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R E S E A R C H

Open Access

Dynamical behaviour of a generalist

predator-prey model with free boundary

Zhi Ling

1†

, Lai Zhang

1,2,3†

, Min Zhu

1,4*

and Malay Banerjee

5

*Correspondence: min_zhuly@163.com

1School of Mathematical Science, Yangzhou University, Si Wangting Road, Yangzhou, 225002, China 4Department of Mathematics, Anhui Normal University, Wuhu, 241000, China

Full list of author information is available at the end of the article †Equal contributors

Abstract

In this paper, we consider a free boundary problem describing the invasion of a generalist predator into a prey population. We analytically derive the conditions guaranteeing the existence and uniqueness of the classical solution by means of the Schauder fixed point theorem, and further study the long-time behaviours of these two species. Finally, we numerically investigate the dynamical behaviour during the early invasion stage. Numerical results show that generalist predators are more likely to succeed in alien invasion by reducing the threshold size of the spatial domain of initial invasion, below which invasion fails.

Keywords: generalist predator-prey model; free boundary; long-time behaviour

1 Introduction

Alien invasion has frequently been reported to cause detrimental impacts on native ecosystems functions by altering population fitness, triggering extinction and secondary extinction of native species []. It has been reported that about % of all species in the United States are at risk because of competition with or predation by an alien species [– ]; in other parts of the world, this figure can be even higher [, ].

The invaders can be a specialist feeding on a particular prey population, but it can also be a generalist feeding on multiple food sources, and thus their impacts are generally un-expected. A convenient approach to understand the impact of alien invasion is to develop an appropriate mathematical modelling. In this regards, two approaches have been widely applied. One is to develop ordinary differential equations based on the mean field theory to describe the temporal population dynamics, and the other is to develop reaction-diffusion equations to describe the spatio-temporal population dynamics. While the former ap-proach ignores the spatial aspects of alien invasion, the later suffers from the drawback that alien species can instantaneously spread over the entire spatial domain even if they start to invade in a small area.

To best describe the invasion process, free boundary problems were introduced to biol-ogy [–]. In fact, free boundary problems have received considerable attention in many fields, such as tumor cure [] and wound healing [] in medicine, vapor infiltration of pyrolytic carbon in chemistry [], and expansion of the area infected by the virus in epi-demiology []. To the best of our knowledge, it was first introduced to biology by Lin to describe the process of a predator invading a prey population []. Since then many mathe-matical models with free boundary have been developed to research biological population

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[–]. For example, Du and Lin in [] investigated a diffusive logistic model with a free boundary and proved a spreading-vanishing dichotomy. Wang in [] studied a diffusive logistic equation with a free boundary and sign-changing coefficient and also derived a spreading-vanishing dichotomy. Additionally, Monobe and Wu in [] introduced the free boundary into a reaction-diffusion-advection logistic model in heterogeneous environ-ment, and obtained the long-time behaviour of the solution and the asymptotic spreading speed.

While predator-prey models with free boundary have attracted great attention, almost all of the studied models assume that the predator is a specialist, which means that the predator will certainly go to extinction in the absence of the focal prey. In reality, it might be not true since most predators are very likely to have alternative food sources [, ]. Predators with multiple food sources are called generalist. A question arises how the invasion of generalist predator into a local system affects the population dynamics in a predator-prey model with free boundary, which remains unexplored.

The aim of this paper is to obtain insight into the question raised above through the following generalist predator-prey model:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

utuxx=u( –u) –+auuv :=f(u,v), t> ,x> ,

vtdvxx=+auuv ++cvbvev:=g(u,v), t> ,  <x<h(t),

v(t,x) = , t≥,xh(t),

ux(t, ) =vx(t, ) = , t≥,

h(t) = –μvx(t,h(t)), t≥,

u(,x) =u(x), ≤x<∞,

h() =h, v(,x) =v(x), ≤xh,

()

where the initial valuesu(x) andv(x) are non-negative and satisfy

⎧ ⎨ ⎩

u(x)∈C([,∞)), u() =  and u(x) >  in [,∞);

v(x)∈C([,h)), v() =v(h) and v(x) >  in [,h).

