# Coexistence in a One-Predator, Two-Prey System with Indirect Effects

Authors: Renato Colucci
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We study the dynamics of a one-predator, two-prey system in which the predator has an indirect effect on the preys. We show that, in presence of the indirect effect term, the system admits coexistence of the three populations while, if we disregard it, at least one of the populations goes to extinction.


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## Body

## 1. Introduction

The importance of indirect effects is well established in Biology (see [1–5]), for example, in the case of predation (see [6]), the predator can alter the morphology (see [7]) or the behavior of the preys. The preys, in order to reduce the possibility of contacts with the predators, could modify their normal conduct by reducing their activity or by hiding themselves for long time. There are many types of indirect effects (see [3] for a detailed discussion); another interesting case is the refuge indirect effect (see [8], e.g.); anyway, it is of great interest trying to describe the indirect interactions in population dynamics. Recently in [9] a model including indirect effects was proposed, modeling the effects of predator Daphnia over two groups of Phytoplankton of different morphology (see [10]), having Phosphorous as a resource (see [11] or [12]). A general model describing indirect effects of predation can be written in the following form:(1)x˙1=x1f1(x1,x2,x3),x˙2=x2f2(x1,x2,x3)-ℐ(x1,x2,x3),x˙3=x3f3(x1,x2,x3)+ℐ(x1,x2,x3),
where x1 is the density of population of the predator while x2 and x3 are the densities of population of the preys. The positive function ℐ(x1,x2,x3) represents the indirect effects of the predator on the prey x3 generated by predation over the prey x2. In [9], a system in which the functions fi are linear and the function ℐ(x1,x2,x3)=ℐ(x1,x2) is quadratic was proposed. The system takes the following form:
(2)Z′=Z(-e+ucC+ugG),C′=C(acP-ucZ)-mCZ,G′=G(agP-ugZ)+mCZ,P′=P(-agG-acC)+eZ,
where ug, uc, ac, ag, e, m∈ℝ are positive parameters and where (Z,C,G,P)⊂ℝ+4.In the above systemZ represent the density of population of a predator (Daphnia-Zooplankton) that predates the preys (Phytoplankton) C and G that are of different size, in particular C is of a smaller size than G. The variable P represents the amount of resources (Phosphorous) for the preys C and G. We will show below that the system can be put in the form of system (1).In absence of preys, the population of predatorsZ will extinguish exponentially while the presence of them brings a positive growth rate. In absence of predator, the populations of preys will increase depending on the amount of available resources, while the presence of predators affects negatively their growth rate. Moreover, the population of predator Z has an effect on the prey C which is described by the term -mCZ which produce a negative growth rate. This effect produces an indirect effect that is positive (the term mCZ) for the population of prey G. The growth rate of the amount of resources P of the system is effected by the presence of the preys while, for its renewable character, the presence of the predator lets it regenerate (the term eZ).We first observe that the system is closed; in fact if we sum (2), we obtain
(3)Z′+C′+G′+P′=0,
and as a consequence we have the following proposition.Proposition 1.
The functionI(t) defined as
(4)I(t)=Z(t)+C(t)+G(t)+P(t)
is a constant of motion.By using the first integralI(t) we can reduce the degree of freedom of the problem; in fact if we fix a value of the the first integral, I(t)=I0, then the system can be rewritten in the following way:
(5)Z′=Z(-e+ucC+ugG),C′=C[acI0-(ac+uc)Z-acC-acG]-mCZ,G′=G[agI0-(ag+ug)Z-agC-agG]+mCZ.Since we are interested only in positive solution and using condition (4), we can limit our analysis to the following region:
(6)Δ={(Z,C,G)∈ℝ+3:Z+C+G≤I0}.First of all we show that the dynamics develops in the regionΔ (see Figure 1).Figure 1
The positive invariant setΔ.Theorem 2.
The setΔ is positively invariant for the solutions of the system (5).Proof.
In order to prove thatΔ is positively invariant, it is sufficient to compute the vector field of the system on the boundary of Δ that is made up by 4 triangular regions.
First of all we consider the vertexes ofΔ, we set
(7)P0=(0,0,0),P1=(I0,0,0),P2=(0,I0,0),P3=(0,0,I0).
The vertexes are all fixed points except the point P1 on which the vector field is (-eZ,0,0).
We study the vector field on each triangular region separately.
The Face
(
T
1
=
Δ
∩
{
G
=
0
}
)
. In this case G′≥0, then, in the interior of the triangle, the vector field points toward the interior of Δ. We check the sides of the triangle T1: if Z=0 we have Z′=G′=0 and C′>0; moreover, C′→0 as C→I0. If C=0 we have that G′=C′=0 and Z→0. If we consider the side on which C+Z=I0 then we have that C′+Z′=-mCZ-eZ<0. Finally we conclude that on T1 the vector field is tangent or point inward Δ.
The Face
(
T
2
=
Δ
∩
{
C
=
0
}
). In this region we have that C′=0, that is, the plane C=0 is invariant and as a consequence we have only to check the vector field on the boundary of T2. If Z+G=I0 we have that the vector field points inward T2 since Z′+G′=-eZ<0, while in the side where G=0 we have that G′=0 and Z→0 so it is positively invariant. Finally, on the side where Z=0 we have that Z′=0 and that G′→0 as G→I0.
The Face
(
T
3
=
Δ
∩
{
Z
=
0
}
)
. In this region we have that Z′=0, then the plane Z=0 is invariant and as a consequence we have only to check the vector field on the boundary of T3. On the side where C+G=I0 we have that C′=G′=0 and as a consequence C(t)+G(t)=I0, for all t≥0. Finally on the side where C=0 we have C′=0 and G′>0 and G′→0 as G→I0, while on the side where G=0 we have that G′=0, C′>0, and C′→0 when C→I0.
The Face
(
T
4
=
Δ
∩
{
Z
+
C
+
G
=
I
0
}
)
. In this case we have that the vector field points inward Δ. In fact, if we introduce the function V(t) that is the sum of the three variables
(8)V(t)=Z(t)+C(t)+G(t)
and we sum the equations of the system, we obtain the following differential equation depending on V:
(9)V′=(I0-V)(acC+agG)-eZ.
From (9) we get that
(10)V=I0⟹V′≤0;
this concludes the proof since if the solutions start on T4 then they enter the set Δ or they remain on T4.The rest of the paper is organized as follows: in Section2 we analyze the stability of the fixed points while in Section 3 we study the dynamics on the boundary of Δ. In Section 4 we analyze the dynamics of the system in absence of indirect effects terms (that is m=0) while in Section 5 we study the asymptotic behavior of the solutions of the system for m>0, in particular we prove the coexistence of the three species under a proper choice of the parameters. In Section 6 we present some numerical simulations while in Section 7 we give some comments and remarks for future investigation.


