Nonlinear Analysis: Modelling and Control, Vol. 25, No.

2, 225–244 ISSN: 1392-5113

https://doi.org/10.15388/namc.2020.25.16514 eISSN: 2335-8963

Bifurcation on diffusive Holling–Tanner predator–prey

model with stoichiometric density dependence

Maruthai Selvaraj Surendara,1 , Muniyagounder Sambatha,2 ,

Krishnan Balachandranb
a
Department of Mathematics, Periyar University, Salem 636 011, India
sambathbu2010@gmail.com
b
Department of Mathematics, Bharathiar University, Coimbatore-641046, India

Received: October 4, 2018 / Revised: July 4, 2019 / Published online: March 2, 2020

Abstract. This paper studies a diffusive Holling–Tanner predator–prey system with stoichiometric
density dependence. The local stability of positive equilibrium, the existence of Hopf bifurcation
and stability of bifurcating periodic solutions have been obtained in the absence of diffusion.
We also study the spatially homogeneous and nonhomogeneous periodic solutions through all
parameters of the system, which are spatially homogeneous. In order to verify our theoretical results,
some numerical simulations are carried out.
Keywords: stability analysis, stoichiometry, food quality, Hopf bifurcation, predator–prey model.

1 Introduction
Nowadays, reaction–diffusion equation modeling of several different systems has attracted
considerable attention in mathematical biology, especially, the predator–prey systems
with many different kinds of functional responses and distinct boundary conditions. In
general, one top predator species are considered feeding on other species, which make up
a food web. At present, mathematical ecology along with experimental ecology represents
an important tool for the evolution of a quantitative theory to describe predator–prey
interactions. Predator–prey interactions play the most important role in the functioning of
ecosystems. The ecological interaction between the species such as lynx and hare, spider
mite and mite, sparrow and sparrow hawk, etc., are modeled through the predator–prey
system by many researchers (see [25,26] to mention only some of them). May modernized
a model known as the Holling–Tanner predator–prey model [14] in which he incorporated
the Holling rate [5,6]. The Holling–Tanner functional is one of the prototypical model for
the predator–prey interactions, which has been studied by many researchers; for more
details, one can refer to May [14] and Murray [15].
1 The
author supported by research fellowship from Periyar University, Salem, India.
2 The
author supported by UGC(BSR)-Start Up grant No. F.30-361/2017(BSR) and the DST-FIST grant
No. SR/FST/MSI-115/2016(Level-I).

c 2020 Authors. Published by Vilnius University Press

This is an Open Access article distributed under the terms of the Creative Commons Attribution Licence,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and
source are credited.
226 M.S. Surendar et al.

Among the most widely used mathematical models in theoretical ecology, the Holling–
Tanner model plays a peculiar role in view of the interesting dynamics it possesses.
The Holling–Tanner predator–prey model, stability and Hopf bifurcation analysis has
been investigated extensively by many authors; see [2–4, 7, 8, 10, 12, 13, 16–24, 27]. In
particular, the authors have concentrated on the study of the local and global stability
of equilibrium and Hopf bifurcation. However, spatial dynamic behavior has been not
well studied. Recently, Hsu and Huang [7] analyze global stability of the positive equilib-
rium of a predator–prey system without diffusivity along with certain conditions on the
parameters. Further, the existence/nonexistence of the nonconstant positive steady state
solutions with cross-diffusion and global stability of the positive constant steady state
studied in [16]. Chen and Shi [4] proved that the unique constant equilibrium of a diffusive
system is globally asymptotically stable under a new, simpler parameter condition, and
Liu [12] studied spatiotemporal behavior of the Holling–Tanner system. Li et al. [10]
studied the Hopf bifurcation and Turing instability of the Holling–Tanner predator–prey
model with diffusion. In [11], authors studied the global stability of a predator–prey
system with Beddington–DeAngelis and Tanner functional response by using the iteration
method and comparison principle of the unique positive equilibrium solution.
Predator–prey interactions have been modeled and analyzed by various authors in
different aspects. In particular, Anderson et al. [1] constructed predator (herbivore)–prey
(autotroph) model using stoichiometric density dependence principles to admit predation
effects on food quantity and food quality. Using this intention, we construct a predator–
prey model with food quality term. We enclose that term via stoichiometry principles [1].
The main goal of this paper is to study the stability and Hopf bifurcation analysis of the
positive equilibrium solution of the diffusive Holling–Tanner predator–prey model with
stoichiometric density dependence.
The structure of this paper is arranged as follows: In Section 2, we introduce a math-
ematical predator–prey model. In Section 3, we analyze the local stability and Hopf
bifurcation of that model (2). In Section 4, bifurcations of spatially homogeneous and
nonhomogeneous periodic solutions are rigorously proved for system (3). To verify our
theoretical results, some numerical simulations are carried out in Section 5, and some
concluding comments are made in Section 6.

2 Mathematical model and analysis

The Lotka–Volterra predator–prey model has been a point of origin for much theoretical
analysis of population dynamics. In recent years, a lot of predator–prey models have been
proposed and analyzed widely since the innovators Lotka and Volterra’s theoretical works.
Population models have a long history dating back to the Malthus model formulated in the
early nineteenth century and then corrected by Verhulst about 50 years later to compensate
for the prediction that either a population grows or dies out exponentially fast. The logistic
correction allows instead for a horizontal asymptote to which the population tends as
time flows, the value of which expresses the carrying capacity of the environment for the
population under scrutiny.

