A diffusive predator prey system with prey refuge and gestation delay

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R E S E A R C H

Open Access

A diffusive predator-prey system with prey

refuge and gestation delay

Ruizhi Yang

*

, Haoyu Ren and Xue Cheng

*Correspondence:

yangruizhi529@163.com Department of Mathematics, Northeast Forestry University, Harbin, Heilongjiang 150040, China

Abstract

The dynamics of a diffusive predator-prey system with prey refuge and gestation delay is investigated in this paper. For a non-delay system, global stability, Turing instability and Hopf bifurcation are studied. For a delay system, time delay induced instability and Hopf bifurcation are discussed. By the theory of normal form and the center manifold method, the direction and stability of the bifurcating periodic solution are discussed.

MSC: 34K18; 35B32

Keywords: predator-prey; prey refuge; delay; Turing instability; Hopf bifurcation

1 Introduction

Population biology is an important subject in both ecology and mathematical ecology. And predator-prey model is one of the most interesting and popular areas in population biology. Many scholars have done a lot of work in it and derived some important results [–].

Generally, prey-predator models can be written as the following form: ⎧

⎨ ⎩

du

dt =uf(u) –g(u,v)v, dv

dt =cg(u,v)vθv,

(.)

whereu(t) andv(t) represent prey and predator densities at timet, respectively.f(u) repre-sents the prey growth law in the absence of predators, andg(u,v) is the functional response of predators to prey density (the average feeding rate of a predator). Parameterscandd

represent the conversion rate from prey to predator and the death rate of predator. In predator-prey models, the functional response of predators to prey density is essen-tial, and it can enrich the dynamics of predator-prey systems. In ecology, many factors, such as prey escape ability, predator hunting ability and the structure of the prey habi-tat, can affect functional responses [, ]. Generally, functional responses can be divided into the following types: prey-dependent (such as Holling I-III []) and prey-predator-dependent (such as Beddington-DeAngelis [], Crowley-Martin [], Hassel-Varley []). Recently, more and more researchers have suggested that prey-predator-dependent functional responses are more suitable and can enrich the dynamics of predator-prey sys-tems [–]. Hsu et al. studied the global property of a general predator-prey model with

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Hassell-Varley type functional response []. Cantrell and Cosner discussed some dynam-ical properties of predator-prey models with Beddington-DeAngelis functional response []. Tripathi et al. considered the permanence, non-permanence, local asymptotic sta-bility and global asymptotic stasta-bility of equilibria of delayed predator-prey models with Crowley-Martin functional response []. Skalski and Gilliam suggest that Beddington-DeAngelis or Hassell-Varley functional responses are suitable for the case where predator feeding rate becomes independent of predator density at high prey density, and Crowley-Martin model is suitable for the case where predator feeding rate is decreased by higher predator density even when prey density is high [].

In predator-prey models, prey refuge is one of the important factors, and many re-searchers have studied it. In [], Sharma and Samanta studied a Leslie-Gower model with disease and refuge in prey, including the positivity and boundedness of solutions, and the stability of equilibria. In [], Tripathi et al. considered a delayed predator-prey model with Beddington-DeAngelis functional response and prey refuge. They studied local and global asymptotic stability of various equilibria and time delay induced Hopf bifurcation. In [], Chen et al. investigated the effect of prey refuge on a Leslie-Gower predator-prey model. They suggest that prey refuge has no influence on the persistent property of this model, but it can affect the prey and predator densities. Most of works suggest that prey refuge has a stabilizing effect on the prey-predator model [–]. In [], Tripathi et al. studied a predator-prey model with Beddington-DeAngelis type functional response incorporating a prey refuge, that is,

⎧ ⎨ ⎩

du

dt =u( –u

a(–m)v +b(–m)u+cv), dv

dt =v(

e(–m)u +b(–m)u+cv–θ),

(.)

with the initial conditionsu() =u> ,v() =v> , which are biologically meaningful. mis the prey refuge rate. Tripathi et al. studied local and global stability of various bound-ary equilibria and coexisting equilibria. Using some data, they also discussed the inverse problem of estimation of a model parameter.

