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R E S E A R C H

Open Access

Hopf bifurcation of a delayed diffusive

predator-prey model with strong Allee effect

Jia Liu and Xuebing Zhang

*

*Correspondence:

zxb1030@163.com Huaian Vocational College of Information Technology, Huaian, 223003, People’s Republic of China

Abstract

The paper is concerned with a delayed diffusive predator-prey system where the growth of prey population is governed by Allee effect and the predator population consumes the prey according to Beddington-DeAngelis type functional response. The situation of bi-stability and the existence of two coexisting equilibria for the proposed model system are addressed. The stability of the steady state together with its dependence on the magnitude of time delay has been obtained. The conditions that guarantee the occurrence of the Hopf bifurcation in presence of delay are demonstrated. Furthermore, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. Finally, some numerical simulations have been carried out in order to validate the assumptions of the model.

MSC: 34C23; 34D23

Keywords: predator-prey model; Allee effect; stability; Hopf bifurcation; delay

1 Introduction

In this paper, we consider the following delayed diffusive predator-prey system with Beddington-DeAngelis functional response and strong Allee effect:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

∂u

∂t =du+ru( – u

K)(um) –

quv(tτ)

a+bu+cv(tτ), (x,t)∈×(, +∞),

∂v

∂t =dv+v(–d+ eu

a+bu+cv), (x,t)∈×(, +∞),

u(x,t) =u(x,t), v(x,t) =v(x,t), (x,t)∈×[–τ, ], ∂u(t,x)

∂n = ∂v(t,x)

∂n = , t> ,x,

()

whereu(x,t) represents the population of the prey andv(x,t) represents the population of the predator. The parametersr,K,m,q,a,b,c,d,e,d,dare positive constants withd

representing the death rate of predator as well asrandKstanding for the intrinsic rate of increase and the carrying capacity for the prey population, respectively. The predator consumes the prey with functional response of Beddington-DeAngelis type [–].dand

d denote the diffusion coefficients of the prey and the predator,respectively.denotes the Laplacian operator,is a bounded domain,nis the normal vector that goes out of the bounded domain. The homogeneous Neumann boundary conditions indicate that there is no population flux across the boundaries. For the initial conditions, we assume

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that

ψj(s,x)∈C=C

[–τ, ],X

andXis defined by

X=

uW,() :∂u(t,x) ∂n =

∂v(t,x)

∂n = ,x

with the inner product·,·. The term

ru

 – u

K

(um)

is the growth function considering Allee effect. The so-called Allee effect dates from , Allee reported that the prey growth rate is negative or an increasing function at low pop-ulation density []. The Allee effect is caused by various kinds of biological and environ-mental factors, such as difficulties in finding mates, low probability of successful mating, depletion in inbreeding rate, anti-predator aggression, predator avoidance due to evolu-tionary change, etc. [–]. Allee effects can be broadly classified into two types: the strong Allee effect and the weak Allee effect []. For model (), the Allee effect is strong or weak asm>  orm≤. In recent years, there have been some excellent papers with the Allee effect in a predator-prey system (see, for example, [–]).

It is well known that delays which occur in the interaction between predator and prey play a complicated role in a predator-prey system. Many researchers have incorporated it into biological models [–] as delays could affect the stability of a predator-prey system by creating instability, oscillation, and chaos phenomena. The reason for introducing a delay into model () is that the predator species may need timeτto possess the ability of predation after it was born.

On the other hand, in real life the species is spatially heterogeneous and hence indi-viduals will tend to migrate towards regions of lower population density to increase the possibility of survival []. Hence, in model (), we consider the factor of diffusion.

In this paper, we will study the Hopf bifurcation of system (), the direction of Hopf bi-furcation and the stability of the bifurcating periodic solutions with the help of the theory of the normal form and center manifold [].

