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LETTERS IN BIOMATHEMATICS, 2016
VOL. 3, NO. 1, 119–139
http://dx.doi.org/10.1080/23737867.2016.1213146
RESEARCH ARTICLE OPEN ACCESS
Potential effects of invasive Pterois volitans in coral reefs
Banamali Majia, Joydeb Bhattacharyyaband Samares Pala
aDepartment of Mathematics, University of Kalyani, Kalyani, India; bDepartment of Wildlife and Fisheries
Sciences, Texas A&M University, College Station, TX, USA
ABSTRACT
The invasion of predatory lionfish (Pterois volitans) represents a major
threat to the western Atlantic coral reef ecosystems. The proliferation
of venomous, fast reproducing and aggressive P. volitans in coral
reefs causes severe declines in the abundance and diversity of reef
herbivores. There is also widespread cannibalism amongst P. volitans
populations. A mathematical model is proposed to study the effects
of predation on the biomass of herbivorous reef fishes by considering
two life stages and intraguild predation of P. volitans population with
harvesting of adult P. volitans. The system undergoes a supercritical
Hopf bifurcation when the invasiveness of P. volitans crosses a certain
critical value. It is observed that cannibalism of P. volitans induces
stability in the system even with high invasiveness of adult P. volitans.
The dynamic instability of the system due to higher invasiveness of
P. volitans can be controlled by increasing the rate of harvesting of
P. volitans.ItisalsoproventhatP. volitans goes extinct when the
harvest rate is greater than some critical threshold value. These results
indicate that the dynamical behaviour of the model is very sensitive to
the harvesting of P. volitans, which in turn is useful in the conservation
of reef herbivores.
ARTICLE HISTORY
Received 14 December 2015
Accepted 10 July 2016
KEYWORDS
Stage-structured
cannibalism; invasiveness;
harvesting; Hopf bifurcation
1. Introduction
The invasion of Pterois volitans in the western Atlantic has brought a major change to
the biodiversity of coral reefs (Morrisetal.,2009). P. volitans are voracious in nature,
spreading rapidly to new marine environments and driving down the populations of reef
herbivores drastically (Benkwitt,2015). The loss of herbivores results in the proliferation
of algae, especially the brown algae Lobophora variegata, Dictyota spp. and Sargassum
spp., which prevent the growth of corals (Acropora spp. and Montastraea spp.) on the
seabed (Bhattacharyya & Pal,2015). With venomous spines, P. volitans are the top-notch
predators in the Atlantic and Caribbean regions. Since top predators like sharks and
groupers typically avoid P. volitans and thus fail to keep the species population in check,
commercial harvesting of the adult P. volitans species seems to be the only way to mitigate
their impact on coral reef ecosystems (Morris & Whitfield,2009).
CONTACT Samares Pal samareshp@klyuniv.ac.in
© 2016 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/
licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
120 B. MAJI ET AL.
Parrotfish (Sparisoma spp.) play an important role in maintaining western Atlantic coral
reef ecosystems by consuming algae to control its growth and promote coral recruitment
(Fishelson,1997;Rotjan & Lewis,2006). P. volitans affect corals by overconsuming Par-
rotfish and other herbivorous fish that keep algae from overgrowing corals. As observed
by Goreau and Hayes (1994), Hare and Whitfield (2003), and Albins and Hixon (2008),
in the presence of predatory P. volitans, there is a rapid loss of herbivorous Parrotfish and
subsequent loss of corals. Apart from preying on reef herbivores, adult P. volitans exhibit
cannibalism, eating juveniles of their own species (Morris et al.,2009). Cannibalism has a
strong impact on population dynamics because it reduces the predation pressure on reef
herbivores (Rudolf,2008).
Almost all organisms have a life history that takes them through multiple stages from
juvenile to adult (Zhang, Chen, & Neumann,2000) and cannibalistic interactions are very
common in stage-structured populations (Rudolf,2007). To model the effect of invasive
P. volitans, we have considered a two-stage-structured system (Bhattacharyya & Pal,2014),
with Parrotfish and P. volitans following a Holling type III functional response (Luwig,
Jones, & Holling,1978). This response function is sigmoid, rising slowly when resources are
rare, accelerating when resources become more abundant and finally reaching a saturated
upper limit (Edwards & Brindley,1999). The rate of loss from the prey population due to
predation is defined as the uptake rate of the predator. In this formulation, the per capita
uptake rate by the predator is given by
f(x)=mx2
a2+x2,
where xis the prey abundance. This functional response is parameterized by the constants
mand a,wheremis the maximal prey uptake rate by the predator, and ais the value of the
prey population level when the uptake rate per unit prey is half their maximum value, i.e.
f(x)|x=a=m/2.
In our model, we have considered the stage structure of the juvenile and adult P. volitans
under the assumption that the adult P. volitans prey on Parrotfish and have reproductive
ability. For effective, widespread control of P. volitans, a non-constant harvesting policy,
introduced by Lenzini and Rebaza (2010), is used in our model. We will examine the inter-
actions of algae, Parrotfish and P. volitans to determine effective strategies for controlling
the growth of P. volitans in coral reefs. We have studied the model analytically as well as
numerically, with all proofs relegated to the Appendix 1.
2. The model
We consider a mathematical model consisting of algae at the first trophic level with
concentration P(t)at time t, and Parrotfish at the second trophic level with concentration
x(t), feeding on the algae. We also consider a two-stage structure for the top predator
P. volitans,withy(t)and z(t)as the concentrations of juvenile and adult P. volitans,
respectively. In our proposed model, it is assumed that adult P. volitans prey both on
Parrotfish and juvenile P. volitans, whereas juvenile P. volitans do not attack Parrotfish,
and have no reproductive ability (Wang & Chen,1997). Adult P. volitans are harvested
with a non-constant harvesting policy that provides diminishing marginal returns of
LETTERS IN BIOMATHEMATICS 121
the harvesting organization (Leard, Lewis, & Rebaza,2008). We make the following
assumptions in formulating the mathematical model:
(H1) In the absence of Parrotfish, macroalgae have only intraspecic competition, and
grow according to the logistic equation with intrinsic growth rate rand carrying
capacity K.
(H2) The death rate of Parrotfish is proportional to the existing Parrotfish population
with a proportionality constant D1.
(H3) The death rate of juvenile P. volitans and the transformation rate from juvenile to
adult P. volitans are proportional to the existing juvenile population with propor-
tionality constants D2and μ, respectively.
(H4) The death rate of adult P. volitans is proportional to the existing adult population
with a proportionality constant D3.
(H5)
m3y2z
a2
3+y2
represents the rate of cannibalism of adult P. volitans (z) by consuming juvenile
P. volitans (y) leading to the growth of new juveniles. The growth rate of new
juveniles is
αm3y2z
a2
3+y2,0<α<1.
Thus,
(1−α)m3y2z
a2
3+y2
represents the reduction in growth rate of juvenile P. volitans due to cannibalism.
