the complex dynamics of a diffusive prey–predator …...predators have to search for food (and...

Ecological Complexity 28 (2016) 123–144

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Ecological Complexity

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Original Research Article

The complex dynamics of a diffusive prey–predator model with an
Allee effect in prey
Feng Rao a, Yun Kang b,*a College of Sciences, Nanjing Tech University, Nanjing, Jiangsu 211816, Chinab Sciences and Mathematics Faculty, College of Integrative Sciences and [2_TD$DIFF]Arts, Arizona State University, Mesa, AZ 85212, USA

A R T I C L E I N F O

Article history:

Received 10 December 2015

Received in revised form 3 June 2016

Accepted 2 July 2016

Available online 5 August 2016

Keywords:

Allee effect

Diffusion

Non-constant positive solution

Pattern formation

Turing instability

A B S T R A C T

This paper investigates complex dynamics of a predator–prey interaction model that incorporates: (a) an

Allee effect in prey; (b) the Michaelis–Menten type functional response between prey and predator; and

(c) diffusion in both prey and predator. We provide rigorous mathematical results of the proposed model

including: (1) the stability of non-negative constant steady states; (2) sufficient conditions that lead to

Hopf/Turing bifurcations; (3) a prior estimates of positive steady states; (4) the non-existence and

existence of non-constant positive steady states when the model is under zero-flux boundary condition.

We also perform completed analysis of the corresponding ODE model to obtain a better understanding

on effects of diffusion on the stability. Our analytical results show that the small values of the ratio of the

prey’s diffusion rate to the predator’s diffusion rate are more likely to destabilize the system, thus

generate Hopf-bifurcation and Turing instability that can lead to different spatial patterns. Through

numerical simulations, we observe that our model, with or without Allee effect, can exhibit extremely

rich pattern formations that include but not limit to strips, spotted patterns, symmetric patterns. In

addition, the strength of Allee effects also plays an important role in generating distinct spatial patterns.

� 2016 Elsevier B.V. All rights reserved.

1. Introduction

Prey–predation interactions have been a central theme forecological studies in the past few decades (Volterra, 1926; Arditiand Ginzburg, 1989; Berryman, 1992; Kuang and Beretta, 1998;Abrams and Ginzburg, 2000; Hsu et al., 2001; Kang and Wedekin,2013). To study the interaction between predators and their prey, itis important to determine the specific form of the functionalresponse between prey and predator that describes the amount ofprey consumed per predator per unit of time since it affects thepopulation dynamics dramatically (Akcakaya et al., 1995). Inliterature, The Holling-Type II functional response (i.e., preydensity dependence) has been considered as a biological mean-ingful functional response (Li and Kuang, 2007). Some researchershave argued that a functional response depending on the ratio ofprey to predator abundance is a suitable representation of somephenomena (Arditi and Ginzburg, 1989). For example, whenpredators have to search for food (and therefore have to share orcompete for food), a more suitable general predator–prey theory

* Corresponding author.

E-mail addresses: [email protected] (F. Rao), [email protected] (Y. Kang).

http://dx.doi.org/10.1016/j.ecocom.2016.07.001

1476-945X/� 2016 Elsevier B.V. All rights reserved.

such as the ratio-dependent theory (i.e., the per capita predatorgrowth rate of the predator depends on a function of the ratio ofprey to predator abundance), should be considered. The work byAkcakaya et al. (1995) and Hsu et al. (2001) supports that the ratio-dependent predation models (also called the prey–predatormodels with Michaelis–Menten type functional responses) arecapable of producing richer and more reasonable dynamicsbiologically. Among all predator–prey models, the predationsystems with the Michaelis–Menten type functional responseshave gained a great interest over many decades (a partial list ofreferences may be found in Kuang and Beretta (1998), Hsu et al.(2001), Xu and Chaplain (2002), Wang et al. (2007), Meng et al.(2010), Rao and Wang (2012), Rao (2014) and the references citedtherein). However, many of these studies have concentrated on theexistence, uniqueness and the stability analysis of predationsystems. In addition to the functional response between prey andpredator, there are many other factors such as Allee effects,dispersal movements of both prey and predator, contributinggreatly to the population dynamics of prey and predator. In thispaper, we propose a reaction–diffusion predator–prey model withAllee effects in prey and diffusion in both prey and predator toinvestigate how the synergy of Allee effects and diffusion affect thespatial temporal dynamics. More specifically, we focus on the rich

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F. Rao, Y. Kang / Ecological Complexity 28 (2016) 123–144124

spatial-temporal dynamics of a Michaelis–Menten type prey–predator model that is distinct from others by incorporating (1) anAllee effect in prey (which can be weak or strong); and (2) diffusionin both prey and predator.

In population dynamics, Allee effect refers to the positivecorrelation between the population density and the per capitagrowth rate (Stephens et al., 1999), which is introduced by Allee(1978). The Allee effect can be caused by a number of sources suchas difficulties in finding mates at small densities, reproductivefacilitation, predation, environmental conditioning and inbreedingdepressions. Allee effect can be classified into strong and weakcases. The strong one introduces a population threshold, and thepopulation must surpass this threshold to grow. In contrast, apopulation with a weak Allee effect does not have such threshold.Detailed investigations relating to the Allee effect can be found inthe papers (Petrovskii et al., 2002; Kent et al., 2003; Zhou et al.,2005; Shi and Shivaji, 2006; Morozov et al., 2006; Celik and Duman,2009; Wang et al., 2011; Kang and Yakubu, 2011; Kang andCastillo-Chavez, 2014).

Allee effect can have a huge impact on the population dynamics(Cai et al., 2014). For example, Petrovskii et al. (2002) showed that afully deterministic predator–prey model with the Allee effects inprey can predict a patchy invasion under certain the parameterrange and the model has interesting dynamical features such as thecoexistence state is unstable but no stable limit cycle exists. In Kentet al. (2003), Kent et al. concluded that the predator–prey systemcan be stabilized by an influx of prey due to a rescue effect, and bedestabilized by an outflux of an Allee effect. Zhou et al. (2005)selected two classical predator–prey systems with the Allee effectsin both predator and prey populations. They showed that the Alleeeffect may be a destabilizing force in these predator–prey systems.Celik and Duman (2009) investigated the impact of the Allee effect(on prey population) on the stability of the positive equilibriumpoint for a discrete-time predator–prey system. Wang et al. (2011)considered the dynamic of a reaction–diffusion Holling type IIpredator–prey system with strong Allee effect in the preypopulation. They showed that the impact of the Allee effectincreases the system spatiotemporal complexity. The Allee effectwas shown to bring essential changes to the population dynamicsand it has drawn considerable attention in almost every aspect ofecology and conservation, however, there is no work that has beendone for the spatio-temporal dynamics of Michaelis–Menten typepredator–prey model with an Allee effect in prey. One of our mainmotivations is to study the synergistic effects of Allee effects inprey and diffusion in both species on the prey–predator populationdynamics.

Understanding the spatial patterns and the related mechanismsof interacting species has been a great interest in conservationbiology and ecology. Theoretically, there are a fair amount work byusing two nonlinear reaction–diffusion equations in two spatialdimensions to study the invasion dynamics of predator–preysystems (Lewis and Kareiva, 1993; Morozov et al., 2006; Wanget al., 2011). To understand the role of random mobility of theindividuals or organisms on the stability and persistence ofinteracting species, the diffusive predator–prey models have beenstudied by many authors (Lin et al., 1988; Murray, 1990; Lou andNi, 1996; Pang and Wang, 2003; Cantrell and Cosner, 2003;Baurmann et al., 2007; Banerjee and Banerjee, 2012) eitherqualitatively or numerically. In Lou and Ni (1996), Lou and Ni dealtwith a strongly-coupled parabolic prey–predator system. Theyinvestigated the effects of diffusion, self-diffusion and cross-diffusion on the dynamics of the system. Baurmann et al. (2007)studied a generalized predator–prey system on a spatial domainwhere diffusion is considered as the principal process of motion.Banerjee and Banerjee (2012) considered a modified spatiotem-poral ecological system originating from the Holling-Tanner model

by incorporating diffusion terms with both numerical andanalytical approaches.

The diffusion has been observed as causes of the spontaneousemergences of ordered structures, called patterns, in a variety ofnonequilibrium situations (Wen, 2013). Patterns generated inhomogeneous environments are particularly interesting becausethey require an explanation based on the individual behavior oforganisms, and they emerge from interactions in spatial scales thatare much larger than the characteristic scale of individuals (Alonsoet al., 2002). In mathematics, pattern formation refers to theprocess that, by changing a bifurcation parameter, the spatiallyhomogeneous steady states lose stability to spatially inhomoge-neous perturbations, and stable inhomogeneous solutions arise(Wang and Hillen, 2007). In recent years, many researchers showthat the reaction–diffusion predator–prey model is an appropriatetool for investigating the fundamental mechanism of complexspatiotemporal predation dynamics (see Alonso et al., 2002;Baurmann et al., 2007; Wang et al., 2007; Banerjee and Petrovskii,2011; Rodrigues et al., 2011 and the references therein). There is alimited work on a reaction–diffusion model with the Michaelis–Menten type functional response incorporating an Allee effect inprey population. In this work, we study a Michaelis–Menten typepredator–prey interaction model where random movements ofboth species are described by the diffusion terms and prey hasAllee effects ranging from weak to strong. Our model uses thetraditional framework of reaction–diffusion equations where thereaction part follows the Michaelis–Menten type interactionbetween prey and predator population. We assume that bothprey and predator are capable to diffuse over a two dimensionallandscape. We aim to answer the following questions through ouranalytic and numerical results:

1. W
hat are the synergistic effects of diffusion and Allee effects onthe spatiotemporal dynamics of our model?
2. C
an our proposed model generate distinct spatial patterns? Andwhat are the mechanisms generating these different patterns?
The organization of this paper is as follows. In Section 2, wederive a reaction–diffusion Michaelis–Menten type predator–preymodel with an Allee effect in prey. In Section 3, we carry out theanalysis of the basic dynamics of the model, and prove a priorestimate for positive steady states of the model. In Section 4, weanalyze the stability of non-negative constant steady states, andprovide sufficient conditions that guarantee the existence of Hopf/Turing bifurcation at positive constant steady states. For thecomparison, we also provide the completed analysis for thecorresponding ODE model. In Section 5, we provide sufficientconditions that can lead to the non-existence and existence of non-constant positive steady states. In Section 6, we use numericalsimulations to reveal the emergence of different patterns and theinfluences of diffusion and Allee effects on the dynamical behaviorof the model. We conclude our work in Section 7 and the detailedproofs of our theoretical work are given in the last section.

2. Model derivation

The two-dimensional Michaelis–Menten type prey–predatorinteraction model can be described as follows:

dN

dT¼ FðNÞ�A

NP

N þ P;

dP

dT¼ mA

NP

N þ P�MP;

8><>: (1)

where N(T) and P(T) are the population densities of prey andpredator at time T � 0, respectively; A is the predation rate, m is the

[(Fig._1)TD$FIG]

Fig. 1. Growth rates of the prey population with strong (b = 0.5) (the red curve) and

weak (b = 0-the black curve; b =�0.45 – the blue curve) Allee effects. The green

curve represents the limiting case of logistic growth. (For interpretation of the

references to color in this figure legend, the reader is referred to the web version of

the article.)

F. Rao, Y. Kang / Ecological Complexity 28 (2016) 123–144 125

food utilization coefficient; and M is the mortality rate of predator;and the function F(N) describes the population dynamics of prey inthe absence of predator.

There are many literature work on the traditional ODEMichaelis–Menten type predator–prey system (1) (Kuang andBeretta, 1998; Hsu et al., 2001; Berezovskaya et al., 2001; Xiao andRuan, 2001; Tang and Zhang, 2005) where F(N) is described by alogistic growth function. These authors have shown that thesystem has very rich dynamics. Kuang and Beretta (1998)presented global qualitative analysis of solutions, and showedthat the ratio-dependent predator–prey models are rich inboundary dynamics. They also established that the system hasno nontrivial periodic solutions provided the positive steady stateis locally asymptotic stable. The work of Berezovskaya et al. (2001)and Xiao and Ruan (2001) showed that the origin is a complicatedequilibrium point whose characteristics determine some impor-tant properties of the system. Hsu et al. (2001) showed that thelimit cycle is always stable and unique once it exists. And the workof Berezovskaya et al. (2001) and Xiao and Ruan (2001) showedthat the heteroclinic bifurcation plays an important role inunderstanding dynamics of the system.