The remaining part of this paper is organised as follows: In Section , we study the ex-istence and boundedness of the system. The exex-istence of possible equilibria are studied in Section . The sufficient conditions ensuring linear stability of equilibria and the ex-istence of Hopf bifurcation of the coexisting equilibrium are obtained in Section . In Section , we investigate the direction and stability of Hopf bifurcation by applying the center manifold method and the normal form theory. Numerical results for the proposed model system are presented in Section . Finally, a conclusion is presented in Section .

2 Existence and boundedness of solution of system (1)

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Theorem . For system(),we have:

(a) Ifu(x)≥,v(x)≥,then system()has a unique solution(u(t,x),v(t,x))such that

u(t,x) > ,v(t,x) > fort∈(, +∞)andx∈ ¯.

(b) Ifu(x,t)≤mand(u,v)≡(m, ),then(u(x,t),v(x,t))tends to(, )uniformly as

t→ ∞.

(c) If K(ecdbd)–ad< ,then(u(x,t),v(x,t))tends to(us(x), )uniformly ast→ ∞,where

us(x)is a non-negative solution of

du+ru

 – u

K

(um) = , x,∂u

∂ν = ,x. ()

(d) IfK(ebdcd)–ad> ,then any solution(u(x,t),v(x,t))of()satisfies

lim sup

t→+∞

u(x,t)≤K, lim sup

t→+∞

v(x,t)≤K(ebd) –ad

cd .

Proof

(a) Define

f(u,v) =ru

 – u

K

(um) – quv(tτ)

a+bu+cv(tτ),

g(u,v) =v

d+ eu

a+bu+cv

.

Thenfv= –(aqu+(bua++bucv)) ≤ andgu=(aev+bu(a++cvcv)) ≥ inR+={u≥,v≥}. Hence, system () is

a mixed quasi-monotone system. Consider the following system:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

du

dt =ru( – u

K)(um), dv

dt =v(–d+ eu a+bu+cv),

u() =u, v() =v.

()

Assumeu(t;u,v),v(t;u,v) is the unique solution to system (). Let

max

¯

×[–τ,]u(x,t) =uM, max

¯

×[–τ,]

v(x,t) =vM.

Obviously, (u(t,x),v(t,x)) = (, ) and (u¯(t),v¯(t)) = (u(t;uM,vM),v(t;uM,vM)) are a pair of lower-solution and upper-solution to system (). Therefore, according to Theorem .. in [] or Theorem .. in [], system () has a unique globally defined solution (u(x,t),v(x,t)) which satisfies

≤u(x,t)≤u(t;uM,vM),

≤v(x,t)≤v(t;uM,vM).

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(b) From the above discussions, we obtain u(x,t)≤u(t;uM,vM) for all t> . From the ODE satisfied by u(t;uM,vM), one can see that u(t;uM,vM)→ if uM <m and

u(t;uM,vM)→KifuM>m. Thus for anyε> , there existsT>  such thatu(x,t)≤K+ε in [T, +∞)× ¯.

(c) If K(ebdcd)–ad< , then there exists aε> , such that (K+ε)(ecdbd)–ad< . Therefore, ac-cording to the equation ofv(t;uM,vM), we obtain ≤v(x,t)≤v(t;uM,vM)→ ast→ ∞ uniformly forx∈ ¯. The limit behavior ofu(x,t) is determined by the semiflow generated by the scalar parabolic equation

⎧ ⎨ ⎩

∂u

∂t =du+ru( – u

K)(um), (x,t)∈×(, +∞), ∂u

∂ν = , x∈ ¯.

()

According to the discussions in [] and [], every orbit of () converges to a steady stateus. Therefore, (u(x,t),v(x,t)) tends to (us(x), ) uniformly ast→ ∞.

(d) By the second equation of (), we easily find that there existsT∈(, +∞) such that v(t,x)≤(K+ε)(ecdbd)–ad in [T, +∞)×for an arbitrary constant > . Therefore,

lim sup

t→+∞ u(x,t)≤K, lim supt→+∞ v(x,t)≤

K(ebd) –ad

cd .

