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International Journal of Bifurcation and Chaos, Vol. 25, No. 5 (2015) 1530014 (16 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0218127415300141

Spatiotemporal Dynamics of a Diffusive Leslie–GowerPredator–Prey Model with Ratio-Dependent

Functional Response*

Hong-Bo ShiSchool of Mathematical Science, Huaiyin Normal University,

Huaian, Jiangsu 223300, P. R. China

Shigui Ruan†Department of Mathematics, University of Miami,

Coral Gables, FL 33124-4250, USAruan@math.miami.edu

Ying SuDepartment of Mathematics, Harbin Institute of Technology,

Harbin, Heilongjiang 150001, P. R. China

Jia-Fang ZhangSchool of Mathematics and Information Science,

Henan University, Kaifeng, Henan 475001, P. R. China

Received April 21, 2014; Revised January 4, 2015

This paper is devoted to the study of spatiotemporal dynamics of a diffusive Leslie–Gowerpredator–prey system with ratio-dependent Holling type III functional response under homoge-neous Neumann boundary conditions. It is shown that the model exhibits spatial patterns viaTuring (diffusion-driven) instability and temporal patterns via Hopf bifurcation. Moreover, theexistence of spatiotemporal patterns is established via Turing–Hopf bifurcation at the degener-ate points where the Turing instability curve and the Hopf bifurcation curve intersect. Variousnumerical simulations are also presented to illustrate the theoretical results.

Keywords : Diffusive predator–prey model; functional response; stability; Turing instability; Hopfbifurcation; Turing–Hopf bifurcation.

1. Introduction

Understanding the nonlinear dynamics of predator–prey systems and determining how the dynamicalbehaviors change along model parameters is animportant subject in theoretical ecology. Because

of the differences in capturing food and consumingenergy, a major trend in theoretical work onpredator–prey dynamics has been launched so asto derive more realistic models and functionalresponses. Consider the following Leslie–Gower

∗Research was partially supported by National Natural Science Foundation of China (Nos. 11461040, 11401245, 11301147 and11201096), Universities Natural Science Foundation of Jiangsu Province (11KJB110003), and National Science Foundation(DMS-1412454).†Author for correspondence

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H.-B. Shi et al.

type predator–prey model [Hsu & Huang, 1995]:

du

dt= r1u

(1 − u

K

)− p(u)v,

dv

dt= r2

(1 − v

hu

),

(1)

where u, v and r1, r2 represent prey and predatordensities and intrinsic growth rates, respectively. Kis the carrying capacity of prey’s environment, whilethe carrying capacity of predator’s environment, hu,is a function on the population size of prey (h/r2 isa measure of the food quality of the prey for conver-sion into predator growth). The form of the preda-tor equation in system (1) was first introduced byLeslie [1948]. The function v

hu is called the Leslie–Gower term [Leslie & Gower, 1960].

The functional response p(u) can be classi-fied into different types [Collings, 1997]. (i) Lotka–Volterra type: p(u) = cu, where c > 0 is theconversion rate of predators. System (1) with theLotka–Volterra type functional response is the so-called Leslie–Gower model [Leslie & Gower, 1960].(ii) Holling type II or Michaelis–Menten type[Holling, 1965]: p(u) = cu

m+u , where m > 0 is thehalf-saturation constant. The Leslie–Gower typepredator–prey model (1) with Holling type II func-tional response is also called the Holling–Tannermodel in the literature [May, 1973; Murray, 1989].(iii) Holling type III [Bazykin, 1998; Smith, 1974]:p(u) = cu2

m+u2 . Hsu and Huang [1995] obtainedsome criteria for the local asymptotic stability ofthe positive equilibrium of system (1) with Hollingtype III functional response and gave conditionsunder which local stability of the positive equilib-rium implies global stability by applying Dulac cri-terion and constructing Lyapunov functions. (iv)Holling type IV: p(u) = cu

m+u2 , which is nonmono-tonic. Li and Xiao [2007] studied system (1) withHolling type IV functional response and performeddetailed qualitative and bifurcation analyses, suchas the classification of equilibria, Hopf bifurcation,Bogdanov–Takens bifurcation, and stable/unstablelimit cycles.

The functional responses mentioned above areonly prey-dependent. Recent biological and physi-ological evidence [Arditi & Ginzburg, 1989; Arditiet al., 1991; Arditi & Saiah, 1992] indicates thatin many situations, especially when predators haveto search for food (and therefore, have to shareor compete for food), a more suitable general

predator–prey theory should be based on the factthat the per capita predator growth rate shouldbe a function of the ratio of prey to predatorabundance, the so-called ratio-dependent functionalresponse. Xiao and Ruan [2001] considered apredator–prey model with ratio-dependent Hollingtype II functional response and provided globalqualitative analysis of the model depending onall parameters and conditions of existence andnonexistence of limit cycles for the model. Ruanet al. [2010] further studied the same predator–prey model as in [Xiao & Ruan, 2001] and con-structed the unfolding and proved its versatilityand degeneracy of codimension-two. They discussedall its possible bifurcations, including transcriti-cal bifurcation, Hopf bifurcation, and heteroclinicbifurcation, gave conditions of parameters for theappearance of closed orbits and heteroclinic loops,and described the bifurcation curves. For more stud-ies on predator–prey systems with ratio-dependentHolling type-II functional response, we refer to[Freedman & Mathsen, 1993; Hsu et al., 2001;Kuang, 1999; Kuang & Beretta, 1998; Li & Kuang,2007; Liang & Pan, 2007; Ruan et al., 2008, 2010;Xiao & Ruan, 2001].