In recent years, since predators and their preys distribute inhomogeneously in differ-ent spatial locations at timet, many researchers have studied predator-prey systems with diffusion term. Predator-prey systems with diffusion term may exhibit richer dynami-cal properties, including Turing instability, pattern formation, spatially inhomogeneous periodic solutions etc. [–]. In [], Tang and Song considered a delayed diffusive predator-prey model with herd behavior and analyzed the stability and Hopf bifurcation. In [] the authors considered the spatial, temporal and spatiotemporal patterns of dif-fusive predator-prey models with mutual interference. In [], Jia and Xue discussed the effects of the self- and cross-diffusion on positive steady states for a generalized predator-prey system. In this paper, we will study the effect of diffusion on system (.).

Motivated by above, we propose the following system:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

∂u(x,t)

∂t =du+u( –u

a(–m)v

+b(–m)u+cv), x,t> , ∂v(x,t)

∂t =dv+sv(

(–m)u(t–τ)

+b(–m)u(t–τ)+cv(t–τ)–d), x,t> , ux(x,t) = , vx(x,t) = , x∂,t> ,

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

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wheres=e,d=θ/e,= (,lπ), (l> ). In system (.), we assume that the regionis closed, with no prey and predator species entering and leaving the region at the boundary. We also assume that the reproduction of predator population after predating the prey will not be instantaneous. There is a time delayτ required for gestation of predator.

The rest of this paper is arranged as follows. We study the stability property of the non-delayed system in the next section and discuss the non-delayed system in Section . Then we give some numerical simulations. At last, we end this paper with a brief conclusion.

2 Stability analysis of the non-delayed system Without delay, system (.) becomes

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

∂u

∂t =du+u( –u

a(–m)v

+b(–m)u+cv), x∈(,lπ),t> , ∂v

∂t =dv+sv( (–m)u

+b(–m)u+cv–d), x∈(,lπ),t> , ux(,t) =vx(,t) = , ux(lπ,t) =vx(lπ,t) = , t> , u(x, ) =u(x,t)≥, v(x, ) =v(x,t)≥, x∈[,lπ].

(.)

Obviously, system (.) has a trivial equilibrium (, ) and a predator-free axial equilibrium (, ). IfE∗(u∗,v∗) is a coexisting equilibrium of system (.), then it is easy to obtain that v∗=u∗(–uad∗). Thenu∗∈(, ) and is a root of

h(u) =cu+a( –bd)( –m) –cuad= . (.)

Obviously,h() = –ad<  andh() =a[( –bd)( –m) –d]. Ifd< ( –bd)( –m), then

h() >  implies that system (.) has at least one coexisting equilibrium.

In this paper, we just suppose system (.) has a coexisting equilibrium pointE∗(u∗,v∗).

2.1 Local stability analysis of the model without diffusion Ifd=d= , the Jacobian matrix atE∗(u∗,v∗) is

J=

a a sa sa

,

where

a=bd( –u∗) –u∗, a=d

c( –u∗)/( –m) –a,

a= ( –bd)( –u∗)/a, a= –cd( –u∗)/a( –m).

Obviously,a< . The characteristic equation is

λ–trλ+= . (.)

The characteristic roots areλ,=[tr± √

]. If

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then the characteristic roots have negative real parts. Make the following hypothesis:

aa–aa> . (H)

If (H) holds, thenE∗(u∗,v∗) is locally asymptotically stable if and only ifa+sa< .

Meanwhile, ifa>  whens near –a/a, Eq. (.) has a pair of complex eigenvalues

α(siω(s), where

α(s) = 

(a+sa), ω(s) =  

s(aa–aa)

and

α(–a/a) = , α(–a/a) =a/, ω(–a/a) > .

Theorem . When d=d= ,assume(H)holds.

(i) Ifa+sa< ,thenP(u∗,v∗)is locally asymptotically stable; (ii) Ifa> ,Hopf bifurcation occurs atP(u∗,v∗)whens= –a/a.