## 2. Stability of the Fixed Points

The first step in studying the dynamics of the system (5) consists in finding all the fixed points in Δ and analyzing their stability character. In Section 1 we have pointed out the existence of three fixed points among the vertexes of Δ:(11)P0=(0,0,0),P2=(0,I0,0),P3=(0,0,I0).
In order to find all possible further fixed points in the faces or in the interior of Δ, we consider the intersections between the nullclines (see [13]). Since it is a standard argument, we do not present the detailed computation. If I0>e/ug, there exists a fixed point
(12)P4=(agag+ug[I0-eug],0,eug)
in the interior of the triangle T2, while if I0=e/ug, then P4=P3.Moreover, there exists a segment of fixed points that we callS:
(13)S={C,G∈[0,I0]:Z=0,C+G=I0}.
The segment S is one of the sides of the face T3 (on the plane Z=0) and its extremes are the fixed points P2 and P3.We also have an interior fixed pointQ=(ZQ,CQ,GQ) if the following two conditions are fulfilled:
(14)M:=acug-aguc-agm>0,(15)euc{(M/mac)+1(M/mac)+(ug/uc)}<I0.In details, the coordinate of the fixed pointQ are(16)ZQ=acmc{I0-(Mmac+1)(e/uc(M/acm)+(ug/uc))},CQ=Mmac{e/uc(M/acm)+(ug/uc)},GQ={e/uc(M/acm)+(ug/uc)},
where mc=ac+uc+m. As a summary of the previous analysis we can state the following theorem.Theorem 3.
The pointsP0, P2, P3 and the points of the segment S are equilibria of the system (5) for any (positive) value of the parameters. The system admits that the fixed point P4 if
(17)I0>eug
and the interior fixed point Q if (14) and (15) hold.Remark 4.
From the above analysis we notice thatI0 is a parameter of bifurcation for the fixed points. In particular we have that fixed points P3 and P4 collapse to a unique fixed point P3 if I0=e/ug, and the fixed point P4∉Δ if I0<e/ug.Then we pass to study the stability character of the fixed points. We consider the functional Jacobian of the vector field:(18)(-e+ucC+ugGucZugZ-mCCac(I0-2C-G)-mCZ-acC-(ag+ug)G+mCmZ-agGag(I0-C-2G)-(ag+ug)Z),wheremc=ac+uc+m.


### 2.1. The Fixed PointP0

We start with the analysis of the originP0=(0,0,0). The matrix of the linearized system at P0 is
(19)A(P0)=(-e000acI0000agI0).
The point P0 is a saddle for any values of the parameters since we have one negative eigenvalue λ1=-e and two positive eigenvalues λ2=acI0 and λ3=agI0. The Z-axis is the stable space of the linearized system at P0 while the instable space is the plane Z=0.It is possible to distinguish two cases: three distinct eigenvalues ifac≠ag (that is λ2≠λ3) and only two distinct eigenvalues if ac=ag (that is λ2=λ3).


### 2.2. The Fixed PointP2

We analyze the pointP2=(0,I0,0), and the matrix of the linearized system at P2 is
(20)A(P2)=(-e+ucI000-(ac+uc+m)I0-acI0-acI0mI000).
The eigenvalues of A(P2) are
(21)λ1=ucI0-e,λ2=-acI0,λ3=0.The eigenvaluesλ2 are always negative, while if we have
(22)I0>euc⟹λ1>0,
then the fixed point P2 is instable. In the case in which λ1≤0, that is, I0≤e/uc, we would need to study the system on the center manifold (see [14]) of the point P2 in order to study its stability.


### 2.3. The Fixed PointP3

The matrix of the linearized system atP3=(0,0,I0) is
(23)A(P3)=(-e+ugI000000-(ag+ug)I0-agI0-agI0).
The eigenvalues of the matrix A(P3) are
(24)λ1=0,λ2=ugI0-e,λ3=-agI0.In this case we have at least one negative eigenvalue, that is,λ3, along the G axes (Z=0,C=0), then the stability depends on the first two eigenvalues. In the case in which
(25)I0>eug⟹λ2>0,
we can conclude that P3 is instable. If λ2≤0 we have to study the system on the center manifold related to the eigenvalue λ1 (and λ2 if it is 0), in order to establish the stability of P3.


### 2.4. The Fixed PointP4

We analyze the pointP4 in the case in which P4∈Δ that is if I0>e/ug. The matrix of the linearized system at P4 is
(26)A(P4)=(0ucZ*ugZ*0acI0-mCZ*-acuge0-(ag+ug)eugmZ*-aguge-ageug),
where we have set Z*=(ag/(ag+ug))(I0-e/ug). The eigenvalues of the matrix A(P4) are
(27)λ1=(I0-eug)Mag+ug,λ2,3=-age2ug±age[eug(1+agug)-4I0],
where M:=acug-aguc-mag. Then if
(28)eug(1+ag4ug)<4I0,
then λ2,3 are complex conjugate with negative real part; otherwise they are both negative (coincident if the inequality holds in (28)). If the eigenvalue λ1 is positive, that is,
(29)M=acug-aguc-mag>0,
we have that P4 is instable.


### 2.5. The Segment of Fixed PointS

On the fixed points that belong to the segmentS we have that Z=0 and C+G=I0, then the matrix of the linearized system at any point of S is of the following form:(30)A(S)=(ugI0-e+C(uc-ug)00-(ac+uc+m)C-acC-acCC(ag+ug+m)-(ag+ug)I0-ag(I0-C)-ag(I0-C)),whereC∈[0,I0]. The eigenvalues of the matrix A(S) are
(31)λ1=ugI0-e+C(uc-ug),λ2=0,λ3=-ag(I0-C)-acC.
In this case we have that the eigenvalue λ3 is always negative since C∈[0,I0]. In the case in which
(32)I0>max{euc,eug},
the eigenvalue λ1>0 and then the fixed points of the segments S are all instable. We note that if uc>ug, then λ1 satisfies
(33)ugI0-e≤λ1≤ucI0-e;
on the contrary λ1 satisfies the opposite inequality.