The corresponding PDE versions of the Michaelis–Menten typepredator–prey systems have gained a lot of attentions. Forexample, Pang and Wang (2003) studied qualitative propertiesof solutions of the system with the homogeneous Neumannboundary condition, including the dissipation, persistence andstability of non-negative constant steady states, and the existenceof non-constant positive steady states of the system. Alonso et al.(2002) investigated how diffusion affects the stability of predator–prey coexistence equilibria and showed a new difference betweenratio- and prey-dependent models. The recent work of Wang et al.(2007) and Banerjee and Petrovskii (2011) studied the spatiotem-poral pattern formations in a ratio-dependent predator–preysystem. Wang et al. (2007) presented a theoretical analysis ofevolutionary processes that involves organisms distribution andtheir interaction of spatially distributed population with localdiffusion. Banerjee and Petrovskii (2011) showed that the systemcan develop patterns both inside and outside of the Turingparameter domain.

The interesting question is how the dynamical properties ofmodel (1) can change when prey experiences Allee effects.Following the work of Lewis and Kareiva (1993), we assume thegrowth function of prey with an Allee effect can be parameterizedas follows:

FðNÞ ¼ 4v

ðK�N0Þ2NðN�N0ÞðK�NÞ; (2)

where K is the prey carrying capacity; v is the maximum per capitagrowth rate; and N0 (N0 < K) is the ‘‘Allee threshold’’ where thegrowth rate F(N) becomes negative when N < N0. The value of N0

can be considered as a measure of the intensity of the Allee effect:the less the value of N0 is, the less prominent is the Allee effect(Petrovskii et al., 2002). More specifically, the Allee effect is called‘strong’ if 0 < N0 < K (when the growth rate becomes negative forN < N0) while it is called ‘weak’ if �K < N0 � 0. When N0 � � K,there is no Allee effect (Morozov et al., 2004).

We aim to investigate the dynamical complexity of a PDEversion of the Michaelis–Menten type predator–prey with Alleeeffects in prey and diffusion in both prey and predator. By choosingappropriate scales for the variables of Eqs. (1) and (2), the numberof parameters can be lessened. Considering dimensionless vari-ables with the following scaling:

u ¼ N

K; v ¼ P

K; t ¼ AT;

we obtain and employ the corresponding diffusive model asfollows:

@uðx; y; tÞ@t

�D1@2

u

@x2þ @2

u

@y2

!¼ auðu�bÞð1�uÞ� uv

uþ v;

@vðx; y; tÞ@t

�D2@2v@x2þ @2v

@y2

!¼ cuv

uþ v�dv;

8>>>><>>>>:

(3)

with

a ¼ 4vK2

AðK�N0Þ2; b ¼ N0

K; c ¼ m; d ¼ M

A;

where u(x, y, t) and vðx; y; tÞ are the densities of prey and predator,respectively, at moment t and position (x, y); a is the relativeintrinsic growth rate of prey; b 2 (�1, 1) denotes the Alleethreshold where b 2 (�1, 0) indicates the weak Allee effects andb 2 (0, 1) indicates the strong Allee effects; c 2 (0, 1] represents theenergy conversion rate from prey to predator; and d denotes therelative death rate of predator. The non-negative constants D1 andD2 are the diffusion coefficients of u and v respectively, whichimply the speed of individual movements.

In model (3), the continuous growth function considering Alleeeffect is expressed by the following equation:

f ðuÞ ¼ auðu�bÞð1�uÞ: (4)

Allee effect expressed by (4) is strong if b 2 (0, 1) while it isweak if b 2 (�1, 0]. Growth curves of the population with strongand weak Allee effect are depicted in Fig. 1. We will mainly focus onthe strong Allee effects for our analytical study, i.e., 0 < b < 1, butwe will include the cases that b � 0 for numerical studies toinvestigate the strength of Allee effects. Model (3) is to be analyzedwith the following initial conditions:

uðx; y; tÞ�0; vðx; y; tÞ�0; ðx; yÞ 2V ¼ ½0; Lx��½0; Ly�; (5)

and zero-flux boundary conditions:

@u

@n¼ @v

@n¼ 0; ðx; yÞ 2@V; (6)

where V is a bounded domain in R2[1_TD$DIFF] with smooth boundary @V; Lx

and Ly give the size of the model in the directions of x and y,respectively; n is the outward unit normal on @V; and zero-fluxboundary conditions imply that no external input is imposed fromoutside (Murray, 1990). In the case that uðx; y; tÞ ¼ vðx; y; tÞ ¼ 0, we


define uvuþv ju¼0;v¼0 ¼ 0. We would like to point out that this

modification is used to treat the singularity at the origin, andthat it was introduced by Kuang and Beretta (1998) as well as Xiaoand Ruan (2001). In the following Sections 3–5, we will provide ourtheoretical results of the PDE model (3) and the corresponding ODE(10). We would like to point out that our theoretical results ofModel (3) are held for the bounded domain V � Rs for s � 2 but wewill focus on the case of s = 2.

3. Basic dynamics

In this section, we investigate the basic dynamics of model (3)such as uniqueness of solution, positive invariance in its statespace, and the boundedness.

Theorem 3.1. Suppose that a, d, D1, D2 > 0, 0 < b < [4_TD$DIFF]1, 0 < c � 1, and

V � R2[3_TD$DIFF] is a bounded domain with smooth boundary. The following

statements hold:

1. If
uðx; y;0Þ ¼ u0ðx; yÞ�0; vðx; y;0Þ ¼ v0ðx; yÞ�0, then model (3)has a unique solution ðuðx; y; tÞ; vðx; y; tÞÞ such that
uðx; y; tÞ�0; vðx; y; tÞ�0 for t 2 (0, 1) and ðx; yÞ 2V;

2. If
u0(x, y) � b and ðu0; v0Þ 6¼ ðb;0Þ, then ðuðx; y; tÞ; vðx; y; tÞÞ tends to
(0, 0) uniformly as t!1;

3. If
d > c, then ðuðx; y; tÞ; vðx; y; tÞÞ tends to (ue(x, y), 0) uniformly as
t!1, where ue(x, y) is a non-negative solution of

D1Duþ auðu�bÞð1�uÞ ¼ 0; ðx; yÞ 2V;

@u

@n¼ 0; ðx; yÞ 2 @V:

(7)

For any solution ðuðx; y; tÞ; vðx; y; tÞÞ of (3), then
4.
limsupt!1

uðx; y; tÞ�1; limsupt!1

vðx; y; tÞ� c

d:

Moreover, if D1 = D2, then for any ðx; yÞ 2V,

limsupt!1

vðx; y; tÞ� að1�bÞ2

4dþ 1:

Remark 1. Theorem 3.1 shows that the compact set G :¼½0;1��½0; c

d� is a global attractor for all solutions of model (3) inthe sense that any non-negative solution ðuðx; y; tÞ; vðx; y; tÞÞ of (3)enters in G for large t and for all (x, y) 2V. This indicates that ourmodel is well-defined biologically. The results on the dynamicalbehavior of model (3) in Theorem 3.1 parts (ii) and (iii) also implythe following results on the steady state solution of (3), whichsatisfy:

�D1Du ¼ auðu�bÞð1�uÞ� uv

uþ v; ðx; yÞ 2V;

�D2Dv ¼ cuv

uþ v�dv; ðx; yÞ 2V;

@u

@n¼ @v

@n¼ 0; ðx; yÞ 2 @V:

8>>>>><>>>>>:

(8)

Therefore, we can have the following Corollary.

Corollary 3.1. Suppose that a, d, D1, D2 > 0, b 2 (0, 1), c 2 (0, 1], and

V is a bounded domain with smooth boundary. Let ðuðx; yÞ; vðx; yÞÞ be

a non-negative solution of model (8):

1. If
u(x, y) � b for all ðx; yÞ 2V, then ðuðx; yÞ; vðx; yÞÞmust be (0, 0) or
(b, 0).
2. If d � c, then ðuðx; yÞ; vðx; yÞÞmust be in form of (ue, 0), which called
a semi-trivial solution.

Proof. From Theorem 3.1 parts (ii) and (iii), we only need to prove
the case when d � c. According to Theorem 3.1 part (iv), we haveu(x, y) � 1 and cu
uþv�d�0. By integrating the second equation of (8),we obtain follows

0�D2

Z ZVjrvj2dL ¼

Z ZV

v2 cu

uþ v�d

� �dL�0;

which indicates that v�0 on V. Therefore, statement holds. &

Now we provide a priori estimates for any non-negativesolutions of model (8). To obtain this aim, we recall two knownresults that are due to Lou and Ni (1996), and to Lin et al. (1988).

Lemma 3.1. Let V be a bounded Lipschitz domain in RN and

g 2CðV�RÞ.

(i) I
f w2C2ðVÞ \C1ðVÞ satisfies
Dwðx; yÞ þ gðx; y;wðx; yÞÞ�0 in V;@w

@n�0 on@V;

and wðx0; y0Þ ¼maxV

wðx; yÞ, then gðx0; y0;wðx0; y0ÞÞ�0.

(ii) I
f w2C2ðVÞ \C1ðVÞ satisfies
Dwðx; yÞ þ gðx; y;wðx; yÞÞ�0 in V;@w

@n�0 on@V;

and wðx0; y0Þ ¼minV

wðx; yÞ, then gðx0; y0;wðx0; y0ÞÞ�0.

The results of Lemma 3.1 and the application of the maximumprinciple theorem allow us to obtain the following estimates.

Theorem 3.2. Suppose that ðuðx; yÞ; vðx; yÞÞ is a non-negative solu-

tion of model (8), then for ðx; yÞ 2V, ðuðx; yÞ; vðx; yÞÞ satisfies

0�uðx; yÞ<1; and 0�vðx; yÞ< cað1�bÞ2

4dþ D1

D2

!; (9)

where a, d, D1, D2 > 0 and 0 < b < 1, 0 < c � 1.

Remark 2. The results of Theorem 3.2 will be used for the proof ofthe non-existence and existence of the non-constant positivesolutions of model (8) in Section 5.

4. Constant steady state solutions

First, we provide a theorem regarding the basins of attraction ofthe extinction equilibrium (0, 0) for model (3).

Theorem 4.1. Suppose that a, d, D1, D2 > 0, b 2 (0, 1), c 2 (0, 1] are

fixed. For a given initial value of the prey population u0(x, y) � 0, there

exists a constant v0 which depends on parameters and u0(x, y), such

that when the initial predator population v0ðx; yÞ� v0, then the

corresponding solution ðuðx; y; tÞ; vðx; y; tÞÞ of model (3) tends to (0,0) uniformly for ðx; yÞ 2V as t!1.

Remark 3. Theorem 4.1 implies that (0, 0) is always a locally stablesteady state with basin of attraction including all large v0 for agiven u0 and parameter values. Our results indicate that for anygiven initial prey population, a large enough initial predator pop-ulation can always lead to the extinction of both species, i.e., theconvergence to the constant steady state (0, 0). To further ourstudy, we will provide complete dynamical analysis of the ODEversion of model (3) in the following subsection.

Table 1

The existence and stability of equilibrium for model (10) where ui ¼

ðbþ1Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðbþ1Þ2�4ðabcþc�dÞ

ac

p2 ¼ ðbþ1Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðb�1Þ2�4ðc�dÞ

ac

p2 ; vi ¼ c�d

d ui and u ¼bþ1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðbþ1Þ2þ8ð1�cÞðc�dÞ

ac2

q4 .

EquilibriumExistence Condition for ExistenceStability Condition

(0, 0) Always exists Always locally stable

(b, 0) Always exists Saddle if c<d; Source if c>d

(1, 0) Always exists Sink if c<d; Saddle if c>d

ðu1; v1Þ c>d and a> 4ðc�dÞcðb�1Þ2

Always saddle

ðu2; v2Þ c>d and a> 4ðc�dÞcðb�1Þ2

Sink if u >u2; Source if u <u2.


4.1. Dynamics of the ODE version of model (3)

Constant steady state solutions of model (3) are the same as itsODE version, i.e., Model (10), which is shown as follows:

du

dt¼ auðu�bÞð1�uÞ� uv

uþ v;

dvdt¼ cuv

uþ v�dv:

8><>: (10)

Define

ui ¼ðbþ 1Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðbþ 1Þ2� 4ðabcþc�dÞ

ac

q2

¼ðbþ 1Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðb�1Þ2� 4ðc�dÞ

ac

q2

;

vi ¼c�d

dui ;

where u1 <u2 and

u ¼bþ 1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðbþ 1Þ2 þ 8ð1�cÞðc�dÞ

ac2

q4

:

Here we provide the global dynamics of model (10) as thefollowing theorem to facilitate our study for model (3).

Theorem 4.2. [Dynamics of the ODE model] Model (10) always has

three boundary equilibria: E0 = (0, 0), E1 = (b, 0), E2 = (1, 0). If c > d,

then model (10) can have two interior equilibria Ei ¼ ðui ; vi Þ; i ¼ 1;2.

Provided that interior equilibria Ei ¼ ðui ; vi Þ; i ¼ 1;2 exist, we can

conclude that b<u1 <bþ1

2 <u2 <1, and the existence and stability

conditions of these equilibria can be listed in Table 1. Moreover, The

global dynamics of model (10) can be summarized as follows:

1. If
c> d and a< 4ðc�dÞcðb�1Þ2
, then model (10) has E0 = (0, 0) as its global

attractors.