On the other hand, in the evolutionary processof the species, the individuals do not remain fixedin space, and their spatial distribution changes con-tinuously due to the impact of many reasons (envi-ronment factors, food supplies, etc.). Therefore,different spatial effects have been introduced intopopulation models, such as diffusion and disper-sal. For example, it has been known since Turing’sclassical work [Turing, 1952] that the interplay ofchemical reaction and diffusion can cause the stableequilibrium of the local system to become unstablefor the diffusive system and lead to the spontaneousformulation of a spatially periodic stationary struc-ture. In particular, this kind of instability is calledTuring instability [Murray, 1989] or diffusion-driveninstability [Okubo, 1980]. The space-dependent sta-tionary solutions induced by diffusion are calledTuring pattern. For reviews and related studies onTuring instability and Turing pattern formation ofreaction–diffusion (R–D) systems from applied sci-ences such as chemistry, biology, ecology and epi-demiology, we refer to [Du et al., 2009; Gambinoet al., 2013; Golovin et al., 2008; Levin & Segel,1985; Li et al., 2013; Malchow et al., 2008; Peng,2013; Peng & Wang, 2008; Ruan, 1998; Wang, 2008;Yi et al., 2009; Zhang et al., 2011], and referencescited therein.

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Spatiotemporal Dynamics of a Diffusive Predator–Prey Model

Recently, studies of dynamics resulting from thecoupling between two different instabilities, Tur-ing instability and Hopf instability (or bifurcation),have become available. Particularly, in some biolog-ical and chemical reaction–diffusion models, focushas been put on the coupling between instabilitiesbreaking temporal and spatial symmetries, respec-tively. For example, Wang et al. [2007] investi-gated the emergence of a diffusive ratio-dependentpredator–prey system with Holling type II func-tional response and obtained conditions of Hopf,Turing, and wave bifurcations in a spatial domain.Furthermore, they presented a theoretical analy-sis of evolutionary processes that involves organ-isms distribution and their interaction of spatiallydistributed population with local diffusion. Formore related works, see [Baurmanna et al., 2007;Camara & Aziz-Alaoui, 2009; Meixner et al., 1997;Wit et al., 1996; Tzou et al., 2011, 2013].

Motivated by the previous works, in this paperby incorporating the diffusion and ratio-dependentHolling type III functional response into system (1),we consider the following partial differential equa-tion (PDE) model under homogeneous Neumannboundary conditions:

∂u

∂t− d1∆u = r1u

(1 − u

K

)− cu2v

u2 + mv2,

x ∈ Ω, t > 0,

∂v

∂t− d2∆v = r2v

(− v

hu

), x ∈ Ω, t > 0,

∂u

∂ν=

∂v

∂ν= 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x) ≥ 0,

v(x, 0) = v0(x) ≥ 0,x ∈ Ω.

(2)

Here, u(x, t) and v(x, t) stand for the densities of theprey and predators at location x ∈ Ω and time t,respectively; Ω ⊂ R

N (N ≤ 3) is a bounded domainwith smooth boundary ∂Ω; ν is the outward unitnormal vector of the boundary ∂Ω. The homoge-neous Neumann boundary conditions indicate thatthe predator–prey system is self-contained with zeropopulation flux across the boundary. The positiveconstants d1, d2 are diffusion coefficients, and theinitial data u0(x), v0(x) are non-negative contin-uous functions. r1,K, c,m, r2, and h are positiveconstants.

By applying the following scaling to (2),

r1t → t,u

K→ u, v → v,

d1

r1→ d1,

d2

r1→ d2,

c

Kr1→ β,

m

K2→ m, Kh → δ,

r2

r1→ r,

it can be simplified as follows (for simplicity, takingδ = 1),

∂u

∂t− d1∆u = u(1 − u) − βu2v

u2 + mv2,

x ∈ Ω, t > 0,

∂v

∂t− d2∆v = rv

(1 − v

u

), x ∈ Ω, t > 0,

∂u

∂ν=

∂v

∂ν= 0, x ∈ ∂Ω, t > 0,

u(x, 0) = u0(x) ≥ 0,

v(x, 0) = v0(x) ≥ 0,x ∈ Ω.