The basic equations with all the parameters are
dP
dt=rP 1−P
K−m1P2x
a2
1+P2
dx
dt=xα1m1P2
a2
1+P2−D1−m2xz
a2
2+x2
dy
dt=α2m2x2z
a2
2+x2−(μ +D2)y−(1−α)m3y2z
a2
3+y2
dz
dt=μy−D3z−hz
c+z,(1)
where P(0)≥0, x(0)≥0, y(0)≥0andz(0)≥0. Here 1/μ represents the total time spent
by P. volitans in its juvenile stage, his the maximum harvesting rate of adult P. volitans,
cis the concentration of adult P. volitans for which the rate of harvesting is exactly half
the maximal harvesting rate, m1is the maximal uptake rate of algae by Parrotfish, m2is
the maximal uptake rate of Parrotfish by the adult P. volitans,m3is the maximal uptake
rate of juvenile P. volitans by the adult P. volitans and aiare the corresponding half
saturation constants (i=1, 2, 3). The parameters αand αirepresent the growth efficiency
(0 <α,αi<1, i=1, 2) of the organisms; all of these are positive quantities. The parameters
122 B. MAJI ET AL.
K,h,care environmental variables, while r,μ,mi,ai,Di,αand αiare properties of the
organisms.
3. Non-dimensionalization of the system
Let us change the variables of the system (1) to non-dimensional ones by substituting
¯
P=P
K,¯x=x
α1K,¯y=y
α1α2K,¯z=z
α1α2K,¯
t=rt,
and defining non-dimensional parameters
¯a1=a1
K,¯a2=a2
α1K,¯a3=a3
α1α2K,¯m1=α1m1
r,¯m2=α2m2
r,¯m3=(1−α)m3
r,
¯
D1=D1
r,¯
D2=D2
r,¯
D3=D3
r,¯μ=μ
r,¯
h=h
α1α2rK ,¯c=c
α1α2K.
After we make the substitutions above and drop the bars for simplicity, the system (1)is
reduced to dX
dt=f(X),(2)
where X=Pxyz
Tand f(X)=F1F2F3F4Twith
F1(P,x)=P(1−P)−m1xP2
a2
1+P2,
F2(P,x,z)=xm1P2
a2
1+P2−m2xz
a2
2+x2−D1,
F3(x,y,z)=m2x2z
a2
2+x2−m3y2z
a2
3+y2−(μ +D2)y,
F4(y,z)=μy−hz
c+z−D3z.
We assume that all initial values are non-negative. The right-hand sides of the equations
in the system (2) are smooth functions of the variables P,x,y,zand the parameters. The
following lemma gives the condition for which the solutions of the system (2) are positive.
Lemma 3.1: If y(t)and z(t)are always positive, then all possible solutions of the system
(2)are positive.
Therefore, as long as y(t)>0andz(t)>0forallt, local existence and uniqueness
properties hold in the region ={(P,x,y,z):P>0, x>0, y>0, z>0}.
4. Boundedness and permanence of the system
We first prove that the solutions of Equation (2) with initial values in are bounded, so
that Equation (2) represents a biologically meaningful system. The proofs of all the lemmas
are given in the Appendix 1.
LETTERS IN BIOMATHEMATICS 123
Lemma 4.1: For all >0, there exists t>0such that all the solutions of (2)with positive
initial values fall into the set (P,x,y,z)∈:P(t)+x(t)+y(t)+z(t)<1/D+,>0
whenever t ≥t, where D =min{1, D1,D2,D3}.
Since P(t)+x(t)+y(t)+z(t)<1/Das t→∞it follows that there exist positive
numbers M1,M2,M3with M1+M2+M3<1/Dsuch that x(t)≤M1,y(t)≤M2and
z(t)≤M3for large values of t.
Let us define
λ=a2
1D1
m1−D1
,m1>D
1.
Then λdenotes the break-even concentration of algae for which the Parrotfish population
is constant in the absence of P. volitans. The following lemma states the condition under
which neither Parrotfish nor P. volitans can survive in the system:
Lemma 4.2: If (i) m1≤D1or (ii) m1>D
1and λ>1/D hold, then
lim
t→∞ x(t)=lim
t→∞ y(t)=lim
t→∞ z(t)=0.
According to Lemma 4.2,wehave
(i) If the maximal uptake rate of Parrotfish is less than or equal to its death rate, then
Parrotfish and P. volitans will not survive in the system.
(ii) If the maximal uptake rate of Parrotfish is greater than its death rate and the break-
even concentration λis greater than 1/D, then Parrotfish and P. volitans will not
survive in the system.
The system (2) will be permanent (Ruan,1993) if there exists u1i,Mi∈(0, ∞)such that
u1i≤lim ui(t)≤Mi
for each organism ui(t)in the system (i=1, ...,4). Permanence represents convergence
on an interior attractor from any positive initial conditions, and hence, can be regarded as
a strong form of coexistence. From a biological point of view, the permanence of a system
ensures the survival of all the organisms in the long run. Without any loss of generality,
we assume that m1>D
1and m2>D
2. The condition given in the following Lemma rules
out the possibility of extinction of any organism in the system.
Lemma 4.3: If there exists p1,0<p
1<λ, then for large t, there exists
x1=m2M3(a2
1+p2
1)
(m1−D1)(p2
1−λ2),y1>0, and z1>M2(a2
2+x2
1)(μ +D2)
(m2−m3)x2
1−m3a2
2
,
such that each solution of the system (2)with positive initial values falls into the compact set
(P,x,y,z):p1≤P(t)≤1/D,x1≤x(t)≤M1,y1≤y(t)≤M2,z1≤z(t)≤M3,
and stays there.
System (2) is called competitive (Smith,1995) if there exists a diagonal matrix
H=diag(1,...,4)such that HJ(X)Hhas non-positive off-diagonal elements,
124 B. MAJI ET AL.
where J(X)is the Jacobian of the system (2)andiis either 1 or –1 (i=1, ...,4).By
choosing H=diag(1, −1, −1, 1), we see that the off-diagonal elements of HJ(X)Hare
non-positive if
m3y2(t)
a2
3+y2(t)<m2x2(t)
a2
2+x2(t)
for all t>0. This leads to the following result:
Lemma 4.4: The system (2)is competitive if
x(t)>x
1>m3a2
2
m2−m3
for large t, where m2>m
3.
5. Equilibria and their stability
The system (2) possesses the following feasible equilibria:
(i) Organism-free equilibrium E0=(0, 0, 0, 0);
(ii) Parrotfish- and P. volitans-free equilibrium E1=(1, 0, 0, 0);
(iii) P. volitans-free equilibrium E2=λ,λ(1−λ)
D1,0,0
;
(iv) The equilibrium of coexistence E∗=(P∗,x∗,y∗,z∗),whereP∗is a positive root of
the equation
m2θ2(P)ψ(P)
a2
2+θ2(P)−m3φ2(P)ψ(P)
a2
3+φ2(P)=(μ +D2)φ(P)
with
θ(P)=(1−P)(a2
1+P2)
m1P,φ(P)=1
μD3+h
c+ψ(P)ψ(P),and
ψ(P)=(m1−D1)(P2−λ2)(a2
2+θ2(P))
m2(a2
1+P2)θ(P);x∗=θ(P∗),y∗=ψ(P∗),z∗=φ(P∗).