### 2.6. The Fixed PointQ

The matrixA(Q) of the linearized system at Q is(34)(0ucacmc{I0-(MmaC+1)D}ugacmc{I0-(MmaC+1)D}-mcMmacD-MmD-MmD-agmcacDmmcac{I0-(MmaC+1+agmcacm)D}-MacZQ-agD),where we have setD=GQ. We consider the trace of the matrix A(Q):
(35)𝒯(A(Q))=-MacZQ-agD-MmD.Then, ifQ∈Δ we have that
(36)𝒯(A(Q))<0,
and then at least one of the eigenvalues of A(Q) is negative.We summarize the results of the previous analysis in the following statements.Theorem 5.
Suppose that the parameters of the system satisfy the following inequalities:(37)I0>max{euc,eug},(38)M=acug-aguc-mag>0.
Then the fixed points P0, P2, P3, P4 and the fixed points that belong to the segment S are all instable.Remark 6.
We note that the instability hypothesis for the pointP3, that is, I0>e/ug, implies the presence of the point P4. Then it would be interesting to prove if it is a necessary condition for the instability of P3, that is, if the presence of the fixed point P4 is necessary for the instability of all fixed points in Δ.Remark 7.
We note that the hypothesis of the previous theorem are not compatible with the inequality(39)I0<euc(M/mac+1)(M/mac+ug/uc),
that is, the condition for which Q∉Δ.In conclusion, the instability of the boundary fixed points implies the existence of the interior fixed pointQ. The presence of the interior fixed point Q represents an obvious case of coexistence of the three species; however, if Q is instable then it is not useful to describe real cases of coexistence.Since we are not able to face the general problem of stability of pointQ, we will follow another strategy in order to prove coexistence of the system (see Section 6 below).Remark 8.
It would be interesting to find if the interior fixed pointQ is instable for some choice of the parameters that are compatible with the instability hypothesis for the boundary fixed points. In that case it would be interesting to look for the existence of a limit cycle surrounding Q or of homoclinic cycle and chaotic attractors (see e.g., [15] or [16] in which the authors proved the chaotic behavior of one-predator, two prey systems without indirect effects).


## 2.1. The Fixed PointP0

We start with the analysis of the originP0=(0,0,0). The matrix of the linearized system at P0 is
(19)A(P0)=(-e000acI0000agI0).
The point P0 is a saddle for any values of the parameters since we have one negative eigenvalue λ1=-e and two positive eigenvalues λ2=acI0 and λ3=agI0. The Z-axis is the stable space of the linearized system at P0 while the instable space is the plane Z=0.It is possible to distinguish two cases: three distinct eigenvalues ifac≠ag (that is λ2≠λ3) and only two distinct eigenvalues if ac=ag (that is λ2=λ3).


## 2.2. The Fixed PointP2

We analyze the pointP2=(0,I0,0), and the matrix of the linearized system at P2 is
(20)A(P2)=(-e+ucI000-(ac+uc+m)I0-acI0-acI0mI000).
The eigenvalues of A(P2) are
(21)λ1=ucI0-e,λ2=-acI0,λ3=0.The eigenvaluesλ2 are always negative, while if we have
(22)I0>euc⟹λ1>0,
then the fixed point P2 is instable. In the case in which λ1≤0, that is, I0≤e/uc, we would need to study the system on the center manifold (see [14]) of the point P2 in order to study its stability.


## 2.3. The Fixed PointP3

The matrix of the linearized system atP3=(0,0,I0) is
(23)A(P3)=(-e+ugI000000-(ag+ug)I0-agI0-agI0).
The eigenvalues of the matrix A(P3) are
(24)λ1=0,λ2=ugI0-e,λ3=-agI0.In this case we have at least one negative eigenvalue, that is,λ3, along the G axes (Z=0,C=0), then the stability depends on the first two eigenvalues. In the case in which
(25)I0>eug⟹λ2>0,
we can conclude that P3 is instable. If λ2≤0 we have to study the system on the center manifold related to the eigenvalue λ1 (and λ2 if it is 0), in order to establish the stability of P3.


## 2.4. The Fixed PointP4

We analyze the pointP4 in the case in which P4∈Δ that is if I0>e/ug. The matrix of the linearized system at P4 is
(26)A(P4)=(0ucZ*ugZ*0acI0-mCZ*-acuge0-(ag+ug)eugmZ*-aguge-ageug),
where we have set Z*=(ag/(ag+ug))(I0-e/ug). The eigenvalues of the matrix A(P4) are
(27)λ1=(I0-eug)Mag+ug,λ2,3=-age2ug±age[eug(1+agug)-4I0],
where M:=acug-aguc-mag. Then if
(28)eug(1+ag4ug)<4I0,
then λ2,3 are complex conjugate with negative real part; otherwise they are both negative (coincident if the inequality holds in (28)). If the eigenvalue λ1 is positive, that is,
(29)M=acug-aguc-mag>0,
we have that P4 is instable.


## 2.5. The Segment of Fixed PointS

On the fixed points that belong to the segmentS we have that Z=0 and C+G=I0, then the matrix of the linearized system at any point of S is of the following form:(30)A(S)=(ugI0-e+C(uc-ug)00-(ac+uc+m)C-acC-acCC(ag+ug+m)-(ag+ug)I0-ag(I0-C)-ag(I0-C)),whereC∈[0,I0]. The eigenvalues of the matrix A(S) are
(31)λ1=ugI0-e+C(uc-ug),λ2=0,λ3=-ag(I0-C)-acC.
In this case we have that the eigenvalue λ3 is always negative since C∈[0,I0]. In the case in which
(32)I0>max{euc,eug},
the eigenvalue λ1>0 and then the fixed points of the segments S are all instable. We note that if uc>ug, then λ1 satisfies
(33)ugI0-e≤λ1≤ucI0-e;
on the contrary λ1 satisfies the opposite inequality.


## 2.6. The Fixed PointQ

The matrixA(Q) of the linearized system at Q is(34)(0ucacmc{I0-(MmaC+1)D}ugacmc{I0-(MmaC+1)D}-mcMmacD-MmD-MmD-agmcacDmmcac{I0-(MmaC+1+agmcacm)D}-MacZQ-agD),where we have setD=GQ. We consider the trace of the matrix A(Q):
(35)𝒯(A(Q))=-MacZQ-agD-MmD.Then, ifQ∈Δ we have that
(36)𝒯(A(Q))<0,
and then at least one of the eigenvalues of A(Q) is negative.We summarize the results of the previous analysis in the following statements.Theorem 5.
Suppose that the parameters of the system satisfy the following inequalities:(37)I0>max{euc,eug},(38)M=acug-aguc-mag>0.
Then the fixed points P0, P2, P3, P4 and the fixed points that belong to the segment S are all instable.Remark 6.
We note that the instability hypothesis for the pointP3, that is, I0>e/ug, implies the presence of the point P4. Then it would be interesting to prove if it is a necessary condition for the instability of P3, that is, if the presence of the fixed point P4 is necessary for the instability of all fixed points in Δ.Remark 7.
We note that the hypothesis of the previous theorem are not compatible with the inequality(39)I0<euc(M/mac+1)(M/mac+ug/uc),
that is, the condition for which Q∉Δ.In conclusion, the instability of the boundary fixed points implies the existence of the interior fixed pointQ. The presence of the interior fixed point Q represents an obvious case of coexistence of the three species; however, if Q is instable then it is not useful to describe real cases of coexistence.Since we are not able to face the general problem of stability of pointQ, we will follow another strategy in order to prove coexistence of the system (see Section 6 below).Remark 8.
It would be interesting to find if the interior fixed pointQ is instable for some choice of the parameters that are compatible with the instability hypothesis for the boundary fixed points. In that case it would be interesting to look for the existence of a limit cycle surrounding Q or of homoclinic cycle and chaotic attractors (see e.g., [15] or [16] in which the authors proved the chaotic behavior of one-predator, two prey systems without indirect effects).