(3)

The rest of this paper is organized as follows. InSec. 2, we first consider the diffusion-driven insta-bility of the positive equilibrium for R–D system (3)when the spatial domain is a bounded interval.In Sec. 3, we study the direction of Hopf bifurca-tion and the stability of the bifurcating periodicsolution, which is a spatially homogeneous periodicsolution of the R–D system (3). In Sec. 4, we presenta detailed investigation of the Turing–Hopf bifur-cation. The paper ends with a brief discussion inSec. 5.

2. Turing Instability

We can see that system (3) has a unique constantpositive steady-state solution E∗ = (u∗, v∗) underthe condition β < 1 + m, where

(u∗, v∗) =(

1 − β

1 + m, 1 − β

1 + m

).

From the viewpoint of ecology, the existence ofconstant positive steady-state solutions implies thecoexistence of both the prey and predators.

In this section, we will derive conditions forthe Turing instability of the spatially homoge-neous equilibrium (u∗, v∗) of the reaction–diffusion

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predator–prey system (3). Here, we consider thespecial case with no-flux boundary conditions in aone-dimensional interval (0, l):

∂u

∂t− d1∆u = u(1 − u) − βu2v

u2 + mv2,

x ∈ (0, l), t > 0,

∂v

∂t− d2∆v = rv

(1 − v

u

), x ∈ (0, l), t > 0,

∂u

∂ν=

∂v

∂ν= 0, x = 0, l, t > 0,

u(x, 0) = u0(x) ≥ 0,

v(x, 0) = v0(x) ≥ 0,x ∈ (0, l),

(4)

where l > 0 is the length of interval. While our cal-culations can be carried over to higher-dimensionalspatial domains, we restrict ourselves to the case ofspatial domain (0, l), for which the structure of the

eigenvalues is clear. To this end, let(u

v

)=(

ρ1

ρ2

)exp(µt + ikx),

where µ is the growth rate of perturbation in time t,ρ1, ρ2 are the amplitudes and k is the wave numberof the solutions.

The linearized system of (4) at (u∗, v∗) has theform: (

ut

vt

)= L

(u

v

):= D

(uxx

vxx

)+ J

(u

v

), (5)

where the Jacobian matrix J is given by

J :=(

r0 σr −r

)=

2β − (1 + m)2

(1 + m)2β(m − 1)(1 + m)2

r −r

and D = diag(d1, d2). L is a linear operator withdomain DL = XC := X ⊕ iX = x1 + ix2 : x1,x2 ∈ X, where

X :=

(u, v) ∈ H2[(0, l)] × H2[(0, l)]

∣∣∣∣∣ ux(0, t) = ux(l, t) = 0

vx(0, t) = vx(l, t) = 0

and H2[(0, l)] denotes the standard Sobolev space.Denote

Jk := J − k2D =

(r0 − k2d1 σ

r −r − k2d2

).

It is clear that the eigenvalues of the operator Lare given by the eigenvalues of the matrix Jk. Thecharacteristic equation of Jk is

Pk(µ) := µ2 − TrJk · µ + Det Jk = 0, (6)

where

TrJk := r0 − r − k2(d1 + d2), (7)

Det Jk := d1d2k4 + (rd1 − r0d2)k2

− r(r0 + σ). (8)

We can check that

−r(r0 + σ) =−r[2β − (1 + m)2]

(1 + m)2− rβ(m − 1)

(1 + m)2

=r(1 + m − β)

(1 + m)2> 0.

The roots of (6) yield the dispersion relation

µk =Tr Jk ±√(Tr Jk)2 − 4Det Jk

2.

If we assume that 2β < λ(1+m)2, then r0 < 0. It iseasy to see that TrJk < 0 and Det Jk > 0. Thus, wecan conclude that the two roots of Pk(µ) = 0 bothhave negative real parts for all k ≥ 0. Therefore, wehave the following result.

Proposition 2.1. Assume that β < min1 +m, 1

2(1 + m)2. Then the unique positive constantsteady state (u∗, v∗) of (4) is locally asymptoticallystable.

Remark 2.2. In Proposition 2.1, we supposed thatβ < min1+m, 1

2(1+m)2. If m = 1, then 1+m =12(1 + m)2 = 2; if m < 1, then 1 + m > 1

2(1 + m)2;if m > 1, then 1 + m < 1

2(1 + m)2.

Next, we investigate the Turing stability of thespatially homogeneous equilibrium (u∗, v∗) of R–Dsystem (4). Turing condition is the one in whichthe uniform steady state of the reaction–diffusionequation is stable for the corresponding ordinary

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Spatiotemporal Dynamics of a Diffusive Predator–Prey Model

differential equations, but it is unstable in the par-tial differential equations with diffusion terms. It iseasy to see that the positive equilibrium (u∗, v∗) forthe corresponding ordinary differential equations(i.e. d1 = d2 = 0) is locally asymptotically stablewhen r > r0 and a family of small amplitude peri-odic solutions can bifurcate from the positive equi-librium (u∗, v∗) when r crosses through the criticalvalue r0.