2. If
c < d, then model (10) has E0 = (0, 0) and E1 = (1, 0) as its global
attractors.

3. If
c > d and
a>max 4ðc�dÞcðb�1Þ2

;

ðc�dÞ cd�c�dþ 4bc

ðb�1Þ2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficd�c�dþ 4bc

ðb�1Þ2

� �2

þ4ðcd�2c�dÞ2

bðb�1Þ2

s24

35

2bc2

8>>>>><>>>>>:

9>>>>>=>>>>>;

,

then model (10) has E0 = (0, 0) and E2 ¼ ðu2; v2Þ as its global

attractors.4. If c > d and

4ðc�dÞcðb�1Þ2

< a<


ðb�1Þ2þ


ðb�1Þ2

� �2

þ4ðcd�2c�dÞ2

bðb�1Þ2

s24

35

2bc2 , then

ðu2; v2Þ is source.

Remark 4. Kang and Castillo-Chavez (2014) studied similar ODEmodels of (10), where their models are in the framework ofSusceptible-Infected (SI) epidemiological models, i.e., they assumethat c = 1. In this case, we can conclude that if E2 exits, then it isalways locally asymptotically stable since u2 >u always holds.Theorems 4.1 and 4.2 imply that the reaction–diffusion model (3)and its ODE version (10) have the same local stability at E0, i.e., E0 isalways locally stable for both cases. The results of Theorem 4.2provide us a starting point to explore the effects of spatial hetero-geneity through the diffusion in both prey and predator. Moreover,in Sen et al. (2012) presented the analysis results about theemerging limit cycle. They obtained that in the model with thelogistic growth, the sustainable oscillations are possible whereas

they are impossible in the same system with the Allee effectregardless whether or not it is strong or weak (see discussion page22 third paragraph in Sen et al. (2012)). Their results are alsosupported by our ODE model (10).

In the remaining part of this section, we assume that d < c � 1

and a> 4ðc�dÞcðb�1Þ2

. Under this assumption, according to Theorem 4.2,

the ODE model (10) has five non-negative constant steady statesolutions: E0 = (0, 0), E1 = (b, 0), E2 = (1, 0) and two positive

constant solutions Ei ¼ ðui ; vi Þ; i ¼ 1;2, where E0 is always a sink,

E1 is a source, E2; E1 are saddles, and E2 can be a sink or source

depending on the values of a. When c > d and

4ðc�dÞcðb�1Þ2

< a<


ðb�1Þ2þ


ðb�1Þ2

� �2

þ4ðcd�2c�dÞ2

bðb�1Þ2

s24

35

2bc2 , then

E2 ¼ ðu2; v2Þ is a source. For example, take the parameters as

a = 4.17, b = 0.2, c = 0.3, d = 0.1, then E1 = (0.2, 0) is a nodal source,E2 = (1, 0) is a saddle point and the positive equilibria are E1 ¼ð0:58869;1:1774Þ which is a saddle point and E2 ¼ð0:61131;1:2226Þ is a spiral source, see Fig. 2(a). If a = 4.2,b = 0.2, c = 0.3, d = 0.1, then the positive equilibria are E1 ¼ð0:56437;1:1287Þ is a saddle point and E2 ¼ ð0:63563;1:2713Þ is

a spiral sink where E2 is locally asymptotically stable (Fig. 2(b)).

Paradox of enrichment and biological control paradox: The‘‘paradox of enrichment’’ means that in one predator-one preysystems, destabilization occurs as the resource supply for the preyspecies increased (enrichment). This paradox was first presentedby Rosenzweig (1971). Biological control is to introduce naturalpredators or parasites to ecosystems for pest control, which workswhen the pest populations are either eradicated or kept at a stableand low density equilibrium or low density cycle. Luck (1990)raised a paradox for the biological control of pests by naturalenemies, when a classical predator–prey model was used to solvethe biological control problem. And Luck mentioned thatcontrolled pests do not remain in a strict equilibrium; they oftenappear to fluctuate because of local extinctions. An interestingproblem is the biological control paradox. The essential paradox isthat if one models a predator-pest system, via the Holling type IIfunctional response, one cannot obtain a stable coexistenceequilibrium, where the pest density is low (Deng et al., 2007).However, in reality many predators introduced for biologicalcontrol purposes, are able to keep pest densities down to lowlevels.

Li and Kuang (2007) studied the dynamics of the Michaelis–Menten ratio-dependent predator–prey system, i.e., model (10)without Allee effect takes the form of

du

dt¼ auð1�uÞ� uv

uþ v;

dvdt¼ cuv

uþ v�dv:

8><>: (11)

[(Fig._2)TD$FIG]

Fig. 2. Phase portraits of model (10): If a = 4.17, b = 0.2, c = 0.3, d = 0.1. E1 = (0.2, 0) is a nodal source and E2 = (1, 0) is a saddle point, E1 ¼ ð0:58869;1:1774Þ is a saddle point and

E2 ¼ ð0:61131;1:2226Þ is a spiral source; If a = 4.2, b = 0.2, c = 0.3, d = 0.1. E1 = (0.2, 0) is a nodal source and E2 = (1, 0) is a saddle point, E1 ¼ ð0:56437;1:1287Þ is a saddle point

and E2 ¼ ð0:63563;1:2713Þ is a spiral sink.


They obtained that the ratio-dependent predator–prey system(11) does not produce the so called ‘‘paradox of enrichment’’
(Rosenzweig, 1971) and the ‘‘paradox of biological control’’ (Arditiand Berryman, 1991). However, in Sen et al. (2012), Sen et al. bothconsidered the case of a strong Allee effect and the case of a weakAllee effect in the prey growth rate of a ratio-dependent predator–prey system, which are described by the following system of ODEs
dn

dt¼ uð1�uÞðn�bÞ� anp

nþ p;

dp

dt¼ a1np

nþ p�dp:

8>><>>: (12)

By clearly analyzing and demonstrating, they obtained that theratio-dependent model with the Allee effect (either strong orweak) does not exhibit ‘the paradox of enrichment’, but does resultin ‘the paradox of biological control’, stating that a stablecoexistence of prey and predator is impossible at prey densitiesmuch smaller than the value of the carrying capacity. Thisconclusion is confirmed by our ODE version model. For Model(3), the constant steady state solutions of (3) are the same as itsODE version, i.e., model (10) (see model (12) in Sen et al. (2012) andthe references therein).

In the following subsections, we will investigate the localstability of the trivial constant solutions (i.e., E2) of reaction–diffusion version model (3), and explore how diffusion coefficientDi, i = 1, 2 affect the local stability by comparing to its ODE model(10).

4.2. Local stability of the interior equilibrium E2 and bifurcations

Let us denote w ¼ ðu; vÞT , D = diag(D1, D2),

FðwÞ ¼ ðauðu�bÞð1�uÞ� uv

uþ v;

cuv

uþ v�dvÞ

T

;

then model (3) can be rewritten as:

@w

@t�DDw ¼ FðwÞ; ðx; yÞ 2V; t>0;

@w

@n¼ 0; ðx; yÞ 2@V; t>0;

wðx; y;0Þ ¼ ðu0ðx; yÞ; v0ðx; yÞÞT ; ðx; yÞ 2V:

8>>>><>>>>:

(13)

Let 0 = m0<m1 <m2 < � � � be the eigenvalues of the operator�Din V with the zero-flux boundary condition, S(mi) be the
eigenspace corresponding to mi in C1ðVÞ, {fijjj = 1, 2, 3, . . .,dimS(mi)} be an orthonormal basis of S(mi), and Xij = {cfijjc 2 R2} (cis a constant vector).
Define

X ¼ w2 ½C2ðVÞ \C1ðVÞ�2j @w

@n¼ 0 on @V

� �; (14)

then X ¼ oplus1i¼1Xi, where Xi ¼ oplusdimSðmiÞj¼1 Xij. The linearization of

model (3) at a constant solution E2 ¼ ðu2; v2Þ can be expressed by

wt ¼ LðwÞ�DDwþ Jw;

where J is defined as (32).For each i � 1, Xi is invariant under the operator L, and l is an

eigenvalue of L if and only if l is an eigenvalue of the matrixJ i ¼ �miDþ J. The characteristic equation of J i is given by

’iðlÞ ¼ l2�trðJ iÞlþ detðJ iÞ ¼ 0;

where

trðJ iÞ ¼ �miðD1 þ D2Þ þ trðJÞ;detðJ iÞ ¼ D1D2m

2i �½ðau2ð1þ b�2u2Þ

þ u2v2ðu2 þ v2Þ

2ÞD2�

cu2v2ðu2 þ v2Þ

2D1�mi þ detðJÞ;

with

trðJÞ¼au2ð1þ b�2u2Þþu2v

2ð1�cÞ

ðu2 þ v2Þ2; detðJÞ¼

acðu2Þ2v2 2u2�ð1þ bÞ� �

ðu2 þ v2Þ2

:

(15)

The interior equilibrium E2 of the ODE model (10) is locallystable if conditions in Part (3) of Theorem 4.2 hold. We areinterested in how the diffusion may affect the local stability of E2.First, we have the following theorem regarding the local stability ofthe nontrivial constant steady state E2 of the PDE model (3).

Theorem 4.3. Assume that conditions in Part (3) of Theorem 4.2 hold.

For model (3), the positive constant steady state E2 ¼ ðu2; v2Þ is locally

asymptotically stable if one of the following inequalities holds:

� d
ðc�dÞc < au2

a

q or


� 0
<dðc�dÞ�ac2u
2


ac

pcdðc�dÞ <

D1D2<1 or

D
� D 1
2>1or !" #
� dðc�dÞ
cþacu2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðb�1Þ2�4ðc�dÞ

ac

r� dðc�dÞD1

D2

2

<4dðc�dÞau2


ac

r:

(16)

Remark 5. Theorem 4.3 implies that whenever the diffusion coef-ficient of predator is smaller than the diffusion rate of prey, or thediffusion coefficient of predator is not too large along with otherconditions, the PDE model (3) reserves the stable stability of E2. Inparticular, the inequality (16) will reverse its direction if the ratioof the prey’s diffusion rate D1 to the predator ’s diffusion rate D2,i.e., D1

D2is too large or too small. In the case that the ratio D1

D2is too

small, then we have

dðc�dÞc

þ acu2


ac

r !� dðc�dÞD1

D2

" #2

dðc�dÞc

þ acu2


ac

r !" #2

�4dðc�dÞau2


ac

r:

Hopf bifurcation and Turing bifurcation are two basic types ofsymmetry-breaking bifurcations for the reaction–diffusion mod-els. They are responsible for the emergence of spatiotemporalpatterns. We will investigate these two bifurcations around theconstant interior equilibrium E2 for our PDE model (3).

The space-independent Hopf bifurcation breaks the temporalsymmetry of a model and gives rise to oscillations that are uniformin space and periodic in time. We derive the condition of Hopfbifurcation for model (3) at the positive constant steady stateE2 ¼ ðu2; v2Þ, where the coexisting homogeneous steady state losesstability through Hopf bifurcation. See the following theorem.

Theorem 4.4. For model (3), the constant steady state E2 ¼ ðu2; v2Þcan lose stability through Hopf bifurcation at the critical value of a,

where

a ¼ aH

¼

ðc�dÞ cd�c�dþ 4bcðb�1Þ2

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficd�c�dþ 4bc

ðb�1Þ2

� �2

þ 4ðcd�2c�dÞ2

bðb�1Þ2

s24

35

2bc2:

In general, E2 can lose stability through Hopf bifurcation at mode k

where mk is the first one that satisfies the following equality for the

given a:

�mkðD1 þ D2Þ�aððbþ 1Þ þ


ac

qÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðb�1Þ2� 4ðc�dÞ

ac

q2

þ dðc�dÞð1�cÞc2

¼ 0:

Remark 6. The Hopf-bifurcation for ODE case occurs if and only ifthere exists a critical value of a, i.e., a = aH, such that (i)½trðJðE2ÞÞ�a¼aH

¼ 0; (ii) ½detðJðE2ÞÞ�a¼aH>0; (iii) when E2 ¼ ðu2; v2Þ

exists, the characteristic equation is l2 þ ½detðJðE2ÞÞ�a¼aH¼ 0,

whose roots are purely imaginary; (iv) dda ½trðJðE2ÞÞ�a¼aH

6¼0. TheHopf bifurcation of the corresponding PDE model (3) occurs at the

branch k2Z when Im(l(k)) 6¼ 0, Re(l(k)) = 0 where l(k) satisfiesthe following equality:

lðkÞ2�trðJ iÞlðkÞ þ detðJ iÞ ¼ 0:

Both the condition of Hopf bifurcation of our PDE model and itsODE version are the same for k = 0. In general, the PDE model (3)can possess any periodic solution of (10) as a spatially homoge-neous periodic solution, including the ones from Hopf bifurcationin Theorem 4.4. In these cases, the diffusion can play an importantrole for the stability of these periodic solutions. However, this isnot our focus here. The related work on the Hopf bifurcation ofspatially homogeneous periodic solutions can be found in therecent work by Li et al. (2011).