## 3. The Dynamics on the Boundary ofΔ

In this section we analyze the dynamics on the boundary ofΔ, in order to do that we first study the dynamics on the coordinate axes (see Figure 2) and then on the faces of Δ.Figure 2
The dynamic on the axes.


### 3.1. Dynamics on the Axes

The three axes are invariant for the dynamics; in particular, on theZ-axes (C=0,G=0) we have
(40)Z′=-eZ,C′=0,G′=0.
Then the solution is Z=Z0e-et and we have that Z(t)→0 as t→∞. On the axes C and G we have logistic growth and in particular C′,G′→0 as C,G→I0.In fact on theC-axes (Z=0,G=0) we have
(41)Z′=0,C′=acC(I0-C),G′=0,
while on the G-axes (Z=0,C=0) we have
(42)Z′=0,C′=0,G′=agG(I0-G).
For both axes we have that C(t),G(t)→I0 as t→∞ and the solutions are of the form
(43)C(t)=I0C0eacI0tI0+C0(eacI0t-1),G(t)=I0G0eagI0tI0+G0(eagI0t-1),
where C0=C(0) and G0=G(0) are the initial conditions.


### 3.2. The Dynamics on the Faces

In this subsection we analyze the dynamics on the faces ofΔ.The Face
T
3 is on the invariant plane Z=0. We have a system of competitive Volterra-Lotka equation:
(44)C′=acC(I0-C-G),G′=agG(I0-C-G).
This is a degenerate case; in fact we have a segment S (see definition (13)) of fixed points that connects P2 and P3. The ω-limit of every orbit starting inside T3 (that is C,G≠0) is a point on S; hence, the preys do not vanish and there are not periodic orbits. The behavior of the system on the face T3 is represented in Figure 3 above.Figure 3
The dynamics on the faceT3.The Face
T
2 is on the invariant plane C=0. The system reduces to a system of Volterra-Lotka equations with intraspecific competitions between the preys:
(45)Z′=Z(-e+ugG),G′=G[agI0-(ag+ug)Z-agG].
We recall that on T2 we have the fixed point P4 if I0>e/ug. In order to understand the behavior of the solution we analyze the nullclines. The Z-nullclines are Z=0 and G=e/ug and the G-nullclines are G=0 and (1+ug/ag)Z+G=I0. We distinguish two cases: with or without the fixed point P4.In the case in whichP4∉Δ or P4=P3 (that is I0≤e/ug), the Z-nullcline G=e/ug is above (or pass through) the fixed point P3, as a consequence Z′<0 inside T2. The G-nullcline (1+ug/ag)Z+G=I0 divide T2 in two regions: below it we have G′>0 while above it we have G′<0. Then, all the solutions with G≠0 tend to the fixed point P3 according to the saddle character of P0. The behavior of the system is represented in Figure 4.Figure 4
The dynamic on the faceT2 without the point P4.In the case in whichP4∈Δ (that is, I0>e/ug) the nullclines intersect at the point P4. The direction of the vector field is represented in Figure 5 below. Hence, the solutions wind in the clockwise direction around P4. To be more specific we consider the linearization of (45) at P4. The eigenvalues of the matrix of the linearized system are
(46)λ±=12{-eagug±(eagug)2-4eag(I0-eug)}.
Then the eigenvalues are both negative and in the complex case they have negative real part. We conclude that P4 is asymptotically stable.Figure 5
The dynamic on the faceT2 in the presence of the point P4.In both cases there are no periodic orbit; in fact two dimensional Volterra-Lotka equations do not admit isolated periodic orbit (see [17]), and they admit a continuum of periodic orbits if and only if the eigenvalues at the interior fixed point (in this case P4) are purely imaginary (if and only if the trace of the matrix of the linearized system is zero, that is, -age/ug=0).The Face
T
4 is on the plane Z+C+G=I0, and we have already shown (see equation (9)) that if the solutions start on it then they enter Δ while if Z=0 they remain on C+G=I0.The Face
T
1 is easy to analyze, if the solutions do not start on the axes (that is if C,Z≠0) we have that G′>0, as a consequence the solutions leave the plane G=0 and go inside Δ.Then we have proved the following.Theorem 9.
The system (5) admits neither limit cycles nor periodic orbits in the boundary of Δ.


## 3.1. Dynamics on the Axes

The three axes are invariant for the dynamics; in particular, on theZ-axes (C=0,G=0) we have
(40)Z′=-eZ,C′=0,G′=0.
Then the solution is Z=Z0e-et and we have that Z(t)→0 as t→∞. On the axes C and G we have logistic growth and in particular C′,G′→0 as C,G→I0.In fact on theC-axes (Z=0,G=0) we have
(41)Z′=0,C′=acC(I0-C),G′=0,
while on the G-axes (Z=0,C=0) we have
(42)Z′=0,C′=0,G′=agG(I0-G).
For both axes we have that C(t),G(t)→I0 as t→∞ and the solutions are of the form
(43)C(t)=I0C0eacI0tI0+C0(eacI0t-1),G(t)=I0G0eagI0tI0+G0(eagI0t-1),
where C0=C(0) and G0=G(0) are the initial conditions.