We shall restrict our discussion to r0 > 0, i.e.2β > (1 + m)2. In this case, m < 1 and thus σ < 0(see Remark 2.2). It is well known that the posi-tive equilibrium (u∗, v∗) of system (4) is unstablewhen (6) has at least one root with positive realpart. Note that Tr Jk < 0 when r > r0. Hence, (6)has no complex roots with positive real part. Forthe sake of convenience, define

ϕ(k2) := Det Jk

= d1d2k4 + (rd1 − r0d2)k2 − r(r0 + σ),

which is a quadratic polynomial with respect to k2.It is necessary to determine the sign of ϕ(k2). Whenϕ(k2) < 0, (6) has two real roots in which one ispositive and another is negative. When

H(d1, d2) := rd1 − r0d2 < 0, (9)

it is easy to obtain that ϕ(k2) will take the mini-mum value

mink

ϕ(k2) = −r(r0 + σ) − (rd1 − r0d2)2

4d1d2

< 0 (10)

at k2 = k2min, where

k2min = −rd1 − r0d2

2d1d2.

Define the ratio θ = d2/d1 and let

Λ(d1, d2) := (rd1 − r0d2)2 + 4r(r0 + σ)d1d2

= r20d

22 + 2r(r0 + 2σ)d1d2 + r2d2

1.

Then

Λ(d1, d2) = 0 ⇔ r20θ

2 + 2r(r0 + 2σ)θ + r2 = 0,

H(d1, d2) = 0 ⇔ θ =r

r0≡ θ∗.

Note that −r(r0 + σ) > 0 and σ < 0, we have

4r2(r0 + 2σ)2 − 4r2r20 = 16r2σ(r0 + σ) > 0.

Then Λ(d1, d2) = 0 has two positive real roots

θ1 =−r(r0 + 2σ) + 2r

√σ(r0 + σ)

r20

(11)

Fig. 1. Numerical simulations of the stable positive equilibrium solution for R–D system (14) with r = 0.08 > r0 = 0.0744,l = 4, (u0(x), v0(x)) = (0.4 + 0.03 cos(πx/2), 0.3 + 0.05 cos(πx/2)), d1 = 1 and d2 = 1.

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0 0.02 0.04 0.06 0.08 0.10

0.05

0.1

0.15

0.2

0.25

d1

d 2

d2=theta

1d

1

d2=theta*d

1

d2=theta

2d

1

Fig. 2. Parameter space for Turing instability for R–D sys-tem (4) with m = 0.1, β = 0.65 and r = 0.08. The unstableregion is the region between the line d2 = θ1d1 and the d2-axis.

and

θ2 =−r(r0 + 2σ) − 2r

√σ(r0 + σ)

r20

. (12)

It is easy to find that 0 < θ2 < θ∗ < θ1. There-fore, when d2

d1> θ1 holds, we have mink ϕ(k2) < 0

and H(d1, d2) < 0, that is, if

d2 >−rd1(r0 + 2σ) + 2rd1

√σ(r0 + σ)

r20

D2, (13)

then (u∗, v∗) is unstable. This indicates that Turinginstability occurs.

Based on the above argument, we have the fol-lowing result about diffusion-driven instability.

Theorem 2.3. Assume that β < 1 + m, 2β > (1 +m)2 and r > r0. Then (u∗, v∗) is unstable for R–Dsystem (4), that is, Turing instability occurs ifd2 > D2, where D2 is given in (13).

Remark 2.4. We can see that d2d1

> 1 under theassumption that r > r0. Hence, for diffusive insta-bility of system (4), the predators must diffusefaster than the prey. When cross-diffusion oranomalous diffusion is incorporated into the model,the restriction on the choice of the diffusion coef-ficients for Turing instability to happen may belightened, see e.g. [Gambino et al., 2014; Vanag &Epstein, 2009], etc.

Fig. 3. Numerical simulations of the Turing instability for R–D system (14) with r = 0.08 > r0 = 0.0744, l = 4,(u0(x), v0(x)) = (0.4 + 0.03 cos(πx/2), 0.3 + 0.05 cos(πx/2)), d1 = 0.008 and d2 = 1.

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Example 2.5. As an example, we consider the R–D system with no-flux boundary conditions on one-dimensional spatial domain (0, l) and change theparameter r and the diffusion coefficients d1 and d2:

∂u

∂t− d1∆u = u(1 − u) − 0.65u2v

u2 + 0.1v2,

x ∈ (0, l), t > 0,

∂v

∂t− d2∆v = rv

(1 − v

u

), x ∈ (0, l), t > 0,

∂u

∂ν=

∂v

∂ν= 0, x = 0, l, t > 0,

u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ (0, l).

