(y∗)2 + β2= 0 (14)

20

x

y

0

y

d2 − a2 d2a1d2−a2d1a1−a2

a1 > a2 > d2

x0d2 − a2 d2

a1d2−a2d1a1−a2

y

x0d2 − a2 d2

a1d2−a2d1a1−a2

a1 > a2 > d2

a1 > a2 > d2

x = d1

y

x0d2 − a2 d2a1d2−a2d1

a1−a2

a1 > a2 > d2

x = d1

(a) (b)

(d)(c)

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 + β2)

(y∗)2 + β2 + a1y∗and z∗ =

(1 + a2 + a3 − y∗)((y∗)2 + β2)− a2β2

((y∗)2 + β2)(1 + a4)

subject to x∗ > d1, z∗ > a3

a4and y∗ < 1− d1. This implies that

(1−d1−y∗)((y∗)2+β2)(y∗)2+β2+a1y∗

= (1+a2+a3−y∗)((y∗)2+β2)−a2β2

((y∗)2+β2)(1+a4)

⇒ (1− d1 − y∗) =[(1+a2+a3−y∗)((y∗)2+β2)−a2β2]((y∗)2+β2+a1y

∗)

(1+a4)((y∗)2+β2)2

21

x

y

0

y

d2 − a2 d2a1d2−a2d1a1−a2 x

0 d2 − a2 d2a1d2−a2d1a1−a2

y

x0 d2 − a2 d2

a1d2−a2d1a1−a2

x = d1

y

x0 d2 − a2 d2

a1d2−a2d1a1−a2

x = d1

(a) (b)

(d)(c)

a1 > a2

d2 > a2

a1 > a2

d2 > a2

a1 > a2

d2 > a2

a1 > a2

d2 > a2

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+β2)−a2β2](y2+β2+a1y)

(1+a4)(y2+β2)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 (yo, 0) (i.e., g2(yo) = 0) due to the fact that (1+a2 +a3−y)(y2 +β2) =

a2β2 has exactly one positive real root.

22

2. g2(1) =[(1+a2+a3−1)(1+β2)−a2β2](1+β2+a1)

(1+a4)(1+β2)2 > 0 and g2(1 + a2 + a3) < 0, thus, we have

1− d1 < 1 < yo < 1 + a2 + a3.

3. g1(0) = 1− d1 and g2(0) = 1+a31+a4

= 1− a4−a31+a4

.

Thus, we can classify System (5) into the following two cases depending on

g1(0) > g2(0)⇐⇒ d1 <a4 − a31 + a4

or g1(0) < g2(0)⇐⇒ d1 >a4 − a31 + a4

1. If the inequality g1(0) > g2(0) ⇐⇒ d1 <a4−a31+a4

holds, then according to Corollary 1, we can conclude

that System (5) is permanent. Since g2(1 − 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−a31+a4

holds, then according to Lemma 2, we can conclude

that System (5) is locally asymptotically stable at the boundary equilibrium(a4−a3a4+1 , 0,

1+a3a4+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 ac-

cording 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

equilibrium

Sufficient Conditions for (4) Sufficient Conditions for (5)

None 1. d1 < d2; or 2. di > 1, i = 1 or 2; or

3. a2 > a1, d2 > d1&d2 < a1d2−a2d1a1−a2

(see Graph (a) of Figure 6-7); or 4. a2 <

a1, d2 > d1&d2 − a2 > a1d2−a2d1a1−a2

.

1. a3 > a4; or 2. No easy expression

(see Graph (a) of Figure 9).

One 1. a2 < a1, d2 > d1, f1(d1) <

f2(d1)&b2 − a2 < a1d2−a2d1a1−a2

< d2 (see

Graph (c) of Figure 4-5); or 2. a2 >

a1, d2 > d1, f1(d1) > f2(d1)&b2 − a2 <

a1d2−a2d1a1−a2

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

Figure 6 and Graph (b) of Figure 7)

d1 < a4−a31+a4

&g2(yc) < g1(yc) (see

Graph (b) of Figure 8).

Two 1. a2 < a1, d2 > d1, f1(xc) >

f2(xc)&d1 + a2 < d2 < a1d2−a2d1a1−a2

(see Graph (b) of Figure 5); or

2. a2 > a1, d2 > d1, f1(xc) >

f2(xc)&a1d2−a2d1a1−a2

< d1 < d2 − a2 (see

Graph (d) of Figure 7).

d1 > a4−a31+a4

&g2(yc) >

g1(yc)& there some y ∈

(yc, yo) such that g2(y) < g1(y)

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

numerical simulations do not find this

case.