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Bifurcation on diffusive Holling–Tanner predator–prey model 227

Predator–prey model has been developed by various authors who considered several
environmental conditions. Also, food quantity and quality effects play a major role in
population dynamics. Basically, stoichiometric models incorporate both food quantity and
food quality effects in a single model. Since prey and predator have shared the same
function in the food chain and the same nutritional relationship to the primary sources
of energy, it is sensible that their population sizes both ought to be constrained by the
total nutrient of the system. Stoichiometric considerations used to modify predator–prey
system through both the density dependence of prey growth rate and the growth efficiency
of predators.
To present the stoichiometric principles in our mathematical model, we need to con-
centrate on two things. First one is, individual growth and population dynamics may be
directly constrained by food quality in terms of the nutrient element content. And another
one implies that release and recycling of elements will be determined by the difference
between ingested nutrients and those incorporated into new consumer biomass. In [1],
Anderson et al. proposed the stoichiometric density dependence functional response α
(1 − Qu/(K − qv)) instead of classical Lotka–Volterra model.
We consider a two-compartment predator–prey system with constant dilution. For
both prey (u) and predator (v), total nutrient K is provided at a constant rate. If the
predators have a constant nutrient content q and the preys have a minimum nutrient
content Q, then the system must be confined to the triangle bounded below by the positive
cone (i.e. u > 0 and v > 0) and above by Qu + qv 6 K. Notice that by this we implicitly
assume that the concentration of dissolved inorganic nutrient is negligible, so that nutrient
in prey is given as the difference between the total nutrient and the nutrient contained in
predator. This simplifying assumption, which is very convenient for maintaining a planar
phase space directly comparable to the Lotka–Volterra model.
When predator has constant stoichiometry q, the quantity of nutrient available to prey
growth will be K −qv. In contrast to logistic density dependence (Qu/K), stoichiometric
density dependence (Qu/(K − qv)) implies that prey growth rate becomes a decreasing
function of both prey (u) and predator (v) biomasses. Introducing stoichiometric density
dependence has a strong effect on the shape of the prey nullcline, and thus also on the
dynamical stability properties of the system.
For this stoichiometric density dependence functional response, the predator–prey
model takes the form
   
Qu βuv δv
u̇(t) = αu 1 − − , v̇(t) = γv 1 − ,
K − qv u+v u (1)
u(0) = u0 > 0, v(0) = v0 > 0,

where the parameters α, β, γ, δ, q, Q and K are positive constants, u and v denote,

respectively, the population densities of the prey and predator at time t. The prey grows
logistically with intrinsic growth rate α and carrying capacity K/Q. The rate at which
predator removes the prey (βuv/(u + v)) is known as a ratio dependent functional re-
sponse [9]. The predator population grows logistically with intrinsic growth rate γ and

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228 M.S. Surendar et al.

carrying capacity proportional to the population size of the prey; δ is the number of preys
involve to sustain one predator at equilibrium when v matches u/δ [25].
For easiness, we nondimensionalize (1) with the following scaling:
Qu
u 7→ , v 7→ v, t 7→ αt,
K
and then obtain the form
   
u suv ev
u̇(t) = u 1 − − , v̇(t) = ρv 1 − ,
1 − av u + cv u (2)
u(0) = u0 > 0, v(0) = v0 > 0,
where a = q/K, s = βQ/(αK), c = Q/K, ρ = γ/α, e = Qδ/K.
In fact, living beings are distributed in space and regularly interface with the physical
environment and different living beings in their spatial neighborhood. Numerous physical
aspects of the earth, for example, atmosphere, substance creation or physical structure
can differ from place to place. So, we need to consider the population dynamic changes
depends upon both space and time (spatial movement) also.
As the predator–prey with their density confined to a fixed open bounded domain Ω
in RN with smooth boundary, system (2) is expressed as the following reaction–diffusion
system (spatial system):
 
u suv
ut = d1 ∆u + u 1 − − , x ∈ Ω, t > 0,
1 − av u + cv
 
ev
vt = d2 ∆v + ρv 1 − , x ∈ Ω, t > 0, (3)
u
∂ν u = ∂ν v = 0, x ∈ ∂Ω, t > 0,
u(x, 0) = u0 (x) > 0, v(x, 0) = v0 (x) > 0, x ∈ Ω.
In the above, ∆ is the Laplacian operator on Ω ∈ RN , where d1 and d2 denote, re-
spectively, diffusivity of prey and predator are kept independent of space and time. The
no-flux boundary condition means that the spatial environment Ω is isolated, and ν is the
outward unit normal to ∂Ω. The initial values u0 (x), v0 (x) are assumed to be positive
and bounded in Ω.

3 Existence of stability and Hopf bifurcation

In this section, we study mainly the local stability of the equilibria and the existence of
the Hopf bifurcation of constant periodic solutions surrounding the positive equilibrium
of system (2).

3.1 Steady states

System (2) has a boundary equilibrium point E 1 and a nontrivial positive equilibrium
point E ∗ , where

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Bifurcation on diffusive Holling–Tanner predator–prey model 229

(i) E 1 = (1, 0) – the prey only survives or extinct of the predator.

(ii) E ∗ = (u∗ , v ∗ ) is a nontrivial stationary state (coexistence of prey and predator),
where
e(c + e − s) u∗
u∗ = > 0 and v ∗ = > 0.
(c + e)(e − a) − as e
The dynamical behavior of the equilibrium points can be studied by computing the eigen-
values of the Jacobian matrix J of system (2), namely,

csv 2 au2 su2

!
2u
1 − 1−av − (u+cv) 2 − (1−av) 2 − (u+cv)2
J= ρev 2
. (4)
u2 ρ(1 − 2ev
u )

The existence and local stability of the equilibrium solutions can be stated as follows:

Theorem 1. The boundary equilibrium point E 1 = (1, 0) is always a saddle point.

Proof. The Jacobian matrix of system (2) evaluated at the equilibrium point E 1 = (1, 0)
is given by  
−1 −(a + s)
J|E =
1
0 ρ
tr J|E 1 = ρ − 1 and det J|E 1 = −ρ < 0. Therefore E 1 = (1, 0) is saddle point.

