**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*)*v*–*θv*,

(.)

where*u*(*t*) and*v*(*t*) represent prey and predator densities at time*t*, respectively.*f*(*u*)
repre-sents the prey growth law in the absence of predators, and*g*(*u*,*v*) is the functional response
of predators to prey density (the average feeding rate of a predator). Parameters*c*and*d*

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

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 conditions*u*() =*u*> ,*v*() =*v*> , which are biologically meaningful.
*m*is 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 time*t*, 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* =*d**u*+*u*( –*u*–

*a(–m)v*

+b(–m)u+cv), *x*∈*,t*> ,
*∂v(x,t)*

*∂t* =*d**v*+*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, ],

where*s*=*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 delay*τ* 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* =*d**u*+*u*( –*u*–

*a(–m)v*

+b(–m)u+cv), *x*∈(,*lπ),t*> ,
*∂v*

*∂t* =*d**v*+*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
(, ). If*E*∗(*u*∗,*v*∗) is a coexisting equilibrium of system (.), then it is easy to obtain that
*v*∗=*u*∗(–u* _{ad}*∗). Then

*u*∗∈(, ) and is a root of

*h*(*u*) =*cu*+*a*( –*bd*)( –*m*) –*cu*–*ad*= . (.)

Obviously,*h*() = –*ad*< and*h*() =*a*[( –*bd*)( –*m*) –*d*]. If*d*< ( –*bd*)( –*m*), then

*h*() > implies that system (.) has at least one coexisting equilibrium.

In this paper, we just suppose system (.) has a coexisting equilibrium point*E∗*(*u∗*,*v∗*).

**2.1 Local stability analysis of the model without diffusion**
If*d*=*d*= , the Jacobian matrix at*E*∗(*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*–*tr**λ*+= . (.)

The characteristic roots are*λ*,=_{}[*tr*±
√

]. If

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

*a**a*–*a**a*> . (H)

If (H) holds, then*E*∗(*u*∗,*v*∗) is locally asymptotically stable if and only if*a*+*sa*< .

Meanwhile, if*a*> when*s* near –*a*/*a*, Eq. (.) has a pair of complex eigenvalues

*α(s*)±*iω(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 at*E*∗(*u*∗,*v*∗) is

*λ*–*trnλ*+*n*(*r*) = , *n*∈N, (.)

where ⎧ ⎨ ⎩

*trn*=*tr*–*n*

*l*(*d*+*d*),

*n*=–*n*

*l*(*d**a*+*d**sa*) +*d**d**n*
*l*,

(.)

and the eigenvalues are

*λ*(n)_{,}(*r*) =*trn*±

*tr*
*n*– *n*

, *n*∈N. (.)

Notice that*a*+*sa*< implies that*trn*< for*n*∈N. Suppose (H) holds, then the

eigenvalues of (.) have negative real parts if*n*> guaranteed by

*d**a*+*d**sa*≤, (.)

or

*d**a*+*d**sa*> , and (*d**a*+*d**sa*)– *d**d*< . (.)

If

*d**a*+*d**sa*> , and (*d**a*+*d**sa*)– *d**d*> (.)

hold, and there exists*k*∈Nsuch that*k*< , then the eigenvalues of (.) have a positive

real part*λ*(k)_{(}_{s}_{), which implies that}_{E}

Assume (H) holds and*a*> . When

*s*=*sn*:= –

*a*

*a*–
*n*

*l*(*d*+*d*)

, *n*∈N (.)

and*n*(*sn*) > , Eq. (.) has purely imaginary values. Obviously,(*s*) = –_{a}a_{}(*a**a*–
*a**a*) > . From (.), we know that there exists*n*∗≥∈Nsuch that*n*(*sn*) > for*n*=

, , . . . ,*n*∗– . Let

*λn*(*s*) =*αn*(*s*)±*iωn*(*s*), *n*= , , . . . ,*n*∗–

be the roots of Eq. (.) satisfying

*αn*(*sn*) = , *ωn*(*sn*) =

*n*(*sn*).

When*s*is near*sn*,

*αn*(*s*) =
*trn*(*s*)

, *ωn*(*s*) =

*n*–*αn*(*s*).

From (.), we know that

*α _{n}*(

*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*≤*n*≤*n*∗– .

Similarly, we can obtain that for system (.), (, ) is unstable and (, ) is locally stable
under condition*d*> ( –*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*

*∂t* –*d*

*∂**u*

*∂x* =*u*

–*u*– *a*( –*m*)*v*
+*b*( –*m*)*u*+*cv*

≤*u*( –*u*).