2.2 Turing instability and Hopf bifurcation

For system (.), the characteristic equation atE∗(u∗,v∗) is

λ–trnλ+n(r) = , n∈N, (.)

where ⎧ ⎨ ⎩

trn=tr–n

l(d+d),

n=–n

l(da+dsa) +ddnl,

(.)

and the eigenvalues are

λ(n),(r) =trn±

trn– n

 , n∈N. (.)

Notice thata+sa<  implies thattrn<  forn∈N. Suppose (H) holds, then the

eigenvalues of (.) have negative real parts ifn>  guaranteed by

da+dsa≤, (.)

or

da+dsa> , and (da+dsa)– dd< . (.)

If

da+dsa> , and (da+dsa)– dd>  (.)

hold, and there existsk∈Nsuch thatk< , then the eigenvalues of (.) have a positive

real partλ(k)(s), which implies thatE

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Assume (H) holds anda> . When

s=sn:= –

a

a– n

l(d+d)

, n∈N (.)

andn(sn) > , Eq. (.) has purely imaginary values. Obviously,(s) = –aa(aa– aa) > . From (.), we know that there existsn∗≥∈Nsuch thatn(sn) >  forn=

, , . . . ,n∗– . Let

λn(s) =αn(siωn(s), n= , , . . . ,n∗– 

be the roots of Eq. (.) satisfying

αn(sn) = , ωn(sn) =

n(sn).

Whensis nearsn,

αn(s) = trn(s)

 , ωn(s) =

nαn(s).

From (.), we know that

αn(sn) = a

 < . (.)

Theorem . Assume(H)holds.

(i) Ifa+sa< and(.) (or(.))hold,thenE∗(u∗,v∗)is locally asymptotically

stable;

(ii) Ifa+sa< and(.)hold,andk> fork∈N,thenE∗(u∗,v∗)is locally

asymptotically stable;

(iii) Ifa+sa< and(.)hold,andk< fork∈N,thenE∗(u∗,v∗)is Turing

unstable;

(iv) Hopf bifurcation occurs atE∗(u∗,v∗)whens=sn,for≤nn∗– .

Similarly, we can obtain that for system (.), (, ) is unstable and (, ) is locally stable under conditiond> ( –bd)( –m).

2.3 Global stability of (1, 0) and (u,v)

Theorem . When d> ( –bd)( –m), (, )is globally asymptotically stable.

Proof From (.), we can obtain that

∂u

∂td

u

∂x =u

 –ua( –m)v  +b( –m)u+cv

u( –u).

Use the comparison principle, then

lim

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From (.), we can obtain

For the sake of completeness, we give the following theorem about the global stability ofE(u∗,v∗) by the proof process similar to that in [].

computation, it follows that

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= –I(t) +

3 Stability analysis of the delayed system

3.1 Stability analysis and the existence of Hopf bifurcation

For simplification of notations, useu(t),v(t),u(tτ),v(tτ) foru(x,t),v(x,t),u(x,tτ),

The characteristic equation is

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Let(ω> ) be a solution of Eq. (.), then

–ω+iωAn+Bn+s(Cniωa)(cosωτisinωτ) = .

Then we obtain ⎧

⎨ ⎩

–ω+B

n+sCncosωτasinωτ= , AnωCnssinωτacosωτ= .

Then

ω+An– Bnas

ω+BnCns= . (.)

Denotez=ω, then (.) becomes

z+An– Bnas

z+BnCns=  = , (.)

and the roots are

z±=  

An– Bnas

±A

n– Bnas

– BnCns

. (.)

Assume (H) and condition (i) or (ii) in Theorem . holds. Then

Bn+sCn=n> ,

An– Bnas=

dn

l –a

+dn

l –a  s,

and

BnsCn=ddn

l + n

l(adsad) –s(aa–aa).

Denote

S={n|BnsCn< ,n∈N}.

ThenSis a finite set sincelimn→∞BnsCn→+∞.