Three Never d1 < a4−a31+a4

&g2(yc) > g1(yc) (see

Graph (d) of Figure 8).

Table 6: Sufficient conditions on the number of interior equilibrium of (4) and (5)

2. Let r1 = 25; r2 = 1;β = .1; a1 = 1; a2 = .01; d1 = .15; d2 = .54, then according to Lemma 2, we know

that System (4) has a locally asymptotically stable boundary equilibrium (d1, 1− d1, 0) = (0.15, 0.85, 0).

In addition, from numerical simulations (see Figure 11), we can see that (4) has another attractor which

is a locally stable interior equilibrium.

Multiple attractors of System (5): Here we only consider the case when System (5) is permanent

since we are not able to find that (5) has an interior equilibrium when it is locally asymptotically stable at(a4−a31+a4

, 0, 1+a31+a4

)numerically (see the case of (5) has two interior equilibria in Theorem 4). Let r1 = 25; r2 =

1;β = .15; a1 = 1; a2 = .01; a3 = 0.1; a4 = 4.5; d1 = .15, then according to Corollary 1, we know that System

(4) is permanent. From numerical simulations (see Figure 12), we can see that (5) has two attractors where

is a locally stable interior equilibrium and the other is a limit cycle.

24

4.1 Comparison with an intraguild predation model with Holling-Type I functional responses

A standard Lotka-Volterra IGP model with specialist IGP predator (1) proposed by Holt and Polis (1997)

can be represented as follows after scaling:

x′ = x(1− x− y − z)

y′ = γ1y (x− a1z − d1)

z′ = γ2z (x+ a2y − d2) .

(15)

Holt and Polis (1997) explored five equilibria that followed from (15):

– All species are at zero density: E0 = (0, 0, 0).

– The shared prey is present at its carrying capacity: Ex = (1, 0, 0).

– Only the shared prey and IG prey are present: Exy = (d1, 1− d1, 0).

– Only the shared prey and IG predator are present: Exz = (d2, 0, 1− d2).

– All species exist: Exyz =(a2a1−a1d2+a2d1a2+a2a1−a1 , a1(d2−1)+d2(1+a1)a2+a2a1−a1 , a2(1−d1)−d2+d1a2+a2a1−a1

).

Holt and Polis (1997) showed the potential for alternative stable states, general criterions for the coexistence

and the increased likelihood of unstable population dynamics with systems involving IGP. Here we would

like to compare the dynamics of our model (4) to the Holt-Polis model (15) in the following three points

that may partially answer the second question proposed in the introduction:

1. Alternative stable states: Holt and Polis (1997) showed the only alternative stable states (i.e., multiple

attractors) are the locally stable equilibria Exy and Exz while the alternative stable states for System

(4) can be two interior attractors or two attractors where one is a boundary attractor (d1, 1− d1, 0) and

the other is an interior attractor. Moreover, the multiple attractors always include one interior attractor

for both System (4) and (5). This big difference is due to the fact that (4) has Holling-Type III function

responses between IGP prey and specialist IGP predator which can also lead to multiple interior attractors

with more complicated dynamical patterns (e.g., it is possible to have multiple stable limit cycles). Here,

we also want to point out that it is impossible to have multiple attractors while (d2, 0, 1− d2) is locally

stable for System (4).

2. The coexistence: One of the conclusions by Holt and Polis (1997) is that for coexistence, the IG prey

has to be superior to the IG predator at competing for the shared resource. Their argument sketched

there is for a fairly general model (not just the Lotka-Volterra formulation). The metric of competitive

superiority is the “R∗-rule, namely, the superior exploitative competitor is the one which persists at the

lower equilibrial level of the shared resource. This conclusion still holds, for our both models (4)-(5).

25

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

boundary equilibria are Exy = (d1, 1 − d1, 0) and Exz =(a4−a3a4+1 , 0,

1+a3a4+1

), the persistence condition for

species y is 1 − d1 > 1+a3a4+1 ⇒ d1 <

a4−a3a4+1 . Therefore, we can conclude that the persistence of species y

requires it being superior competitor to IG predator. One big difference between our models and the Holt-

Polis model is that our models can have both boundary and interior attractors under certain conditions

while the existence of an interior attractor indicates the permanence in the Holt-Polis model.