3.2 Interior equilibrium

The Jacobian evaluated at the coexistence equilibrium E ∗ (u∗ , v ∗ ) is
!
−u∗ su∗ v ∗ au ∗2
su ∗2

1−av ∗ + (u∗ +cv ∗ )2 − (1−av ∗ )2 − (u∗ +cv ∗ )2

J|E ∗ = ρ . (5)
e −ρ

Then trace and determinant of the Jacobian matrix (5) is

u∗ su∗ v ∗
T = tr J|E ∗ = + ∗ −ρ
av ∗ − 1 (u + cv ∗ )2
and
au∗
  
1
D = det J|E ∗ = ρu∗ − 1 .
av ∗ − 1 e(av ∗ − 1)

Therefore the characteristic equation of the linearized system of (5) at E ∗ = (u∗ , v ∗ ) is

λ2 − T λ + D = 0. (6)
p
The two roots are given as λ1,2 = T ± (T )2 − 4D/2. Therefore if D > 0, then the real
part of the eigenvalues (λ1,2 ) have the same sign. Therefore local stability of E ∗ entirely
depend upon the sign of T , that is, E ∗ is stable when T < 0 and unstable when T > 0.

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230 M.S. Surendar et al.

So for our convenience, consider the condition

(H) v ∗ < 1/a holds.
It is clear that if (H) holds, then D > 0. Therefore equilibrium point E ∗ is locally
asymptotically stable when

u∗ (u∗ + cv ∗ )2 < (av ∗ − 1) ρ(u∗ + cv ∗ )2 − su∗ v ∗ .

 

Further, we analyze the Hopf bifurcation occurring at E ∗ . For the sake of convenience, let
ρ0 = u∗ /(av ∗ − 1) + su∗ v ∗ /(u∗ + cv ∗ )2 . We note that the parameter ρ represents pre-
dation efficiency, and we analyze the Hopf bifurcation occurring at (u∗ , v ∗ ) by choosing
ρ as the bifurcation parameter.
When ρ > ρ0 (i.e. T = ρ0 − ρ < 0), the roots of characteristic equation (6) have
negative real part, then the positive equilibrium E ∗ is asymptotically stable. When ρ0 > ρ
(i.e. T = ρ0 − ρ > 0), the roots of (6) have positive real part, then the positive equilibrium
E ∗ is unstable.
When ρ = ρ0 (i.e. T = ρ0 − ρ = 0), the Jacobian matrix J|E ∗ has a pair of imaginary
eigenvalues. Let λ1,2 = p(ρ) ± iω(ρ) be the roots of (6), where
ρ0 − ρ T 1p 1p
p(ρ) = = , ω(ρ) = 4D − (ρ0 − ρ)2 = 4D − (T )2
2 2 2 2
and we get
dp 1
=− < 0.
dρ ρ=ρ0 2

This shows that the transversality condition holds. By the Poincaré–Andronov–Hopf

bifurcation theorem, we know that system (2) undergoes a Hopf bifurcation at E ∗ when
ρ = ρ0 .
Theorem 2. Assume that condition (H) holds. The equilibrium (u∗ , v ∗ ) of system (2)
is locally asymptotically stable when ρ > ρ0 and unstable when ρ < ρ0 ; system (2)
undergoes a Hopf bifurcation at the positive equilibrium (u∗ , v ∗ ) when ρ = ρ0 .

3.3 Stability of bifurcated solutions

However, the detailed nature of the Hopf bifurcation needs further analysis of the normal
form of the system. Now we investigate the direction of Hopf bifurcation and stability
of bifurcated solutions arising through Hopf bifurcation. We translate the positive equi-
librium E ∗ = (u∗ , v ∗ ) to the origin by the translation û = u − u∗ , v̂ = v − v ∗ . For
convenience, we denote û and v̂ again by u and v, respectively. Thus the local system (2)
becomes
u + u∗ s(u + u∗ )(v + v ∗ )
 
u̇(t) = (u + u∗ ) 1 − ∗
− ,
1 − a(v + v ) (u + u∗ ) + c(v + v ∗ )
(7)
e(v + v ∗ )
 
v̇(t) = ρ(v + v ∗ ) 1 − .
(u + u∗ )

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Bifurcation on diffusive Holling–Tanner predator–prey model 231

Rewrite (7) as      
ut u f (u, v, ρ)
= J(ρ) + , (8)
vt v g(u, v, ρ)
where J(ρ) is defined as in (5)

f (u, v, ρ) = (a1 − b1 )u2 − (2u∗ (a1 + b2 ))uv + a2 (2u∗ − cv ∗ ) − b2 u2 v

− a2 (3 − v ∗ )u3 + · · · ,
c1 c2
g(u, v, ρ) = − u2 + 2c1 uv − 2c2 u2 v + u3 + · · · ,
e e
where
csv ∗ 1 ρ
a1 = , b1 = , c1 = ,
(u∗ + cv ∗ )3 1 − av ∗ u∗
csv ∗ a ρ
a2 = ∗ , b2 = , c2 = ∗ 2 .
(u + cv ∗ )4 (1 − av ∗ )2 u
Therefore the characteristic roots of J(ρ) are λ1,2 = p(ρ) ± iω(ρ), where

1
q
2
p(ρ) = tr J(ρ) , ω(ρ) = det J(ρ) − p(ρ) .
2
The characteristic roots λ1 , λ2 are a pair of complex conjugates when det J(ρ) − (p(ρ))2
is positive and λ1 , λ2 are purely imaginary when ρ = ρ0 , that is, p(ρ0 ) = 0, and we get
λ1,2 = ±iω(ρ0 ).
Set the following matrix:  
1 0
B= ,
M N
1


where M −iN is the eigenvector corresponding to λ = p(ρ) + iω(ρ) with

(1 − av ∗ )2 (u∗ + cv ∗ )2 −u∗ su∗ v ∗

 
M = ∗2 ∗ + − p(ρ) ,
au (u + cv ∗ )2 + su∗ 2 (1 − av ∗ )2 1 − av ∗ (u∗ + cv ∗ )2
(1 − av ∗ )2 (u∗ + cv ∗ )2
N = ∗2 ∗ ω(ρ).
au (u + cv ∗ )2 + su∗ 2 (1 − av ∗ )2
Clearly,  
1 0
B −1 = −M 1 .
N N
By the transformation    
u x
=B
v y
system (7) becomes      
ẋ(t) x F (x, y, b)
= Jρ + , (9)
ẏ(t) y G(x, y, b)