()

In this model,u(x,t) andv(x,t) indicate the population biomass density of the prey and predator at timetand spacex. In the absence of predator, the prey follows the logistic growth and is initially distributed in the whole domain. In the presence of predator, the prey suffers from a predation, following the Holling type II functional response wherea represents the handling time of prey individuals by the predator. The predator feeds on both the focal prey and other food sources, and it suffers at the same time from a back-ground mortality ratee. The parametersbandcindicate the predator’s attack rate and handling time of other preys. To mimic the invading process, we assume the predator is initially distributed only in the rangex∈[,h]. The spreading front of the predator is

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In this model we assume that the alternative food has no dynamics evolution of its own, which means that the alternative food is fixed in constant amounts, availability is signifi-cantly high, and hence unaffected due to consumption (see the references [, ]). This is apparently a simplification to reduce the dimension of the system from three to two, and thus allows the use of the theory of Schauder fixed point to study the consequences of the availability of alternative food. However, this simplification is justified for many arthropod predators because they can rely on plant-provided alternative food sources such as pollen or nectar, the availability of which is unlikely to be influenced by the predator’s consump-tion [–]. The paper is structured as follows. In Secconsump-tion  we derive the condiconsump-tions guaranteeing the existence and uniqueness of the classic solution to the model (). In Sec-tion , we analyse theoretically the long term behaviour of prey and predator. In SecSec-tion , we perform numerical analysis of the population behaviour. The paper ends with a brief conclusion.

2 Global existence, uniqueness and estimate of the solution

In this section, we prove the local existence and uniqueness results to problem () by ap-plying contraction mapping theorem, and then we show the global existence using some suitable estimates.

Theorem . For any given(u,v)satisfying()and anyα∈(, ),there is a T> such

that the problem()admits a unique solution

(u,v,h)C(+α)/,+αDT×C(+α)/,+α(DTC+α/

[,T],

moreover,

uC(+α)/,+α(D

T)+vC(+α)/,+α(DT)+hC+α/[,T]≤C,

where DT={(t,x)R:t∈[,T],x∈[,h(t))},DT ={(t,x)R:t∈[,T],x∈[,∞)},C

and T only depend on h,α,minx∈[,h]u(x),uC([,)),vC([,h ]).

Proof The idea of this proof comes from []. First, we want to keep off the difficulty caused by the free boundary. Takingζ(y) to be a function inC[,∞) and to satisfy

ζ(y) =  if|y–h|<h/,

ζ(y) =  if|y–h|>

h

, ζ

(y)< /h

 ∀y,

and considering the following simple Chang case:

(t,x)→(t,y), wherex=y+ζ(y)h(t) –h

, ≤y<∞,

we find the following. As long as

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the above transformationxyis a diffeomorphism from [,∞) onto [,∞). Moreover, it helps us straighten the free boundary. We obtain from standard calculations that

∂y

∂x=

 +ζ(y)(h(t) –h)≡

k

h(t),y,

y

∂x = –

ζ(y)(h(t) –h)

[ +ζ(y)(h(t) –h)]

k

h(t),y,

–  h(t)

∂y

∂t =

ζ(y)  +ζ(y)(h(t) –h)

k

h(t),y.

Set

u(t,x) =ut,y+ζ(y)h(t) –h

=w(t,y),

v(t,x) =vt,y+ζ(y)h(t) –h

=z(t,y),

and the free boundary problem () can be rewritten as

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

wtkwyy– (k+hk)wy=f(w,z), ≤y<∞,t> ,

ztkdzyy– (kd+hk)zy=g(w,z), ≤y<h,t> ,

z(y,t) = , yh,t≥,

h(t) = –μ∂zy, y=h,t> , ∂w

∂y(,t) = ∂z

∂y(,t) = , t> ,

h() =h,

w(y, ) =u(y) :=w(y), ≤yh,

z(y, ) =v(y) :=z(y), ≤yl,

()

wherek=k(h(t),y),k=k(h(t),y),k=k(h(t),y). It is easy to see

f(w,z) = ⎧ ⎨ ⎩

w( –w) – wz

+aw,  <y<h,

w( –w), yh.