Use the comparison principle, then

lim

From (.), we can obtain

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

computation, it follows that

= –*I*(*t*) +

**3 Stability analysis of the delayed system**

**3.1 Stability analysis and the existence of Hopf bifurcation**

For simpliﬁcation of notations, use*u*(*t*),*v*(*t*),*u*(*t*–*τ*),*v*(*t*–*τ*) for*u*(*x*,*t*),*v*(*x*,*t*),*u*(*x*,*t*–*τ*),

The characteristic equation is

Let*iω*(ω> ) be a solution of Eq. (.), then

–ω+*iωAn*+*Bn*+*s*(*Cn*–*iωa*)(cos*ωτ*–*i*sin*ωτ*) = .

Then we obtain ⎧

⎨ ⎩

–ω_{+}_{B}

*n*+*sCn*cos*ωτ*–*a**sω*sin*ωτ*= ,
*Anω*–*Cns*sin*ωτ*–*a**sω*cos*ωτ*= .

Then

*ω*+*A** _{n}*–

*Bn*–

*a*

*s*

*ω*+*B** _{n}*–

*C*

*= . (.)*

_{n}sDenote*z*=*ω*, then (.) becomes

*z*+*A** _{n}*–

*Bn*–

*a*

*s*

*z*+*B** _{n}*–

*C*

_{n}*s*= = , (.)

and the roots are

*z*±=

–*A** _{n}*–

*Bn*–

*a*

*s*

±*A*

*n*– *Bn*–*a**s*

– *B*
*n*–*C**ns*

. (.)

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

*Bn*+*sCn*=*n*> ,

*A** _{n}*–

*Bn*–

*a*

*s*=

*d*
*n*

*l* –*a*

+*d*_{}*n*

*l* –*a*
*s*,

and

*Bn*–*sCn*=*d**d*
*n*

*l* +
*n*

*l*(*a**d**s*–*a**d*) –*s*(*a**a*–*a**a*).

Denote

S={n|B*n*–*sCn*< ,*n*∈N}.

ThenSis a ﬁnite set sincelim_{n}_{→∞}*Bn*–*sCn*→+∞.

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

j*π*

*ωn*,*j*∈N,*where*

*τ _{n}*=

*ωn*

arccos*Cnω*

_{–}_{B}

*nCn*+*a**Anω*
*s*(*C*

*n*+*a**ω*)

, *and* *ωn*=

√

*z*+ _{(}_{given in}_{(.)).} _{(.)}

*Proof* Diﬀerentiating two sides of (.) with respect to*τ*, we have

*dλ*

*dτ*
–

=λ+*An*–*sa**e*

–*λτ*

*s*(*Cn*–*λa*)λ*e*–*λτ*

–*τ*
*λ*.

Then

Re

*dλ*

*dτ*
–

*τ*=*τnj*
=Re

λ+*An*–*sa**e*–*λτ*
*s*(*Cn*–*λa*)λ*e*–*λτ*

–*τ*
*λ*

=

*ω*

_{ω}_{+}* _{A}*

*n*– *Bn*–*a**s*

=

*ω*

* _{A}*

*n*– *Bn*–*a**s*

– *B*
*n*–*Cn**s*

> ,

where=*C _{n}*

*s*

*ω*+

*a*

_{}

*s*

*ω*> . ThereforeRe

*λ*(τ

_{n}*nj*) > .

Denote*τ*_{∗}=min*i*∈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
ﬁxed*j*∈Nand*n*∈S, we denote*τ*˜=*τnj*. Let*u*¯(*x*,*t*) =*u*(*x*,*τt*) –*u*∗and*v*¯(*x*,*t*) =*v*(*x*,*τt*) –*v*∗.

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

⎨ ⎩

*∂u*

*∂t* =*τ*[*d**u*+ (*u*+*u∗*)(( –*u*–*u∗*) –

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

*∂t* =*τ*[*d**v*+*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*Lμ*(φ) and*F*(φ,*μ) are given respectively by*

*Lμ*(φ) =*μ*

*a**φ*() +*a**φ*()
*sa**φ*(–) +*sa**φ*(–)

,

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*(–)

for*φ*= (φ,*φ*)*T*∈C.

Consider the linear equation

*dU*(*t*)

*dt* =*τ*˜*DU*(*t*) +*Lτ*˜(*Ut*). (.)

From the previous discussion, we know that±i*ωn*are simply purely imaginary

character-istic values of the linear functional diﬀerential equation

*dz*(*t*)

*dt* = –*τ*˜*D*
*n*

*l**z*(*t*) +*Lτ*˜(*zt*). (.)

From the Riesz representation theorem, there exists a bounded variation function
*ηn*(σ,*τ*˜)–≤*σ*≤ such that –*τ*˜*Dn _{l}*

*φ()+Lτ*˜(φ) =

–*dη*

*n*_{(σ}_{,}_{τ}_{)φ(σ}_{) for}_{φ}_{∈}_{C}_{([–, ],}** _{R}**

_{).}

In fact, we can choose

*ηn*(σ,*τ*) =
⎧
⎪
⎪
⎨
⎪
⎪
⎩

*τE*, *σ*= ,

, *σ*∈(–, ),

–τ*F*, *σ*= –,

(.)

where

*E*=

*a*–*d**n*
*l* *a*

–*d**n*

*l*

, *F*=

*sa* *sa*

. (.)