3 The existence of equilibria

In this section, we will find all possible non-negative equilibria. Clearly, system () has four feasible non-negative equilibria, namely:

() the trivial pointE= (, );

() the boundary equilibriumE= (m, ), representing the state corresponding to the extinction of predator;

() the boundary equilibriumE= (K, ), representing the state corresponding to the extinction of predator;

() the coexisting equilibrium point(s)E∗= (u∗,v∗).

At the coexisting equilibrium, we must have

v=(ebd)uad

cd ,

andusatisfies

Au+Au+Au+A= , ()

where

A=cer;

A= –cer(m+K);

A=K(qebdq+cemr);

A= –Kadq.

()

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Figure 1 The blue curves are the prey-nullclines and the red lines are the predator-isoclines. The four figures are the possible plots of predator-nullcline for four different values ofc. (a)Forc= 0.2, two nullclines do not intersect at any point in the interior of the feasible domain (the prey nullcline is below the predator nullcline), suggesting there is no equilibrium point;(b)as we increase the slope of

predator-nullcline, the two equilibria approach each other and collide forc= 0.28 and consequently, there is one equilibrium point;(c)forc= 0.3, both the nullclines cross twice, suggesting there are two equilibrium points;(d)forc= 2, the prey nullcline is unbounded and has two vertical asymptotesx=x±shown by black lines. The other parameter values arer= 0.8,K= 5,q= 0.2,a= 2,b= 0.4,d= 0.1,e= 0.2,m= 2.

The possible number of equilibria can be better analysed by studying the intersections of the nullclines which is one of great feature of planar systems. Letf(u,v) =  andg(u,v) = , we show the existence of non-negative equilibria in Figure .

4 Local stability and bifurcation

In this section, we discuss the local stability of non-negative equilibria. Before developing our argument, let us set up the following notations.

Notation . Let  =μ<μ<μ<· · ·<μn<· · · → ∞are the eigenvalues of –on under homogeneous Neumann boundary condition. We define the following space de-composition:

(i) S(μn)is the space of eigenfunctions corresponding toμiforn= , , , . . .; (ii) Xij:={c·φij: c∈R}, where{φij}are orthonormal basis ofS(μn)for

j= , , . . . ,dim[S(μn)];

(iii) X:={u= (u,v)∈[C(¯)]: ∂u ν =

∂v

ν= }, and soX=

i=Xi, where

Xi=

dim[S(μj)]

j= Xij.

The linearization of system () at a constant solutionE∗= (u∗,v∗) can be expressed by

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whereD=diag(d,d), u = (u(x,t),v(x,t))T, and u

τ= (u(x,tτ),v(x,tτ)),

J=

a

a a

, J=

 –a

 

,

where

a= 

r+rm

K

u∗–r

Ku

∗mr qv∗(a+cv∗)

(a+bu∗+cv∗);

a=

qu∗(a+bu∗)

(a+bu∗+cv∗); a=

ev∗(a+cv∗) (a+bu∗+cv∗);

a= –d+ eu

(a+bu)

(a+bu∗+cv∗).

()

In view of Notation ., we can induce the eigenvalues of system () confined on the subspace Xi. Ifλis an eigenvalue of () on Xi, it must be an eigenvalue of the matrix –μnD+J∗for∀n∈ {, , , . . .}:=N, where

J∗=

aaeλτ

a a

.

It is easy to see thatλsatisfies the following characteristic equation:

λ+Anλ+Bn+Ceλτ= , ()

where

An=dμn+dμnaa,

Bn= (dμna)(dμna),

C=aa.

()

4.1 Stability of constant steady state as

τ

= 0

() ForE= (, ), the corresponding characteristic equation is

(λ+dμn+mr)(λ+dμn+d) = .

Clearly, we obtain

λ= –mrdμn,λ= –ddμn.

Hence,Eis always stable.

() ForE= (m, ), the corresponding characteristic equation is

λ+dμnmr

 –m

K

λ+dμn+d

em a+bm

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Obviously,

λ=mr

 –m

K

dμn, λ= –d+ em

a+bmdμn.