The Turing bifurcation breaks spatial symmetry, leading to theformation of patterns that are stationary in time and oscillatory inspace. Turing instability refers to the diffusion-induced instability(Turing, 1952), which means the stability of a positive constantsteady-state solution E2 changing from stable for the spatiallyhomogeneous version of model (25) to unstable for model (3). Theinstability arises due to the small heterogeneous perturbationaround the homogeneous steady state. We provide the conditionsof Turing instability of the solutions to model (3) in the followingtheorem.

Theorem 4.5. Assume that conditions in Part (3) of Theorem 4.2 hold.

If the following two inequalities hold

dðc�dÞc

þ acu2


ac

r !�dðc�dÞD1

D2

" #2

>4dðc�dÞau2


ac

r;

D1

D2<

dðc�dÞ�ac2u2


ac

qcdðc�dÞ ;

and there is a wavenumber k2 2 (m�, m+) where

mþðD1;D2; aÞ ¼

trðJÞD2 þdðc�dÞðD2�D1Þ

c

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitrðJÞD2 þ

dðc�dÞðD2�D1Þc

� �2

�4D1D2detðJÞ

s

2D1D2;

m�ðD1;D2; aÞ ¼


c

�



� �2

�4D1D2detðJÞ

s

2D1D2

with

trðJÞ ¼ au2ð1þ b�2u2Þ þu2v

2ð1�cÞ

ðu2 þ v2Þ2;

detðJÞ ¼acðu2Þ

2v2 2u2�ð1þ bÞ� �

ðu2 þ v2Þ2

:

then the positive constant steady-state solution E2 ¼ ðu2; v2Þ of model

(3) is Turing unstable.

omplexity 28 (2016) 123–144

Remark 7. Theorem 4.5 implies that Turing instability more likely
to occur if the ratio of the prey’s diffusion rate D1 to the predator ’sdiffusion rate D2, i.e., D1
D2is small enough, e.g., D1

D2< < 1

c. This resultagrees with the study of predator–prey systems by Timm andOkubo (1992), which suggest that the existence of diffusive insta-bility in such systems may require a predator’s diffusivity beingsufficiently larger when compared to the prey’s. Notice that theexact expression of Turing bifurcation curves are extremely com-plex, and only there exits fixed number of k2 2 (m�, m+) whereTuring instability can occur. For demonstration, we omit the exactexpression of these coefficients and give a numerical example ofthe Turing instability curves in Fig. 3.

We fixed values of c = 0.3, d = 0.1, D1 = 0.01 for Model (3).Examples of Hopf-bifurcation and Turing instability curves of (3)for b = 0.2, 0, � 0.1 are shown in Fig. 3(b)–(d): the red curve is thevalue of aH where the Hopf-bifurcation at E2 occurs for k = 0, i.e.,when a is less than the value of aH, there is a limit cycle emerged fora < aH; the blue curve is the value of a and D2 where E2 loses itsstability of the PDE model (3), i.e., below the blue curve and theabove the red curve, the E2 becomes unstable for the PDE model (3)while it is locally stable for the ODE model (10). Therefore, abovethe Turing (blue) curve, it is stable for all pairs of (D2, a), that is,there is no diffusive instability. While below the blue curve andabove Hopf (red) curve, it is unstable for (D2, a), and diffusiveinstability emerges. And if below the red curve, the PDE system (3)has periodic solutions.[(Fig._3)TD$FIG]

F. Rao, Y. Kang / Ecological C130

Fig. 3. Bifurcation diagram for model (22) without Allee effect and model (3) with Allee

curves denote the Turing curve and Hopf curve, respectively. (a) is for Model (22) without

P4 are points corresponding to Fig. 6(a) and (b), respectively. (b): Points P1 and P2 are co

Fig. 9(a) and (b), respectively. (d): Points P1 and P2 are corresponding to Fig. 10(a) and (b)

reader is referred to the web version of the article.)

5. Non-constant positive steady states

The focus of this section is to provide sufficient conditions on thenon-existence and existence of non-constant steady states of model(8). We first discuss the non-existence of non-constant positivesteady states to model (8) by the effect of diffusions. For convenience,we write Q instead of the collective constants (a, b, c, d).

Theorem 5.1. Let m1 be the smallest positive eigenvalue of the

operator �D on V with zero-flux boundary condition, and D2 be a

fixed positive constant such that D2 >c

m1. Then there exists a positive

constant D1 ¼ D1ðQ; D2Þ such that model (8) has no positive non-

constant steady state provided that D1�D1; D2�D2.

Remark 8. The result of model (8) implies the nonexistence ofnon-constant positive solution when the diffusion coefficients arelarge. In the next theorem, we provide sufficient conditions of theexistence of non-constant positive solutions for model (8).

To show the existence of non-constant positive solutions formodel (8), we will use the priori estimates from Theorem 3.2 andthe Leray–Schauder degree theory (Nirenberg, 2001). Let w ¼ E2,X be the space defined in (14), and define

Xþ ¼ fw2Xju; v>0 on Vg:

effects in the D2-a parameter space when c = 0.3, d = 0.1, D1 = 0.01. The blue and red

Allee effect: Points P1 and P2 are corresponding to Figs. 4 and 5, respectively; and P3,

rresponding to Figs. 7 and 8, respectively. (c): Points P1 and P2 are corresponding to

, respectively. (For interpretation of the references to color in this figure legend, the


Then model (8) can be written as:

�Dw ¼ GðwÞ; w2Xþ;@nw ¼ 0; on @V;

�(17)

where

GðwÞ ¼ u

D1aðu�bÞð1�uÞ� v

uþ v

� �;

vD2

cu

uþ v�d

� �� T

:

Define a compact operator F : Xþ!Xþ by

FðwÞ :¼ ðI�DÞ�1fGðwÞ þwg;

where (I �D)�1 is the inverse operator of I �D subject to the zero-flux boundary condition. Then w is a positive solution of model(17) if and only if w satisfies

ðI�FÞw ¼ 0; in Xþ:

In order to apply the index theory, we consider the eigenvalueproblem

�ðI�FwðwÞÞC ¼ lC; C 6¼0; (18)

where C = (c1, c2)T and FwðwÞ ¼ ðI�DÞ�1ðIþ JÞ with

J ¼

1

D1au2ð1þ b�2u2Þ þ

u2v2

ðu2 þ v2Þ2

!� u2

2

D1ðu2 þ v2Þ2

cv22

D2ðu2 þ v2Þ2

� cu2v2

D2ðu2 þ v2Þ2

0BBBB@

1CCCCA :

¼ D�11 J11 D�1

1 J12

D�12 J21 D�1

2 J22

� �:

After some calculations, (18) can be rewritten as:

�ðlþ 1ÞDCþ ðlI�JÞ ¼ 0; in V;@nC ¼ 0; on @V;C 6¼0:

8<: (19)

One can see that (19) has a non-trivial solution if and only ifdet(lI + (mi + 1)�1(miI � J)) = 0 for some l � 0 and i � 0. That is, lis an eigenvalue of (18), and so (19), if and only if l is an eigenvalueof the matrix (mi + 1)�1(miI � J) for i � 0. Thus, we note thatI�FwðwÞ is invertible if and only if, for any i � 0 the matrix

Mi :¼ miI�J ¼ mi�D�11 J11 �D�1

1 J12

�D�12 J21 mi�D�1

2 J22

� �

is invertible. By a direct calculation,

detðMiÞ ¼ D�11 D�1

2 ðD1D2m2i �ðD1J22 þ D2J11Þmi

þ J11J22�J12J21Þ: (20)

For convenience, we denote

MðD1;D2;mÞ ¼ D1D2m2i �ðD1J22 þ D2J11Þmi þ J11J22�J12J21;

then M(D1, D2, m) = D1D2 det(Mi). If (D1J22 + D2J11)2 >

4D1D2(J11J22 � J12J21), then M(D1, D2, m) = 0 has two real rootsm given by:

mþðD1;D2Þ ¼D1J22 þ D2J11 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðD1J22 þ D2J11Þ

2�4D1D2ðJ11J22�J12J21Þq

2D1D2;

m�ðD1;D2Þ ¼D1J22 þ D2J11�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðD1J22 þ D2J11Þ

2�4D1D2ðJ11J22�J12J21Þq

2D1D2:

In order to compute indexðI�F ;wÞ, we need the followingresults by Pang and Wang (2003).

Lemma 5.1. Suppose M(D1, D2, mi) 6¼ 0 for all mi 2 S(m). Then

indexðI�F ;wÞ ¼ ð�1Þ%;

where

% ¼

Xm2B\ SðmÞ

mðmiÞ; if B\ SðmÞ 6¼ ;;

0; if B\ SðmÞ ¼ ;:

8<:

In particular, if M(D1, D2, m) > 0 for all m � 0, then % = 0.

By applying the results of Lemma 5.1, calculatingindexðI�F ;wÞ, and determining the range of m such that M(D1,D2, m) < 0, we have the following theorem by applying theapproach of Pang and Wang (2003):

Lemma 5.2. Assume that the conditions of Theorem 3.2 hold. Then the

following statements hold:

1. M
odel (8) has at least one non-constant solution if the inequali-
ty(21) holds, m� 2 (mi, mi+1), m+ 2 (mj, mj+1) for some 0 � i < j, andPjk¼iþ1mðmkÞ is odd.

dðc�dÞc

þ acu2


ac

r !� dðc�dÞD1

D2

" #2

>4d

�ðc�dÞau2


ac

r: (21)

dðc�dÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 4ðc�dÞq

1 dðc�dÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 4ðc�dÞq� �

2.
If
c2 > ðb�1Þ � ac ; D1 c2 �au2 ðb�1Þ � ac

2 ðmj;mjþ1Þ for some j � 1, andPj

i¼1mðmiÞ is odd, then there exists

a positive large constant D* such that model (8) has at least one

non-constant solution whenever D2 > D*.

Remark 9. Theorem 5.2 demonstrates that under certain paramet-ric restrictions diffusion terms can generate non-constant positivesteady states, especially when the predator diffusive coefficient D2

is large enough.

6. Pattern formations

In this section, we use numerical simulations to explore theimpact of diffusion and the Allee effects in prey on thespatiotemporal dynamical patterns by comparing the spatiallyextended model (3) with Allee effects and the traditionaldiffusion–reaction prey–predator model (22) without Allee effects:

@uðx; y; tÞ@t

¼ D1ð@2

u

@x2þ @2

u

@y2Þ þ auð1�uÞ� uv

uþ v;

@vðx; y; tÞ@t

¼ D2ð@2v@x2þ @2v

@y2Þ þ cuv

uþ v�dv:

8>>><>>>:

(22)

Both models are solved numerically by finite differences. Allsimulations run in a square spatial domain 256 � 256 where theexplicit scheme is used. The numerical integration is performed bymeans of the forward Euler integration, with a time step of Dt ¼ 1

100

and a space step of Dx ¼ Dy ¼ 13. We use the standard five-point

approximation for the two-dimensional Laplacian with the zero-flux boundary conditions (Garvie, 2007). In the simulations, weobserved that the distributions of prey and predator are always ofthe same type. Thus, we show our analysis of pattern formations of


the distribution of prey u in this paper where we take a series ofsnapshots with red (blue) corresponding to the high (low) value ofu.

Generally, the invasion of an alien species starts with a localintroduction of exotic species; thus, relevant initial conditions forModel (3) and (22) should be described by functions with acompact support where the density of one or both species at theinitial moment of time is non-zero and is only inside a certaindomain (Wang et al., 2011). According to the work (Garvie, 2007;Medvinsky et al., 2002), the choice of initial conditions can affectthe spatiotemporal dynamics of a reaction–diffusion model inecosystems. In this section, we choose the following forms of theinitial conditions for Model (3) and (22) to investigate theevolution of the spatial pattern of prey u at a different evolutiontime in a square:

Model ð3Þuðx; y;0Þ ¼ u2; vðx; y;0Þ ¼ v2; 90< x; y<110; (23)

Model ð22Þuðx; y;0Þ ¼ u; vðx; y;0Þ ¼ v; 90< x; y<110; (24)

where ðu2; v2Þ ¼ðbþ1Þ


ac

p2 ; c�d

d u2

� �and ðu; vÞ ¼

ac�cþdac ; ðc�dÞu

d

� �.