We see that E0and E1always exist, E2exists if λ<1, and E∗exists if P∗>λ.
We analyse the local stability of system (2) using eigenvalue analysis of the Jacobian
matrix, J(X), evaluated at the appropriate equilibrium. The eigenvalues of J(X)at E0are
1, −D1,−μ−D2and −D3−h/c. This gives the following result:
Lemma 5.1: The organism-free equilibrium E0of the system (2)is always a saddle point.
Therefore, the system (2) is very unlikely to collapse.
Lemma 5.2: The critical point E1of the system (2)is locally asymptotically stable if D1>
m1/(a2
1+1).
Therefore, with a high mortality rate of Parrotfish, the system (2) stabilizes at the
Parrotfish- and P. volitans-free equilibrium E1. The decrease in the mortality rate of
Parrotfish changes the stability of the system (2) from an algae-dominated state in the
absence of Parrotfish to an algae-Parrotfish coexistence state. The following Lemma gives
LETTERS IN BIOMATHEMATICS 125
the condition for which coexistence of Parrotfish and algae is possible in the absence of
P. volitans.
Lemma 5.3: The critical point E2=λ,λ(1−λ)
D1,0,0
of the system (2)is locally asymptot-
ically stable if
λ<1and λ(1−λ)
a2cμm2
(μ +D2)(cD3+h)−1<D
1<m1
2(1−λ) .
The system is persistent at E∗if all the boundary equilibria repel interior trajectories
(Ruan,1993). The following Lemma gives the condition of persistence of the system (2)
at E∗:
Lemma 5.4: All the organisms will persist if λ<1and
D1≤min m1
a2
1+1,λ(1−λ)
a2cμm2
(μ +D2)(cD3+h)−1.
Therefore, with a low mortality rate of Parrotfish, all the organisms in the system (2)
coexist.
Lemma 5.5: The system (2)has no periodic solution around the positive equilibrium E∗if
1+μ+m11+2x∗
P∗+m21+2z∗
x∗+m3<L,
where L is the minimum of the following six quantities:
(i) D1+2P∗+2m1P∗x∗a2
1
(a2
1+P∗2)2+2m2x∗z∗a2
2
(a2
2+x∗2)2,
(ii) 2P∗+μ+D2+2m3y∗z∗a2
3
(a2
3+y∗2)2+2m1P∗x∗a2
1
(a2
1+P∗2)2,
(iii) 2P∗+D3+2m1P∗x∗a2
1
(a2
1+P∗2)2+ch
(c+z∗)2,
(iv) μ+D1+D2+2m2x∗z∗a2
2
(a2
2+x∗2)2+2m3y∗z∗a2
3
(a2
3+y∗2)2,
(v) D1+D3+2m2x∗z∗a2
2
(a2
2+x∗2)2+ch
(c+z∗)2,
(vi) μ+D1+D3+ch
(c+z∗)2+2m3y∗z∗a2
3
(a2
3+y∗2)2.
Corollary 5.1: If the conditions stated in Lemmas 5.4 and 5.5 both hold, then the positive
equilibrium is locally asymptotically stable.
Now we use the Routh–Hurwitz criterion to find the necessary and sufficient conditions
for stability of the system (2)atE∗.
Lemma 5.6: The positive equilibrium E∗of the system (2)is locally asymptotically stable if
Q1>0, Q1Q2>Q
3,and Q1Q2Q3>Q
2
3+Q2
1Q4,
where
Q1=−F1
P|E∗+F2
x|E∗+F3
y|E∗+F4
z|E∗,
Q2=F1
S|E∗+F2
x|E∗F3
y|E∗+F4
z|E∗+F1
P|E∗F2
x|E∗+F3
y|E∗F4
z|E∗
−μF3
z|E∗−F1
x|E∗F2
P|E∗,
126 B. MAJI ET AL.
Q3=μF3
z|E∗−F3
y|E∗F4
z|E∗)(F1
P|E∗+F2
x|E∗
+F1
x|E∗F2
P|E∗−F1
P|E∗F2
x|E∗F3
y|E∗+F4
z|E∗−μF3
x|E∗F2
z|E∗,
Q4=F3
y|E∗F4
z|E∗−μF3
z|E∗F1
P|E∗F2
x|E∗−F1
x|E∗F2
P|E∗+μF1
P|E∗F2
z|E∗F3
x|E∗.
Due to the complexity in the algebraic expressions involved, it is difficult to interpret
the results in ecological terms; however, numerical simulations are used to illustrate the
dynamical behaviour of the system about E∗.
6. Hopf bifurcation
We will study the Hopf bifurcation of the system (2)atE∗,takingm2as a bifurcation
parameter.
The characteristic equation of the Jacobian matrix at E∗is G(λ) =u4+Q1u3+Q2u2+
Q3u+Q4=0. Solving Q1Q2Q3−Q2
3−Q2
1Q4=0, the critical value of m2can be obtained,
say, m2=m2cr .
Lemma 6.1: The system undergoes a Hopf bifurcation at m2=m2cr if and only if
(i) f1(m2cr )=f2(m2cr ),
(ii) M(m2)K(m2)−N(m2)L(m2)m2=m2cr = 0,
where
f1(m2)=Q1(m2)Q2(m2)Q3(m2),f2(m2)=Q2
3(m2)+Q2
1(m2)Q4(m2),
K(m2)=4β3
1(m2)−12β1(m2)β2
2(m2)+3β2
1(m2)−β2
2(m2)Q1(m2)
+2β1(m2)Q2(m2)+Q3(m2),
L(m2)=12β2
1(m2)β2(m2)−4β3
2(m2)+6β1(m2)β2(m2)Q1(m2)+2β2(m2)Q2(m2),
M(m2)=β3
1(m2)Q
1(m2)−3β1(m2)β2
2(m2)Q
1(m2)+β2
1(m2)−β2
2(m2)Q
2(m2)
+β1(m2)Q
3(m2),
N(m2)=3β2
1(m2)β2(m2)Q
1(m2)−β3
2Q
1(m2)+2β1(m2)β2(m2)Q
2(m2)
+β2(m2)Q
3(m2);
β1(m2)and β2(m2)are real and imaginary parts, respectively, of a pair of eigenvalues for
all m2∈(m2cr −,m2cr +).
The condition (ii) is equivalent to dg(m2)/dm2|(m2=m2cr )= 0. Thus, using numerical
methods, condition (i) can be verified by showing that the curves y=f1(m2)and
y=f2(m2)intersect at m2=m2cr , whereas the condition (ii) can be verified by showing
that the tangent to the curve y=g(m2)at m2=m2cr is not parallel to the m2axis
(Siekmann, Malchow, & Venturino,2008).