Leth∗= –μv(h),z(y) =  fory>h, andδ=min≤yhu(y) and for  <T

h

(+h∗),

and define

WT=

wCT:w(y, ) =w(y),wwC(T)≤δ/

,

ZT=

zC(T),z(y, )≡ foryh, ≤tT,

z(y) =z(y) for ≤yh,zzC(T)≤δ/

,

HT=

hC[,T],h() =h,h() =h∗,hhC[,T]≤

,

whereT= [,T]×[,h],T = [,T]×[,∞). Clearly,DT=WT×ZT×HTis a

com-plete metric space with metric

D(w,z,h), (w,z,h)

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Observing that, forhandh∈HT,h() =h() =hleads to

h–hC([,T])Th–hC([,T]). ()

By standardLptheory and the Sobolev imbedding theorem, for any (w,z,h)D

Tthe

fol-lowing diffraction problem:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˜

wtkw˜yy– (k+hk)w˜y=f(w,z), t> , ≤y<∞,

˜

ztkd˜zyy– (kd+hk)˜zy=g(w,z), t> , ≤y<h,

˜

wy(,t) =z˜y(,t) = , z(y,t)≡, t> ,h≤y<∞,

˜

w(y, ) =w(y) :=u(y), ≤y<∞,

˜

z(y, ) =z(y) :=v(y), ≤yh,

()

admits a unique bounded solution (u,˜ v)˜ ∈C(+α)/,+α(

TC(+α)/,+α(T) and

˜wC(+α)/,+α(

T)≤C, ()

˜zC(+α)/,+α(

T)≤C, ()

whereCis a constant depending onα,h,δ,uC[,)andvC[,h ].

Defining

˜

h(t) =h–μ

t

 ˜

zy(τ,h), ()

it follows that

˜

h(t) = –μz˜y(t,h),

and subsequently, we haveh˜(t)∈/([,T]) and

h˜(t)C(+α)/[,T]C:=μC. ()

Introducing a mappingF:DTC(TC(TC([,T]) byF(w,z,h) = (w,˜ z,˜ h), we˜

can show thatFhas a fixed point, which is just a solution to the system ().

Taking into accounth˜(t) –h∗=μvy(h, ) –v˜y(h,t)) and using the estimates (), ()

and (), we have

h˜–hC([,T])μh˜Cα/([,T])T

α/μC/,

˜ww(T)≤ ˜wwC(+α)/,(

T)T

(+α)/C

T(+α)/,

˜zz(T)≤ ˜zzC(+α)/,(

T)T

(+α)/C

T(+α)/.

If we takeT≤min{(μC)–/α,C–/(+ α)

 }, the mappingFmapsDTinto itself.

Next we prove that, forT>  sufficiently small,Fis a contraction mapping onDT. Let

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(w˜i,z˜i,h˜i) =F(wi,zi,hi), it follows from (), () and () that

˜wiC(+α)/,+α(

T)≤C, ˜ziC(+α)/,+α(T)≤C, h˜

i(t)/[,T]C.

SettingW=w–w, we find thatW(y,t) satisfies

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Wtk(y,h)Wyy– (k(y,h) +hk(y,h))Wy

= [k(y,h) –k(y,h)]w˜,yy+ [k(y,h) –k(y,h)]w˜,y

+ [hk(y,h) –hk(yh)]w˜,y+f(w,z) –f(w,z), t> ,  <y<∞,

Wy= (t, ) = , W(t,h) = , t> ,

W(,y) = , ≤yh.