Initially, the entire system is placed in the stationary stateðu2; v2Þ for Model (3) with Allee effects and the unique interiorequilibrium ðu; vÞ for Model (22) without Allee effects. Thepropagation velocity of the initial random perturbations is of theorder of 5 � 10�4 space units per time unit. Then the system isintegrated for 105 time steps with a series of images saved. We fixthe values for the following set of parameters:

c ¼ 0:3; d ¼ 0:1; D1 ¼ 0:01:

The intrinsic growth rate of prey a and the diffusion of predatorD2 are chosen as the controlling parameter to investigate theireffects on the regimes of biological invasion in Model (22). And thediffusion of predator D2 is chosen as the controlling parameter toinvestigate its effect for Model (3).
[(Fig._4)TD$FIG]
Fig. 4. The processes of pattern formations of prey u in model (22) with (D2, a) = (0.6, 0.8)

(e) t = 1010 and (f) t = 3000.

In Fig. 3, we provide the bifurcation diagram and the relatedpoints where we perform simulations for Model (3) and (22). TheD2�a parameter space in Fig. 3 is separated by the Turinginstability curve (blue) and Hopf-bifurcation curve (red) whenc = 0.3, d = 0.1, D1 = 0.01. Fig. 3(a) is for Model (22) without Alleeeffect: Points P1 and P2 are corresponding to Figs. 4 and 5,respectively; and P3, P4 are points corresponding to Fig. 6(a) and(b), respectively. Fig. 3(b): Points P1 and P2 are corresponding toFigs. 7 and 8, respectively. Fig. 3(c): Points P1 and P2 arecorresponding to Fig. 9(a) and (b), respectively. Fig. 3(d): PointsP1 and P2 are corresponding to Fig. 10 (a) and (b), respectively.

6.1. Without Allee effects

We first consider the case when there is no Allee effect in prey,i.e., Model (22). We are interested in how the synergistic effects ofthe intrinsic growth rate in prey a and the diffusion D2 affectpattern formations of the PDE model (22) by varying the values of a

and D2 and fixing c = 0.3, d = 0.1, D1 = 0.01. Initially, the entiresystem is placed in the stationary state ðu; vÞ ¼ ðac�cþd

ac ; ðc�dÞud Þ

and the propagation velocity of the initial random perturbations isof the order of 5 � 10�4 space units per time unit:

1. I

an

n Fig. 4, we show the evolution of the spatial pattern of preypopulation u by taking the value of parameters D2 = 0.6 anda = 0.8 at a different evolution time. From Fig. 4(a)–(b), one cansee that a target wave pattern emerges after perturbation of thestationary solution u* and v of model (22). Some iterations laterit breaks from the corners and wave patterns occur (c.f., Fig. 4(c)and (d)). Then, it grows slightly and the wave increases withtime (c.f., Fig. 4(e)). When the iteration time is large enough, weobserve that the time-independent striped-like patterns settledown and staying stable (c.f., Fig. 4(f)). It finally prevails over thewhole domain at t = 3000.

2. W
hen the parameters are taken as (D2, a) = (0.6, 0.86), Fig. 5shows snapshots of the spatial distribution of model (22) at timet = 0, 200, 2450, 3050, 5000 and 9000. After perturbing thestationary solution u* and v, a spot pattern breaks towards the
d c = 0.3, d = 0.1, D1 = 0.01. Time steps: (a) t = 0, (b) t = 160, (c) t = 290, (d) t = 500,

[(Fig._6)TD$FIG]

Fig. 6. Snapshots of contour pictures of the time evolution of prey u in model (22) at t = 3000 and c = 0.3, d = 0.1, D1 = 0.01. (a) (D2, a) = (1.2, 0.8) and (b) (D2, a) = (1.2, 0.86).

[(Fig._5)TD$FIG]

Fig. 5. The processes of pattern formation of prey u in model (22) with (D2, a) = (0.6, 0.86) and c = 0.3, d = 0.1, D1 = 0.01. Time steps: (a) t = 0, (b) t = 200, (c) t = 2450, (d) t = 3050,

(e) t = 5000 and (f) t = 9000.


core (c.f., Fig. 5(b)). As time increasing, the time-independentregular spots pattern takes a while to settle down and dominatesthe domain eventually (c.f., Fig. 5(f)).

3. W
hen increasing D2 = 0.6 to D2 = 1.2, Fig. 6 show two typicalpatterns for model (22) at t = 3000 with a = 0.8, and a = 0.86,respectively. In the case of (D2, a) = (1.2, 0.8), a symmetrical stripepattern appears (c.f., Fig. 6(a)), and the dynamics of the model donot undergo any further changes. While increasing a = 0.8 toa = 0.86, i.e., (D2, a) = (1.2, 0.86), the spots-target wave mixturespattern emerges and dominates the domain, see Fig. 6(b).
6.2. With Allee effects

In this subsection, we explore the spatiotemporal dynamics ofmodel (3) with an Allee effect (weak or strong Allee effect). Oursimulations show that the dynamical patterns are sensitive withrespect to the values of a, b, c, i.e., we can observe a large variety ofdistinct patterns by making small changes in their values. We areinterested in how diffusion affects pattern formations of Model (3)

with Allee effects for fixed deterministic parameters and its extentto which diffusion is capable of changing the patterns exhibited bythe model. We vary the values of parameters a, D2, and fix the otherparameters as c = 0.3, d = 0.1, D1 = 0.01:

1. C
ase (I) 0 < b = 0.2. In this case, the ODE model (10) has as strongAllee effect in prey with a locally asymptotically stable interiorequilibrium.
Choose (D2, a) = (0.6, 4.2). Fig. 7 shows that the spontaneousformation of complex pattern structure emerges: one can seethat for model (3), the random initial distribution around thesteady state ðu2; v2Þ leads to the formation of a strongly regulartransient pattern in the domain (c.f., Fig. 7(c)).

After the regular pattern forms (c.f., Fig. 7(d) and (e)), thedynamics of the model do not undergo any further changes (c.f.,Fig. 7(f)).

As D2 is increased from 0.6 to 1.2, we found that thetransition of different type patterns of the prey population intwo dimensional space will emerge. In Fig. 8, we show thesnapshots of contour pictures of the time evolution of prey

[(Fig._8)TD$FIG]

Fig. 8. The processes of pattern formations of prey u in model (3) with (D2, a) = (1.2, 4.2) and b = 0.2, c = 0.3, d = 0.1, D1 = 0.01. Time steps: (a) t = 0, (b) t = 200, (c) t = 600, (d)

t = 700, (e) t = 1050 and (f) t = 3000.

[(Fig._7)TD$FIG]

Fig. 7. The processes of pattern formations of prey u in model (3) with (D2, a) = (0.6, 4.2) and b = 0.2, c = 0.3, d = 0.1, D1 = 0.01. Time steps: (a) t = 0, (b) t = 200, (c) t = 800, (d)

t = 950, (e) t = 1600 and (f) t = 3000.


u with (D2, a) = (1.2, 4.2). Although the dynamics of the modelpreceding the formation of spatial structure is somewhatregular like the previous case, it follows a different scenario.

2. C
ase (II) b = 0. Fig. 9 shows two typical patterns for model (3) att = 3000 with different values of a and D2. In Fig. 9(a), (D2,a) = (0.6, 2.703), and Fig. 9(b), (D2, a) = (1.2, 2.703).
3. C
ase (III) b =�0.1 < 0 (weak Allee effects) In Fig. 10, we showtwo typical patterns for model (3) at t = 3000 with differentvalues of a and D2. In the case of (D2, a) = (0.6, 2.25), asymmetrical block pattern appears and dominates the domain
(c.f., Fig. 10(a)). While increasing D2 = 0.6 to D2 = 1.2, i.e., (D2,a) = (1.2, 2.25), the cyclic pattern emerges and the dynamics ofthe model do not undergo any further changes, see Fig. 10(b).

Summary on pattern formations: In this section, we usenumerical simulations to explore how Allee effects, the diffusionrate, and the growth rate of the prey affect pattern formations.Our simulations show that the PDE model (22) without Alleeeffects exhibits complicated pattern formation such as striped-like patterns, spots patterns and spots-target wave mixtures

[(Fig._10)TD$FIG]

Fig. 10. Snapshots of contour pictures of the time evolution of prey u in model (3) at t = 3000 and b =�0.1, c = 0.3, d = 0.1, D1 = 0.01. (a) (D2, a) = (0.6, 2.25) and (b) (D2, a) = (1.2,

2.25).

[(Fig._9)TD$FIG]

Fig. 9. Snapshots of contour pictures of the time evolution of prey u in model (3) at t = 3000 and b = 0, c = 0.3, d = 0.1, D1 = 0.01. (a) (D2, a) = (0.6, 2.703) and (b) (D2, a) = (1.2,

2.703).


patterns. More specifically, we have the following two observa-tions:

1. T
he effects of the prey growth rate a: By comparing (1) Figs. 4(f)–5(f) and (2) Fig. 6(a) and (b), it seems that increasing the value ofa could generate more regular patterns.
2. T
he effects of the diffusion rate of predator D2: By comparing (1)Figs. 4(f)–6(a) and (2) Figs. 5(f)–6(a), it seems that increasing thevalue of D2 could generate mixture patterns (e.g., strippedpatterns mixing with spots patterns).
The PDE model (3) with Allee effects in prey also exhibits richand distinct pattern formations when we vary the key parameters.More specifically, we have the following observations:

1. T
he effects of the prey growth rate a: By comparing (1) Figs. 7(f)–8(f), it seems that increasing the value of a could generatesimilar but more smooth patterns.
2. T
he effects of the diffusion rate of predator D2: By comparing (1)Fig. 9(a) and (b) and (2) Fig. 10(a) and (b), it seems thatincreasing the value of D2 could generate more regular andsmooth patterns.
Our simulations show that our PDE model (3) with Allee effectsin prey exhibits rich and distinct pattern formations than the PDEmodel (22) without Allee effects. In addition, the strength of Allee

effects, i.e., the value of b, also plays important role in shapingvaried dynamical patterns: In both positive and negative Alleethreshold b cases, our PDE model (3) exhibits the different spatialpatterns when the values of a and D2 varied. But in the zero Alleethreshold b case, we find that under different values of D2 thepatterns seem to share many similarities. These numerical findingsmay enrich the research of the pattern formations arising from theinterplay between the Allee effects and diffusion rates inMichaelis–Menten predator–prey model. And we could infer thatthe impact of the Allee effects and diffusion significantly increasesthe model complexity.

7. Conclusion

In this paper, we investigate the dynamical complexity of areaction diffusion Michaelis–Menten type predator–prey interac-tion model with an Allee effect in prey with the zero-flux boundaryconditions. Our study is distinct from others (Hilker et al., 2007;Sen et al., 2012; Aguirre et al., 2014; Flores and Gonzalez-Olivares,2014) as our model incorporates: (a) An Allee effect in prey; (b) TheMichaelis–Menten type functional response between prey andpredator; and (c) Diffusion in both prey and predator. Our workfocuses on the case when the densities of prey and predator arespatially inhomogeneous in a bounded domain subject to the zero-flux boundary condition. Our results provide useful insights in therich spatial-temporal dynamics of the proposed model throughrigorous analysis and carefully designed numerical simulations.


The values of this study have many-fold in both aspects ofmathematics and biology. Our analytical results partially provideanswers to the two questions proposed in the introduction: Whatare the effects of diffusion and Allee effects for the species on thespatiotemporal dynamics of our model? How our proposed modelmay generate distinct spatial patterns?

1. W
e first obtain the basic behavior of time dependent solutions ofthe model. In Theorem 3.1(iii), we show that the predator ofmodel (3) is destined to go extinct when the death rate ofpredator is too large, i.e. d > c. We provide a priori estimates forthe positive solutions that will be used for results on the non-existence and existence of non-constant positive solution of themodel (see Theorem 3.2).
2. I
n Theorem 4.1, we show that for large predator initial values,both species go extinct. This indicates the importance of theinitial condition in the dynamical outcomes of Model (3). Wefurther provide the completed analysis of the ODE version of ourmodel (see Theorem 4.2). Correspondingly, for all possiblehomogeneous steady states involved with model (3), we haveobtained the local stability conditions dependent upon theparametric restrictions (c.f., Theorem 4.3). Our results show thatthe coexistence steady state loses stability through Hopf andTuring bifurcation (c.f., Theorems 4.4 and 4.5) when the predatordiffusive coefficient is large while the prey diffusive coefficientis small.
3. W
e derive results on the non-existence and existence of non-constant positive solution of the model. Theorem 5.1 impliesthat the nonexistence of non-constant positive solution whenthe diffusion coefficients are large. Existence of non-constantsteady state depends upon certain parametric restrictionsinvolving intrinsic rate constants and diffusivity of both species(c.f., Theorem 5.2).
We illustrate spatial patterns of our proposed model when theparameter values are close to the onset of instability bifurcation vianumerical simulations. We observe that the trajectories of ourmodel are sensitive to disturbance of both initial conditions andparameters. The increase of the predator diffusion intensity has adrastic impact on the diffusion pattern formations of the specieswith or without the Allee effect. By fixing the values of b, c, d and D1

in the model, we perform a large number of computer simulationswith different values of the intrinsic prey growth rate a and thepredator diffusive coefficient D2. Our results show that the modelwith or without Allee effect has complicated spatiotemporalpatterns; the Allee effect strength b plays import roles in shapingvaried pattern formations; and different patterns appear when thediffusion coefficient D2 varies. In addition, some of our mainfindings can be summarized as follows:

1. D
iffusion and Allee effect can induce Turing instability: FromTheorem 4.5, we know that the positive constant steady stateE2 ¼ ðu2; v2Þ of model (3) can be Turing unstable due to the Alleeeffects and the diffusion rates. More specifically, under theproper values of the Allee effect strength b, Turing instability ismore likely to occur when the intrinsic growth rate of prey isrelatively small, the predator diffusive coefficient is large andthe prey diffusive coefficient is small. The proper diffusion rateof prey and predator can de-stabilize the symmetric solutions sothat the model with diffusion can form the stationary Turingpattern.
2. M
odel (3) exhibits complex pattern formations, and distinctpattern formations than the PDE model (22) without Alleeeffects. The different patterns shown in Figs. 4–6 (no Alleeeffects), and Figs. 7–10 (with Allee effects) illustrate the effectsof Allee effects in pattern formations. We also observe that the
predator diffusion rate can control spatial dynamical patterns:by varying the predation diffusion rate, the spatial pattern canbe stripes (Fig. 4), spots (Fig. 5), spots-target wave mixturespattern (Fig. 6(b)), and regular patterns showed in Figs. 7–10.

