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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: [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
3
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-
4
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).
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
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).
8
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:
9
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)
10
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.
11
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.