Due to|wi–w| ≤δ/ and|zi–z| ≤δ/,i= , , we have +awi> aδ/ and +czi> cδ/,

which results in

wz

 +aw

wz  +aw

a|ww|+|w|

( +aw)( +aw)|

z–z|+ |z|

( +aw)( +aw)|

w–w|

≤ L aδ

|z–z|+|w–w|

and bz

 +cz

bz  +cz

b

( +cz)( +cz)|z

z| ≤

b

cδ|z–z|,

where the constantLdepends onuC([,∞)) andvC([,h]). This implies thatf(w,z)

andg(w,z) are local Lipschitz continuous functions of (w,z). Using theLp estimates for parabolic equations and Sobolev’s imbedding theorem, we obtain

˜w–w˜C(+α)/,+α(

T)

C

w–wC(T)+z–zC(T)+h–hC([,T])

. ()

Similarly, we have the following estimate:

˜z–z˜C(+α)/,+α(

T) ≤C

w–wC(T)+z–zC(T)+h–hC([,T])

, ()

whereCandCdepend onCandC, the local Lipschitz coefficients off andg, as well

as the functionsA,BandCin the definition of transformation (t,y)→(t,x). Taking the difference of the equations forh˜,h˜results in

h˜h˜Cα/([,T])μ˜z,y–z˜,y/,(

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Combining inequalities (), (), () and (), and assuming thatT≤, we obtain

˜w–w˜C(+α)/,+α(

T)+˜z–z˜C(+α)/,+α(T)+h˜

–h˜/([,T])

C

w–wC(T)+z–zC(T)+h

–hC[,T]

,

withCdepending onC,C, andμ. Hence, denoting

T:=min

,

 C

/α

, (μC)–/α,C–/(+α),

h

( +h∗)

,

we have

˜w–w˜C(T)+˜z–˜zC(T)+h˜

–h˜C([,T])

T(+α)/ ˜w–w˜C(+α)/,+α(

T)+˜z–˜zC(+α)/,+α(T)

+/h˜h˜Cα/([,T])

C/

w–wC(T)+z–zC(T)+h

–hC([,T])

≤ 

w–wC(T)+z–zC(T)+h

–hC([,T])

.

This shows thatF is a contraction mapping inDT ifT is small enough. It follows from

the contraction mapping theorem thatFhas a unique fixed point (w,z,h) inDT. That is,

(w,z,h) is the unique solution of the problem () and accordingly (u,v,h) is the unique so-lution of the problem (). Moreover, utilising theLpestimate, we have additional regularity

of the solutionu(t,x)C(+α)/,+α(D

T),v(t,x)C(+α)/,+α(DT) andh(t)C+α/([,T]).

The proof of Theorem . is completed.

It is observed that there exists a timeT such that the solution exists in time interval [,T]. Since the global existence theorem depends on a prior estimate with respect to h(t), in what follows, we aim to derive a prior estimate for any solution of the problem (). Theorem . If a–<e<b,the solution(u,v;h)of the free boundary problem()satisfies

 <u(t,x)Mfor≤tT, ≤x<∞,

 <v(x,t)Mfor≤tT, ≤x<h(t),

and

 <h(t)≤Mfor <tT,

where the constant Miare independent of T for i= , , .

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Sinceusatisfies ⎧

⎪ ⎪ ⎨ ⎪ ⎪ ⎩

utuxx=u( –u) –+auuv ,  <tT,x> ,

ux(t, ) = , u(t,h(t)) = , t≥,

u(,x) =u(x), ≤x<∞,

we getu≤max{u∞, }:=Mby the maximum principle. Similarly, asvsatisfies

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

vtdvxx=+auuv ++cvbvev,  <tT,  <x<h(t),

vx(t, ) = , t≥,

v(,x) =v(x), ≤x<∞,

we also havev≤max{v∞,c( b e–+MaM

– )}:=M, where b e–+MaM

–  >  thanks toa–<

b<e.