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232 M.S. Surendar et al.

where  
p(ρ) −ω(ρ)
Jρ =
ω(ρ) p(ρ)
with
F (x, y, ρ) = (a1 − b1 ) − M 2u∗ (a1 + b2 ) x2 − N 2u∗ (a1 + b2 ) xy
 

+ N a2 (2u∗ − cv ∗ ) − b2 x2 y + M a2 (2u∗ − cv ∗ ) − b2

 

+ a2 (3 − v ∗ ) x3 + · · · ,

−M 1
G(x, y, ρ) = F (x, y, ρ) + g1 (x, y, ρ)
N N
and
 
−c1
g1 (x, y, ρ) = + 2M c1 x2 + (2N c1 )xy − (2N c2 )x2 y
e
 
c2
+ − 2M c2 x3 + · · · .
e
Rewrite (9) in the polar coordinates as
ṙ = p(ρ)r + a(ρ)r3 + · · · ,
(10)
θ̇ = ω(ρ) + c(ρ)r2 + · · · .
Then the Taylor expansion of (10) at ρ = ρ0 yields
ṙ = p0 (ρ0 )(ρ − ρ0 )r + a(ρ0 )r3 + . . . ,
(11)
θ̇ = ω(ρ0 ) + ω 0 (ρ0 )(ρ − ρ0 ) + c(ρ0 )r2 + · · · .
To find the stability of Hopf bifurcation periodic solution, we need to calculate the sign
of the coefficient a(ρ0 ) given by
Fxxx + Fxyy + Gxxy + Gyyy
a(ρ0 ) =
16 (0,0,ρ0 )
Fxy (Fxx + Fyy ) − Gxy (Gxx + Gyy ) − Fxx Gxx + Fyy Gyy
+ ,
16ω(ρ0 ) (0,0,ρ0 )

where
Fxxx = 6 M a2 (2u∗ − cv ∗ ) − b2 + a2 (3 − v ∗ ) ,
 

Fxyy = Gyyy = Fyy = Gyy = 0,

Fxx = 2 (a1 − b1 ) − M 2u∗ (a1 + b2 ) , Fxy = −N 2u∗ (a1 + b2 ) ,
 

Gxxy = −2M a2 (2u∗ − cv ∗ ) − b2 − 4c2 , Gxy = M 2u∗ (a1 + b2 ) + 2c1 ,

 

M2
   
−M 2M c1
au∗ (a1 + b2 ) + 2


Gxx = 2 (a1 − b1 ) + c1 − .
N N N Ne

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Bifurcation on diffusive Holling–Tanner predator–prey model 233

Thus we obtain
a(ρ0 )
Λ=− .
p0 (ρ0 )
Now, from the Poincaré–Andronov–Hopf bifurcation theorem p0 (ρ)|ρ=ρ0 = −1/2 < 0,
and from the above calculations of a(ρ0 ) we have the following conclusion:
Theorem 3. Assume that condition (H) holds. When a(ρ0 ) < 0, the direction of Hopf bi-
furcation is supercritical, and the bifurcated periodic solutions are stable; when
a(ρ0 ) > 0, the direction of Hopf bifurcation is subcritical, and the bifurcated periodic
solutions are unstable.

4 Stability and direction of spatial Hopf bifurcation

Now we discuss the existence of spatially homogeneous and nonhomogeneous periodic
solutions bifurcating from the Hopf bifurcation of the reaction–diffusion system
 
u suv
ut = d1 uxx + u 1 − − , x ∈ (0, lπ), t > 0,
1 − av u + cv
 
ev
vt = d2 vxx + ρv 1 − , x ∈ (0, lπ), t > 0, (12)
u
∂ν u = ∂ν v = 0, x ∈ 0, lπ, t > 0,
u(x, 0) = u0 (x) > 0, v(x, 0) = v0 (x) > 0, x ∈ (0, lπ).
To cast our focus into the frame work of the Hopf bifurcation theorem, by the tran-
sition û = u − u∗ , v̂ = v − v ∗ we translate (12) into the following system. For the
sake of convenience, we still indicate û and v̂ by u and v, respectively. Thus the reaction–
diffusion system (12) becomes
ut − d1 uxx = F(ρ, u, v), x ∈ (0, lπ), t > 0,
vt − d2 vxx = G(ρ, u, v), x ∈ (0, lπ), t > 0,
(13)
ux (0, t) = ux (lπ, t) = 0, vx (0, t) = vx (lπ, t) = 0, t > 0,
u(x, 0) = u0 (x), v(x, 0) = v0 (x), x ∈ (0, lπ).
Define
(u + u∗ ) s(u + u∗ )(v + v ∗ )
 
F(ρ, u, v) = (u + u∗ ) 1 − − ,
1 − a(v + v ∗ ) (u + u∗ ) + c(v + v ∗ )
e(v + v ∗ )
 
∗
G(ρ, u, v) = ρ(v + v ) 1 − ,
(u + u∗ )
where F, G : R × R2 → R are C ∞ smooth with F(ρ, 0, 0) = G(ρ, 0, 0) = 0.
Now we define the real-valued Sobolev space
2
X = (u, v) ∈ H 2 (0, lπ) : (ux , vx )|x=0,lπ = 0 ,
 

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234 M.S. Surendar et al.

and the complexification of X:

XC := X ⊕ iX = {u1 + iu2 : u1 , u2 ∈ X}.