Corollary 6.1: The period τof the bifurcating periodic orbits close to m2=m2cr is given by
τ(m2cr )=2πQ1(m2cr )
Q3(m2cr ).
LETTERS IN BIOMATHEMATICS 127
6.1. Stability of bifurcating periodic solution
We investigate the orbital stability of the Hopf-bifurcating periodic solution using Poore’s
sufficient condition (Poore,1976). The supercritical/subcritical nature of Hopf-bifurcating
periodic solution is determined by the positive/negative sign of the real part of , respec-
tively, where
=−alFl
ujumusbjbm¯
bs+2alFl
ujumbj(J−1
E∗)mrFr
upuqbp¯
bq
+alFl
ujuk
¯
bj(JE∗−2iω0)−1kr Fr
upuqbpbq,
the repeated indices within each term imply a sum from 1 to 4, all the derivatives of Fl
are evaluated at the equilibrium E∗with u1=P,u2=x,u3=y,u4=z,andJE∗is the
Jacobian matrix of (2)calculatedatE∗.(JE∗)−1mr denotes the element in row m,column
rof (JE∗)−1.
The left and right normalized eigenvectors of JE∗with respect to the eigenvalues ±iω0
at m2=m2cr are given by
a=ξ1
F2
PF3
xF4
yF2
PF3
xF4
y,a2F3
xF4
y,a3F4
y,a4and
b=ξ2
F1
xF2
zF4
yF1
xF2
zF4
y,b2F2
zF4
y,b3,b4F4
yT,
where ξ1and ξ2are complex numbers,
a2=iω0−F1
P,
a3=F4
y[(F1
PF2
x−F1
xF2
P−ω2
0)−iω0(F1
P+F2
x)],
a4=ω2
0(F1
P+F2
x)−F3
y(F1
PF2
x−F1
xF2
P−ω2
0)+iω0{F1
PF2
x−F1
xF2
P−ω2
0+F3
P(F1
P+F2
x)},
b2=iω0−F1
P,
b3=ω2
0(F1
P+F2
x)−F4
z(F1
PF2
x−F1
xF2
P−ω2
0)+iω0{F1
PF2
x−F1
xF2
P−ω2
0+F4
z(F1
P+F2
x)},
b4=[ω2
0(F1
P+F2
x)−F4
z(F1
PF2
x−F1
xF2
P−ω2
0)+iω0{F1
PF2
x−F1
xF2
P−ω2
0+F4
z(F1
P+F2
x)}].
Using a.b=1, we obtain ξ1ξ2.
If ()m2=m2cr >0, then the system (2) undergoes a supercritical Hopf bifurcation as m2
is increased through m2cr , so that the bifurcating periodic orbit is asymptotically orbitally
stable.
7. Numerical simulations
In this section, we investigate the effects of various parameters on the qualitative behaviour
of our system using the numerical approach of Bhattacharyya and Pal (2013)using
MATLAB. The default set of parameter values, mostly taken from Bhattacharyya and
Pal (2013), is given in Table 1. Under this set of parameter values, it is observed that the
system becomes locally asymptotically stable at E∗(cf. Figure 1).
We will now verify the feasibility of the criteria of stability in Section 5.
128 B. MAJI ET AL.
Table 1. Default parameter values used in the numerical analysis.
Parameters Description of parameters Default value Dimension
rIntrinsic growth rate of Algae 4 1/time
KCarrying capacity of the system 5 mass/volume
m1Maximal uptake rate of Parrotfish on Algae 3 1/time
m2Maximal uptake rate of adult Pterois volitans on Parrotfish 5 1/time
m3Maximal uptake rate of adult Pterois volitans on juvenile Pterois volitans 5 1/time
a1Half saturation const. for uptake of Algae by Parrotfish 2 mass/volume
a2Half saturation const. for uptake of Parrotfish by adult Pterois volitans 2 mass/volume
a3Half saturation const. for uptake of juvenile Pterois volitans by adult Pterois volitans 3 mass/volume
α1Growth efficiency of Parrotfish on Algae 0.3 –
α2Growth efficiency of adult Pterois volitans on Parrotfish 0.6 –
αGrowth efficiency of adult Pterois volitans growth on juvenile Pterois volitans 0.4 –
D1Death rate of Parrotfish 0.3 1/time
D2Death rate of juvenile Pterois volitans 0.2 1/time
D3Death rate of adult Pterois volitans 0.1 1/time
1
μTotal time spent by Pterois volitans in its juvenile stage 0.2 time
hMaximal harvesting rate of adult Pterois volitans 2 1/time
cHalf saturation constant for harvesting of adult Pterois volitans 1 1/time
0100 200 300 400 500 600 700 800
0
0.5
1
Time
Algae
0100 200 300 400 500 600 700 800
1
2
3
Time
Parrotfish
0100 200 300 400 500 600 700 800
0
0.1
0.2
Time
Juvenile Lionfish
0100 200 300 400 500 600 700 800
0
0.2
0.4
Time
Matured Lionfish
Figure 1. Time series analysis of the system for the parameters given in Table 1, the system has a stable
focus at E∗.
Example 1: If the maximal harvesting rate of adult P. volitans is increased (viz. h=2.5),
and all other parameters are as given in Table 1,thenweobtain
λ<1and0.0503 =λ(1−λ)
a2cμm2
(μ +D2)(cD3+h)−1<D
1<m1
2(1−λ) =0.1569,
satisfying the condition of stability at E2=(0.2828, 2.7046, 0, 0)asgiveninLemma5.3.E2
is a stable focus with eigenvalues −0.0477, −1.9023 and −0.1195 ±i0.2396 (cf. Figure 2).
LETTERS IN BIOMATHEMATICS 129
0 50 100 150
0
0.5
1
Time
Algae
0 50 100 150
0
2
4
Time
Parrotfish
0 50 100 150
0
0.2
0.4
Time
Juvenile Lionfish
0 50 100 150
0
0.5
1
Time
Matured Lionfish
Figure 2. Time series analysis of the system for h=2.5 and other parameter values as given in Table 1,
the system has a stable focus at E2.
0
0.5
1
1.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
E
1
Algae (P)
E
0
Parrotfish (x)
Matured Lionfish (z)
Figure 3. Phase plane diagram of the system projected on Pxz-space for D1=0.85 and other parameter
values as given in Table 1, the system has a stable node at E1.
Example 2: If the death rate of Parrotfish is increased (viz. D1=0.85), leaving all other
parameters unaltered, then the system approaches a stable node at E1=(1, 0, 0, 0),with
eigenvalues −1, −0.0185, −0.525 and −1.3. In this case, we obtain D1>m
1/(a2
1+1)=
0.194, satisfying Lemma 5.2 (cf. Figure 3).