Lemma . Assume(H)and condition(i)or(ii)in Theorem.holds.Then Eq. (.)has a pair of purely imaginary roots±iωn(n∈S)atτnj=τn+

jπ

ωn,j∈N,where

τn=  ωn

arccosCnω

B

nCn+aAnωs(C

n+aω)

, and ωn=

z+ (given in(.)). (.)

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Proof Differentiating two sides of (.) with respect toτ, we have

–

=λ+Ansae

λτ

s(Cnλa)λeλτ

τ λ.

Then

Re

–

τ=τnj =Re

λ+Ansaeλτ s(Cnλa)λeλτ

τ λ

= 

ω

ω+A

n– Bnas

= 

ω

A

n– Bnas

– BnCns

> ,

where=Cnsω+asω> . ThereforeReλnnj) > .

Denoteτ=mini∈S{τi}. According to the above analysis, we have the following theorem.

Theorem . Assume(H)and condition(i)or(ii)in Theorem.holds. (i) E∗(u∗,v∗)is locally asymptotically stable forτ∈[,τ∗).

(ii) E∗(u∗,v∗)is unstable forτ >τ∗.

(iii) τ=τnj(n∈S,j∈N)are Hopf bifurcation values of system(.).

3.2 Stability and direction of Hopf bifurcation

We give detailed computation about Hopf bifurcation using the method in [, ]. For fixedj∈Nandn∈S, we denoteτ˜=τnj. Letu¯(x,t) =u(x,τt) –u∗andv¯(x,t) =v(x,τt) –v∗.

For convenience, we drop the bar. Then (.) can be written as ⎧

⎨ ⎩

∂u

∂t =τ[du+ (u+u∗)(( –uu∗) –

a(–m)(v+v∗) +b(–m)(u+u∗)+c(v+v∗))], ∂v

∂t =τ[dv+s(v+v∗)(

(–m)(u(t–)+u∗)

+b(–m)(u(t–)+u∗)+c(v(t–)+v∗)–d)].

(.)

Let

τ=τ˜+μ, u(t) =u(·,t),

u(t) =v(·,t) and U= (u,u)T.

In the phase spaceC:=C([–, ],X), (.) can be rewritten as

dU(t)

dt =τ˜DU(t) +L˜τ(Ut) +F(Ut,μ), (.)

where(φ) andF(φ,μ) are given respectively by

(φ) =μ

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

,

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with

f(φ,μ) = (τ˜+μ)F(φ,μ),F(φ,μ)

T

,

F(φ,μ) =

φ() +u

 –φ() –u

a( –m)(φ() +v∗)

 +b( –m)(φ() +u∗) +c(φ() +v∗)

aφ() –aφ(),

F(φ,μ) =s

φ() +v∗

( –m)(φ(–) +u∗)

 +b( –m)(φ(–) +u∗) +c(φ(–) +v∗)–d

saφ(–) –saφ(–)

forφ= (φ,φ)T∈C.

Consider the linear equation

dU(t)

dt =τ˜DU(t) +˜(Ut). (.)

From the previous discussion, we know that±iωnare simply purely imaginary

character-istic values of the linear functional differential equation

dz(t)

dt = –τ˜D n

lz(t) +˜(zt). (.)

From the Riesz representation theorem, there exists a bounded variation function ηn(σ,τ˜)–≤σ≤ such that –τ˜Dnlφ()+Lτ˜(φ) =

 –

n,τ)φ(σ) forφC([–, ],R).

In fact, we can choose

ηn(σ,τ) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

τE, σ= ,

, σ∈(–, ),

–τF, σ= –,

(.)

where

E=

a–dnla

 –dn

l

, F=

 

sa sa

. (.)