3. Unstable dynamics: Holt and Polis (1997) observed that their model could get unstable dynamics

under certain conditions, even though all species could when rare invade the stable sub-system comprised

of the other species. This has to do with the destabilizing impact of long feedback loops permitted by the

three-way interaction which still holds for our models. Moreover, it has long been known that saturating

functional responses (such as those in (4)-(5)) can lead to stable limit cycles, basically because there

is positive density dependence in a prey species emerging from the functional response. This is another

factor that can cause the unstable dynamics of our models. One big difference between our models and

the Holt-Polis model is that our models can possibly have multiple stable limit cycles.

5 Conclusion

An analysis of simple community modules such as three-species IGP can help us provide a useful bridge

between the thoroughly analyzed two-species dynamics and the excessively complex multi-species world that

ecologists hope to understand (Holt and Polis 1997). Mathematical models have been useful in identifying

sufficient conditions for coexistence and persistence which may provide some insights and help conservation

biologists to identify the leading causes for species extinction. In this article, we develop and study IGP models

with either specialist predator or generalist predator to answer the following two ecological questions: 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?

Our theoretical analyses and simulations related to the above two questions can be summarized as follows:

1. IGP with generalist predator can have potential “top down” regulation. This is suggested by our sufficient

conditions for the extinction and persistence for both System (4) and (5) summarized in Figure 2 and

Figure 3. Our results indicate that the persistence of IGP with specialist predator is more likely controlled

by the relative death rates of IG prey and predator di, while IGP model with generalist predator is more

likely determined by the availability of outside resource, i.e., a3 and a4.

26

2. The persistence of species y requires it being superior competitor to IG predator. However, there is a big

difference between (4) and (5) when species y persists: The permanence of (5) holds whenever species y

persists, while (4) can either be permanent or exhibit one boundary attractor and one interior attractor.

3. By comparing Figure 2 to Figure 3, we can see that (5) has much larger region being permanent (see

Region V in Figure 2 for (4) and Region III in Figure 3 for (5)). In addition, our numerical simulations

suggest that (5) can not have both boundary attractor and interior attractor like IGP model specialist

predator (4). This may suggest that IGP model with generalist predator is prone to have coexistence of

three species.

4. Holling-Type III functional response between IG prey and IG predator in IGP models lead to much more

complicate dynamics than IGP models with only Holling-Type I functional response (e.g., a simple IGP

model proposed by Holt and Polis 1997). For example, from our numerical simulations, we can see that

our IGP models have multiple attractors where at least one attractor is an interior attractor. In order to

explore whether the complicated dynamics is due to the included Type III response (predator saturation),

or it more essentially reflects the increasing return to scale of foraging by the IG predator with increasing

of density IG prey, at low levels of the latter. We consider a limiting case by setting β to zero. In this

case, IG predator inflicts a constant rate of mortality upon the IG prey, so the per capita mortality rate

of IG prey go up steeply as the IG predator density increases. Model (4) becomes (16) in the case that

β = 0:

x′ = x(1− x− y − z)

y′ = γ1y (x− d1)− γ1a1z

z′ = γ2z (x+ a2 − d2) .

(16)

where its dynamics can be summarized as follows:

– The extinction state (0, 0, 0) is always unstable.

– If a2 > 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).

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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 Conditions for stability Conditions for instability

(0, 0, 0) Never Always

(1, 0, 0) dg > egagKg and dm > emamKp dg < egagKg ; or dm < emamKp(dm

emamKp, 0, 1− dm

emamKp

)dm

emam<

dgegag

and dm < emamKpdm

emam>

dgegag

; or dm > emamKp(dg

egagKp, 1− dg

egagKp, 0

)dgegag

+a(

dgegagKp

−1)2

(−1+dg

egagKp)2+

(rpb

ag

)2 < dmemam

and

dg < egagKg

dgegag

+a

(dg

egagKp−1

)2

(−1+

dgegagKp

)2+

(rpb

ag

)2 > dmemam

or

dg > egagKg

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

BE point Conditions for stability Conditions for instability

(0, 0, 0) Never Always(0, 0, amKm

rp

)amKmrp

> 1 amKmrp

< 1

(1, 0, 0) Never Always(rm(rp−Kmam)

rprm+KmKpa2mem, 0,

Kmam(rm+amemKp)

rprm+KmKpa2mem

)0 <

rm(rp−Kmam)

rprm+KmKpa2mem<

dgegagKp

a4−a31+a4

>dg

egagKp; or amKm

rp> 1(

dgegagKp

, 1− dgegagKp

, 0)

Never Always

Table 8: Generalist Predator (3): Local stability conditions for boundary equilibria (BE)