The linearized operator of system (12) evaluated at (u∗ , v ∗ ) is

!
∂2
d1 ∂x2 + A(ρ) B(ρ)
L(ρ) = ∂2
C(ρ) d2 ∂x2 + D(ρ)

with the domain DL(ρ) = XC , where

u∗ su∗ v ∗
A(ρ) = Fu (ρ, 0, 0) = − + ,
1 − av ∗ (u∗ + cv ∗ )2
−au∗ 2 su∗ 2
B(ρ) = Fv (ρ, 0, 0) = − ,
(1 − av ∗ )2 (u∗ + cv ∗ )2
ρ
C(ρ) = Gu (ρ, 0, 0) = , D(ρ) = Gv (ρ, 0, 0) = −ρ
e
with (u∗ , v ∗ ) as defined in Section 3.1.
The following condition is necessary to ensure that the Hopf bifurcation occurs:
(H1) There exists a number ρH ∈ R and a neighborhood O of ρH such that for
ρ ∈ O, L(ρ) has a pair of complex, simple, conjugate eigenvalues p(ρ) ± iω(ρ),
continuously differentiable in ρ with p(ρH ) = 0, ω0 = ω(ρH ) > 0 and
p0 (ρH ) 6= 0; all other eigenvalues of L(ρ) have nonzero real parts for ρ ∈ O.
Now we apply Hopf bifurcation result appearing in [28] to our model. It is well known
that the eigenvalue problem
−ϕ00 = µϕ, x ∈ (0, lπ),
0 0
ϕ (0) = ϕ (lπ) = 0,

has eigenvalues µn = n2 /l2 (n = 0, 1, 2, . . .) with corresponding eigenfunctions ϕn (x) =

cos(nx/l). Let
  X ∞  
φ an nx
= cos
ψ bn l
n=0

be an eigenfunction of L(ρ) corresponding to an eigenvalue σ(b). That is, L(ρ)(φ, ψ)T =

σ(ρ)(φ, ψ)T . Then from a straightforward analysis we obtain the following relation:
   
a a
Ln (ρ) n = σ(ρ) n , n = 0, 1, 2, . . . ,
bn bn

where !
2
−d1 nl2 + A(ρ) B(ρ)
Ln (ρ) = 2 .
C(ρ) −d2 nl2 + D(ρ)

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Bifurcation on diffusive Holling–Tanner predator–prey model 235

It follows that eigenvalues of L(ρ) are given by the eigenvalues of Ln (ρ) for n = 0, 1,
2, . . . . The characteristic equation of Ln (ρ) is

σ 2 − Tn (ρ)σ + Dn (ρ) = 0, n = 0, 1, 2, . . . ,
where
−u∗ su∗ v ∗ (d1 + d2 )n2
 
Tn (ρ) = ∗
+ ∗ ∗ 2
− ρ − ,
1 − av (u + cv ) l2
 ∗ ∗ (u∗ +cv∗ )2
u (sv − 1−av∗ ) − ρ(u∗ + cv ∗ )2 (d1 + d2 )n2

= ∗ ∗ 2
−
(u + cv ) l2
u∗ au∗ u∗
   
ρ ∗ ∗
Dn (ρ) = e − + (su − esv )
e 1 − av ∗ 1 − av ∗ (u∗ + cv ∗ )2 (14)
n2 n2 n2 u∗ su∗ v ∗
 
+ d1 2 ρ + d2 2 d1 2 + −
l l l 1 − av ∗ (u∗ + cv ∗ )2
au∗ 2 su∗ 2 u∗ su∗ v ∗
   
ρ
= + + ρ −
e (1 − av ∗ )2 (u∗ + cv ∗ )2 1 − av ∗ (u∗ + cv ∗ )2
n2 n2 n2 u∗ su∗ v ∗
 
+ d1 2 ρ + d2 2 d1 2 + − .
l l l 1 − av ∗ (u∗ + cv ∗ )2

Therefore the eigenvalues are established by

p
Tn (ρ) ± Tn2 (ρ) − 4Dn (ρ)
σ(ρ) = , n = 0, 1, 2, . . . .
2

If condition (H1) holds, we see that, at ρ = ρH , L(ρ) has a pair of simple purely
imaginary eigenvalues ±iω0 if and only if there exists a unique n ∈ N ∪ {0} such
that ±iω0 are the purely imaginary eigenvalues of Ln (ρ). In such a case, denote the
associated eigenvector by q = qn = (an , bn )T cos(nπ/l), with an , bn ∈ C, such that
Ln (ρ)(an , bn )T = iω0 (an , bn )T or L(ρH )q = iω0 q.
We discover the Hopf bifurcation value ρH , which satisfies condition (H1) taking the
following form if there exists n ∈ N ∪ {0} such that, for any j 6= n,

Tn (ρH ) = 0, Dn (ρH ) = 0 and Tj (ρH ) 6= 0, Dj ρH 6= 0,

(15)

and for the unique pair of complex eigenvalues p(ρ) ± iω(ρ), near the imaginary axis,
p0 (ρH ) 6= 0. It is easy to derive from (14) that Tn (ρ) < 0 and Dn (ρ) > 0 if sv ∗ 6
(u + cv ∗ )2 /(1 − av ∗ ), which implies that (0, 0) is a locally asymptotically stable steady
state of system (12).
If sv ∗ > (u + cv ∗ )2 /(1 − av ∗ ), we define
∗
(u∗ +cv ∗ )2
su∗ v ∗ − ( u 1−av ∗ ) −u∗ su∗ v ∗
ρ∗0 = = + ∗ > 0. (16)
(u∗ + ∗
cv ) 2 1 − av ∗ (u + cv ∗ )2

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236 M.S. Surendar et al.

Hence the potential Hopf bifurcation point lives in the interval (0, ρ∗0 ]. For any Hopf
bifurcation ρH in (0, ρ∗0 ], p(ρH ) ± iω(ρH ) are the eigenvalues of L(ρH ), where

Tn (ρH )
q
ρH = ω ρH = Dn ρH − p2 ρH



,
2
and

1 0
p0 ρH = Tn ρH < 0.