Considering the transformation

y=x/h(t), w(t,y) =u(t,x), z(t,y) =v(t,x),

similar to the proof of Lemma . in [], we obtainzy(t, ) <  for  <tT. Thus,h(t) =

μvx(t,h(t)) >  in (,T].

Now we demonstrate thath(t)≤Mwith someMindependent ofT. To see this, let

Mbe a positive constant,

M=

(t,x) :  <t<T,h(t) –M–<x<h(t),

and construct an auxiliary function

ω(x,t) =M

Mh(t) –xMh(t) –x.

In the following, we will chooseMto guaranteeω(x,t)v(t,x) inM.

Straightforward calculation shows that

ωt= MMh(t)

 –Mh(t) –x≥, –ωxx= MM,

u  +au+

b  +cve

v≤(/a+be)M.

It follows that

ωtωxx≥MM≥(/a+be)M≥

u  +au

b  +cve

v,

ifMa+b–e

 . On the other hand,

ω

t,h(t) –M

=M≥v

t,h(t) –M

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Next, we will further choose someMsuch thatω(,x)v(x) forx∈[h–M–,h]. We

divide [h–M–,h] into two subsets: [h–M–,h– (M)–] and [h– (M)–,h]. For

x∈[h– (M)–,h],

ωx(,x) = –MM

 –M(h–x)

≤–MM≤v(x).

withM≥vC([,h])

M . This implies

ω(,x)v(x), forx

h– (M)–,h

,

due toω(,h) =v(h) = . Forx∈[h– (M)–,h– (M)–], we also have

ω(,x)≥

M≥ vC([,h])M–≥v(x).

Therefore, by choosing

M=max

h

,

a+be

 ,

vC([,h ])

M

,

we can apply the comparison principle toωandv, and conclude thatωvinM.

Recall-ing the free boundary condition in () yields

h(t) = –μvx

t,h(t)≤–μωx

t,h(t)M:= μMM,

whereMis independent ofT. The proof is completed.

Theorem . shows that the free boundary is strictly monotonically increasing with timet, which indicates that the domain invaded by the predatorvis gradually expand-ing with time.

Moreover, the other inequalities in Theorem ., in whichMiis independent ofT, imply

that we can extend the solution of problem () to the global range. The proof of the fol-lowing theorem is omitted here and the interested reader can refer to that of Theorem . in [] or Theorem . in [].

Theorem . Problem()admits a unique solution(u,v;h),which exists globally in[,∞) with respect to t.

3 Long-time behaviour of(u,v)

To begin with, we present the definitions of spreading or vanishing of the predator popu-lation and the related comparison principle.

Definition . Ifh∞=∞andlim inft→+∞v(t,·)C([,h(t)])> , we call the predator

spread-ing, which means that the predator can survive and spread to the whole domain [,∞). While ifh∞<∞andlim inft→+∞v(t,·)C([,h(t)])= , we call the predatorvanishing, which

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Lemma .(Comparison principle) Assume that T∈(,∞),hC[,T],uC,(D T)

and vC(GT)∩C,(GT),where

DT=

(t,x)R,  <tT,  <x<∞,

GT=

(t,x)R,  <tT,  <x<h(t).

If(u,v;h)satisfies

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

utuxxu( –u),  <tT,  <x<∞,

vtdvxxv(a +be),  <tT,  <x<h(t),

v(t,h(t)) = ,  <tT,

ux(t, )≤, vx(t, )≤,  <tT,

h(t)≥–μvx(t,h(t)), h()h,  <tT,

u(,x)u(x),  <x<∞,

v(,x)v(x), ≤xh,

then the solution(u,v,h)of problem()satisfies

h(t)h(t),  <tT,

u(t,x)u(t,x),  <tT,  <x<∞,

v(t,x)v(t,x),  <tT, ≤xh(t).

In fact, (u,v,h) is called an upper solution of problem (). We continue to exhibit the following two lemmas, whose proof can be referred to Lemma . in [] and Theorem . in [], respectively. We omit the details here.