Consequently, ifm<Kord<a+embm, thenE= (m, ) is unstable. On the contrary, ifm>K

andd>a+embm, thenE= (m, ) is stable.

() ForE= (K, ), the corresponding characteristic equation is

(λ+dμn+Krmr)

λ+dμn+d

eK a+bK

= .

Obviously,

λ= –Kr+mrdμn, λ= –d+ eK

a+bKdμn.

Therefore, ifm<Kandd>aeK+bK, thenEis stable.

() ForE∗= (u∗,v∗), whenτ= , the corresponding characteristic equation is

λ+ (dμn+dμnaa)λ+ (dμna)(dμna) +aa= .

Obviously,

λ+λ=a+adμndμn,λλ= (dμna)(dμna) +aa. ()

All roots of () have negative real parts if

(H) a< , a<  and aa+aa> . ()

Therefore, the coexisting equilibriumE∗= (u∗,v∗) of system () is locally asymptotically stable forτ =  when condition () holds.

4.2 Hopf bifurcation

In this section, we are going to analyze the conditions about the parameters under which the Hopf bifurcation occurs at the coexisting equilibrium.

Assume that(ω> ) is a root of equation (). Thenωshould satisfy the following equation for somen≥:

ω+iAnω+Bn+C

cos(ωτ) –isin(ωτ)= , ()

which implies that

⎧ ⎨ ⎩

ω+Bn= –Ccos(ωτ),

Anω=Csin(ωτ).

()

From (), adding the squared terms for both equations yields

ω+An– Bn

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Letz=ω, equation () becomes

z+An– Bn

z+BnC= , ()

where

An– Bn=

d+dμn– (ad+ad) +a+a,

BnC= (Bn+C)(BnC)

=ddμn– (ad+ad)μn+aa+aa

×ddμn– (ad+ad)μn+aaaa.

()

For further discussions, we make the following assumptions:

(H) aaaa> ;

(H) aa–aa< .

Theorem . If(H)and(H)hold,then all roots of equation()have negative real parts for all τ ≥.Furthermore,the coexisting equilibrium Eof system()is asymptotically stable for allτ≥.

Proof From equation (), we know that

An– Bn> .

By (H), we getBn+C> . Obviously, if (H) and (H) hold, then

BnC=ddμn– (ad+ad)μn+aa–aa> 

for anyn≥.

These imply that equation () has no positive roots, and hence the characteristic equa-tion () has no purely imaginary roots. Therefore, all roots of equaequa-tion () have negative

real parts asτ≥.

Denote

μ∗=ad+ad+

(ad+ad)– dd(aaaa)

dd .

Thus, there must exist someN∗∈N, such thatμ∗=μN∗orμN∗<μ∗<μN∗+. Hence, we

have the following lemma.

Lemma . If(H)and(H)hold,then equation()has a pair of purely imaginary roots

±iωn(≤nN∗)at

τ=τnj=τn+ ωn

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where condition of equation () having positive roots. For ≤nN∗a unique positive rootzn of equation () is

is the imaginary part of the purely imaginary root, at

τ=τnj=τn+

and hence we have a complete ordering of the bifurcation valuesτnj.

Lemma . If(H)and(H)hold,then

τNj∗≥τNj–≥ · · · ≥τj>τ

j

for jN.

Proof From the above analysis, we know

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whereA

ωn is strictly increasing inn.

Namely,

Proof Differentiating the two sides of equation () with respect toτyields

Substitutingτnj into the above equation, we obtain

From the above analysis, we have the following conclusion.

Theorem . If(H)and(H)hold,then the following statements are true:

(i) Whenτ∈[,τ),the coexisting equilibrium of system()is locally asymptotically stable.

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5 Direction and stability of Hopf bifurcation

In the previous section, we have shown that system () admits a series of periodic solu-tions bifurcating from the coexisting equilibrium at the critical valueτnj(n,jN). In this section, we derive explicit formulas to determine the properties of the Hopf bifurcation at the critical valueτnj (n,jN). By using the normal form theory and center manifold reduction for PFDEs developed by [].