(17)
2
From the above discussion the determination of Hopf bifurcation point reduces to describ-
ing the set
Λ1 = ρH ∈ (0, b∗0 ]: for some n ∈ N ∪ {0}, (15) is satisfied


when a set of parameters (d1 , d2 , a, s, c, e) are given. In the following, we fix (d1 , d2 , a, s,
∗
c, e) > 0 and choose l appropriately. First, for all l > 0, ρH 0 = ρ0 is always an element
of Λ1 . (Since T0 (ρ0 ) = 0, Tj (ρ0 ) < 0 for all j > 1 and Dm (ρH
H H
0 ) > 0 for all m ∈
N ∪ {0}.) This corresponds to the Hopf bifurcation of spatially homogeneous periodic
solution. Obviously, ρH0 is also the unique value for the Hopf bifurcation of the spatially
homogeneous periodic solution for any l > 0. Hence, in the following, we look for
spatially nonhomogeneous Hopf bifurcation points.
Note that, when ρ < ρ∗0 , it is easy to show that Tn (ρ) = 0 is equivalent to
(d1 + d2 )n2
ρ = ρ∗0 − .
l2
Substituting it in Dn (ρ) of (14), we have
 2 2 " 
n2 u∗ av ∗
  
2 n 1
Dn (ρ) = −d1 2 + 2 d1 ρ∗0 − e −
l l e 1 − av ∗ 1 − av ∗
u∗ u∗ av ∗
    
∗ ∗ 1
+ ∗ (su − esv ) − d2 e−
(u + cv ∗ )2 e 1 − av ∗ 1 − av ∗
∗ ∗ ∗ ∗
  
u ∗ ∗ u su v
+ ∗ (su − esv ) − − ∗
(u + cv ∗ )2 1 − av ∗ (u + cv ∗ )2
∗ ∗ ∗ ∗
   
ρ0 u av u ∗ ∗
+ e− + ∗ (su − esv ) .
e 1 − av ∗ 1 − av ∗ (u + cv ∗ )2
Let
u∗ av ∗ u∗
    
1
B0 = d1 ρ∗0 − e − + (su∗
− esv ∗
)
e 1 − av ∗ 1 − av ∗ (u∗ + cv ∗ )2
u∗ av ∗ u∗
    
1 ∗ ∗
− d2 e − + (su − esv )
e 1 − av ∗ 1 − av ∗ (u∗ + cv ∗ )2
u∗ su∗ v ∗
 
− ∗
− ∗ ,
1 − av (u + cv ∗ )2

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Bifurcation on diffusive Holling–Tanner predator–prey model 237

then Dn (ρ) > 0 if and only if

q
ρ∗ u∗ av ∗ u∗
n2 −B0 + B02 + 4d21 e0 [ 1−av ∗ (e − 1−av ∗ ) + (u∗ +cv ∗ )2 (su
∗ − esv ∗ )]
< .
l2 2d21
So all the potential Hopf bifurcation points can be tagged as Λ1 = {ρH N
n }n=0 for some
N ∈ N ∪ {0}, where
∗ (d1 + d2 )n2
ρH
n = ρ0 − . (18)
l2
That is,
∗
0 < ρH H H H
n < ρN −1 < · · · < ρ1 < ρ0 = ρ0 (19)
satisfying
q
ρ∗ u∗ av ∗ u∗
ρ∗0
− ρH −B0 + B02 +4d21 ∗ ∗
e [ 1−av ∗ (e− 1−av ∗ )+ (u∗+cv ∗ )2 (su −esv )]
0
n
06 < .
d1 + d2 2d21
Now we only need to verify whether Di (ρH n ) 6= 0 for i 6= n. Here we derive
a condition on the parameters so that Di (ρH
n ) > 0 for each i = 0, 1, 2, . . . . Since

i4 i2 ∗
D i ρH H



n = d1 d2 4 + 2 d1 ρn − d2 ρ0
l l
∗
ρH av ∗ u∗
   
n u ∗ ∗
+ e− + ∗ (su − esv ) ,
e 1 − av ∗ 1 − av ∗ (u + cv ∗ )2
∗
we choose the diffusion coefficient d2 as small as possible so that d1 ρH
n − d2 ρ0 > 0, that
is, given the fixed N defined by (19) for every 0 < n 6 N , d2 < (l, a, s, c, e, N ), where
N2
b∗0 − l2
(l, a, s, c, e, N ) := N2
> 0. (20)
b∗0 + l2

Therefore Di (ρH
n ) > 0.
Then summarizing our analysis above and using Hopf bifurcation theorem in [28], we
have the main result of this section on the existence of both spatially homogeneous and
nonhomogeneous periodic solutions bifurcating from Hopf bifurcation.
Theorem 4. Assume that (u + cv ∗ )2 /(1 − av ∗ ) < sv ∗ . For any ρH n , defined by (18), if
there exists  = (l, a, s, c, e, N ) defined by (20) such that 0 < d2 < , then system (12)
undergoes a Hopf bifurcation at each ρ = ρH n (0 6 n 6 N ). With s sufficiently small,
for ρ = ρ(s), ρ(0) = ρH n , there exists a family of T (s)-periodic continuously differ-
entiable solutions (u(s)(x, t), v(s)(x, t)), and the bifurcating periodic solutions can be
parametrized in the form
nx
u(s)(x, t) = s an e2πit/T (s) + ān e−2πit/T (s) cos + o s2 ,



l
(21)
2πit/T (s) −2πit/T (s) nx
+ o s2 ,



v(s)(x, t) = s bn e + b̄n e cos
l

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238 M.S. Surendar et al.