30

Persistent Species Sufficient Conditions for (2) Sufficient Conditions for (3)

species x Always amKmrp

< 1

species zdgegag

+a

(dg

egagKp−1

)2

(−1+

dgegagKp

)2+

(rpb

ag

)2 >

dmemam

&dg

egagKp< 1 or dm

emamKp<

1 &dg

egagKp> 1

Always

species x, ydg

egagKp< min{1, dm

emamKp} rm(rp−Kmam)

rprm+KmKpa2mem>

dgegagKp

& amKmrp

< 1

species x, zdgegag

+a

(dg

egagKp−1

)2

(−1+

dgegagKp

)2+

(rpb

ag

)2 >

dmemam

&dg

egagKp< 1 or dm

emamKp< 1 <

dgegagKp

amKmrp

< 1

species x, y, zdgegag

+a

(dg

egagKp−1

)2

(−1+

dgegagKp

)2+

(rpb

ag

)2 >

dmemam

>dgegag

& dmemamKp

< 1

rm(rp−Kmam)

rprm+KmKpa2mem>

dgegagKp

& amKmrp

< 1

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

Extinct Species Sufficient Conditions for (2) Sufficient Conditions for (3)

species x Never amKmrp

> 1

species y dmemamKp

< min{1, dgegagKp

} or

dgegagKp

> 1

dgegagKp

> 1 or amKmrp

> 1

species zdg

egagKp> 1 + a

amKpNever

species x, y Never amKmrp

> 1

species y, z dieiaiKp

> 1, i = m, g Never

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

31

x

y

0

y

d2 − a2 d2a1d2−a2d1a1−a2 x

0d2 − a2 d2a1d2−a2d1a1−a2

y

x = d1

y

x0d2 − a2 d2

a1d2−a2d1a1−a2

x = d1

(a) (b)

(d)(c)

a2 > a1

d2 < a2

a2 > a1

d2 < a2

x0d2 − a2 d2

a1d2−a2d1a1−a2

x = d1

a2 > a1

d2 < a2a2 > a1

d2 < a2

y

x0d2 − a2 d2

a1d2−a2d1a1−a2

x = d1

a2 > a1

d2 < a2

(e)

y

x0b2 − a2 b2

a1d2−a2d1a1−a2

a2 > a1

d2 < a2

(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

x

y

0

y

d2 − a2 d2a1d2−a2d1a1−a2

x0 d2 − a2 d2

a1d2−a2d1a1−a2

y

x = d1

y

x0 d2 − a2 d2

a1d2−a2d1a1−a2

x = d1

(a) (b)

(d)(c)

x0 d2 − a2 d2

a1d2−a2d1a1−a2

x = d1

y

x0 d2 − a2 d2a1d2−a2d1a1−a2

x = d1

(e)

y

x0 d2 − a2 d2

a1d2−a2d1a1−a2

(f)

d2 > a2 > a1 d2 > a2 > a1

d2 > a2 > a1d2 > a2 > a1

d2 > a2 > a1d2 > a2 > a1

x = d1

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

y0

0

(a) (b)

(d)(c)

1+a31+a4

< 1− d1

1− d1

1− d1

1+a31+a4

yo

x

y0

1+a31+a4

< 1− d1

1− d1

1− d1 yo

1+a31+a4

y

1− d1

1− d1 yo

x 1+a31+a4

< 1− d1

1+a31+a4

0 y

1− d1

1− d1 yo

x 1+a31+a4

< 1− d1

1+a31+a4

Fig. 8: Graph (b) and (d) are the schematic cases when the inequality g1(0) > g2(0)⇐⇒ d1 <a4−a31+a4

holds for System 5. The dashed line is g1 while the solid curve is g2. Graph (a) and (c) are impossible

due to the restriction yo > 1− d1.

34

x

y0

0

(a) (b)

(d)(c)

1− d1

1− d1

1+a31+a4

yo

x

y0

1− d1

1− d1yo

1+a31+a4

y

1− d1

1− d1 yo

x

1+a31+a4

0 y

1− d1

1− d1yo

x

1+a31+a4

1+a31+a4

> 1− d11+a31+a4

> 1− d1

1+a31+a4

> 1− d11+a31+a4

> 1− d1

Fig. 9: Graph (a) and (c) are the schematic cases when the inequality g1(0) < g2(0)⇐⇒ d1 >a4−a31+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 yo > 1− d1.