## 3.2. The Dynamics on the Faces

In this subsection we analyze the dynamics on the faces ofΔ.The Face
T
3 is on the invariant plane Z=0. We have a system of competitive Volterra-Lotka equation:
(44)C′=acC(I0-C-G),G′=agG(I0-C-G).
This is a degenerate case; in fact we have a segment S (see definition (13)) of fixed points that connects P2 and P3. The ω-limit of every orbit starting inside T3 (that is C,G≠0) is a point on S; hence, the preys do not vanish and there are not periodic orbits. The behavior of the system on the face T3 is represented in Figure 3 above.Figure 3
The dynamics on the faceT3.The Face
T
2 is on the invariant plane C=0. The system reduces to a system of Volterra-Lotka equations with intraspecific competitions between the preys:
(45)Z′=Z(-e+ugG),G′=G[agI0-(ag+ug)Z-agG].
We recall that on T2 we have the fixed point P4 if I0>e/ug. In order to understand the behavior of the solution we analyze the nullclines. The Z-nullclines are Z=0 and G=e/ug and the G-nullclines are G=0 and (1+ug/ag)Z+G=I0. We distinguish two cases: with or without the fixed point P4.In the case in whichP4∉Δ or P4=P3 (that is I0≤e/ug), the Z-nullcline G=e/ug is above (or pass through) the fixed point P3, as a consequence Z′<0 inside T2. The G-nullcline (1+ug/ag)Z+G=I0 divide T2 in two regions: below it we have G′>0 while above it we have G′<0. Then, all the solutions with G≠0 tend to the fixed point P3 according to the saddle character of P0. The behavior of the system is represented in Figure 4.Figure 4
The dynamic on the faceT2 without the point P4.In the case in whichP4∈Δ (that is, I0>e/ug) the nullclines intersect at the point P4. The direction of the vector field is represented in Figure 5 below. Hence, the solutions wind in the clockwise direction around P4. To be more specific we consider the linearization of (45) at P4. The eigenvalues of the matrix of the linearized system are
(46)λ±=12{-eagug±(eagug)2-4eag(I0-eug)}.
Then the eigenvalues are both negative and in the complex case they have negative real part. We conclude that P4 is asymptotically stable.Figure 5
The dynamic on the faceT2 in the presence of the point P4.In both cases there are no periodic orbit; in fact two dimensional Volterra-Lotka equations do not admit isolated periodic orbit (see [17]), and they admit a continuum of periodic orbits if and only if the eigenvalues at the interior fixed point (in this case P4) are purely imaginary (if and only if the trace of the matrix of the linearized system is zero, that is, -age/ug=0).The Face
T
4 is on the plane Z+C+G=I0, and we have already shown (see equation (9)) that if the solutions start on it then they enter Δ while if Z=0 they remain on C+G=I0.The Face
T
1 is easy to analyze, if the solutions do not start on the axes (that is if C,Z≠0) we have that G′>0, as a consequence the solutions leave the plane G=0 and go inside Δ.Then we have proved the following.Theorem 9.
The system (5) admits neither limit cycles nor periodic orbits in the boundary of Δ.


## 4. Analysis form=0

In this section we study the behavior of the system in absence of indirect effects that is in the casem=0.In this case we have the presence of a further boundary fixed points:(47)P5=(ac(I0-e/uc)ac+uc,euc,0),
in the case in which I0>e/uc. If I0=e/uc, then P5=P2, while if I0<e/uc we have that P5∉Δ. Then we have the following.Theorem 10.
The pointsP0, P2, P3 and the points of the segment S are equilibria of the system (5) with m=0, for any value of the other parameters. The system admits a further fixed point P4 if
(48)I0>eug,
and a fixed point P5 if
(49)I0>euc.The dynamics on the boundary is similar to that of the casem>0. In particular on the faces T2 and T3 there are no differences between the two cases m=0 and m>0.To analyze the dynamics on the faceT4 we can use the same method of the analysis of the case m>0 and it is easy to see that we obtain the same conclusions.The remarkable difference is that, in the casem=0, the plane G=0 is invariant. The dynamics on it (the faces T1) is similar to that on T2 in the case in which m>0 (see Figures 6 and 7).Figure 6
The dynamic on the faceT1 in the case in which m=0 and in which I0>e/uc.Figure 7
The dynamic on the faceT1 in the case in which m=0 and in which I0≤e/uc.We can distinguish two cases: with or without the fixed pointP5. The matrix of the linearized system at P5 is the following:
(50)A(P5)=(0ucac(I0-e/uc)ac+ucugac(I0-e/uc)ac+uc-(ac+uc)euc-aceuc-aceuc00aguc-acugac+uc),This matrix has at least one real eigenvalue, that is,λ3=(aguc-acug)/(ac+uc), while the other two eigenvalues cannot be purely imaginary since the trace, of the matrix formed by the first two columns and rows of A(P5), is negative. Then, as in the case m>0, we can conclude the following.Theorem 11.
Ifm=0 the system (5) admits neither limit cycles nor periodic orbits on the boundary of the set Δ.In the casem=0 the asymptotic behavior of the solution can be studied by using a well-known result (see [17, Theorem 5.2.1, page 43.]) that states that if in the interior of the forward invariant set Δ there are not fixed points then the interior of Δ does not contain α-limit or ω-limit sets of the solutions. In particular if there are not fixed points in the interior of Δ then there are not periodic orbits too (see [18], e.g., of system without periodic orbit). Then we can state the following.Theorem 12.
Ifm=0, the system (5) admits neither limit cycles nor periodic orbit in the set Δ; that is, for each solution starting in Δ at least one of the species goes to extinction.Remark 13.
It is interesting to note that three-dimensional predator-prey systems (without indirect effects) admit nontrivial cases of coexistence; see, for example, the paper [19] where the authors studied oscillations for a class of singularly perturbed three-dimensional predator-prey systems.


## 5. Asymptotic Behavior form>0

In Section4 we have shown that the asymptotic dynamics on Δ does not admit coexistence of populations if m=0. In this section we analyze the case m>0 and show that the system admits coexistence of the three species by using persistence theory and in particular an acyclicity approach (see [20]).First of all we give a remark on the evolution of volume elements and then we pass to the proof that the system admits coexistence of the three populations under a proper choice of the parameters.


### 5.1. Volume Evolution

We consider the evolution of volumes under the flowϕt of the system (5). Let D be a region of ℝ3 with regular boundary and define D(t)=ϕt(D), that is, the image of D under the flow at time t. Let 𝒱(t) be the volume of D(t), then by Liouville's theorem we have that the evolution of 𝒱(t) is described by the following differential equation:
(51)d𝒱dt=∭D(t)div(F)dZdCdG,
where F is the vector field of the system (5). In details we have that
(52)divF=(ac+ug)(I0-Z-C-G)-e-(uc+ug+m)Z-(ac-uc)C-G(ag-ug).The above expression is quite difficult to study and suggests a complex behavior. However, there exist values of the parameter for whichdiv(F) is negative. In fact at least in the case in which
(53)ac>uc,(54)ag>ug,(55)I0≤eac+ug,
we have that divF≤0 so the 3-dimensional volume elements contract in Δ.In this case, if an attractor (or limit cycle) exists, we expect that it would be contained in a small subset ofΔ. On the contrary the attractor could be contained in a bigger subset of Δ (see the numerical simulations below).Remark 14.
From condition (53)-(55), we have that the volume element contracts in the whole region Δ; however, these conditions are incompatible with condition (37). Since (37) could be not necessary conditions, it would be possible to have both volume decay and instability of the fixed points.