where
2π
1 + τ2 s2 + o s4 ,



T (s) = n
ω0
1

Re(c1 (ρHn )) 0 H
τ2 = − n Im c1 ρH



n − 0 H
ω ρn ,
ω0 p (ρn )
and
4π 4π

Re(c1 (ρH
n )) 0 H
T 00 (0) = τ2 = − n 2 Im c1 ρH



n n − 0 H
ω ρn .
w0 (ω0 ) p (ρn )
If all eigenvalues (except ±iω0n ) of L(ρH
n ) have negative real parts, then the bifurcating
periodic solutions are stable (resp., unstable) if Re(c1 (ρH n )) < 0 (resp., > 0). The
bifurcation is supercritical (resp., subcritical) if −(1/p0 (ρH H
n ))Re(c1 (ρn )) < 0 (resp.,
> 0). Moreover
(i) The bifurcating periodic solutions from ρH 0 are spatially homogeneous, which
coincide with the periodic solutions of the corresponding ODE system.
(ii) The bifurcating periodic solutions from ρH n , n > 0, are spatially nonhomoge-
neous.
Next, we follow the methods in [28] to calculate the direction of Hopf bifurcation
and the stability of the bifurcating periodic orbits bifurcating from ρ = ρ∗0 . We have the
following result.
Theorem 5. For system (12), the bifurcating (spatially homogeneous) periodic solu-
tions bifurcating from ρ = ρH 0 are locally asymptotically stable (resp., unstable) if
Re(c1 (ρH0 )) < 0 (resp. > 0). Furthermore, the direction of Hopf bifurcation at ρH
0 is
subcritical (resp., supercritical) if Re(c1 (ρH
0 )) < 0 (resp. > 0).
Proof. Following the notations and calculation in [28], we set
q = (a0 , b0 )T
(1 − av ∗ )[u∗ (u∗ + cv ∗ )2 − su∗ v ∗ (1 − av ∗ )]

= 1, −
au∗ 2 (u∗ + cv ∗ )2 + su∗ 2 (1 − av ∗ )2
T
iω0 (1 − av ∗ )2 (u∗ + cv ∗ )2
− ∗2 ∗ ,
au (u + cv ∗ )2 + su∗ 2 (1 − av ∗ )2
A∗
T
q ∗ = (a∗0 , b∗0 )T = −Im(b0 ) + iRe(b0 ), −i ,
lπ
where
−(au∗ 2 (u∗ + cv ∗ )2 + su∗ 2 (1 − av ∗ )2 )
A∗ =
(1 − av ∗ )2 (u∗ + cv ∗ )2
such that hq ∗ , qi = 1 and hq ∗ , q̄i = 0. Then, by direct computation, we get
c0 = x1 + iy1 , d0 = x2 + iy2 , e0 = x3 ,
f0 = x4 , g0 = x5 + iy5 , h0 = x6 + iy6 ,

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Bifurcation on diffusive Holling–Tanner predator–prey model 239

where
x1 = 2(a1 − b1 ) − 4u∗ (a1 + b2 )Re(b0 ), y1 = −4u∗ (a1 + b2 )Im(b0 ),
−2c1
x2 = + 4c1 Re(b0 ), y2 = 4c1 Im(b0 ),
e
−2c1
x3 = 2(a1 − b1 ) + 4u∗ (a1 + b2 )Re(b0 ), x4 = + 4c1 Re(b0 ),
e
x5 = 6 a2 (3 − v ∗ ) + a2 (2u∗ − cv ∗ ) − b2 Re(b0 ) ,



y5 = 2 a2 (2u∗ − cv ∗ ) − b2 Im(b0 ),
 

6c2
x6 = − 12c2 Re(b0 ), y6 = −4c2 Im(b0 ).
e
Then
hq ∗ , Qqq i = A∗ −Im(b0 )x1 + Re(b0 )y1 − y2 + i −Im(b0 )y1 + x2 − Re(b0 )x1 ,



hq ∗ , Qqq̄ i = A∗ −Im(b0 )x3 + i x4 − Re(b0 )x3 ,




hq¯∗ , Qqq i = A∗ −Im(b0 )x1 − Re(b0 )y1 + y2 + i −Im(b0 )y1 + Re(b0 )x1 − x2 ,



hq¯∗ , Qqq̄ i = A∗ −Im(b0 )x3 + i Re(b0 )x3 − x4 ,




hq ∗ , Cqqq̄ i = A∗ −Im(b0 )x5 − y6 + Re(b0 )y5 + i −Img(b0 )y5 + x6 − Re(b0 )x5 .




Direct computation gives

     
c a ā
H20 = 0 − hq ∗ , Qqq i 0 − hq¯∗ , Qqq i 0 = 0,
d0 b0 b̄0
     
e a ā
H11 = 0 − hq ∗ , Qqq̄ i 0 − hq¯∗ , Qqq̄ i 0 = 0.
f0 b0 b̄0
Then, by Yi et al. [28], it implies that w20 = w11 = 0; hence
hq ∗ , Qw20 q̄ i = hq ∗ , Qw11 q̄ i = 0.
After elementary but lengthy computations, we obtain
 
H


i ∗ ∗ 1 ∗
Re c1 ρ0 = Re hq , Qqq ihq , Qqq̄ i + hq , Cqqq̄ i
2ω0 2
−1 ∗ 2 


= A −Im(b0 )x1 + Re(b0 )y1 − y2 x4 − Re(b0 ) x3
2ω0


+ −Im(b0 )y1 + x2 − Re(b0 )x1 ) −Im(b0 )x3
1 
+ A∗ −Im(b0 )x5 − y6 + Re(b0 )y5 .
2
It follows from (17) that p0 (ρH 0 ) < 0, and then, by Theorem 2.1 in [28], the periodic
solutions bifurcating from ρ = ρH 0 are locally asymptotically stable (resp., unstable) if
Re(c1 (ρH0 )) < 0 (resp., > 0). Furthermore, the direction of Hopf bifurcation at ρ0 is
H
H
subcritical (resp., supercritical) if Re(c1 (ρ0 )) < 0 (resp., > 0).