LetA(τ˜) denote the infinitesimal generators of a semigroup included by the solutions of Eq. (.) andA∗be the formal adjoint ofA(τ˜) under the bilinear pairing

(ψ,φ) =ψ()φ() –

–

σ

ξ=

ψ(ξ–σ)dηn(σ,τ˜)φ(ξ)

=ψ()φ() +τ˜

–

ψ(ξ+ )Fφ(ξ)dξ (.)

forφC([–, ],R),ψC([–, ],R).±iωnτ˜is a pair of simple purely imaginary

eigen-values of A(τ˜) and A∗. Let P and P∗ be the center subspace, that is, the generalized eigenspace ofA(τ˜) andA∗associated with±iωn, respectively. ThenP∗is the adjoint space

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Letp(θ) = (,ξ)Teiωnτ σ˜ (σ∈[–, ]),q(r) = (,η)e–iωnτ˜r(r∈[, ]) be the eigenfunctions

ofA(τ˜) andA∗corresponding toiωnτ˜, –iωnτ˜, respectively. By direct calculations, we chose

ξ= 

a

iωna+dnl

, η=–e

–iτ ωn

sa

iωn+a–dn

l

.

Let= (,) and∗= (∗,∗)Twith

(σ) =

p(σ) +p(σ)

 =

Re(eiωnτ σ˜ )

Re(ξeiωnτ σ˜ ) ,

(σ) =

p(σ) –p(σ)

i =

Im(eiωnτ σ˜ )

Im(ξeiωnτ σ˜ )

forθ∈[–, ], and

∗(r) = q(r) +q(r)

 =

Re(e–iωnτ˜r)

Re(ηe–iωnτ˜r) ,

∗(r) = q(r) –q(r)

i =

Im(e–iωnτ˜r)

Im(ηe–iωnτ˜r)

forr∈[, ]. Then we can compute by (.)

D:=∗,

, D:=∗,

, D:=∗,

, D:=∗,

.

Define (∗,) = (j∗,k) =

D D D∗D∗

and construct a new basisforP∗by

= (,)T=

∗,–∗.

Then (,) =I. In addition, definefn:= (βn,βn), where

βn=

cosnlx

 , β

n=

 cosnlx .

We also define

c·fn=cβn+cβn, forc= (c,c)T∈C.

Thus the center subspace of linear equation (.) is given byPCNC⊕PSC, andPSC

denotes the complement subspace ofPCNCinC,

u,v:= 

uvdx+

uvdx

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Let˜denote the infinitesimal generator of an analytic semigroup induced by the linear

system (.), and Eq. (.) can be rewritten as the following abstract form:

dU(t)

dt =˜Ut+R(Ut,μ), (.)

where

R(Ut,μ) =

⎧ ⎨ ⎩

, θ∈[–, );

F(Ut,μ), θ= .

(.)

By the decomposition ofC, the solution above can be written as

Ut=

xx

fn+h(x,x,μ), (.)

where

xx

=,Ut,fn

,

and

h(x,x,μ)PSC, h(, , ) = , Dh(, , ) = .

In particular, the solution of (.) on the center manifold is given by

Ut=

x(t) x(t)

fn+h(x,x, ). (.)

Letz=x–ix, and notice thatp=+i. Then we have

xx

fn= (,)

z+zi(z–z)

fn=

(pz+pz)fn

and

h(x,x, ) =h

z+z

 ,

i(zz)

 , 

.

Hence, Eq. (.) can be transformed into

Ut=

(pz+pz)fn+h

z+z

 ,

i(zz)

 , 

= 

(pz+pz)fn+W(z,z), (.)

where

W(z,z) =h

z+z

 ,

i(zz)

 , 

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(14)
(15)

κ=e–iτ ωnW()(–)(guu+ξguv) +e–iτ ωnW()(–)(guv+ξgvv)

+ e

iτ ωnW()

(–)(guu+ξguv) +

 e

iτ ωnW()

(–)(guv+ξgvv),

κ=

 e

–iτ ωng

uuu+ (ξ+ ξ)guuv+ξ

(ξ+ξ)guvv+ ξ ξgvvv

.

Denote

() –i() := (γ γ).

Notice that

cosnx

l dx= , n= , , , . . . ,

and we have

() –i()

F(Ut, ),fn

=z

(γχ+γς)τ˜+zz(γχ+γς)τ˜+

z

(γχ+γς)τ˜

+z

z

τ˜[γκ+γκ] +· · ·. (.)