(14)

Thus, when r = 0.08 > r0 = 0.0744, we have θ1 =−r(r0+2σ)+2r

√σ(r0+σ)

r20

= 25.7558. In this case, thepositive equilibrium E∗ is stable for the ODE localsystem. If r = 0.08 > r0 = 0.0744 and d2/d1 < θ1,then the positive equilibrium E∗ is locally asymp-totically stable (see Fig. 1). The instability regionfor system (14) is given in Fig. 2. According toTheorem 2.3, when r > r0 and d2

d1> θ1 = 25.7558,

Turing instability occurs (see Fig. 3).

3. Hopf Bifurcation

In the following, we analyze the Hopf bifurcationoccurring at the positive equilibrium (u∗, v∗) bychoosing r as the bifurcation parameter. In fact,r can be regarded as the intrinsic growth rate ofpredators and plays an important role in determin-ing the stability of the positive equilibrium and theexistence of Hopf bifurcation.

Let u = u − u∗, v = v − v∗. For the sake ofconvenience, we still denote u and v by u and v,respectively. Thus, system (3) is transformed into

∂u

∂t= d1∆u + (u + u∗) − (u + u∗)2

− β(u + u∗)2(v + v∗)(u + u∗)2 + m(v + v∗)2

,

∂v

∂t= d2∆v + r(v + v∗)

(1 − v + v∗

u + u∗

).

(15)

Thus the constant positive steady-state solution(u∗, v∗) of system (3) is transformed into the zeroequilibrium of system (15).

Using the Taylor expansion at (u, v) = (0, 0),system (15) can be expressed as the followingsystem:

∂u

∂t= d1∆u + r0u + σv + f1(u, v, r),

∂v

∂t= d2∆v + ru − rv + f2(u, v, r),

(16)

where

f1(u, v, r) = A20u2 + A11uv + A30u

3 + A21u2v

+ o(|x|4, |x|3|y|),f2(u, v, r) = B20u

2 + B11uv + B02v2

+ B30u3 + B21u

2v + B12uv2

+ o(|x|4, |x|3|y|, |x|2|y|2)and

A20 = −1 − βm(m − 3)(1 + m)3u∗ , A11 =

2βm(m − 3)(1 + m)3u∗ ,

A30 =4βm(m − 1)(1 + m)4u∗2 , A21 =

βm(m4 − 14m + 9)(1 + m)4u∗2 ,

B20 = − r

u∗ , B11 =2ru∗ , B02 = − r

u∗ ,

B30 =r

u∗2 , B21 = − 2ru∗2 , B12 =

r

u∗2 .

Suppose that iω is a pure imaginary root ofcharacteristic equation (6). Substituting iω into (6),we get Tr Jk = r0 − r − k2(d1 + d2) = 0. Denoterk = r0 − k2(d1 + d2), k ∈ N0 = 0, 1, 2, . . ..Then, the only value of r at which the homoge-neous Hopf bifurcation occurs is at r = r0. Letλ(r) = α(r) ± iω(r) be a pair of complex rootsof Pk(µ) = 0 when r is near r0. Then we haveα′(r0) = −1

2 < 0. This shows that the transver-sality condition holds. Based on the expression ofRe(c1(r0)) given in the Appendix, we can obtainthe following results.

Theorem 3.1. Suppose that β < 1 + m and 2β >(1 + m)2.

(i) The constant positive steady state (u∗, v∗) ofsystem (4) is locally asymptotically stable whenr > r0 and unstable when r < r0;

(ii) System (4) undergoes a Hopf bifurcation atthe constant positive steady state (u∗, v∗) whenr = r0. Furthermore, the direction of the Hopfbifurcation is subcritical and the bifurcating

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Fig. 4. Numerical simulations of the stable time periodic solutions for R–D system (14) with r = 0.02 < r0 = 0.0744, l = 4,(u0(x), v0(x)) = (0.4 + 0.03 cos(πx/2), 0.3 + 0.05 cos(πx/2)), d1 = 1 and d2 = 1.

(spatially homogeneous) periodic solutions areorbitally asymptotically stable if Re(c1(r0)) <0; the direction of the Hopf bifurcation is super-critical and the bifurcating periodic solutionsare unstable if Re(c1(r0)) > 0.

Remark 3.2. In Theorem 3.1, we require that β <1 + m and 2β > (1 + m)2 hold simultaneously. Inthis case, we need the inequality 1+m > 1

2 (1+m)2

to hold, i.e. m < 1, and thus σ < 0.

Example 3.3. To perform some numerical simu-lations on Hopf bifurcation, we continue to con-sider the R–D system (14). We knew that whenr = 0.08 > r0 = 0.0744 and d2/d1 < θ1, thepositive equilibrium E∗ is locally asymptoticallystable (Fig. 1). By Theorem 3.1, Hopf bifurcationoccurs at r = r0 and the bifurcating periodic solu-tions exist when r < r0. Choosing r = 0.02 <0.0744, we have Re(c1(r0)) = −0.9927 < 0, whichindicates that the bifurcating periodic solutions areorbitally asymptotically stable (see Fig. 4).