35

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6IGP−specialist

time t

Bla

ck−

reso

urc

e;

Blu

e−

Pre

y;

Re

d−

Spe

cia

list

Pre

dato

r

r1=25;r

2=1;b=0.1;a

1=1;

a2=0.6;d

1=0.15;d

2=0.54;

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

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

0 20 40 60 80 100 120 140 160 180 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7IGP−specialist−time series

time t

Bla

ck−

reso

urc

e;

Blu

e−

Pre

y;

Re

d−

Sp

ec

ialist

Pre

da

tor

r1=25;r

2=1;b=0.1;a

1=1;

a2=0.6;d

1=0.15;d

2=0.54;

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

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

0

0.1

0.2

0.3

0.4

0.5

0

1

2

3

4

5

60

0.01

0.02

0.03

0.04

0.05

0.06

Resource

IGP−specialist−3D Phase Plane

IGP−Prey

IGP

−P

red

ato

r

r1=25;r

2=1;b=0.1;a

1=1;

a2=0.6;d

1=0.15;d

2=0.54;

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

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

0.35

0.4

0.45

0.5

0.55

0.6

0

0.05

0.1

0.15

0.20.4

0.42

0.44

0.46

0.48

0.5

0.52

Resource

IGP−specialist−3D Phase Plane

IGP−Prey

IGP

−P

red

ato

r

r1=25;r

2=1;b=0.1;a

1=1;

a2=0.6;d

1=0.15;d

2=0.54;

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

(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

0 20 40 60 80 100 120 140 160 180 2000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5IGP−specialist

time t

Bla

ck−

reso

urc

e;

Blu

e−

Pre

y;

Re

d−

Sp

ec

ialist

Pre

da

tor

r1=25;r

2=1;b=0.1;a

1=1;

a2=0.01;d

1=0.15;d

2=0.54;

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

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

0 20 40 60 80 100 120 140 160 180 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7IGP−specialist−time series

time t

Bla

ck−

reso

urc

e;

Blu

e−

Pre

y;

Re

d−

Sp

ec

ialist

Pre

da

tor

r1=25;r

2=1;b=0.1;a

1=1;

a2=0.01;d

1=0.15;d

2=0.54;

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

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

0

0.1

0.2

0.3

0.4

0.5

0

1

2

3

4

50

0.01

0.02

0.03

0.04

0.05

Resource

IGP−specialist−3D Phase Plane

IGP−Prey

IGP

−P

red

ato

r

r1=25;r

2=1;b=0.1;a

1=1;

a2=0.011;d

1=0.15;d

2=0.54;

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

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

0.05.

0.35

0.4

0.45

0.5

0.55

0.6

0

0.05

0.1

0.15

0.20.38

0.4

0.42

0.44

0.46

0.48

0.5

Resource

IGP−specialist−3D Phase Plane

IGP−Prey

IGP

−P

red

ato

r

r1=25;r

2=1;b=0.1;a

1=1;

a2=0.01;d

1=0.15;d

2=0.54;

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

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

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

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7IGP−generalist−time series

time t

Bla

ck

−re

so

urc

e;

Blu

e−

Pre

y;

Re

d−

Gen

era

list

Pre

da

tor

r1=25;r

2=1;b=0.15;a

1=1;

a2=0.01;a

3=0.1;a

4=4.5;d

1=0.15;

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

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

0 20 40 60 80 100 120 140 160 180 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8IGP−generalist−time series

time t

Bla

ck

−re

so

urc

e;

Blu

e−

Pre

y;

Re

d−

Gen

era

list

Pre

da

tor

r1=25;r

2=1;b=0.15;a

1=1;

a2=0.01;a

3=0.1;a

4=4.5;d

1=0.15;

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

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

00.1

0.20.3

0.40.5

0.60.7

0

1

2

3

4

5

60.05

0.055

0.06

0.065

0.07

0.075

0.08

0.085

0.09

0.095

Resource

IGP−generalist−3D Phase Plane

IGP−Prey

IGP

−P

red

ato

r

r1=25;r

2=1;b=0.15;a

1=1;

a2=0.01;a

3=0.1;a

4=4.5;d

1=0.15;

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

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

0.40.45

0.50.55

0.60.65

0.70.75

0

0.02

0.04

0.06

0.08

0.1

0.12

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Resource

IGP−generalist−3D Phase Plane

IGP−Prey

IGP

−P

red

ato

r

r1=25;r

2=1;b=0.15;a

1=1;

a2=0.01;a

3=0.1;a

4=4.5;d

1=0.15;

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

(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.