Lemma . If h∞<∞,then there exists a constant K> such that the solution(u,v,h)of

()satisfies

v(t,·)C[,h(t)]K, ∀t> ; lim

t→∞h

(t) = .

Lemma . Suppose that a–<e<b.If h

∞<∞,then h∞≤,where=π

d/(be).

The following theorem presents the sufficient condition for the vanishing of predator species.

Theorem . Suppose a–<e<b and h<,then there existsμ> such that h∞<∞

whenμμ,and moreover,the solution(u,v,h)of problem()satisfies

lim

t→∞v(t,·)C([,h(t)])= , ()

lim

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Proof To apply Lemma ., we will construct the suitable upper solution (u,v;h). Define

h(t) =h

 +δδ

e

σt

, t≥,

u(t,x) =et

et–  +  u∞

–

, t≥,

v(t,x) =MeδtV

x h(t)

, ≤xh(t),

whereV(y) =cos(π

y),δis sufficiently small and meets withh( +δ) <, bothσandM

are positive and to be determined later. Obviously, it follows that

ut=u( –u), t> , u() =u∞≤u(x).

Meanwhile, we carry out some direct calculations

vtdvxxv

a+be

MVeσt

π

 

d ( +δ)h

a+be

σ

. ()

Once we choose

σ:= 

π

 

d ( +δ)h

a+be

,

we will get () non-negative.

By choosingMlarge enough, one hasv(,x)v(x) for allx∈[,h]. Additionally, there

existsμsuch that

h(t)≥–μvx

t,h(t), ∀μμ.

According to Lemma ., it follows that h(t)h(t),u(t,x)u(t,x) andv(t,x)v(t,x). Thush∞≤h(∞) =h( +δ) <∞. Naturally, conclusion () can be deducted from

Propo-sition . in [] and Lemma .. It remains to prove ().

Sincelimt→∞u(t) = , by the comparison principle¯ u(t,x)≤ ¯u(t) for allt∈[,∞) and

x∈[,∞), we havelim supt→∞u(t,x)≤ uniformly in [,∞).

On the other hand, with the help of conclusion () and the condition thatv(t,x)≡ fort>  andx∈/[,h(t)), we see that, for any given  <ε, there exists such that

v(t,x) <εand

v  +au<

ε

 +au<ε,

fort>,x∈[,∞). For any givenεandH> , letandbe determined by Lemma A.

in [], where>. Then the functionu(t,x) satisfies

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

utuxxu( –uε), t>,  <x<,

ux(t, ) = , u(t,lε) > , t>,

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Owing to Lemma A. in [],lim inft→∞u(t,x)≥( –ε) –εuniformly on [,H]. Then

from the arbitrariness of the ε and H one derives that lim inft→∞u(t,x)

 –ε uniformly in the compact subset of [,∞). Since ε >  is arbitrary, we deduce

that lim inft→∞u(t,x)≥  uniformly on any bounded subset of [,∞). The proof is

complete.

The following theorem exhibits the sufficient condition for the spreading of predator species.

Theorem . Suppose that a–<e<b and h

<. Then there existsμ>  such that

h∞=∞whenμ>μ,and,moreover,if b<min{ce( –a), ( +c)(e+a )},then the solution

(u,v,h)of problem()satisfies

lim

t→∞u(t,x) =u

, lim

t→∞v(t,x) =v

,

uniformly on any compact subset of [,∞), where(u∗,v∗)is the positive solution of the following system:

⎧ ⎨ ⎩

 –u+auv = ,

u +au+

b

+cv–e= .

Proof The second equation in problem () implies

vtdvxx≥–ev,  <x<h(t).

Considering the following auxiliary problem:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

wtdwxx= –ew, t> ,  <x<g(t),

w(t,g(t)) = , t> ,

w(t,x) = , g(t) = –μwx(t,x), t> ,x=g(t),

w(,x) =v(x),  <x<h,

g() =h.