Comparison with the literature work: McLellan et al. (2010)reevaluated the potential for predator-mediated Allee effects in amulti-prey system using Hollings disc equation. In addition, theyused empirical data on a large herbivore to examine how groupingbehavior may influence the potential for predation-mediated Alleeeffects. Their work predicts that Allee effects caused by predationon relatively rare secondary prey may not occur because handlingtime of the abundant prey dominates the functional response suchthat secondary prey are largely ‘‘bycatch’’. However, a predator-mediated Allee effect can occur if secondary prey live in groups andif, as the population declines, their average group size declines (arelationship seen in several species) Wang et al. (2011) studiedglobal bifurcation analysis of a general predator–prey ODE modelwith strong Allee effects in prey population. They showed theexistence of a point-to-point heteroclinic orbit loop, considered theHopf bifurcation, and proved the existence/uniqueness and thenonexistence of limit cycle for appropriate ranges of parameters.We have applied their results to our corresponding ODE model (10)through the work of (Kang and Castillo-Chavez, 2014).

Wang et al. (2011) investigated the dynamics of a reaction–diffusion predation model with strong Allee effects in prey andthe Holling type II functional response between prey and predator(i.e., uv

uþa) which takes the following form:

@u

@t¼ d1Duþ uð1�uÞðu

b�1Þ� muv

aþ u;

@v@t¼ d2Dv�dvþ muv

aþ u:

8>><>>:

9>>=>>;

with b > 0, i.e., the model of Wang et al. (2011) focuses on the preywith strong Allee effects. They obtained interesting mathematicalanalysis results on the global existence of solutions, the extinctionof both population, a prior bounds of the dynamic and steady statesolutions, the existence of nonconstant steady states and time-periodic orbits.

In our work, we study a reaction–diffusion predator–preymodel with the Michaelis–Menten ratio-dependent functionalresponse (i.e., uv

uþv ¼ uu=vþ1) whose per capita predator growth rate

depends on the ratio of prey to predator abundances. In addition,our model allows prey population has Allee effects being weak orstrong depending on the sign of b, see our scaled model below

@u

@t¼ D1Duþ auð1�uÞðu�bÞ� uv

uþ v;

@v@t¼ D2Dv�dvþ cuv

uþ v:

8>><>>:Our assumption on the Michaelis–Menten ratio-dependentfunctional response is following the work of Akcakaya et al.(1995) which focused on the ratio-dependent predation work.They showed that most natural systems are closer to the ratio-dependent functional response than to the prey dependence (i.e.,the Holling type-II functional response), and the ratio-dependentmodels are capable of producing richer and more reasonabledynamics biologically.

In our paper, we study that the complex dynamics of theMichaelis–Menten type reaction–diffusion predation model withan Allee effect in prey. Our work focuses on the case when thedensities of prey and predator are spatially inhomogeneous in abounded domain subject to the zero-flux boundary condition. Ourstudy provides useful insights in the rich spatial-temporal


dynamics of the proposed model through rigorous analysis andcarefully designed numerical simulations. We would like to pointout that some of our theoretical results using the similarapproaches of Wang et al. (2011), however, we also give theboundary condition of Turing bifurcation; determine the Turingspace; and investigate the pattern formations controlled bydiffusion and Allee effects in the Michaelis–Menten type predationmodel through novel numerical simulations. The numerical resultsindicate that the Michaelis–Menten type predation model withdiffusion and Allee effects exhibits rich dynamical transitions fromthe wave growth not only to striped-like/spots patterns, but also toregular patterns. This implies that the synergistic effects ofdiffusion and Allee effects can contribute to the patterns of thesolutions in different ways.

8. Proofs

Proof of Theorem 3.1

Proof. (1) Define

Fðu; vÞ ¼ auðu�bÞð1�uÞ� uv

uþ v; Gðu; vÞ ¼ cuv

uþ v�dv;

then Fv�0; Gu�0 in R2þnfð0;0Þg. Let ðuðx; y; tÞ;v ðx; y; tÞÞ ¼ ð0;0Þ and

ðuðx; y; tÞ; vðx; y; tÞÞ ¼ ðuðtÞ; vðtÞÞ, where ðuðtÞ; vðtÞÞ is a constantsolution of the following ODE model:

du

dt¼ auðu�bÞð1�uÞ;

dvdt¼ cuv

uþ v�dv;

uð0Þ ¼ u ¼ supV

u0ðx; yÞ and vð0Þ ¼ v ¼ supV

v0ðx; yÞ:

8>>>>><>>>>>:

(25)

Then ðuðx; y; tÞ;v ðx; y; tÞÞ ¼ ð0;0Þ and ðuðx; y; tÞ; vðx; y; tÞÞ ¼ðuðtÞ; vðtÞÞ are the lower-solution and upper-solution to (3),respectively, since

@u

@t�Duðx; y; tÞ�Fðuðx; y; tÞ;v ðx; y; tÞÞ�

@u

@t�Duðx; y; tÞ

�Fðuðx; y; tÞ; vðx; y; tÞÞ;

and

@v@t�Dvðx; y; tÞ�Gðuðx; y; tÞ; vðx; y; tÞÞ� @v

@t�Dvðx; y; tÞ�Gðuðx; y; tÞ;v ðx; y; tÞÞ;

the boundary conditions are satisfied, and 0 � u0(x, y) � u* and0�v0ðx; yÞ�v. Using the definition of lower/upper-solution inPao (1992) and Ye and Li (1994), it shows that model (3) has aunique globally defined solution ðuðx; y; tÞ; vðx; y; tÞÞ whichsatisfies

0�uðx; y; tÞ�uðtÞ; 0�vðx; y; tÞ�vðtÞ; t�0:

The strong maximum principle implies that uðx; y; tÞ; vðx; y; tÞ>0

when t > 0 for all ðx; yÞ 2V. Moreover if u0(x, y)� u* < b, thenapparently u*(t)! 0 and consequently vðtÞ!0 as t!1. Thiscompletes the proof of parts (i) and (ii).

(2) From the above proof, we obtain that u(x, y, t) � u*(t) for allt > 0. From model (25) satisfied by u*(t), one can see that u*(t)! 0if u* < b and u*(t)! 1 if u* > b. Thus for any e > 0, there existsT0 > 0 such that u(x, y, t) � 1 + e in V�½T0;1Þ.

If d > c, we choose e > 0 such that d > c(1 + e), then for t > T, u(x,y, t) � 1 + e. We use the comparison argument above again withu(T) = u* � 1 + e. Then the equation of vðtÞ implies that0�vðx; y; tÞ�vðtÞ!0 as t!1 uniformly for ðx; yÞ 2V. The limit

behavior of the equation of u(x, y, t) is determined by the semiflowgenerated by the scalar parabolic equation

ut ¼ D1Duþ auðu�bÞð1�uÞ; ðx; yÞ 2V; t>0;@u

@n¼ 0; ðx; yÞ 2@V:

8<: (26)

It is known that (26) is a gradient system, and every orbit of (26)converges to a steady state ue. From the theory of asymptoticallyautonomous dynamical systems, the solution ðuðx; y; tÞ; vðx; y; tÞÞ of(3) converges to (ue, 0) as t!1. This proves part (iii).

(3) Since both u(x, y, t) and vðx; y; tÞ are positive, vðx; y; tÞ satisfies

vt ¼ D2Dvþ cuv

uþ v�dv�D2Dvþ cu�dv:

By using limsupt!1 uðx; y; tÞ�1 proved above, then, for any e > 0,there exists a large T > 0 such that

vt�D2Dvþ cð1þ eÞ�dv for ðx; y; tÞ 2V�ðT;1Þ:

Therefore, the comparison theorem yields limsupt!1 vðx; y; tÞ�cð1þ eÞ=d for (x, y) 2V. This estimate implies

limsupt!1

vðx; y; tÞ� c

dfor ðx; yÞ 2V:

(4) If D1 = D2, we can add the two equations in (3) and obtain

wt ¼ D1Dwþ auðu�bÞð1�uÞ� ð1�cÞuv

uþ v�dv; ðx; yÞ 2V; t> T;

@w

@n¼ 0; ðx; yÞ 2@V; t> T;

wðx; y; TÞ ¼ uðx; y; TÞ þ vðx; y; TÞ; ðx; yÞ 2V;

8>>><>>>:where wðx; y; tÞ ¼ uðx; y; tÞ þ vðx; y; tÞ. When t > T, u(x, y, t) � 1 + e,then we get

auðu�bÞð1�uÞ� ð1�cÞuv

uþ v�dv ¼ auðu�bÞð1�uÞ� ð1�cÞuv

uþ v

þ du�dw� að1�bÞ2

4þ d

!u�dw;� að1�bÞ2

4þ d

!ð1þ eÞ�dw;

and for the equation

@f@t¼ D1Dfþ ð að1�bÞ2

4þdÞð1þ eÞ�df; ðx; yÞ 2V; t> T;

@f@n¼ 0; ðx; yÞ 2@V; t> T;

8>><>>:

(27)

the solution fðx; y; tÞ! ðað1�bÞ24d þ 1Þð1þ eÞ as t!1, then the

comparison argument shows that

limsupt!1

vðx; y; tÞ� limsupt!1

wðx; y; tÞ�ð að1�bÞ2

4dþ 1Þð1þ eÞ;

which implies the last part of (iv). &


Proof. Let ðuðx; yÞ; vðx; yÞÞ be a non-negative solution of (8). If thereexists ðx0; y0Þ 2V such that vðx0; y0Þ ¼ 0, then vðx; yÞ�0 from themaximum principle and u(x, y) satisfies

D1Duþ auðu�bÞð1�uÞ ¼ 0; ðx; yÞ 2V;@u

@n¼ 0; ðx; yÞ 2@V:

8<: (28)

Similarly if u(x0, y0) = 0 for some (x0, y0) 2V, we also have u(x,y) � 0 which implies v�0. Otherwise u(x, y) > 0 and vðx; yÞ>0 forðx; yÞ 2V.


From the maximum principle, u(x, y) < 1 for all ðx; yÞ 2V. Byadding the two equations in (8) with parameter k � 0, if 0�k� 1

c, weobtain

�ðD1Duþ kD2DvÞ¼auðu�bÞð1�uÞ� ð1�kcÞuv

uþ v�kdv�auðu�bÞð1�uÞ

�kdv ¼ u aðu�bÞð1�uÞ þ dD1

D2

� �� d

D2ðD1uþ kD2vÞ

� að1�bÞ2

4þ dD1

D2

!� d

D2ðD1uþ kD2vÞ:

Then,

D1uþ kD2v<að1�bÞ2D2

4dþ D1:

And if choosing k ¼ 1c, we can get the desired estimate. &


Proof. For a fixed e > 0, there exists T1 > 0 such that u(x, y,t) � 1 + e and vðx; y; tÞ� c

d for t > T1 from Theorem 3.1 (iv). Thereforeu(x, y, t) satisfies

ut ¼ D1Duþ auðu�bÞð1�uÞ� uv

uþ v; ðx; yÞ 2V; t> T1;

uðx; y; T1Þ�1þ e:

((29)

Let v1ðx; y; tÞ be the solution to

vt ¼ D2Dv�dv; ðx; yÞ 2V; t>0;@v@n¼ 0; ðx; yÞ 2 @V; t>0;

vðx; y;0Þ ¼ v0ðx; yÞ; ðx; yÞ 2V:

8>><>>: (30)

From the comparison principle of parabolic equation, thenvðx; y; tÞ� v1ðx; y; tÞ for any t > 0. Moreover, if v0ðx; yÞ� v0, thenvðx; y; tÞ� v0e�dðT1þT2Þ when t 2 [0, T1 + T2] for some T2 > 0.