Example 3: Under the set of parameter values are as given in Table 1, the system is locally
asymptotically stable at E∗=(0.5803, 1.5968, 0.095, 0.2801), with eigenvalues −1.7192,
130 B. MAJI ET AL.
0
0.2
0.4
0.6
0.8
1
0
0.5
1
1.5
2
2.5
3
0
0.05
0.1
0.15
0.2
0.25
E1
Algae (P)
E2
E0
E*
Parrotfish (x)
Matured Lionfish (z)
I1
Figure 4. Phase plane diagram of the system projected on Pxz-space for m1=2.5 and other parameters
as given in Table 1with initial value I1. The system is oscillatory around E∗(in black). For m1=1.2
and other parameters as given in Table 1, the system is locally asymptotically stable at E2(in blue). For
m1=0.5 and other parameters as given in Table 1, the system is locally asymptotically stable at E1(in
red).
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.5
1
1.5
2
2.5
3
0
0.05
0.1
0.15
0.2
0.25
0.3
E1
Algae (P)
E*
E0
Parrotfish (x)
E2
Matured Lionfish (z)
I1 = (0.1, 1, 0.01)
Figure 5. Phase plane diagram of the system projected on Pxz-space for m2=7 and other parameters
as given in Table 1with initial value I1=(0.1, 1, 0.01). The system is oscillatory around E∗(in blue). For
m2=7andh=3.2, other parameters as given in Table 1, the system is locally asymptotically stable at
E2(in black).
−0.3302 and −1.0076 ±i0.1422. In this case we obtain
λ<1andD1≤λ(1−λ)
a2cμm2
(μ +D2)(cD3+h)−1=0.093,
satisfying the analytical conditions of persistence as given in Lemma 5.4.Wealsoobtain
Q1=1.2282, Q1Q2−Q3=1.2282, and Q1Q2Q3−Q2
3−Q2
1Q4=0.0126,
LETTERS IN BIOMATHEMATICS 131
0100 200 300 400 500 600 700 800
0.4
0.6
0.8
1
Time
Algae
0100 200 300 400 500 600 700 800
0
1
2
3
Time
Parrotfish
0100 200 300 400 500 600 700 800
0
0.1
0.2
Time
Juvenile Lionfish
0100 200 300 400 500 600 700 800
0
0.2
0.4
0.6
Time
Matured Lionfish
Figure 6. Time series analysis of the system for m2=7 and other parameter values as given in Table 1,
the system is oscillatory around E∗(solid). For m2=7, a3=0.5 and other parameter values as given in
Table 1, the system is LAS at E∗(dotted).
4.6 55.15 5.2 5.25 5.5 6
0.02
0.04
0.06
0.08
0.1
0.12
2
f
1
(m
2
), f
2
(m
2
)
4.6 55.15 5.2 5.25 5.5 6
−0.04
−0.02
0
0.02
0.04
0.06
2
g(m
2
) = f
1
(m
2
) − f
2
(m
2
)
f
2
(m
2
)
f
1
(m
2
) = f
2
(m
2
)
g(m
2
)
f
1
(m
(a)
(b)
2
)
Unstable at E
*
(Q
1
Q
2
Q
3
< Q
3
2
+ Q
1
2
Q
4
)
Stable at E
*
(Q
1
Q
2
Q
3
> Q
3
2
+ Q
1
2
Q
4
)
Figure 7. The relative position of f1(m2),f2(m2)and g(m2)showing that a Hopf bifurcation occurs when
the two curves intersect at m2cr =5.2.
satisfying the analytical conditions of stability at E∗as given in Lemma 5.6 (cf. Figure 1).
Example 4: For m1=2.5, the system is oscillatory around E∗(cf. Figure 4). When we
decrease the value of m1(viz. m1=1.2), the system becomes stable at E2. On further
lowering the value of m1(viz. m1=0.5), we find that the system stabilizes at E1.
132 B. MAJI ET AL.
4.6 4.8 55.2 5.4 5.6 5.8 6
0.4
0.6
0.8
m2
Algae
4.6 4.8 55.2 5.4 5.6 5.8 6
0.5
1
1.5
2
2.5
m2
Parrotfish
4.6 4.8 55.2 5.4 5.6 5.8 6
0
0.1
0.2
m2
Juvenile Lionfish
4.6 4.8 55.2 5.4 5.6 5.8 6
0
0.2
0.4
0.6
m2
Matured Lionfish
Figure 8. The system undergoes a supercritical Hopf bifurcation as m2is increased through m2cr =5.2
with other parameter values as given in Table 1.
Example 5: We observe that the system is oscillatory around E∗for m2=7(cf.
Figure 5). In this case by increasing the maximal harvesting rate of adult P. volitans (viz.
h=3.2), the system becomes locally asymptotically stable at E2.
Example 6: For high invasiveness of adult P. volitans, the system becomes oscillatory
around E∗(cf. Figure 5). When the value of a3is lowered (viz. a3=0.5), the system
stabilizes at E∗(cf. Figure 6).
7.1. Hopf bifurcation
We observe that the system becomes oscillatory when the maximal uptake rate of adult
P. volitans is high. We therefore consider m2as a bifurcation parameter. For m2<5.2we
see that f1(m2)>f
2(m2), satisfying the Routh–Hurwitz condition, so that the system is
locally asymptotically stable at E∗.Form2>5.2weseethatf1(m2)<f
2(m2)and so the
system is unstable at E∗(cf. Figure 7(a)). Moreover, we observe that the tangent to the graph
of g(m2)at m2=5.2 is not parallel to the m2axis (cf. Figure 7(b)), satisfying the condition
dg/dm2|(m2=5.2)= 0. In Figure 8, we observe that a supercritical Hopf bifurcation
occurs when the parameter m2is increased through the critical value m2cr =5.2.
8. Discussion
We have considered a tri-trophic food chain model consisting of algae and Parrotfish in
the first two trophic levels, respectively, while juvenile and adult P. volitans reside in the
third trophic level. We analyse the effect of predation with stage-structured cannibalism
and study the effect of harvesting of the adult P. volitans on the dynamics of the system.
The threshold values for the existence and stability of various steady states of the system
LETTERS IN BIOMATHEMATICS 133
are worked out. In order to keep sustainable development of the coral reef ecosystem, it is
desirable to have a positive equilibrium which is asymptotically stable. Keeping this view
in mind, we have established some strong criteria for existence of the positive equilibrium.
We studied bifurcation with respect to the parameter representing the invasiveness of
adult P. volitans in the system. The critical parameter value at which bifurcation occurs
is determined to preserve the system under consideration in its natural state. We observe
that when the maximal uptake rate of adult P. volitans on Parrotfish crosses a certain
critical value, the system enters into Hopf bifurcation that induces oscillation around the
positive equilibrium. The stability as well as the direction of Hopf bifurcation near the
interior equilibrium is obtained by applying the algorithm due to Poore (1976). We have
also provided numerical simulations to substantiate our analytic results. From analytical
and numerical observations we obtain the following conclusions:
(i) If the growth rate of Parrotfish is low, then the Parrotfish would become extinct.