By (.), (.) and (.), we obtain thatg=g=g= , forn= , , , . . . . Ifn= , we

have

g=γτ χ˜ +γτ ς˜ , g=γτ χ˜ +γτ ς˜ , g=γτ χ˜ +γτ ς˜ .

And forn∈N,g=τ˜(γκ+γκ).

From [], we have

˙

W(z,z) =Wz˙z+Wzz˙ +Wzz˙+Wzz˙+· · ·,

˜W(z,z) =˜W z

 +˜Wzz+˜W

z

 +· · ·,

andW(z,z) satisfies

˙

W(z,z) =˜W+H(z,z),

where

H(z,z) =H z

 +Wzz+H

z

 +· · · =XF(Ut, ) –

,XF(Ut, ),fn

·fn

. (.)

Hence, we have

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(17)

That is,

Similarly, from (.), we have

(18)

That is,

W(θ) = i

iωnτ˜

p(θ)g–p(θ)g

+E.

Similarly, we have

E=τ˜E

χ

ς

cos

nx l

,

where

E∗=

dn

l –a –a

sa dnl –sa

–

.

Thus, we can obtain the following quantities which determine the property of Hopf bifur-cation:

c() = i

ω˜

gg– |g|–| g|

+

g, μ= –

Re(c())

Re(λ(τnj))

,

T= –

ωnτ˜

Imc()

+μIm

λτnj, β= Re

c()

.

(.)

Theorem . For any critical valueτnj,we have the following results.

(i) Whenμ> (resp.<),the Hopf bifurcation is forward(resp.backward).

(ii) Whenβ< (resp.>),the bifurcating periodic solutions on the center manifold are

orbitally asymptotically stable(resp.unstable).

(iii) WhenT> (resp.T< ),the period increases(resp.decreases).

4 Numerical simulations

4.1 The case of

τ

= 0

Consider the following system:

⎧ ⎨ ⎩

∂u

∂t = .u+u( –u

.(–m)v +(–m)u+.v), ∂v

∂t = v+ v(

(–m)u

+(–m)u+.v– .),

(.)

where= .π. To study the effect of prey refuge, vary the parametermin system (.). The stabilities ofE∗(u∗,v∗) for system (.) are shown in Table . It suggests that prey refuge

has a stabilizing effect on system (.) without and with diffusion. But whenm= ., ., there are differences in these two kinds of systems.

Table 1 Stability ofE(u,v) for system (4.1)

m Without diffusion With diffusion

0, 0.05, 0.1 Unstable Unstable

0.15, 0.2 Stable Unstable

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Figure 1 Phase portraits of system (4.1) whend1=d2= 0.Left:s= 10, and the initial condition (0.5, 8.2).

Right:s= 15 and the initial condition (0.5, 8.2).

Figure 2 For system (4.1),s= 15 and the initial condition is (0.5, 8.2).Left component:u(x,t) (Turing unstable). Right component:v(x,t) (Turing unstable).

Now, fixm= ., vary the parametersin system (.).E∗(., .) is the unique positive equilibrium, and

a≈. > , aa–aa≈. > ,

then (H) holds. Ifd=d=  anda+sa< , thenE∗(u∗,v∗) is locally asymptotically

stable, and the system undergoes Hopf bifurcation atE∗(u∗,v∗) whens= –a/a(shown

in Figure ). For system (.), if we chooses= , then by Theorem .(iii),E∗(u∗,v∗) is

Turing unstable, this is shown in Figure . For system (.), by Theorem .(v), Hopf bi-furcation occurs whens= –a/a, this is shown in Figure .

4.2 The case of

τ

= 0

Consider the following system: ⎧

⎨ ⎩

∂u

∂t = .u+u( –u

.(–.)v +(–.)u+.v), ∂v

∂t = .v+ .v(

(–.)u(t–τ)

+(–.)u(t–τ)+.v(t–τ)– .),

(.)

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Figure 3 For system (4.1),s= 10 and the initial condition is (0.5, 8.2).Left component:u(x,t) (Stable). Right component:v(x,t) (Stable).