Since aðu�bÞð1�uÞ� að1�bÞ24 ¼: M1 for all u � 0, and uþ v�1þ

eþ cd for t > T1, then u(x, y, t) satisfies that

ut�D1Duþ M1�dv0e�dðT1þT2Þ

dð1þ eÞ þ c

!u;

uðx; y; T1Þ�1þ e:

8><>: (31)

for (x, y) 2V, T1 < t < T1 + T2. Thus the comparison principle showsthat for t 2 [T1, T1 + T2] and ðx; yÞ 2V,

uðx; y; tÞ�ð1þ eÞexp M1�dv0e�dðT1þT2Þ

dð1þ eÞ þ c

!ðt�T1Þ

!:

If choosing

v0�2e2dT1 M1 1þ eþ c

d

� � 1þ eb

� � dM1

;

and

T2� T1 þ1

M1ln

1þ eb

;

then

M1�dv0e�dðT1þT2Þ

dð1þ eÞ þ c��M1;

and for any ðx; yÞ 2V,

uðx; y; T1 þ T2Þ�ð1þ eÞexpððM1�dv0e�dðT1þT2Þ

dð1þ eÞ þ cÞðT2�T1ÞÞ< b:

Therefore, ðuðx; y; tÞ; vðx; y; tÞÞ tends to (0, 0) for as t!1 fromTheorem 3.1 (ii). Since e is chosen arbitrarily, then v0 depends onlyon the fixed parameters and T1 which depends on u0(x, y). &


Proof. First of all, we can easily calculate that model (25) has onlythree boundary equilibria: E0 = (0, 0), E1 = (b, 0), E2 = (1, 0) if one ofthe following two conditions hold: (1) c � d or, (2)c> d and a< 4ðc�dÞ

cðb�1Þ2. By evaluating the Jacobian matrix of model

(25) at these three boundary equilibria, we have the following twocases:

� If
c < d, then both E0 and E2 are locally asymptotically stablewhile E1 is a saddle. Since model (25) is a two dimensional ODE,then it has E0 and E2 as its global attractor according to Poincare–Bendison theorem (Walter, 1998). � If c> d and a< 4ðc�dÞ
cðb�1Þ2, then E0 is the only locally asymptotically

stable equilibrium while E1 is a source and E2 is a saddle. Thus,according to Poincare-Bendison theorem (Walter, 1998), model(25) has E0 as its global attractor.

Now let us focus on the case when c> d and a> 4ðc�dÞcðb�1Þ2

. Underthis condition, model (25) has E0 = (0, 0), E1 = (b, 0), E2 = (1, 0) as itsboundary equilibria where E0 is always locally asymptoticallystable, E1 is a source and E2 is a saddle. Model (25) has two interior

equilibria Ei ¼ ðui ; vi Þ; i ¼ 1;2 where u1 <u2 and

ui ¼ðbþ 1Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðbþ 1Þ2� 4ðabcþc�dÞ

ac

q2

¼ðbþ 1Þ


ac

q2

;

vi ¼c�d

dui :

The expression of Ei indicates that b<u1 <bþ1

2 <u2 <1: The localstability of Ei can be determined by the Jacobian matrix of model(25) evaluating at Ei which has the following form:

JjEi¼

aui ð1þ b�2ui Þ þui v

i

ðui þ vi Þ2�

ui2

ðui þ vi Þ2

cvi2

ðui þ vi Þ2

�cui v

i

ðui þ vi Þ2

0BBBB@

1CCCCA: (32)

Let li, i = 1, 2 be the eigenvalues of (32), then we have thefollowing two equalities:

l1 þ l2 ¼ aui ð1þ b�2ui Þ þui v

i ð1�cÞ

ðui þ vi Þ2;

l1l2 ¼acðui Þ

2vi 2ui�ð1þ bÞ� �

ðui þ vi Þ2

:

(33)

Notice that b<u1 <bþ1

2 <u2 <1, thus from (33), we canconclude

l1l2ðu1; v1Þ<0 and l1l2ðu2; v2Þ>0:

This implies that E1 is always a saddle while E2 could be a sink orsource depending the value of u2. To further investigate thestability of E2, we let v2 ¼ c�d

d u2 in l1 + l2 which gives

l1 þ l2 ¼ 2a �ðu2Þ2 þ ðbþ 1Þu2

2þ ð1�cÞðc�dÞ

2ac2

:

Let f l ¼ �ðuÞ2 þ ðbþ1Þu

2 þ ð1�cÞðc�dÞ2ac2 and u ¼

bþ1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðbþ1Þ2þ8ð1�cÞðc�dÞ

ac2

q4 ,

then we can conclude follows:

f l>0 when 0<u<u; f l<0 when u>u:

This indicates that fl > 0 if u2 <u while fl < 0 if u2 >u. Therefore,the interior equilibrium E2 is locally asymptotically stable if u <u2while E2 is a source if u>u2 provided that E2 exists.

To further explore sufficient conditions that lead to E2 being a sink or a source, we let u2�u and decide how the value of a effects its sign.We define f as a function of a, i.e.,

f ðaÞ ¼ a2�ðc�dÞ cd�d�dþ 4bc

ðb�1Þ2

� �bc2

a�ðc�dÞ2ðcd�2c�dÞ2

bc4ðb�1Þ2:

Through algebraic calculations, we can conclude that

1. f(a) < 0 if 0< a<


ðb�1Þ2þ


ðb�1Þ2

� �2

þ4ðcd�2c�dÞ2

bðb�1Þ2

s24

35

2bc2 ; while f(a) > 0 if a>


ðb�1Þ2þ


ðb�1Þ2

� �2

þ4ðcd�2c�dÞ2

bðb�1Þ2

s24

35

2bc2 .

2. If f(a) > 0, then u2 >u, therefore, E2 is a sink. While if f(a) < 0, then u2 <u, therefore, E2 is a source.

Therefore, we can conclude that

� If c > d and a>maxf 4ðc�dÞcðb�1Þ2

;


ðb�1Þ2þ


ðb�1Þ2

� �2

þ4ðcd�2c�dÞ2

bðb�1Þ2

s24

35

2bc2 g, then model (25) has E0 = (0, 0) and E2 ¼ ðu2; v2Þ as its globalattractors.

� If c > d and 4ðc�dÞcðb�1Þ2

< a<


ðb�1Þ2þ


ðb�1Þ2

� �2

þ4ðcd�2c�dÞ2

bðb�1Þ2

s24

35

2bc2 , then ðu2; v2Þ is source.&


Proof. From the proof of Theorem 4.2, we have det(J) > 0. Since we assume that conditions in Part (3) of Theorem 4.2 hold, thus, we can

conclude that when the inequality a>


ðb�1Þ2þ


ðb�1Þ2

� �2

þ4ðcd�2c�dÞ2

bðb�1Þ2

s24

35

2bc2 holds, we have tr(J) < 0.

Based on the discussion on Page 8–9, we can conclude that detðJ iÞ>0> trðJ iÞ holds for all i � 1 if the following inequality holds

au2ð1þ b�2u2Þ þu2v

2

ðu2 þ v2Þ2

!D2�

cu2v2ðu2 þ v2Þ

2D1

" #2

<4D1D2acðu2Þ

2v2 2u2�ð1þ bÞ� �

ðu2 þ v2Þ2

: (34)

From the Routh–Hurwitz criterion, the two roots li1 and li2 of wi(l) = 0 have negative real parts for i � 1.

Notice that u2 ¼ðbþ1Þþ


ac

p2 ; v2 ¼ c�d

d u2, then we have

2u2�ð1þ bÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðb�1Þ2�4ðc�dÞ

ac

r; and

u2v2

ðu2 þ v2Þ2¼ dðc�dÞ

c2:

Therefore, the inequality (34) can be rewritten as follows:

u2v2

ðu2 þ v2Þ2

D2�cD1ð Þ�aD2u2 2u2�ð1þ bÞ �" #2

<4D1D2acu2 2u2�ð1þ bÞ� � u2v2

ðu2 þ v2Þ2

dðc�dÞc2

D2�cD1ð Þ�aD2u2


ac

r" #2

<4D1D2acu2


ac

rdðc�dÞ

c2

dðc�dÞc2

�au2


ac

r !D2�

dðc�dÞD1

c

" #2

<4D1D2adðc�dÞu2

c


ac

r

au2


ac

r�dðc�dÞ

c2

!D2 þ

dðc�dÞD1

c

" #2

<4D1D2adðc�dÞu2

c


ac

r

which is equivalent to

au2


ac

r�dðc�dÞ

c2

!2

D22 þ

d2ðc�dÞ2D21

c2þ 2

dðc�dÞD1D2

c

� au2


ac

r� dðc�dÞ

c2

!<

4D1D2adðc�dÞu2c


ac

r


which gives:

dðc�dÞc2

�au2


ac

r !2

D22 þ

d2ðc�dÞ2D21

c2�2

dðc�dÞD1D2

c

dðc�dÞc2

þau2


ac

r !<0

dðc�dÞc2

þau2


ac

r !2

D22 þ

d2ðc�dÞ2D21

c2�2

dðc�dÞD1D2

c

dðc�dÞc2

þau2


ac

r !< 4

dðc�dÞD22au2


ac

rc2

:

Therefore, we have

dðc�dÞc2

þ au2


ac

r !D2�

dðc�dÞD1

c

" #2

<4dðc�dÞD2

2au2


ac

qc2

which is equivalent to

dðc�dÞc

þ acu2


ac

r !� dðc�dÞD1

D2

" #2

<4dðc�dÞau2


ac

r:

This implies that large ratios of the prey diffusion coefficient D1 to the predator diffusion coefficient D2 can reserve the stability of E2.

On the other hand, we have trðJÞ ¼ au2ð1þ b�2u2Þ þu

2v

2ð1�cÞ

ðu2þv

2Þ2; detðJÞ ¼ acðu

2Þ2v

22u

2�ð1þbÞ½ �

ðu2þv

2Þ2

: Then we have:

ðau2ð1þ b�2u2Þ þu2v

2

ðu2 þ v2Þ2ÞD2�

cu2v2

ðu2 þ v2Þ2

D1 ¼ trðJÞD2 þdðc�dÞðD2�D1Þ

c:

Thus, we can conclude that E2 is locally stable for the PDE model (3) if au2ð1þ b�2u2Þ þu

2v

2

ðu2þv

2Þ2

� �D2�

cu2v

2

ðu2þv

2Þ2

D1 <0 which is equivalent tothe inequality


c<0

since under this condition we have

D1D2m2i � trðJÞD2 þ


mi þ detðJÞ>0: (35)

The condition of trðJÞD2 þ dðc�dÞðD2�D1Þc <0 can be held in the following three cases

1. Since Conditions in Part (3) of Theorem 4.2 hold, then we have tr(J) < 0 and det(J) > 0 which implies that the inequality (35) holds for allmi > 0 if D2 < D1 holds.