### 5.2. Coexistence of the Three Populations

The biological problem of the coexistence of the three species can be put in mathematical terms by looking for the conditions which prevent the positive solutions starting in the interior ofΔ from converging to ∂Δ∩∂ℝ+3 as t→∞. These ideas can be made rigorous in the context of persistence theory (see [20]). There are many definitions of persistence; the most useful for biological applications (see [21]) is uniform persistence.Definition 15.
The system (5) is uniformly ρ-persistent if there exists ε>0 such that
(56)liminft→∞ρ(Z(t),C(t),G(t))>ε,
with
(57)ρ(Z(0),C(0),G(0))>0,(Z(0),C(0),G(0))∈Int(Δ),
and where
(58)ρ(x,y,z):=min{x,y,z}.In order to prove uniform persistence we use an acyclicity approach (see [20]), then (see Theorem 8.17 page 188, hypothesis (H) page 185 and Theorem 5.2 page 126 in [20]) the following conditions are needed.(C1)
There exists a compact attractor of bounded set.(C2)
The invariant sets of∂Δ∩∂ℝ+3 are weakly ρ-repelling.(C3)
The invariant sets of∂Δ∩∂ℝ+3 are acyclic.In the following lines we give the rigorous definitions of the above ideas and prove conditions (C1)–(C3).Theorem 16.
The system (5) admits a unique compact attractor 𝒜⊂Δ that attracts all bounded sets of ℝ+3.Proof.
By Theorem 2.33 page 43 in [20], we get the existence of a compact attractor of bounded sets if the flow φ associated with the system (5) is point dissipative, asymptotically smooth, and eventually bounded on every bounded set in ℝ+3.
In order to prove the thesis we consider Theorem2; in particular we write again the system (5) in function of V=Z+C+G, that is,
(59)V′=(I0-V)(acC+agG)-eZ.
If V≥I0+r with r>0 and (Z,C,G)∈ℝ+3, then
(60)V′≤-min{rac,rag,e}V,
and we conclude that there exists T>0 such that V(t)≤I0+r for t≥T. This proves the point dissipative property of the flow.
We consider the following family of sets:(61)ΔR={(Z,C,G)∈ℝ+3:V=Z+C+G≤I0+R},
where R≥0 and Δ0=Δ. Every bounded set of ℝ+3 is contained in a set ΔR for R sufficiently large. The family of sets ΔR is forward invariant (see Theorem 2 for the set Δ0), in fact, again we have
(62)V′≤-min{acR,agR,e}V.
This proves that bounded sets have bounded orbits. Finally we get the asymptotically smoothness of the flow by noting that it is asymptotically compact on every forward invariant bounded closed set (see remark 2.26, page 39 in [20]).Before proving condition (C2) we consider the following definition (see [20, chapter 8]).Definition 17.
Letφ(t,X*) be the solution of (5) starting at X*=(Z*,C*,G*). A set M in ℝ+3 is called weakly ρ-repelling if there is no X*∈ℝ+3 such that ρ(Z*,C*,G*)>0 and φ(t,X*)→M as t→∞.In order to prove weaklyρ-repelling property it is sufficient to show that the stable manifolds of the invariant sets of ∂Δ are contained in ∂Δ. From the analysis of the stability of fixed points and of the dynamics on ∂Δ we have that there are no periodic orbits on ∂Δ and the only invariant set are P0, P2, P3, P4, and S.Theorem 18.
Suppose that hypothesis of Theorem5 holds. Then the stable manifolds of all the fixed points of Δ are contained in ∂Δ∩∂ℝ+3.Proof.
Following the results of Section2 we have that the stable manifold of the fixed point P0 is the Z axes: Ws(P0)={(Z,C,G)∈ℝ3:C=G=0}, while (see also Section 3) the stable manifold of the fixed point P4 satisfies Ws(P4)⊂{(Z,C,G)∈ℝ3:C=0,Z,G>0}.
The points belonging to the segmentS have a stable manifold inside the face T3 on the invariant plane Z=0. In particular the extremes P2 and P3 of the segment S satisfy Ws(P2)={(Z,C,G)∈ℝ3:Z=G=0} and Ws(P3)={(Z,C,G)∈ℝ3:Z=C=0}. Finally all the points PS belonging to the segment S have the same center manifold
(63)Wc(PS)={(Z,C,G)∈ℝ3:Z=0,C+G=I0},
and this concludes the proof.In order to prove condition (C3) we first consider the following definitions (see [20, chapter 8]).Definition 19.
LetA,B⊂∂Δ∩∂ℝ+3. A is chained to B in ∂Δ∩∂ℝ+3, written A⇛B, if there exists a total trajectory ϕ in ∂Δ∩∂ℝ+3 with ϕ(0)∉A∪B, ϕ(-t)→A and ϕ(t)→B as t→∞.Definition 20.
A finite collection{M1,…,Mk} of subsets of ∂Δ∩∂ℝ+3 is called cyclic if after possibly renumbering M1⇛M1or M1⇛M2⇛⋯⇛Mj⇛M1 in ∂Δ∩∂ℝ+3 for some j∈2,…,k. Otherwise it is called acyclic.Theorem 21.
Suppose that hypothesis of Theorem5 holds. Then there set {P0,P2,P3,P4,S}, of invariant sets of ∂R+3, is acyclic.Proof.
By the analysis of stability of the fixed points and of the dynamics on the boundary ofΔ it follows that there are no periodic orbits in ∂Δ. Moreover, the only invariant sets P0, P2, P3, P4, and S are all contained in ∂Δ∩∂ℝ+3. By the stability analysis of fixed points we get that only the following chains are admissible:
(64)P0⇛P3⇛P4;P0⇛P2;P0⇛PS.
Then the set {P0,P2,P3,P4,S} is acyclic.
From Theorems16, 18, and 21, we get the final result.Theorem 22.
Suppose that hypothesis of Theorem5 holds. Then the system (5) is uniformly persistent.From the previous theorem we obtain coexistence of the three populations under the hypothesis of Theorem5 without analyzing the stability of the fixed point Q.


## 5.1. Volume Evolution

We consider the evolution of volumes under the flowϕt of the system (5). Let D be a region of ℝ3 with regular boundary and define D(t)=ϕt(D), that is, the image of D under the flow at time t. Let 𝒱(t) be the volume of D(t), then by Liouville's theorem we have that the evolution of 𝒱(t) is described by the following differential equation:
(51)d𝒱dt=∭D(t)div(F)dZdCdG,
where F is the vector field of the system (5). In details we have that
(52)divF=(ac+ug)(I0-Z-C-G)-e-(uc+ug+m)Z-(ac-uc)C-G(ag-ug).The above expression is quite difficult to study and suggests a complex behavior. However, there exist values of the parameter for whichdiv(F) is negative. In fact at least in the case in which
(53)ac>uc,(54)ag>ug,(55)I0≤eac+ug,
we have that divF≤0 so the 3-dimensional volume elements contract in Δ.In this case, if an attractor (or limit cycle) exists, we expect that it would be contained in a small subset ofΔ. On the contrary the attractor could be contained in a bigger subset of Δ (see the numerical simulations below).Remark 14.
From condition (53)-(55), we have that the volume element contracts in the whole region Δ; however, these conditions are incompatible with condition (37). Since (37) could be not necessary conditions, it would be possible to have both volume decay and instability of the fixed points.