Figure 4 For system (4.2),τ= 0 and the initial condition is (0.5, 12).Left component:u(x,t) (Stable). Right component:v(x,t) (Stable).

Figure 5 For system (1.3),τ= 8 and the initial condition is (0.5, 12).Left component:u(x,t) (Stable). Right component:v(x,t) (Stable).

By computation, we obtain τ ≈.. By Theorem ., we know that ifτ ∈[,τ), thenE∗(u∗,v∗) is locally asymptotically stable, this is shown in Figure . By Theorem ., Hopf bifurcation occurs at theE∗(u∗,v∗) when τ =τ∗. By Theorem ., we haveμ≈

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Figure 6 For system (1.3),τ= 9.2 and the initial condition is (0.5, 12).Left component:u(x,t) (Periodic solution). Right component:v(x,t) (Periodic solution).

5 Conclusion

In this paper, a diffusive predator-prey system with prey refuge and gestation delay is in-vestigated. For a non-delay system, the conditions inducing Turing instability and Hopf bifurcation are given. The global stabilities of equilibria (, ) and (u∗,v∗) are also

consid-ered. By numerical simulation, we conclude that prey refuge has a stabilizing effect on the reaction-diffusion system similar to the ODE system. But, when prey refuge is equal to some values, Turing instability may occur. For a delayed system, time induced instability and Hopf bifurcation are investigated. We conclude that time delayτ may affect the sta-bility of the positive equilibrium. When it is smaller than the critical valueτ∗, then prey and predator will coexist and tend to the interior equilibriumE∗(u∗,v∗). When the delay passes through some critical values, the positive equilibrium loses its stability and Hopf bifurcations occur. Then prey and predator will exhibit oscillatory behavior.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The idea of this research was introduced by RY. All authors contributed to the main results and numerical simulations.

Acknowledgements

The authors wish to express their gratitude to the editors and reviewers for the helpful comments. This research is supported by the National Nature Science Foundation of China (No. 11601070) and Fundamental Research Funds for the Central Universities (No. 2572017BB16).

Publisher’s Note

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

Received: 20 February 2017 Accepted: 3 May 2017

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Figure

Table 1 Stability of E∗(u∗,v∗) for system (4.1)

Table 1.

Stability of E u v for system 4 1 . View in document p.18
Figure 1 Phase portraits of system (4.1) when d1 = d2 = 0. Left: s = 10, and the initial condition (0.5,8.2).Right: s = 15 and the initial condition (0.5,8.2).

Figure 1.

Phase portraits of system 4 1 when d1 d2 0 Left s 10 and the initial condition 0 5 8 2 Right s 15 and the initial condition 0 5 8 2 . View in document p.19
Figure 2 For system (4.1), s = 15 and the initial condition is (0.5,8.2). Left component: u(x,t) (Turingunstable)

Figure 2.

For system 4 1 s 15 and the initial condition is 0 5 8 2 Left component u x t Turingunstable . View in document p.19
Figure 3 For system (4.1), s = 10 and the initial condition is (0.5,8.2). Left component: u(x,t) (Stable).Right component: v(x,t) (Stable).

Figure 3.

For system 4 1 s 10 and the initial condition is 0 5 8 2 Left component u x t Stable Right component v x t Stable . View in document p.20
Figure 4 For system (4.2), τ = 0 and the initial condition is (0.5,12). Left component: u(x,t) (Stable)

Figure 4.

For system 4 2 0 and the initial condition is 0 5 12 Left component u x t Stable . View in document p.20
Figure 5 For system (1.3), τ = 8 and the initial condition is (0.5,12). Left component: u(x,t) (Stable)

Figure 5.

For system 1 3 8 and the initial condition is 0 5 12 Left component u x t Stable . View in document p.20
Figure 6 For system (1.3), τsolution). Right component: = 9.2 and the initial condition is (0.5,12)

Figure 6.

For system 1 3 solution Right component 9 2 and the initial condition is 0 5 12 . View in document p.21

References

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