Let*A*(*τ*˜) denote the inﬁnitesimal generators of a semigroup included by the solutions of
Eq. (.) and*A*∗be the formal adjoint of*A*(*τ*˜) under the bilinear pairing

(ψ,*φ) =ψ()φ() –*

–

*σ*

*ξ*=

*ψ*(ξ–*σ*)*dηn*(σ,*τ*˜)φ(ξ)*dξ*

=*ψ()φ() +τ*˜

–

*ψ*(ξ+ )*Fφ(ξ)dξ* (.)

for*φ*∈*C*([–, ],**R**),*ψ*∈*C*([–, ],**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 of*A*(*τ*˜) and*A*∗associated with±i*ωn*, respectively. Then*P*∗is the adjoint space

Let*p*(θ) = (,*ξ)Teiωnτ σ*˜ (σ∈[–, ]),*q*(*r*) = (,*η)e*–i*ωnτ*˜*r*(*r*∈[, ]) be the eigenfunctions

of*A*(*τ*˜) and*A*∗corresponding to*iωnτ*˜, –*iωnτ*˜, respectively. By direct calculations, we chose

*ξ*=

*a*

*iωn*–*a*+*d*
*n*
*l*

, *η*=–*e*

–i*τ ωn*

*sa*

*iωn*+*a*–*d*
*n*

*l*

.

Let= (,) and∗= (∗,∗)*T*with

(σ) =

*p*(σ) +*p*(σ)

=

Re(*eiωnτ σ*˜ _{)}

Re(ξ*eiωnτ σ*˜ _{)} ,

(σ) =

*p*(σ) –*p*(σ)

*i* =

Im(*eiωnτ σ*˜ _{)}

Im(ξ*eiωnτ σ*˜ _{)}

for*θ*∈[–, ], and

_{}∗(*r*) = *q*(*r*) +*q*(*r*)

=

Re(*e*–i*ωnτ*˜*r*_{)}

Re(η*e*–i*ωnτ*˜*r*_{)} ,

_{}∗(*r*) = *q*(*r*) –*q*(*r*)

*i* =

Im(*e*–i*ωnτ*˜*r*_{)}

Im(η*e*–i*ωnτ*˜*r*_{)}

for*r*∈[, ]. Then we can compute by (.)

*D*∗_{}:=_{}∗,

, *D*∗_{}:=_{}∗,

, *D*∗_{}:=_{}∗,

, *D*∗_{}:=_{}∗,

.

Deﬁne (∗,*) = ( _{j}*∗,

*k*) =

*D*∗_{} *D*∗_{}
*D*∗*D*∗

and construct a new basisfor*P*∗by

= (,)*T*=

∗,–∗.

Then (,*) =I*. In addition, deﬁne*fn*:= (β*n*,*βn*), where

*β _{n}*=

cos*n _{l}x*

, *β*

*n*=

cos*n _{l}x* .

We also deﬁne

*c*·*fn*=*c**βn*+*c**βn*, for*c*= (*c*,*c*)*T*∈C.

Thus the center subspace of linear equation (.) is given by*PCN*C⊕*PS*C, and*PS*C

denotes the complement subspace of*PCN*CinC,

u,*v*:=

*lπ*
*lπ*

*u**v**dx*+

*lπ*
*lπ*

*u**v**dx*

Let*Aτ*˜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* =*Aτ*˜*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*=

*x*
*x*

*fn*+*h*(*x*,*x*,*μ),* (.)

where

*x*
*x*

=,*Ut*,*fn*

,

and

*h*(*x*,*x*,*μ)*∈*PS*C, *h*(, , ) = , *Dh*(, , ) = .

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

*Ut*=

*x*(*t*)
*x*(*t*)

*fn*+*h*(*x*,*x*, ). (.)

Let*z*=*x*–*ix*, and notice that*p*=+*i*. Then we have

*x*
*x*

*fn*= (,)

*z+z*
*i(z–z)*

*fn*=

(*p**z*+*p**z*)*fn*

and

*h*(*x*,*x*, ) =*h*

*z*+*z*

,

*i*(*z*–*z*)

,

.