## 5.2. Coexistence of the Three Populations

The biological problem of the coexistence of the three species can be put in mathematical terms by looking for the conditions which prevent the positive solutions starting in the interior ofΔ from converging to ∂Δ∩∂ℝ+3 as t→∞. These ideas can be made rigorous in the context of persistence theory (see [20]). There are many definitions of persistence; the most useful for biological applications (see [21]) is uniform persistence.Definition 15.
The system (5) is uniformly ρ-persistent if there exists ε>0 such that
(56)liminft→∞ρ(Z(t),C(t),G(t))>ε,
with
(57)ρ(Z(0),C(0),G(0))>0,(Z(0),C(0),G(0))∈Int(Δ),
and where
(58)ρ(x,y,z):=min{x,y,z}.In order to prove uniform persistence we use an acyclicity approach (see [20]), then (see Theorem 8.17 page 188, hypothesis (H) page 185 and Theorem 5.2 page 126 in [20]) the following conditions are needed.(C1)
There exists a compact attractor of bounded set.(C2)
The invariant sets of∂Δ∩∂ℝ+3 are weakly ρ-repelling.(C3)
The invariant sets of∂Δ∩∂ℝ+3 are acyclic.In the following lines we give the rigorous definitions of the above ideas and prove conditions (C1)–(C3).Theorem 16.
The system (5) admits a unique compact attractor 𝒜⊂Δ that attracts all bounded sets of ℝ+3.Proof.
By Theorem 2.33 page 43 in [20], we get the existence of a compact attractor of bounded sets if the flow φ associated with the system (5) is point dissipative, asymptotically smooth, and eventually bounded on every bounded set in ℝ+3.
In order to prove the thesis we consider Theorem2; in particular we write again the system (5) in function of V=Z+C+G, that is,
(59)V′=(I0-V)(acC+agG)-eZ.
If V≥I0+r with r>0 and (Z,C,G)∈ℝ+3, then
(60)V′≤-min{rac,rag,e}V,
and we conclude that there exists T>0 such that V(t)≤I0+r for t≥T. This proves the point dissipative property of the flow.
We consider the following family of sets:(61)ΔR={(Z,C,G)∈ℝ+3:V=Z+C+G≤I0+R},
where R≥0 and Δ0=Δ. Every bounded set of ℝ+3 is contained in a set ΔR for R sufficiently large. The family of sets ΔR is forward invariant (see Theorem 2 for the set Δ0), in fact, again we have
(62)V′≤-min{acR,agR,e}V.
This proves that bounded sets have bounded orbits. Finally we get the asymptotically smoothness of the flow by noting that it is asymptotically compact on every forward invariant bounded closed set (see remark 2.26, page 39 in [20]).Before proving condition (C2) we consider the following definition (see [20, chapter 8]).Definition 17.
Letφ(t,X*) be the solution of (5) starting at X*=(Z*,C*,G*). A set M in ℝ+3 is called weakly ρ-repelling if there is no X*∈ℝ+3 such that ρ(Z*,C*,G*)>0 and φ(t,X*)→M as t→∞.In order to prove weaklyρ-repelling property it is sufficient to show that the stable manifolds of the invariant sets of ∂Δ are contained in ∂Δ. From the analysis of the stability of fixed points and of the dynamics on ∂Δ we have that there are no periodic orbits on ∂Δ and the only invariant set are P0, P2, P3, P4, and S.Theorem 18.
Suppose that hypothesis of Theorem5 holds. Then the stable manifolds of all the fixed points of Δ are contained in ∂Δ∩∂ℝ+3.Proof.
Following the results of Section2 we have that the stable manifold of the fixed point P0 is the Z axes: Ws(P0)={(Z,C,G)∈ℝ3:C=G=0}, while (see also Section 3) the stable manifold of the fixed point P4 satisfies Ws(P4)⊂{(Z,C,G)∈ℝ3:C=0,Z,G>0}.
The points belonging to the segmentS have a stable manifold inside the face T3 on the invariant plane Z=0. In particular the extremes P2 and P3 of the segment S satisfy Ws(P2)={(Z,C,G)∈ℝ3:Z=G=0} and Ws(P3)={(Z,C,G)∈ℝ3:Z=C=0}. Finally all the points PS belonging to the segment S have the same center manifold
(63)Wc(PS)={(Z,C,G)∈ℝ3:Z=0,C+G=I0},
and this concludes the proof.In order to prove condition (C3) we first consider the following definitions (see [20, chapter 8]).Definition 19.
LetA,B⊂∂Δ∩∂ℝ+3. A is chained to B in ∂Δ∩∂ℝ+3, written A⇛B, if there exists a total trajectory ϕ in ∂Δ∩∂ℝ+3 with ϕ(0)∉A∪B, ϕ(-t)→A and ϕ(t)→B as t→∞.Definition 20.
A finite collection{M1,…,Mk} of subsets of ∂Δ∩∂ℝ+3 is called cyclic if after possibly renumbering M1⇛M1or M1⇛M2⇛⋯⇛Mj⇛M1 in ∂Δ∩∂ℝ+3 for some j∈2,…,k. Otherwise it is called acyclic.Theorem 21.
Suppose that hypothesis of Theorem5 holds. Then there set {P0,P2,P3,P4,S}, of invariant sets of ∂R+3, is acyclic.Proof.
By the analysis of stability of the fixed points and of the dynamics on the boundary ofΔ it follows that there are no periodic orbits in ∂Δ. Moreover, the only invariant sets P0, P2, P3, P4, and S are all contained in ∂Δ∩∂ℝ+3. By the stability analysis of fixed points we get that only the following chains are admissible:
(64)P0⇛P3⇛P4;P0⇛P2;P0⇛PS.
Then the set {P0,P2,P3,P4,S} is acyclic.
From Theorems16, 18, and 21, we get the final result.Theorem 22.
Suppose that hypothesis of Theorem5 holds. Then the system (5) is uniformly persistent.From the previous theorem we obtain coexistence of the three populations under the hypothesis of Theorem5 without analyzing the stability of the fixed point Q.


## 6. Numerical Simulations

The strategy used for proving coexistence of the three species does not exclude quasi-periodic or chaotic behavior of the system. The uniform persistence only lets us conclude the existence of a bounded attractor contained inΔ and with positive distance from ∂Δ. In this section we provide several numerical experiments in order to describe the structure of the attractor.


### 6.1. Experiment 1

We consider the following values of the parameters:(65)Z(0)=C(0)=G(0)=23,uc=ug=e=m=1,ac=32,ag=12.
In a first numerical experiment (represented in Figures 8, 12, 13, and 14 below) we considered the value I0=8 for which the attractor is the fixed point Q. In a second numerical experiment (represented in Figures 9, 15, 16, and 17) we considered the value I0=11. In this case the attractor appears to be bidimensional.Figure 8
The solution of the system with the choice of parameters as in (65) and I0=8.Figure 9
The solution of the system with the choice of parameters as in (65) and I0=11.