4. Turing–Hopf Bifurcation

Ecologically speaking, the Turing instability breaksthe spatial symmetry leading to the pattern

formation that is stationary in time and oscilla-tory in space, while the Hopf bifurcation breaks thetemporal symmetry of the system and gives rise tooscillations which are uniform in space and periodicin time. In this part, we will investigate the cou-pling between these two different instabilities, i.e.Turing–Hopf bifurcation, in the (r, d1) parameterspace.

Assume that β < 1 + m and 2β > (1 + m)2.Thus, −(r0 + σ) > 0 and σ < 0. We choose r as thebifurcation parameter. From Theorem 3.1, we knowthat the critical value of Hopf bifurcation parameterr is

rH = r0 =2β − (1 + m)2

(1 + m)2.

At the bifurcation point, the frequency of these tem-poral oscillations is given by

ωH = Im(µk) =√

Det Jk|k=0

=√

−r(r0 + σ).

Based on the analysis in Sec. 2, we know thatthe Turing instability occurs when d1 d2. In the

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0 0.02 0.04 0.06 0.08 0.10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

d1

r

II

IV

III

I

TH

TuringHopf

Fig. 5. Turing–Hopf bifurcation diagram for R–D system(4) with m = 0.1, β = 0.65 and d2 = 1.

following, we fix d2 = 1. From (11), the critical valueof Turing bifurcation parameter r takes the form,

rT =1d1

(r0√−σ +√−(r0 + σ)

)2

.

At the Turing instability threshold, the bifurca-tion of stationary spatially periodic patterns is

characterized by the wavenumber kT with

kT =

√−r(r0 + σ)

d1.

In Fig. 5, the curves at which Hopf and Turinginstabilities occur are plotted in the (r, d1) param-eter space for fixed m = 0.1, β = 0.65 and d2 = 1.The Hopf bifurcation curve and the Turing insta-bility curve divide the parametric space into fourdistinct regions. In region I, the upper part ofthe displayed parameter space, the positive equilib-rium is the only stable solution of R–D system (4).Region II is the region of pure Turing bifurcation,while region III is the region of pure Hopf bifur-cation. In region IV, located below the two bifur-cation curves, both Turing and Hopf bifurcationsoccur. This can give rise to an interaction of bothtypes of bifurcations, producing particularly com-plex spatiotemporal patterns if the thresholds forboth instabilities occur close to each other. This isthe case in the neighborhood of a degenerate point(marked by TH), where the Turing and the Hopfbifurcations coincide: it is called a codimension-twoTuring–Hopf point, since the two control variablesare necessary to fix these bifurcation points in ageneric system of equations.

Fig. 6. Numerical simulations of the spatiotemporal Turing–Hopf structures for R–D system (14) with r = 0.05 < r0 = 0.0744,l = 80, (u0(x), v0(x)) = (0.4 + 0.03 cos(πx/2), 0.3 + 0.05 cos(πx/2)), d1 = 0.005 and d2 = 1.

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H.-B. Shi et al.

Fig. 7. Numerical simulations of the uniformly convergent solutions of R–D system (14) in the (x, t) plane with r = 0.1 >r0 = 0.0744, l = 80, (u0(x), v0(x)) = (0.4 + 0.1 cos(πx/2), 0.4 + 0.1 cos(πx/2)), d1 = 0.04 and d2 = 1.

Fig. 8. Numerical simulations of the spatially inhomogeneous time periodic solutions of R–D system (14) in the (x, t) planewith r = 0.05 < r0 = 0.0744, l = 80, (u0(x), v0(x)) = (0.4 + 0.1 cos(πx/2), 0.4 + 0.1 cos(πx/2)), d1 = 0.04 and d2 = 1.

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(a)

(b)

(c)

Fig

.9.

Num

eric

alsim

ula

tions

ofTuring–H

opfst

ruct

ure

sfo

rR

–D

syst

em(1

4)

inth

e(x

,t)

pla

ne

wit

hdec

reasing

valu

esof

d1.

Her

er

=0.0

5<

r 0=

0.0

744,

l=

80,

(u0(x

),v 0

(x))

=(0

.4+

0.1

cos(

πx/2),

0.4

+0.1

cos(

πx/2))

,and

d2

=1.

(a)

d1

=0.1

;

(b)

d1

=0.0

2;and

(c)

d1

=0.0

05.

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H.-B. Shi et al.

At the Turing–Hopf point, we have rH = rT ; inother words,

r0 =1d1

(r0√−σ +√−(r0 + σ)

)2

.

This condition is satisfied for the critical valueof d1:

d∗1 =r0

(√−σ +

√−(r0 + σ))2.

If d1 < d∗1, then rH < rT . With increasingr, the Hopf threshold is the first to be crossedand thus the Hopf bifurcation will be the first tooccur near the criticality. On the contrary, if d1 >d∗1, the first bifurcation will occur toward Turingpattern.