Hence, Eq. (.) can be transformed into

*Ut*=

(*p**z*+*p**z*)*fn*+*h*

*z*+*z*

,

*i*(*z*–*z*)

,

=

(*p**z*+*p**z*)*fn*+*W*(*z*,*z*), (.)

where

*W*(*z*,*z*) =*h*

*z*+*z*

,

*i*(*z*–*z*)

,

*κ*=*e*–i*τ ωnW*()(–)(*guu*+*ξguv*) +*e*–i*τ ωnW*()(–)(*guv*+*ξgvv*)

+
*e*

*iτ ωn _{W}*()

(–)(*guu*+*ξguv*) +

*e*

*iτ ωn _{W}*()

(–)(*guv*+*ξgvv*),

*κ*=

*e*

–i*τ ωn*_{}_{g}

*uuu*+ (ξ+ ξ)*guuv*+*ξ*

(ξ+*ξ*)*guvv*+ ξ ξ*gvvv*

.

Denote

() –*i*() := (γ *γ*).

Notice that

*lπ*
*lπ*

cos*nx*

*l* *dx*= , *n*= , , , . . . ,

and we have

() –*i*()

*F*(*Ut*, ),*fn*

=*z*

(γ*χ*+*γ**ς*)*τ*˜+*zz*(γ*χ*+*γ**ς*)*τ*˜+

*z*

(γ*χ*+*γ**ς*)*τ*˜

+*z*

_{z}

*τ*˜[γ*κ*+*γ**κ*] +· · ·. (.)

By (.), (.) and (.), we obtain that*g*=*g*=*g*= , for*n*= , , , . . . . If*n*= , we

have

*g*=*γ**τ χ*˜ +*γ**τ ς*˜ , *g*=*γ**τ χ*˜ +*γ**τ ς*˜ , *g*=*γ**τ χ*˜ +*γ**τ ς*˜ .

And for*n*∈N,*g*=*τ*˜(γ*κ*+*γ**κ*).

From [], we have

˙

*W*(*z*,*z*) =*W**z˙z*+*W**zz*˙ +*W**zz*˙+*W**zz*˙+· · ·,

*Aτ*˜*W*(*z*,*z*) =*Aτ*˜*W*
*z*

+*Aτ*˜*W**zz*+*Aτ*˜*W*

*z*

+· · ·,

and*W*(*z*,*z*) satisﬁes

˙

*W*(*z*,*z*) =*Aτ*˜*W*+*H*(*z*,*z*),

where

*H*(*z*,*z*) =*H*
*z*

+*W**zz*+*H*

*z*

+· · ·
=*X**F*(*Ut*, ) –

*,X**F*(*Ut*, ),*fn*

·*fn*

. (.)

Hence, we have

That is,

Similarly, from (.), we have

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*∗=

*d**n*

*l* –*a* –*a*

–*sa* *d**n*
*l* –*sa*

–

.

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

*c*() =
*i*

ω*nτ*˜

*g**g*– |*g*|–|
*g*|

+

*g*, *μ*= –

Re(*c*())

Re(λ(τ*nj*))

,

*T*= –

*ωnτ*˜

Im*c*()

+*μ*Im

*λτ _{n}j*,

*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**

**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 parameter*m*in system (.).
The stabilities of*E*∗(*u*∗,*v*∗) for system (.) are shown in Table . It suggests that prey refuge

has a stabilizing eﬀect on system (.) without and with diﬀusion. But when*m*= ., .,
there are diﬀerences in these two kinds of systems.

**Table 1 Stability of****E**_{∗}(**u**_{∗},**v**_{∗}) for system (4.1)

**m****Without diffusion** **With diffusion**

0, 0.05, 0.1 Unstable Unstable

0.15, 0.2 Stable Unstable

**Figure 1 Phase portraits of system (4.1) when****d****1=****d****2= 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, ﬁx*m*= ., vary the parameter*s*in system (.).*E∗*(., .) is the unique
positive equilibrium, and

*a*≈. > , *a**a*–*a**a*≈. > ,

then (H) holds. If*d*=*d*= and*a*+*sa*< , then*E*∗(*u*∗,*v*∗) is locally asymptotically

stable, and the system undergoes Hopf bifurcation at*E*∗(*u*∗,*v*∗) when*s*= –*a*/*a*(shown

in Figure ). For system (.), if we choose*s*= , then by Theorem .(iii),*E*∗(*u*∗,*v*∗) is

Turing unstable, this is shown in Figure . For system (.), by Theorem .(v), Hopf
bi-furcation occurs when*s*= –*a*/*a*, this is shown in Figure .

**4.2 The case of**

**τ**

**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*)– .),

(.)

**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 if*τ* ∈[,*τ*_{∗}),
then*E∗*(*u∗*,*v∗*) is locally asymptotically stable, this is shown in Figure . By Theorem .,
Hopf bifurcation occurs at the*E*∗(*u*∗,*v*∗) when *τ* =*τ*∗. By Theorem ., we have*μ*≈

**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 delay*τ* may aﬀect the
sta-bility of the positive equilibrium. When it is smaller than the critical value*τ*∗, then prey
and predator will coexist and tend to the interior equilibrium*E∗*(*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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