(ii) Higher mortality rate of Parrotfish can lead to the extinction of both Parrotfish
and P. volitans from the system. This represents the fact that rapid elimination of
herbivorous fish can be fatal for the coral reef ecosystem.
(iii) High rate of predation of adult P. volitans on Parrotfish induces oscillation around
the positive equilibrium leading to dynamic instability, representing the phenomenon
of ecological imbalance due to high invasiveness of P. volitans in the coral reef
ecosystem. This dynamic instability can be controlled by increasing the rate of
harvesting of adult P. volitans. Moreover, a high harvesting rate of adult P. volitans
can eliminate P. volitans from the system.
(iv) The increase of cannibalism of P. volitans stabilizes the system even with high
invasiveness of adult P. volitans.
Throughout the article we focus on searching for a suitable way to control the growth of
algae, Parrotfish and P. volitans, and maintain a stable coexistence of all the species. Our
numerical simulations suggest that the maximal harvesting rate of adult P. volitans can be
used as a control parameter to maintain the stability of the system at the coexistence steady
state.
Our results are based on a model that has no growth equation for the corals. It would
be interesting to incorporate corals in our system to study the dynamics of coral reefs in
the presence of invasive P. volitans.
Disclosure statement
No potential conflict of interest was reported by the authors.
Funding
This work was supported by Science and Engineering Research Board [grant number SR/S4/MS:863/13].
ORCID
Samares Pal http://orcid.org/0000-0002-8792-0031
134 B. MAJI ET AL.
References
Albins,M.A.,&Hixon,M.A.(2008). Invasive Indo-Pacific lionfish (Pterois volitans)reduce
recruitment of Atlantic coral-reef fishes. Marine Ecology Progress Series, 367, 233–238.
doi:10.3354/meps07620
Benkwitt, C. E. (2015). Non-linear effects of invasive lionfish density on native coral-reef fish
communities. Biological Invasions, 17, 1383–1395. doi:10.1007/s10530-014-0801-3
Bhattacharyya, J., & Pal, S. (2013). Stage-structured cannibalism with delay in maturation
and harvesting of an matured predator. Journal of Biological Physics, 39, 37–65.
doi:10.1007/s10867-012-9284-6
Bhattacharyya, J., & Pal, S. (2014). Dynamics of a stage-structured system with harvesting and
discrete time delays. Systems Science & Control Engineering: An Open Access Journal, 2, 192–215.
doi:10.1080/21642583.2014.886973
Bhattacharyya, J., & Pal, S. (2015). Hysteresis in coral reefs under macroalgal toxicity and overfishing.
Journal of Biological Physics, 41, 151–172. doi:10.1007/s10867-014-9371-y
Edwards,A.M.,&Brindley,J.(1999). Zooplankton mortality and the dynamical behaviour of
plankton population models. Bulletin of Mathematical Biology, 61, 303–339.
Fishelson, L. (1997). Experiments and observations on food consumption growth and starvation in
dendrochirus brachypterus and Pterois volitans (Pteroinae, Scorpaenidae). Environmental Biology
of Fishes, 50, 391–403.
Goreau, T. J., & Hayes, R. (1994). Coral bleaching and ocean “hot spots”. Ambio, 23, 176–180.
Hare, J. A., & Whitfield, P. E. (2003). An integrated assessment of the introduction of lionfish
(Pterois volitans/miles complex) to the western Atlantic Ocean. NOAA Technical Memorandum,
NOS NCCOS, 2, 1–21.
Leard, B., Lewis, C., & Rebaza, J. (2008). Dynamics of ratio-dependent predator-prey models with
nonconstant harvesting. Discrete and Continuous Dynamical Systems – Series S, 1, 303–315.
Lenzini, P., & Rebaza, J. (2010). Nonconstant predator harvesting on ratio-dependent predator-prey
models. Applied Mathematical Sciences, 4, 791–803.
Li, Y., & Muldowney, S. (1993). On Bendixson criteria. Journal of Differential Equations, 106, 27–39.
Luwig, D., Jones, D. D., & Holling, C. S. (1978). Qualitative analysis of insect outbreak systems: The
spruce budworm and forest. The Journal of Animal Ecology, 47, 315–332.
Morris, J. A., Akins, J. L., Barse, A., Cerino, D., Freshwater, D. W., Green, S. J., ... Whitefield, P. E.
(2009). Biology and ecology of invasive lionfishes, Pterois miles and Pterois volitans. Gulf and
Caribbean Fisheries Institute, 61, 1–6.
Morris, Jr., J. A., & Whitfield, P. E. (2009). Biology, ecology, control and management of the invasive
Indo-Pacific lionfish: An updated integrated assessment. NOAA Technical Memorandum NOS-
NCCOS, 99, 1–65.
Poore, A. B. (1976). On the theory and applications of Hopf-Friedrichs bifurcation theory. The
Archive for Rational Mechanics and Analysis, 60, 371–393.
Rotjan,R.D.,&Lewis,S.M.(2006). Parrotfish abundance and selective corallivory on a Belizean
coral reef. Journal of Experimental Marine Biology and Ecology, 335, 292–301.
Ruan,S.(1993). Persistence and coexistence in zooplanktonphytoplankton-nutrient models
with instantaneous nutrient recycling. Journal of Mathematical Biology, 31, 633–654.
doi:10.1007/bf00161202
Rudolf,V.H.W.(2007). The interaction of cannibalism and omnivory: Consequences for
community dynamics. Ecology, 88, 2697–2705. doi:10.1890/06-1266.1
Rudolf,V.H.W.(2008). The impact of cannibalism in the prey on predator-prey dynamics. Ecology,
89, 3116–3127. doi:10.1890/07-0709.1
Siekmann, I., Malchow, H., & Venturino, E. (2008). An extension of the Beretta–Kuang model of
viral diseases. Mathematical Biosciences and Engineering, 5, 549–565. doi:10.3934/mbe.2008.5.549
Smith, H. L. (1995). Monotone dynamical systems: An introduction to the theory of competitive
and cooperative systems. Mathematical surveys and monographs. Providence, RI: American
Mathematical Society.
Wang, W., & Chen, L. (1997). A predator-prey system with stage structure for predator. Computers
& Mathematics with Applications, 33, 83–91. doi:10.1016/S0898-1221(97)00056-4
LETTERS IN BIOMATHEMATICS 135
Zhang, X., Chen, L., & Neumann, A. U. (2000). The stage-structured predator-
prey model and optimal harvesting policy. Mathematical Biosciences, 168, 201–210.
doi:10.1016/s0025-5564(00)00033-x
Appendix 1.
A.1. Proof of Lemma 3.1
Proof: From the first two equations of the system (2) we have,
P(t)=P(0)et
01−P−m1xP
a2
1+P2dτand x(t)=x(0)et
0α1m1P2
a2
1+P2−m2xz
a2
2+x2−D1dτ
.