Example 4.1. Once again consider, as an example,R–D system (14). Figure 6 gives the Turing–Hopfstructures for the system.

Remark 4.2. We provide more numerical simula-tions in the (x, t) plane to see how the patternforms strictly depend on the two instability mech-anisms — one occurs between the two instabilities.From Fig. 5, we know that neither Turing nor Hopfbifurcation occurs when d1 = 0.04 and r = 0.1. Insuch a situation, solutions approach the steady stateuniformly in space (see Fig. 7). When r is decreasedto 0.05, both Turing and Hopf bifurcations occur.Figure 8 provides a spatially inhomogeneous timeperiodic solution. Finally, when r = 0.05 is fixed,choose d1 = 0.1, d1 = 0.02 and d1 = 0.005, respec-tively, Fig. 4 shows that the Turing effect is strongerwhen d1 is smaller.

5. Discussion

In this paper, we have considered a diffusiveLeslie predator–prey system with ratio-dependentHolling type III functional response under homo-geneous Neumann boundary conditions. For thereaction–diffusion model, we first investigatedTuring instability which induces spatially inhomo-geneous solutions. Next, we performed a detailedHopf bifurcation analysis of the model and derivedconditions to determine the direction of Hopf bifur-cation and stability of the bifurcating temporalperiodic solutions by applying the normal formtheory and the center manifold reduction. Then weshowed that at the intersecting points of the Turing

bifurcation and Hopf bifurcations curves, the modelexhibits Turing–Hopf bifurcation, which producesspatiotemporal patterns for the reaction–diffusionpredator–prey system.

The positive equilibrium and periodic solu-tions of the local system are spatially homogeneoussolutions of the diffusive system (4). Therefore,we can regard the dynamics of ODE model (i.e.d1 = d2 = 0) as subdynamics of the PDE model (4).Moreover, the direction of Hopf bifurcation forsystem (4) at r = r0 is the same as that of ODEsystem. However, the stability of the positive equi-librium (u∗, v∗) can change due to the effect ofdiffusion.

It is well known that predator–prey modelswith ratio-dependent Holling type II functionalresponse have very rich and complex dynamicalbehaviors (see [Hsu et al., 2001; Kuang, 1999;Kuang & Beretta, 1998; Li & Kuang, 2007; Liang &Pan, 2007; Ruan et al., 2008, 2010; Xiao & Ruan,2001]). If both the predators and their prey can dis-perse randomly in their habits but do not crossthe boundary, our results demonstrate that thediffusive Leslie–Gower predator–prey system withratio-dependent Holling type III functional responsecan exhibit spatial patterns (via Turing instabil-ity), temporal patterns (via Hopf bifurcation), aswell as spatiotemporal patterns (via Turing–Hopfbifurcation).

Acknowledgments

Part of this work was performed when the firstauthor was visiting the University of Miami in 2013,he would like to thank the faculty and staff inthe Department of Mathematics at the Universityof Miami for their warm hospitality. The authorsalso thank the referees for their valuable commentswhich have led to a much improved paper.

References

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Appendix: Calculation of Re(c1(r0))

In the Appendix, following the techniques and pro-cedure in [Hassard et al., 1981] we give the expres-sion of Re(c1(r0)), which is used to determine thedirection of the Hopf bifurcation and stability ofbifurcating periodic solutions. Let L∗ be the conju-gate operator of L defined as (5):

L∗(

u

v

):= D

(uxx

vxx

)+ J∗

(u

v

), (A.1)

where

J∗ := JT =

2β − (1 + m)2

(1 + m)2r

m − 1(1 + m)2

−r

,

with the domain DL∗ = XC . Let

q :=

(q1

q2

)=

1

−r0

σ+

ω0

σi

,

q∗ :=

(q∗1q∗2

)=

σ

2πω0

ω0

σ+

r0

σi

i

.

For any a ∈ DL∗ , b ∈ DL, it is not difficult to verifythat

〈L∗a, b〉 = 〈a,Lb〉, L(r0)q = iω0q,

L∗(r0)q∗ = −iω0q∗, 〈q∗, q〉 = 1, 〈q∗, q〉 = 0,

where 〈a, b〉 =∫ π0 aT bdx denotes the inner product

in L2[(0, l)] × L2[(0, l)].Following Hassard et al. [1981], we decompose

X = XC ⊕ XS with XC = zq + zq : z ∈ C,XS = w ∈ X : 〈q∗,w〉 = 0. For any (u, v) ∈ X,there exists z ∈ C and w = (w1,w2) ∈ XS suchthat (u, v)T = zq+zq+(w1,w2)T , z = 〈q∗, (u, v)T 〉.Thus,

u = z + z + w1,

v = z

(−r0

σ+ i

ω0

σ

)+ z

(−r0

σ− i

ω0

σ

)+ w2.