### 6.2. Experiment 2

We consider the following values of the parameters:(66)Z(0)=C(0)=G(0)=23,uc=810,ug=210,e=110,ac=910,ag=110,m=45.
Again, we perform two numerical experiments with different values of I0. In a first simulation we considered the value I0=3/2 (the solution is represented in Figure 10) and the attractor is the fixed point Q. In a second simulation we considered the value I0=3 (the solution is represented in Figure 11) and the attractor appears to be a limit cycle.Figure 10
The solution of the system with the choice of parameters as in (66) and I0=3/2.Figure 11
The solution of the system with the choice of parameters as in (66) and I0=3.Figure 12
The functionZ(t) in the case of (65) and I0=8.Figure 13
The functionC(t) in the case of (65) and I0=8.Figure 14
The functionG(t) in the case of (65) and I0=8.Figure 15
The functionZ(t) in the case of (65) and I0=11.Figure 16
The functionC(t) in the case of (65) and I0=11.Figure 17
The functionG(t) in the case of (65) and I0=11.


### 6.3. Remarks on the Numerical Experiments

These experiments suggested that the parameterI0 could be considered as a bifurcation parameter for the attractor. Moreover, we have effectuated many simulations with different choices of the function ℐ(Z,C,G) with different values of parameters and in m any cases a limit cycle arises. This suggests that the attractor may have some structural stability properties.


## 6.1. Experiment 1

We consider the following values of the parameters:(65)Z(0)=C(0)=G(0)=23,uc=ug=e=m=1,ac=32,ag=12.
In a first numerical experiment (represented in Figures 8, 12, 13, and 14 below) we considered the value I0=8 for which the attractor is the fixed point Q. In a second numerical experiment (represented in Figures 9, 15, 16, and 17) we considered the value I0=11. In this case the attractor appears to be bidimensional.Figure 8
The solution of the system with the choice of parameters as in (65) and I0=8.Figure 9
The solution of the system with the choice of parameters as in (65) and I0=11.


## 6.2. Experiment 2

We consider the following values of the parameters:(66)Z(0)=C(0)=G(0)=23,uc=810,ug=210,e=110,ac=910,ag=110,m=45.
Again, we perform two numerical experiments with different values of I0. In a first simulation we considered the value I0=3/2 (the solution is represented in Figure 10) and the attractor is the fixed point Q. In a second simulation we considered the value I0=3 (the solution is represented in Figure 11) and the attractor appears to be a limit cycle.Figure 10
The solution of the system with the choice of parameters as in (66) and I0=3/2.Figure 11
The solution of the system with the choice of parameters as in (66) and I0=3.Figure 12
The functionZ(t) in the case of (65) and I0=8.Figure 13
The functionC(t) in the case of (65) and I0=8.Figure 14
The functionG(t) in the case of (65) and I0=8.Figure 15
The functionZ(t) in the case of (65) and I0=11.Figure 16
The functionC(t) in the case of (65) and I0=11.Figure 17
The functionG(t) in the case of (65) and I0=11.


## 6.3. Remarks on the Numerical Experiments

These experiments suggested that the parameterI0 could be considered as a bifurcation parameter for the attractor. Moreover, we have effectuated many simulations with different choices of the function ℐ(Z,C,G) with different values of parameters and in m any cases a limit cycle arises. This suggests that the attractor may have some structural stability properties.


## 7. Conclusions

We have shown that in absence (that ism=0) of the terms that describe indirect effects, the system (5) does not admit coexistence of the three populations. The present work suggests the importance of indirect effects in describing cases of coexistence (with or without an interior fixed point); in particular the parameter m is a bifurcation parameter for coexistence. It has been already pointed out (see for instance [22]) that a presence of limit cycle, oscillations, or chaotic fluctuations allows the coexistence of many species and could be beneficial for the functioning of the ecosystem (see [23]).It would be interesting to consider indirect effects in different contexts such as spatial model (see [24]) or evolution problems (see [25]). In fact it is an important ecological problem (see [26]) to determine the consequences of indirect effects in evolution.In the context of population dynamics, it would be interesting to consider a different form of the functionℐ(Z,C,G); in particular if we sum the second and the third equations we get
(67)(C+G)′=(acC+aGG)(I0-Z-C-G)-Z(ucC+ugG),
then the indirect effect terms have no effects on the whole population of preys; they effect only the proportion between them. In order to consider a different situation it would be sufficient to put different constants m1 and m2 in the function ℐ(·) for the two preys (see Figure 18).Figure 18
The solution withm1=1 (in the range of the admissible values) and m2=3 (out of the range).Moreover, it would be interesting to consider the choiceℐ(C,Z)=mCαZβ, or at least ℐ(C,Z)=mCZβ. This would be an improvement in description of indirect effects; in fact in the present work we have considered that the indirect effect of the predator over the prey G has the same order of magnitude of the direct effect (compare the terms -(ac+uc)CZ and mCZ). In the case considered, if m>0, the form of the third equation is the most effected while the second equation can be rewritten in the following way:
(68)C′=C[acI0-(ac+uc+m)Z-acC-acG],
that is simply a change of the coefficient of Z. By considering a term
(69)ℐ(Z,C,G)=mCZβ,
also the form of the second equation would be drastically effected. Moreover, it is an interesting question (see [3]) to establish if direct and indirect effects are of the same magnitude. Then it would be useful to consider different values of β in (69) in order to describe cases in which direct effects are bigger (resp., less) than indirect effects.Another possible improvement would be to consider the nonautonomous case for whichm(t) is a function of time; for example, in the case of seasonal indirect effects, we could make the following choice:
(70)m(t)=K|sint|,
where K is a positive constant (see Figures 19 and 20). Some analysis of that data (see [3]) suggests that indirect effects could take a long time to become apparent since they occur at a slower time scale than any direct effect (see [4]). For this reason it could be interesting to consider a delayed version (see [27], e.g., of existence of periodic solutions for a predator-prey model with delay) of the system (5) where, for example, in the third equation we put
(71)ℐ(Z,C,G)=mC(t)Z(t-τ).
Moreover, for more general forms of the function ℐ(·), it could be interesting to look for chaotic behavior (see [28]) of the solutions (see [29], for example, in population dynamics and [30] for a discrete time case). In conclusion we can note that, beside the system (2) which is a simplification of the description of indirect effects, from the numerical experiments, it seems that it shares the main features with more sophisticated models.Figure 19
The solution withm replaced by m|sint| and I0=15.Figure 20
The solution withm replaced by m|sint| and I0=30.



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*Source: 625391-2013-04-16.xml*