For fixedn,jN, denoteτnj byτ∗and introduce the new parameterμ=ττ∗. Nor-malizing the delayτby the time-scalingtt/τ. Then () can be rewritten as

dU(t)

dt =τ

DU(t) +Lτ(U

t) +F(Ut,μ), ()

whereL(μ)(ϕ) :CXandF(·,μ) :CXare given by

L(μ)(ϕ) =μ

aϕ() –aϕ(–) aϕ() +aϕ()

, ()

F(ϕ,μ) =μDϕ() +L(μ)ϕ+f(ϕ,μ), ()

and

f(ϕ,μ) =τ∗+μ

bϕ() +bϕ()ϕ(–) +bϕ(–) cϕ

() +cϕ()ϕ() +cϕ()

+h.o.t., ()

where

b= 

r+rm

K

br

Ku

+bqv∗(a+cv∗)

(a+bu∗+cv∗),

b= –aq(a+bu

+cv) + qcbuv

(a+bu∗+cv∗) , b=

qcu∗(a+bu∗) (a+bu∗+cv∗),

c=–ebv

(a+cv)

(a+bu∗+cv∗), c= –

ea(a+bu∗+cv∗) + ecbuv

(a+bu∗+cv∗) ,

c= – ecu(a+bu

)

(a+bu∗+cv∗),

andϕ= (ϕ,ϕ,ϕ)TC.

Then the linearized system of () at (, ) is

dU(t)

dt =τ

DU(t) +Lτ(U

t). ()

Based on the discussion in Section , we can easily see that, forτ=τ∗, the characteristic equation of () has a pair of simple purely imaginary eigenvalues={τ∗, –τ∗}.

LetC:=C([–, ],R), consider the following FDE onC:

˙

z=∗(zt). ()

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whose elements are of bounded variation such that

In fact, we can choose

ηθ,τ∗=τ

whereδis the Dirac delta function.

LetA(τ∗) denote the infinitesimal generator of the semigroup induced by the solutions of () andA∗be the formal adjoint ofA(τ∗) under the bilinear pairing

the discussion in Section , we know that A(τ∗) has a pair of simple purely imaginary eigenvalues±τ∗, and they are also eigenvalues ofA∗sinceA(τ∗) andA∗are a pair of

adjoint operators. LetPandP∗be the center spaces, that is, the generalized eigenspaces ofA(τ∗) andA∗ associated with, respectively. ThenP∗ is the adjoint space ofPand dimP=dimP∗= . Direct computations give the following results.

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forθ∈[–, ], and

Then the center space of linear equation () is given byPCNC, where

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whereX: [–, ]→B(X,X) is given by

X(θ) = ⎧ ⎨ ⎩

, –≤θ< ,

I, θ= . ()

Then ∗ is the infinitesimal generator induced by the solution of () and () can be rewritten as the following operator differential equation:

˙

Ut=Ut+XF(Ut,μ). ()

Using the decompositionC=PCNCPSCand (), the solution of () can be rewritten as

Ut=

x(t)

x(t)

·f+h(x,x,μ), ()

where

x(t)

x(t)

=,Ut,f

, ()

andh(x,x,μ)∈Pscwithh(, , ) =Dh(, , ) = . In particular, the solution of () on the center manifold is given by

Ut∗=

x(t) x(t)

·f+h(x,x, ). ()

Settingz=xixand noticing thatp=+i, then () can be rewritten as

Ut∗= 

zz i(zz¯)

·f+w(z,z¯) = 

(pz+p¯z¯)·f+W(z,z¯), ()

whereW(z,z¯) =h(z+z¯, –ziz¯, ). Moreover, by [],zsatisfies

˙

z=τz+g(z,z¯), ()

where

g(z,z¯) =() –i()

FUt∗, ,f

. ()

Let

W(z,z¯) =Wz

 +Wzz¯+W ¯

z

 +· · · ()

and

g(z,z¯) =gz

 +gz¯z+g ¯

z

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From (), we have follows easily from () that

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(17)

By the definition of∗, we get from ()

From the above expression, we can see easily that

E=

In a similar way, we have

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Similar to the above, we obtain

F=  

a a

aa –

×

b+b(α¯eiωτ

+αeτ) + b

αα¯

c+c(α¯+α) + cαα¯

.