This implies, if P(0)>0andx(0)>0, then P(t)>0andx(t)>0 for all t>0. Hence, if y(t)>0
and z(t)>0 for all t, it may be concluded that all the solutions of the system (2) are always positive.
A.2. Proof of Lemma 4.1
Proof: Since dP/dt≤P(1−P), it follows that for >0, there exists t>0suchthatP(t)≤1+for
all t≥t.Wehaved(t)/dt<1+−D(t)for all t≥t,where(t)=P(t)+x(t)+y(t)+z(t)
and D=min{1, D1,D2,D3}.Letu(t)be the solution of du/dt+uD =1, satisfying u(0)=
(0).Thenu(t)=1/D+(0)−1/De−tD →1/Das t→∞. By comparison, it follows that
P(t)+x(t)+y(t)+z(t)<1/D+, for all t≥t, proving the Lemma.
A.3. Proof of Lemma 4.2
Proof: If possible, let limt→∞ P(t)=0. Then if P(t)decreases monotonically to zero, there exists
T1>0suchthatP(t)+m1x(t)P(t)/(a2
1+P2(t)) <1 for all t>T
1. This gives dP/dt>0 for all t>T
1,
contradicting to our assumption. Therefore, there exists 0 <p
1≤1/Dsuch that p1≤P(t)≤1/D
for all t>T
1,whereD=min{1, D1,D2,D3}. Since limt→∞ sup[x(t)+y(t)+z(t)]<1/D,it
follows that there exists T2>0suchthatx(t)≤M1,y(t)≤M2,andz(t)≤M3,whereM1,M2,
M3are finite positive constants satisfying M1+M2+M3<1/D. For t>max{T1,T2},wehave
dx/dt≥xm1p2
1/(a2
1+p2
1)−D1−m2M3/x. This implies dx/dt|x=x1≥0 for t>max{T1,T2},
where x1=m2M3(a2
1+p2
1)/{(m1−D1)(p2
1−λ2)}. It is also seen that, x1>0ifp1>λ.This
implies that if p1>λis satisfied, then there exists T3>0suchthatx1≤x(t)≤M1for all
t>T
3. For t>T
3,wehavedy/dt≥m2x2
1z/(a2
2+x2
1)−(μ +D2)M2−m3z,andsoifz(t)>
M2(a2
2+x2
1)(μ +D2)/{(m2−m3)x2
1−m3a2
2}holds, then dy/dt>0 for all t>T
3. Let there exist
z1>0suchthatM2(a2
2+x2
1)(μ +D2)/{(m2−m3)x2
1−m3a2
2}<z
1<M
3.Ifz(t)≥z1>0, we
have dy/dt>0 for all t≥T3, and so in this case, there exists T4>0and0 <y
1<M
2such
that y(t)≥y1for all t≥T4. Therefore, for all t≥T4,ifz(t)≥z1holds, then y1≤y(t)≤M2
and z1≤z(t)≤M3.LetT=max{T1,T2,T3,T4}. Then for t>T, there exists finite positive real
numbers p1,x1,y1,z1,M1,M2,M3with
M1+M2+M3<1
D,p1>λ,x1=m2M3(a2
1+p2
1)
(m1−D1)(p2
1−λ2)and z1>M2(a2
2+x2
1)(μ +D2)
(m2−m3)x2
1−m3a2
2
,
such that p1≤P(t)≤1/D,x1≤x(t)≤M1,y1≤y(t)≤M2,andz1≤z(t)≤M3.
A.4. Proof of Lemma 4.3
Proof: (i) If m1≤D1,thendx/dt<−a2
1xD1/(a2
1+P2)<0. This implies limt→∞ x(t)exists and
is non-negative. If possible, let limt→∞ x(t)=η>0. Since dP/dt≤P(1−P), it follows that for
136 B. MAJI ET AL.
>0, there exists t>0suchthatP(t)≤1+, for all t≥t. Thus for all t≥t,weget
x(t)≤x(t)e
−a2
1D1(t−t)
a2
1+(1+)2→0ast→∞,
this leads to a contradiction. Therefore, for m1≤D1,wemusthavelim
t→∞ x(t)=0, and so
limt→∞ y(t)=0=limt→∞ z(t).
(ii) Since limt→∞ sup[P(t)+x(t)+y(t)+z(t)]≤1/D, it follows that for all >0, there
exists t>0suchthatP(t)≤1/D+for all t≥t.Ifm1>D
2, then for all t≥twe have
dx/dt≤x(m1−D1)(1/D+)2−λ2/a2
1. Therefore, if λ>1/D+,thendx/dt<0 for all
t≥t. This implies limt→∞ x(t)=0, and consequently limt→∞ y(t)=0=limt→∞ z(t).
A.5. Proof of Lemma 4.4
Proof: The Jacobian of the system (2)at(P,x,y,z)is given by
J=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1−2P−2m1Pxa2
1
(a2
1+P2)2−m1P2
a2
1+P200
2m1Pxa2
1
(a2
1+P2)2
m1P2
a2
1+P2−D1−2m2xza2
2
(a2
2+x2)20−m2x2
a2
2+x2
02m2xza2
2
(a2
2+x2)2−μ−D2−2m3yza2
3
(a2
3+y2)2
m2x2
a2
2+x2−m3y2
a2
3+y2
00 μ−D3−ch
(c+z)2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
Let us consider
H=⎡
⎢
⎣
10 00
0−100
00−10
00 01
⎤
⎥
⎦.
Then we have
HJH =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1−2P−2m1Pxa2
1
(a2
1+P2)2−m1P2
a2
1+P200
−2m1Pxa2
1
(a2
1+P2)2−m1P2
a2
1+P2+D1+2m2xza2
2
(a2
2+x2)20−m2x2
a2
2+x2
0−2m2xza2
2
(a2
2+x2)2−μ−D2−2m3yza2
3
(a2
3+y2)2
m3y2
a2
3+y2−m2x2
a2
2+x2
00−μ−D3−ch
(c+z)2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
All the off-diagonal elements of the matrix HJH are negative if m3y2/(a2
3+y2)<m
2x2/(a2
2+x2).
Therefore, if m2>m
3and
x(t)>x
1>m3a2
2
m2−m3
for large t,
all the off-diagonal elements of HJH are negative, and consequently, the system (2)is
competitive.
A.6. Proof of Lemma 5.1
Proof: The Jacobian matrix at E0is
V(E0)=⎡
⎢
⎢
⎣
10 0 0
0−D100
00−μ−D20
00 μ−D3−h
c
⎤
⎥
⎥
⎦
.
LETTERS IN BIOMATHEMATICS 137
At E0, the eigenvalues of the Jacobian matrix are 1, −D1,−μ−D2and −D3−h/c. Therefore, the
system (2) is always unstable at E0.
A.7. Proof of Lemma 5.2
Proof: The Jacobian matrix at E1is
V(E1)=⎡
⎢
⎢
⎢
⎣
−1−m1
a2
1+100
0m1
a2
1+1−D100
00−μ−D20
00 μ−D3−h
c
⎤
⎥
⎥
⎥
⎦
.