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240 M.S. Surendar et al.

5 Numerical example

Here we present some numerical simulation to verify our theoretical analysis by using
MATLAB. We consider system (2) with a = 1, s = 2.1, c = 0.2, e = 2. We change only
the predation efficiency ρ.
System (2) has unique positive equilibrium E ∗ (u∗ , v ∗ ) = (0.0444, 0.0222). Under
the set of parameters in (2), we have the critical point ρ0 = 0.822314, and it follows
from Theorem (2) that E ∗ (u∗ , v ∗ ) = (0.0444, 0.0222) is locally asymptotically stable
when ρ > ρ0 = 0.822314 and unstable when ρ < ρ0 = 0.822314. Also, when ρ passes
through ρ0 from side of ρ0 , E ∗ (u∗ , v ∗ ) = (0.0444, 0.0222) will lose its stability, and a
Hopf bifurcation occurs, that is, a family of periodic solutions bifurcate from the interior
equilibrium E ∗ (u∗ , v ∗ ) = (0.0444, 0.0222). These facts are shown by the numerical
simulations; see Fig. 1.
In system (12), let Ω = (0, 60π), l = 60, a = 1, s = 2.1, c = 0.2, e = 2,
d1 = 0.1 and d2 = 0.2. Then it follows from Theorem (5) that there exist five Hopf

(a) (b)

(c) (d)
Figure 1. Time series and phase portrait of system (2) with initial data (u0 , v0 ) = (0.1, 0.1): (a), (b) ρ =
0.85 > ρ0 = 0.822314; (c), (d) ρ = 0.8 < ρ0 = 0.822314.

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Bifurcation on diffusive Holling–Tanner predator–prey model 241

Figure 2. Numerical simulations showing the constant steady state of system (4) is locally asymptotically stable
with ρ = 0.85 > ρH0 and initial value (0.5, 0.5).

Figure 3. Numerical simulations showing a spatially homogeneous periodic of system (12) emerges when ρ
across the first Hopf bifurcation point ρH H
0 with ρ = 0.821 < ρ0 and initial value (0.5, 0.5).

∗
bifurcation points: ρH H H H
0 = ρ0 ≈ 0.8223, ρ1 ≈ 0.8222, ρ2 ≈ 0.8219, ρ3 ≈ 0.8215 and
H
ρ4 ≈ 0.8209. Hence we know from Theorem (5) that the bifurcating periodic solutions
from ρH H
0 are locally asymptotically stable and the direction of Hopf bifurcation at ρ0 is
supercritical. Next, we present numerical simulations near Hopf bifurcation points ρH 0 :
for ρ = 0.85 and ρ = 0.821; the solution (u(t, x), v(t, x)) tends to a constant steady state
and spatially homogeneous periodic solution, respectively; see Figs. 2 and 3.

6 Conclusion
In this paper, we have studied a diffusive Holling–Tanner predator–prey model with
stoichiometric density dependence. Last few decades, many authors have studied the
predator–prey model with logistic growth instead of exponential growth term of the prey.
This article encloses food quality term via stoichiometric principles. We incorporated the
stoichiometric density dependence functional response in Holling–Tanner predator–prey
system and discussed its stability and Hopf bifurcation.

Nonlinear Anal. Model. Control, 25(2):225–244

242 M.S. Surendar et al.

The distribution of the roots of the characteristic equations of the local system (2) at
each of the feasible equilibria and stability of the positive equilibrium point are studied.
Biologically, Theorem (2) states that, whenever the ratio between the growth rate of
predator and prey (ρ = γ/α) is greater than the critical value ρ0 , the predator–prey
population will be stable for any positive initial population. That is, when time t tending
to infinity, we can predict population size precisely. That population size is nothing but
converge to the positive equilibrium (u∗ , v ∗ ). Also, whenever the ratio between the growth
rate of predator and prey is less than the critical value of ρ0 , the predator–prey popula-
tion will be unstable. That is, when time t tending to infinity, we cannot conclude the
population size exactly. Whenever the ratio between the growth rate of predator and prey
is exactly equaled to the critical value ρ0 , the population dynamics change periodically.
This notion is addressed by occurrence of Hopf bifurcation. That is, system (2) undergoes
a Hopf bifurcation at the positive equilibrium (u∗ , v ∗ ) when ρ = ρ0 . Moreover, when
the direction of the Hopf bifurcation is supercritical, the bifurcating periodic solution
is stable, and when the direction of the Hopf bifurcation is subcritical, the bifurcating
periodic solution is unstable. The main results are presented in Theorem 3.
In Section 4, we studied the dynamical changes in predator–prey population according
to both space (environment atmosphere) and time movements. For this spatial move-
ments, we considered the diffusion system (12). We analyzed stability conditions and
the direction of spatial Hopf bifurcation in detail. We derived conditions for which the
occurrence of Hopf bifurcation is due to the ratio between the growth rate of predator and
prey and space Ω = (0, lπ).
The positive constant steady state solutions of system (12) are locally asymptotically
stable when sv ∗ 6 (u + cv ∗ )2 /(1 − av ∗ ). When sv ∗ > (u + cv ∗ )2 /(1 − av ∗ ), ρ∗0
is the unique homogeneous Hopf bifurcation point, where spatially homogeneous orbits
bifurcate from (u∗ , v ∗ ) for any l > 0. Further, there exists multiple nonhomogeneous
Hopf bifurcation points ρH H H
n , with 1 6 n 6 N , satisfying 0 < ρn < ρN −1 < · · · <
H H ∗
ρ1 < ρ0 = ρ0 . At these points, spatially nonhomogeneous periodic orbits bifurcate
from (u∗ , v ∗ ) for suitable l > 0.

Acknowledgment. We would like to thank the anonymous reviewers and the editors for
their valuable suggestions in order to improve the quality of the paper.

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