So far,W(θ) andW(θ) have been expressed by the parameters of system (). There-fore,gcan be expressed explicitly.

Theorem . System()has the following Poincaré normal form:

˙

ξ=τξ+c()ξ|ξ|+o

|ξ|,

where

c() = iωτ

gg– |g|–|g|

 +

g

 ,

so we can compute the following results:

σ= –

Re(c())

Re(λ(τ∗)),

β= Re

c(),

T= –Im(c()) +σIm(λ

(τ))

ωτ

,

which determine the properties of bifurcating periodic solutions at the critical valuesτ∗,

i.e.,σdetermines the directions of the Hopf bifurcation:ifσ>  (σ< ),then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for

τ >τ∗;βdetermines the stability of the bifurcating periodic solutions:the bifurcating pe-riodic solutions on the center manifold are stable(unstable),ifβ<  (β> );and T de-termines the period of the bifurcating periodic solutions:the period increases(decreases),if T>  (T< ).

6 Numerical results and discussions

In this section we present results of the numerical simulations to facilitate the interpreta-tion of our mathematical findings in system () proved in the previous secinterpreta-tions.

6.1 Stability of the interior equilibrium for all

τ

≥0

Consider system () with the following parameters:r= .,K= ,q= .,a= ,b= .,

d= .,e= .,m=  andc= .. Choose= (,π) and the diffusion coefficientsd= .,

d= .. According to the discussions in Section .,E= (, ) andE= (, ) are both stable, andE= (, ) is unstable. In addition, by a direct calculation, we find that system () has a coexisting equilibriumE∗= (., .). Clearly, the conditions (H) and

(H) hold, according to Theorem ., system () is locally asymptotically stable atE∗for

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Figure 2 Phase portraits of the model system (1).E0= (0, 0),E1= (5, 0) andE∗= (4.7296, 2.7926) are all

locally asymptotically stable for allτ≥0.

Figure 3 The coexisting equilibriumE∗is locally stable whereτ= 3 <τ0.

6.2 Hopf bifurcation

Let the parameters be the same as above exceptd= . andm= .. Choose= (,π) and the diffusion coefficientsd= .,d= .. According to the discussions in Section .,

E= (, ) is stable yet, butE= (, ) andE= (, ) become unstable. By a direct

calcula-tion, we find that system () has two the coexisting equilibriaE∗= (., .) and

E= (., .). ForE, condition (H) is not satisfied. Therefore,E∗is unstable. For E= (., .), by simple calculation, we find that the critical value isτ= .. According to Theorem ., system () is locally asymptotically stable forτ= ∈[,τ

), as

shown in Figure  and Figure . Asτincreases through the critical valueτ, the coexisting equilibriumE∗loses its stability permanently and a family of periodic solutions bifurcate from the coexisting equilibriumE∗ caused by the phenomenon of Hopf bifurcation, as shown in Figure .

In addition, whenτ =τ= ., we getc() = –. + .i,σ= –ReRe((λc(())τ∗))=

. > ,β= Re(c()) = –. < . According to Theorem . in Section , the

bifurcated periodic solutions of system () whenτ

 = . in the whole phase space are

both orbitally asymptotically stable, and the Hopf bifurcations are supercritical forσ> .

However, asτ increases further, with the same initial values, the solution converges to

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Figure 4 The periodic solutions bifurcating from the coexisting equilibriumE∗whereτ= 4.2 >τ0.

Figure 5 The solution of system (1) converges toE0withτ= 4.5 >τ0.

Figure 6 Stability region exploring the dynamics of the system in the (m,τ) parameter space.