The eigenvalues of the Jacobian matrix evaluated at E1are −1, −μ−D2,−D2−h/cand m1/(a2
1+
1)−D1. Therefore, the system (2) is locally asymptotically stable at E1if and only if D1>m
1/(a2
1+1).
A.8. Proof of Lemma 5.3
Proof: The Jacobian matrix at E2is
V(E2)=⎡
⎢
⎢
⎢
⎢
⎢
⎣
(1−λ)(2D1−m1)
m1−D100
2D1(1−λ)a2
1
m1λ200−m2λ2(1−λ)2
a2
2D2
1+λ2(1−λ)2
00−μ−D2m2λ2(1−λ)2
a2
2D2
1+λ2(1−λ)2
00μ−D3−h
c
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
The eigenvalues are obtained from the two quadratic equations:
η2−η(1−λ)(2D1−m1)
m1
+2D2
1(1−λ)a2
1
m1λ2=0(A1)
and
η2+ημ+D2+D3+h
c+(μ +D2)D3+h
c−μm2λ2(1−λ)2
a2
2D2
1+λ2(1−λ)2=0.(A2)
All the roots of Equation (A1) and Equation (A2) have negative real parts if
λ(1−λ)
a2cμm2
(μ +D2)(cD3+h)−1<D
1<m1
2(1−λ) .
Therefore, under the aforesaid conditions, the system (2) is locally asymptotically stable at E2.
A.9. Proof of Lemma 5.4
Proof: In order to prove the persistence of the system, we shall show that all the boundary equilibria
of the system are repellers. It is observed that the system is always unstable at E0.IfD1≤m1/(a2
1+1),
then the system is unstable at E1. The system is unstable at E2if
D1≤λ(1−λ)
a2cμm2
(μ +D2)(cD3+h)−1orD1≥m1
2(1−λ) .
138 B. MAJI ET AL.
Since m1/(a2
1+1)<m
1/{2(1−λ)}, it follows that all the boundary equilibria are repellers if
D1≤λ(1−λ)
a2cμm2
(μ +D2)(cD3+h)−1.
We have also proved that the system is bounded. Therefore, the system is persistent under the
aforesaid conditions.
A.10. Proof of Lemma 5.5
Proof: The second additive compound matrix of the Jacobian of the system (2)is
J(2)=⎡
⎢
⎢
⎢
⎢
⎢
⎣
FP+Gx0Gz000
HxFP+HyHzFx00
0μFP+Iz0Fx0
0GP0Gx+HyHz−Gz
00GPμGx+Iz0
0000HxHy+Iz
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
Let |X|∞=supi|Xi|. The logarithmic norm μ∞(J(2))of J(2)endowed by the vector norm |X|∞is
the supremum of the following:
FP+Gx+|Gz|,FP+Hy+|Fx|+|Hx|+|Hz|,FP+Iz+μ+|Fx|,
Gx+Hy+|GP|+|Gz|+|Hz|,Gx+Iz+μ+|GP|,andHy+Iz+|Hx|.
Now, (FP+Gx+|Gz|)E∗<0if
1+m1+m2<D
1+2P∗+2m1P∗x∗a2
1
(a2
1+P∗2)2+2m2x∗z∗a2
2
(a2
2+x∗2)2=L1,
(FP+Hy+|Fx|+|Hx|+|Hz|)E∗<0if
1+m1+m21+2z∗
x∗+m3<2P∗+μ+D2+2m3y∗z∗a2
3
(a2
3+y∗2)2+2m1P∗x∗a2
1
(a2
1+P∗2)2=L2,
(FP+Iz+μ+|Fx|)E∗<0if
1+μ+m1<2P∗+D3+2m1P∗x∗a2
1
(a2
1+P∗2)2+ch
(c+z∗)2=L3,
(Gx+Hy+|GP|+|Gz|+|Hz|)E∗<0if
m11+2x∗
P∗+2m2+m3<μ+D1+D2+2m2x∗z∗a2
2
(a2
2+x∗2)2+2m3y∗z∗a2
3
(a2
3+y∗2)2=L4,
(Gx+Iz+μ+|GP|)E∗<0if
m11+2x∗
P∗+μ<D
1+D3+2m2x∗z∗a2
2
(a2
2+x∗2)2+ch
(c+z∗)2=L5,
LETTERS IN BIOMATHEMATICS 139
and (Hy+Iz+|Hx|)E∗<0if
2m2z∗
x∗<μ+D1+D3+ch
(c+z∗)2+2m3y∗z∗a2
3
(a2
3+y∗2)2=L6.
Therefore, if 1 +μ+m11+2x∗/P∗+m21+2z∗/x∗+m3<L,thenμ∞(J(2)(E∗)) <0,
where L=min{L1,...,L6}. As a direct application of the method adopted by Li and Muldowney
(1993), we can say that under the aforesaid conditions, the system (2) has no periodic solutions
around E∗.
A.11. Proof of Lemma 6.1
Proof: The necessary and sufficient conditions for a Hopf bifurcation to occur at m2=m2cr are
(i) f1(m2cr )=f2(m2cr ),
(ii) Re dλj
dm2m2=m2cr
= 0,
where λj(j=1, ...,4)aretherootsofG(λ) =0. Let g:(0, ∞)→Rbe a continuously differentiable
function of m2defined by g(m2)=f1(m2)−f2(m2).Theexistenceofm2cr is ensured by solving the
equation g(m2cr )=0. At m2=m2cr , the characteristic equation G(λ) =0 can be expressed as
λ2+Q3(m2cr )
Q1(m2cr )λ2+λQ1(m2cr )+Q1(m2cr )Q4(m2cr )
Q3(m2cr )=0.(A3)
Equation (A3) has the pair of purely imaginary roots λ1=iω0and λ2=¯
λ1,whereω0=
Q3(m2cr )/Q1(m2cr ).Ifλ3and λ4are not real, then Reλ3=−Q1(m2cr )/2<0. If λ3and λ4
are real, then λ3+λ4<0andλ3λ4=Q4(m2cr )/ω2
0>0. This implies λ3,λ4<0. Since gis a
continuously differentiable function of m2, there exists an open interval (m2cr −,m2cr +),such
that λ1(m2)=β1(m2)+iβ2(m2)and λ2(m2)=β1(m2)−iβ2(m2)for all m2∈(m2cr −,m2cr +).
Therefore, for all m2∈(m2cr −,m2cr +),dG(λ)/dm2=0 gives
dλ
dm2
=−
{M(m2)K(m2)−N(m2)L(m2)}+i{N(m2)K(m2)−M(m2)L(m2)}
K2(m2)+L2(m2).
Hence, if M(m2)K(m2)−N(m2)L(m2)m2=m2cr = 0holds,thenRe dλj/dm2m2=m2cr = 0, and
consequently, the system (2) undergoes a non-degenerate Hopf bifurcation at m2=m2cr .