System (4) is reduced to the following system in the(z,w)-coordinates:

dz

dt= iω0z + 〈q∗, f〉,

dwdt

= Lw + H(z, z,w),

(A.2)

where

H(z, z,w) = f − 〈q∗, f〉q − 〈q∗, f〉qand f = (f1, f2)T [f1 and f2 are defined as (16)].It is easy to obtain that

〈q∗, f〉 =1

2ω0[ω0f

1 − i(r0f1 + σf2)],

〈q∗, f〉 =1

2ω0[ω0f

1 + i(r0f1 + σf2)],

〈q∗, f〉q =1

2ω0

ω0f1 − i(r0f

1 + σf2)

ω0f2 + i

(ω2

0

σf1 +

r20

σf1 + r0f

2

),

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Spatiotemporal Dynamics of a Diffusive Predator–Prey Model

〈q∗, f〉q =1

2ω0

ω0f1 + i(r0f

1 + σf2)

ω0f2 − i

(ω2

0

σf1 +

r20

σf1 + r0f

2

).

Furthermore, we have H(z, z,w) = (0, 0)T . Let

H =H20

2z2 + H11zz +

H02

2z2 + o(|z|3).

It follows from Appendix A of [Hassard et al., 1981]that system (A.2) possesses a center manifold, andwe can write w in the form:

w =w20

2z2 + w11zz +

w02

2z2 + o(|z|3).

Thus, we have

w20 = (2iω0I − L)−1H20,

w11 = (−L)−1H11,

w02 = w20.

This implies that w20 = w02 = w11 = 0. For lateruses, denote

σ1 := f1uuq2

1 + 2f1uvq1q2 + f1

vvq22

= 2A20 + 2A11q2,

σ2 := f2uuq2

1 + 2f2uvq1q2 + f2

vvq22

= 2B20 + 2B11q2 + 2B02q22,

ν1 := f1uu|q1|2 + f1

uv(q1q2 + q1q2) + f1vv|q2|2

= 2A20 + A11(q2 + q2),

ν2 := f2uu|q1|2 + f2

uv(q1q2 + q1q2) + f2vv|q2|2

= 2B20 + B11(q2 + q2) + 2B02|q2|2,τ1 := f1

uuu|q1|2q1 + f1uuv(2|q1|2q2 + q2

1q2)

+ f1uvv(2q1|q2|2 + q1q

22) + f1

vvv |q2|2q2,

= 6A30 + 2A21(2q2 + q2),

τ2 := f2uuu|q1|2q1 + f2

uuv(2|q1|2q2 + q21q2)

+ f2uvv(2q1|q2|2 + q1q

22) + f2

vvv |q2|2q2

= 6B30 + 2B21(2q2 + q2) + 2B12(2|q2|2 + q22),

where all the partial derivatives evaluated at thepoint (u, v, r) = (0, 0, r0). Therefore, the reaction–diffusion system restricted to the center manifold inz, z coordinates is given by

dz

dt= iω0z +

12g20z

2 + g11zz +12g02z

2

+12g21z

2z + o(|z|4),

where g20 = 〈q∗, (σ1, σ2)T 〉, g11 = 〈q∗, (ν1, ν2)T 〉,g21 = 〈q∗, (τ1, τ2)T 〉. Note that B02 = B20, B11 =−2B20, B21 = −2B12 and ω2

0 = −r0(r0 + σ). Then,straightforward but tedious calculations show that

g20 =σ

2ω0

[(ω0

σ− r0

σi)

σ1 − iσ2

]

= A20 − 2B20 − 2B20

σr0 − i

ω0

(2B20

σr20 + (A20 + A11 + 3B20)r0 + σB20

),

g11 =σ

2ω0

[(ω0

σ− r0

σi)

ν1 − iν2

]= A20 − A11

σr0 − i

ω0

(−A11

σr20 + (A20 + B20)r0 + σB20

),

g21 =σ

2ω0

[(ω0

σ− r0

σi)

τ1 − iτ2

]

= 3A30 − 2B12 − 2(A21 + B12)σ

r0 +i

ω0

(2(A21 − B12)

σr20 − (3A30 + A21 + 5B12)r0 − 3σB30

).

According to Hassard et al. [1981], we have

Re(c1(r0)) = Re

i

2ω0

(g20g11 − 2|g11|2 − 1

3|g02|2

)+

12g21

= − 12ω0

[Re(g20) Im(g11) + Im(g20)Re(g11)] +12

Re(g21)

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H.-B. Shi et al.

= −A211 + 2A11A20 + A11B20 + 2B2

20

2ω20σ

r20 +

2(A220 + A20B20 − 2B2

20) + A11(A20 − B20)2ω2

0

r0

− A21

σr0 − B12

σr0 +

σB20(A20 − B20)ω2

0

+32A30 − B12

= − 1r0 + σ

A220 +

r0

2(r0 + σ)σA2

11 +2r0 − σ

2(r0 + σ)σA20A11 − 1

r0A20B20 +

12σ

A11B20

+r0 + σ

r0σB2

20 +32A30 − r0

σA21 − r0 + σ

σB12.

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