2. trðJÞ þ dðc�dÞc <0, dðc�dÞ

c < au2


ac

q since we have

trðJÞ ¼ au2ð1þ b�2u2Þ þu2v

2ð1�cÞ

ðu2 þ v2Þ2¼ dðc�dÞð1�cÞ

c2�au2


ac

r" #:

3. The inequalities hold D1 <D2 <dðc�dÞD1

dðc�dÞþctrðJÞ ¼dðc�dÞD1

dðc�dÞc �acu

2


a

ph i which are equivalent to

0<dðc�dÞ�ac2u2


ac

qcdðc�dÞ <

D1

D2<1:

To obtain the local stability of E2, we prove that there exists a constant d > 0 such that

Refli1g; Refli2g��d for all i�1: (36)

Let l = miz, then

’iðlÞ ¼ m2i z

2�trðJ iÞmiz þ detðJ iÞ :¼ ’iðzÞ:Since mi!1 as i!1, we can get that

limi!1

’iðzÞm2

i

¼ z2 þ ðD1 þ D2Þz þ D1D2 :¼ ’ðzÞ:

Therefore, ’ðzÞ ¼ 0 has two negative roots �D1 and �D2. Let D ¼minfD1;D2g, then Refz1g; Refz2g��D. By continuity, there exists i0such that the two roots zi1, zi2 of ’iðzÞ ¼ 0 satisfy Refzi1g; Refzi2g�� D

2 ; 8 i� i0. Thus, Refli1g; Refli2g�� Dmi2 �� D

2 ; 8 i� i0. Let

�d ¼ max1�i�i0

fRefli1g; Refli2gg:

Then d>0 and (36) holds for d ¼ minfd; D2g. The proof is completed. &



Proof. For the emergency of Hopf bifurcation at E2 ¼ ðu2; v2Þ, the matrix J i ¼ �miDþ J must have an eigenvalue on the imaginary axis(Mei, 2000), i.e., trðJ iÞ ¼ 0. Assume that a(k) is the critical value of a that the Hopf bifurcation occurring at the branch k when the followingequality holds

�mkðD1 þ D2Þ�aðkÞððbþ 1Þ þ


aðkÞc

qÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðb�1Þ2� 4ðc�dÞ

aðkÞc

q2

þ dðc�dÞð1�cÞc2

¼ 0

When k = 0, the critical value of a at which Hopf bifurcation hypotheses are satisfied is

a ¼ að0Þ :¼


ðb�1Þ2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficd�c�dþ 4bc

ðb�1Þ2

!2

þ 4ðcd�2c�dÞ2

bðb�1Þ2

vuut264

375

2bc2:

Near a(0), the complex conjugate pair a(a) ib(a) is given by:

aðaÞ ¼ 1

2trðJÞ; b2ðaÞ ¼ detðJÞ�a2ðaÞ:

Since a(a(0)) = 0 and det(J) > 0, we get b2(a) 6¼ 0. Moreover, it is easy to check dda aðaÞ 6¼0, and therefore the statement holds. &


Proof. In the presence of diffusion, introducing small perturbations U ¼ u�u2; V ¼ v�v2, where jUj, jVj � 1. To study the effect of diffusionon model (3), we have considered the linearized form of system as follows:

@U

@t¼ J11U þ J12V þ D1r2U;

@V

@t¼ J21U þ J22V þ D1r2V : (37)

Any solution of model (37) can be expanded into a Fourier series so that

Uðr; tÞ ¼X1

m;n¼0

umnðr; tÞ ¼X1

m;n¼0

amnðtÞcoskr; Vðr; tÞ ¼X1

m;n¼0

vmnðr; tÞ ¼X1

m;n¼0

bmnðtÞcoskr;

where k = (km, kn) and km = mp/Ly, kn = np/Lx are the corresponding wavenumbers.Having substituted um and vmn into (37), we obtain

damn

dt¼ ðJ11�D1k2Þamn þ J12bmn;

dbmn

dt¼ J21amn þ ðJ22�D2k2Þbmn; (38)

where k2 ¼ k2m þ k2

n.A general solution of (38) has the form C1 exp(l1t) + C2 exp(l2t), where the constants C1 and C2 are determined by the initial conditions

(5) and the exponents l1 and l2 are the eigenvalues of the following matrix:

AjE2¼ J11�D1k2 J12

J21 J22�D2k2

!: (39)

Correspondingly, l1 and l2 arise as the solution of the following equation l2 � trkl + detk = 0, or solving for l

lðkÞ ¼ 1

2ðtrk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitr2

k�4detk

qÞ; (40)

where

trk ¼ J11 þ J22�k2ðD1 þ D2Þ ¼ au2ð1þ b�2u2Þ þu2v

2ð1�cÞ

ðu2 þ v2Þ2�k2ðD1 þ D2Þ� trðJÞ�k2ðD1 þ D2Þ;

detk ¼ ðJ11�D1k2ÞðJ22�D2k2Þ�J12J21 ¼ D1D2k4� au2ð1þ b�2u2Þ þu2v2

ðu2 þ v2Þ2

!D2�

cu2v2

ðu2 þ v2Þ2

D1

" #k2 þ


ðu2 þ v2Þ2

�D1D2k4� au2ð1þ b�2u2Þ þu2v2

ðu2 þ v2Þ2

!D2�

cu2v2

ðu2 þ v2Þ2

D1

" #k2 þ detðJÞ:

If the homogeneous steady state of model (3) is unstable, then one of the eigenvalues of matrix A has positive real part, which depends

on the signs of trace trk and determinant detk of A. From Theorem 4.3, tr(J) < 0 if a>


ðb�1Þ2þ


ðb�1Þ2

� �2

þ4ðcd�2c�dÞ2

bðb�1Þ2

s24

35

2bc2 , then


trk < 0 is always true. If A has an eigenvalue with positive real part, then it must be a real value one and the other eigenvalue must be anegative real one. Thus, a necessary condition is

ðau2ð1þ b�2u2Þ þu2v

2

ðu2 þ v2Þ2ÞD2�

cu2v2

ðu2 þ v2Þ2

D1 >0; (41)

which is equivalent to the following inequality

D1

D2<

dðc�dÞ�ac2u2


ac

qcdðc�dÞ :

Otherwise the equilibrium E2 is locally stable according to Theorem 4.3 since trk < 0 for k2 > 0 and det(J) > 0.The Turing instability, mathematically speaking, as D1� D2, occurs where Im(l(k)) = 0, Re(l(k)) = 0 at k = kT 6¼ 0. We need detk < 0 for

some k2 > 0 and detk achieves its minimum below zero,

mink2

detk ¼4D1D2detðJÞ� au2ð1þ b�2u2Þ þ

u2v

2

ðu2þv

2Þ2

� �D2�

cu2v

2

ðu2þv

2Þ2

D1

� �2

4D1D2<0; (42)

which is equivalent to the inequality of (21) holds, i.e.,

dðc�dÞc

þ acu2


ac

r !� dðc�dÞD1

D2

" #2

>4dðc�dÞau2


ac

r

at the wavenumber kT satisfies

k2T ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffidetðJÞD1D2

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiacðu2Þ

2v2 2u2�ð1þ bÞ� �

ðu2 þ v2Þ2D1D2

vuut >0:

At the Turing bifurcation threshold, the spatial symmetry of the system is broken and the patterns are stationary in time and oscillatoryin space with the corresponding wavelength lT = 2p/kT. If (41) holds and min

k2 detk <0 with certain k2 in the interval of (m�, m+) where

mþðD1;D2; aÞ ¼trðJÞD2 þ


þ



� �2

�4D1D2detðJÞ

s

2D1D2;

m�ðD1;D2; aÞ ¼trðJÞD2 þ


�



� �2

�4D1D2detðJÞ

s

2D1D2

with

trðJÞ ¼ au2ð1þ b�2u2Þ þu2v2ð1�cÞðu2 þ v2Þ

2; detðJÞ ¼


ðu2 þ v2Þ2

:

Then E2 is an unstable equilibrium with respect to model (3). Therefore, the statement holds. &


Proof. Let ðu; vÞ be any positive solution of model (8) and denote h ¼ jVj�1R RVhdL. Multiplying the first and second equations of (8) by

ðu�uÞ and ðv�vÞ integrating over V, respectively, and since u and v have a uniform upper bound, we have

Z ZV

D1jruj2 þ D2jrvj2� �

dL ¼Z Z

Vðu�uÞ auð1�uÞðu�bÞ� uv

uþ v�auð1�uÞðu�bÞ þ uv

uþ v

� �dL

þZ Z

Vðv�vÞ cuv

uþ v�dv� cuv

uþ vþ dv

� �dL ¼

Z ZV

ðu�uÞ2 að1þ bÞðuþ uÞ�ab�aðu2 þ uuþ u2Þ� vv

ðuþ vÞðuþ vÞ

� �

þ ðv�vÞ2 cuu

ðuþ vÞðuþ vÞ�d

� �þ ðu�uÞðv�vÞ cvv�uu

ðuþ vÞðuþ vÞ

dL�

Z ZV

ðu�uÞ2 að1þ bÞuþ að1þ bÞ2

4�aðu�1þ b

2Þ

2

�ab

!

þ ðv�vÞ2 cu

uþ vð cu

uþ v�1Þ þ cvðu�uÞ

ðuþ vÞðuþ vÞ

� �þ cv2�u2

ðuþ vÞðuþ vÞ ðu�uÞðv�vÞ

dL�Z Z

V

aðb2 þ 2bþ 5Þ

4ðu�uÞ2 þ c2

bCðv�vÞ2

þ cC1

bCju�ujjv�vj

dL�

Z ZVðu�uÞ2 aðb2 þ 2bþ 5Þ

4þ cC1

2ebC

!þ ðv�vÞ2 c

bCc þ eC1

2

� �" #dL

for some positive constants C1, C* and an arbitrary positive constant e. Synthetically, we obtain

Z ZV

D1m1ðu�uÞ2 þ D2m1ðv�vÞ2� �

dL�Z Z

Vðu�uÞ2 aðb2 þ 2bþ 5Þ

4þ cC1

2ebC

!þ ðv�vÞ2 c

bCc þ eC1

2

� �" #dL


by using Poincare inequality (Wu et al., 2003). Since D2m1 > c from the assumption, we can find a sufficiently small e0 such thatD2m1� c2

bC ðc þe0C1

2 Þ. Finally, by taking D1 :¼ 1m1ðaðb

2þ2bþ5Þ4 þ cC1

2e0bCÞ, one can conclude that u ¼ u and v ¼ v which complete the proof. &


Proof. Since the condition (21) holds, that is, (D1J22 + D2J11)2 > 4D1D2(J11J22 � J12J21), it follows that m exists. On the contrary, supposethat model (8) has no non-constant positive solution. By Theorem 5.1, we can fixed D1 >D1 and D2 >D2 such that

(i) model (8) with diffusion coefficients D1 and D2 has no non-constant solutions;(ii) MðD1;D2;mÞ>0 for all m � 0.

From Theorem 3.2, there exists a positive constant C* such that for D1�D1; D2�D2, any solution ðu; vÞ of model (8) with diffusioncoefficients D1 and D2 satisfies 0<u; v<C; ðx; yÞ 2V.

Set

P ¼ fðu; vÞ 2CðVÞ�CðVÞ : 0<u; v<C in Vg;and define

F : P�½0;1�!CðVÞ�CðVÞby

Fðw;$Þ ¼ ðI�DÞ�1fGðw;$Þ þwg;where

Gðw; $Þ ¼$D1 þ ð1�$ÞD1

��1auðu�bÞð1�uÞ� uv

uþ v

� �

$D2 þ ð1�$ÞD2

��1 cuv

uþ v�dv

� �0BB@

1CCA:

We can see that finding the positive solution of model (17) becomes equivalent to finding the fixed point of F(w, 1) in P. By virtue of thedefinition of P, we obtain that F(w, ) = 0 has no fixed point in @P for 0 � � 1.

Since F(w, t) is compact, the Leray–Schauder topological degree deg(I �F(w, ), P, 0) is well defined. From the invariance of Leray-Schauder degree at the homotopy (Wang, 2003), we deduce

degðI�Fðw; 1Þ; P; 0Þ ¼ degðI�Fðw; 0Þ; P; 0Þ: (43)

In view of m� 2 (mi, mi+1) and m+ 2 (mj, mj+1), we get B(D1, D2) \ S(m) = {mi+1, mi+2, � � �, mj} and I�Fðw; 1Þ ¼ I�F . Therefore, if model (8)has no other solutions except the constant w*, then Lemma 5.1 shows that

degðI�Fðw; 1Þ; P; 0Þ ¼ indexðI�F ; wÞ ¼ ð�1ÞSjk¼iþ1mðmkÞ ¼ �1: (44)

On the contrary, by the choice of D1; D2 and (ii), we have BðD1;D2Þ \ SðmÞ ¼ ; and w* is the only fixed point of F(w, 0). It follows fromLemma 5.1 that

degðI�Fðw; 0Þ; P; 0Þ ¼ indexðI�F ; wÞ ¼ ð�1Þ0 ¼ 1: (45)

From (43) to (45), we have a contradiction. Thus, there exists a non-constant solution of model (8).

Sincev

2

ðu2þv

2Þ2þ 1þ b>2u2,

dðc�dÞc2 >


ac

qand when D2 is large enough, then we have (D1J22 + D2J11)2 > 4D1D2

(J11J22 � J12J21) and 0 <m�(D1, D2) <m+(D1, D2). Moreover,

m�ðD1;D2Þ!0; mþðD1;D2Þ!1

D1ðau2ð1þ b�2u2Þ þ

u2v2ðu2 þ v2Þ

2Þ; as D2!1:

Due to the fact that 1D1ðau2ð1þ b�2u2Þ þ

u2v

2

ðu2þv

2Þ2Þ 2 ðmj;mjþ1Þ for some j � 1, there exists D0� 1 such that

mþðD1;D2Þ 2 ðmj;mjþ1Þ; 0<m�ðD1;D2Þ<m1; 8D2�D0:

Therefore, for large D2 > 0,Pj

i¼1mðmiÞ is odd which implies (i) and (ii) in the earlier arguments. This completes the proof. &



Acknowledgements

This research of F.R. is supported by Jiangsu Provincial NaturalScience Foundation (BK 20140927) and Tianyuan Fund forMathematics of NSFC (11426132). The research of Y.K. is partiallysupported by NSF-DMS (1313312), Simons Collaboration Grantsfor Mathematicians (208902 [5_TD$DIFF]), NSF-IOS/DMS (1558127), The JamesS. McDonnell Foundation-UHC Scholar Award (220020472)[6_TD$DIFF], andthe research scholarship from College of [1_TD$DIFF]Integrative Sciences and[7_TD$DIFF]Arts, ASU.

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