6.3 The effect of Allee effect

In order to investigate the effect of Allee effect, we let the parameters be the same as above except formvarying in [., .]. The stability and instability regions of the solution of the system () is shown by plotting the Allee thresholdmversus the critical value of the time delayτ in Figure . Figure  depicts also show the locus of the Hopf bifurcation changes with the change of the two parametersmandτ. We see that as the Allee threshold

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7 Conclusion

In this paper, we considered a delayed diffusive predator-prey system with Beddington-DeAngelis functional response and strong Allee effect. With delayτ as a bifurcation pa-rameter, we showed that a Hopf bifurcation occurs at the critical valueτ. The bifurcating periodic solutions were analyzed in light of the normal form and center manifold. Theo-retical analysis shows that with the strong Allee effect in prey, extinction for both species is always a locally stable equilibrium. Ifu(x,t) <m, then both prey and the predator are destined to go extinct (Theorem .(b)) and ifK(ebdcd)–ad< , then the predator is destined to go extinct (Theorem .(c)). Theoretical analysis and numerical simulations also show that that the increasing delay may cause the prey and predator to go extinct. However, there are still many problems about model () that need to be studied, such as Turing sta-bility, spatiotemporal patterns.

Acknowledgements

The work is sponsored by Natural Science Foundation of Jiangsu Province(CN)(BK20150420).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 30 April 2017 Accepted: 25 June 2017

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2. Deangelis, DL, Goldstein, RA, O’Neill, RV: A model for trophic interaction. Ecology56(4), 881-892 (1975) 3. Wang, W, Wang, H, Li, Z: The dynamic complexity of a three-species Beddington-type food chain with impulsive

control strategy. Chaos Solitons Fractals32(5), 1772-1785 (2007)

4. Allee, WC: Animal Aggregations. University of Chicago Press, Chicago (1931)

5. Courchamp, F, Clutton-Brock, T, Grenfell, B: Inverse density dependence and the Allee effect. Trends Ecol. Evol.14(10), 405-410 (1999)

6. Branzei, R, Tijs, S: A delayed predator-prey model with strong Allee effect in prey population growth. Nonlinear Dyn. 68(1-2), 23-42 (2013)

7. Banerjee, M, Takeuchi, Y: Maturation delay for the predators can enhance stable coexistence for a class of prey-predator models. J. Theor. Biol.412, 154-171 (2017)

8. Hadjiavgousti, D, Ichtiaroglou, S: Allee effect in a prey-predator system. Chaos Solitons Fractals36(2), 334-342 (2008) 9. Wang, M, Kot, M: Speeds of invasion in a model with strong or weak Allee effects. Math. Biosci.171(1), 83-97 (2001) 10. Parshad, RD, Quansah, E, Black, K, Upadhyay, RK, Tiwari, SK, Kumari, N: Long time dynamics of a three-species food

chain model with Allee effect in the top predator. Comput. Math. Appl.71(2), 503-528 (2015)

11. Tobin, PC, Berec, L, Liebhold, AM: Exploiting Allee effects for managing biological invasions. Ecol. Lett.14(6), 615-624 (2011)

12. Wang, J, Shi, J, Wei, J: Predator-prey system with strong Allee effect in prey. J. Math. Biol.62(3), 291-331 (2011) 13. Barton, NH, Turelli, M: Spatial waves of advance with bistable dynamics: cytoplasmic and genetic analogues of Allee

effects. Am. Nat.178(3), 48-75 (2011)

14. Wang, W, Zhu, YN, Cai, Y, Wang, W: Dynamical complexity induced by Allee effect in a predator-prey model. Nonlinear Anal., Real World Appl.16(1), 103-119 (2014)

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Figure

Figure 1 The blue curves are the prey-nullclines and the red lines are the predator-isoclines
Figure 2 Phase portraits of the model system (1). E0 = (0,0), E1 = (5,0) and E∗ = (4.7296,2.7926) are alllocally asymptotically stable for all τ ≥ 0.
Figure 5 The solution of system (1) converges to E0 with τ = 4.5 > τ0.

References

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