The comparison principle yieldsg(t)h(t) andw(t,x)v(t,x) on [,∞)×[,h(t)]. Sim-ilar to the proof of Lemma . in [], there exists a constantμ>  such thatg()≥

for allμμ. Therefore,

h∞= lim

t→∞h(t)tlim→∞g(t)g()≥, ∀μμ,

from which together with Lemma . one deduces thath∞=∞.

This idea of the remainder proof comes from []. We will construct the following iter-ation sequences.

Step . The constructions ofuandv.

SetM=max{M,M}, whereMiis determined by Theorem .,i= , . For any given

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that ⎧ ⎨ ⎩

utuxxu( –u), t>T,  <x<,

ux(t, ) = , u(t,lε)≤M, t>T.

Noting thatu(T,x) >  in [,lε], we deduce from Lemma A. in [] thatlimt→∞u(t,x)

 +εuniformly on [,L]. Then the arbitrariness ofεandLshows

lim sup

t→∞ u(t,x)≤ :=u uniformly on the compact subset of [,∞). ()

Consequently, for any givenL> ,  <δ,ε, we combine Lemma A. in [] with (), h∞=∞, to find that there exist>LandT>Tsuch that

u(t,x)u+δ, h(t) >lε, for∀t>T,  <x<.

Hence,vobeys ⎧ ⎨ ⎩

vtdvxxv(+cvb – (e–+a(uu+ε+ε))), tT,  <x<,

vx(t, ) = , v(t,lε)≤M, t>T.

Sincev(T,x) >  in [,lε], we can apply Lemma A. in [] to deducelimt→∞v(t,x)

c(

b e– u+δ

+a(u+δ)

– ) +εuniformly on [,L]. Because of the arbitrariness ofε,δandL, we have

lim sup

t→∞

v(t,x)≤  c

b eu

+au

– 

:=v uniformly on the compact subset of [,∞). ()

Meanwhile, the condition thata–<e<bmeans that  <v<.

Step . The constructions ofuandv.

Letbe determined by Lemma A. in []. By (), there isT>Tsuch that

v(t,x)v+δ, ∀t>T,  <x<.

Thus,usatisfies ⎧

⎨ ⎩

utuxxu( –v–δu), tT,  <x<,

ux(t, ) = , u(t,lε)≥, tT.

Due tou(T,x) >  in [,lε], it follows from Lemma A. in [] thatlim inft→∞u(t,x)

 –v–δεuniformly on [,L]. Similarly,

lim inf

t→∞ u(t,x)≥ –v:=u uniformly on the compact subset of [,∞). ()

Additionally, for any givenL> ,  <δ,ε, owing to Lemma A. in [], () and h∞=∞, we know that there exist>LandT>Tsuch that

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Therefore,vsatisfies ⎧

⎨ ⎩

vtdvxxv(+cvb – (e–+a(uu–δ

–δ))), t>T,  <x<,

vx(t, ) = , v(t,lε)≥, t>T.

Asv(T,x) >  in [,lε], based on Lemma A. in [], one can obtain

lim inf

t→∞ v(t,x)

c

b eu–δ

+a(u–δ)

– 

ε,

uniformly on [,L]. Similarly, we have

lim inf

t→∞ v(t,x)

c

b eu

+au

– 

:=v,

uniformly on the compact subset of [,∞). Step . The constructions ofuandv.

For any givenL> ,  <δ,ε, letbe selected by Lemma A. in []. With the help

of () and (), one acquire that there isT>Tsuch that

v(t,x)v+δ, u(t,x)u–δ, ∀t>T,  <x<,

which makesusatisfy ⎧

⎨ ⎩

utuxxu( –+a(uv+δ

–δ)–u), t>T,  <x<,

ux(t, ) = , u(t,lε)≥, t>T.

Similar to Step , we have

lim inf

t→∞ u(t,x)≥ –

v

 +au :=u uniformly on the compact subset of [,∞). ()

Next, for any givenL> ,  <δ,ε, by Lemma A. in [], () andh∞=∞, there

existsT>Tsuch that

u(t,x)uδ, h(t) >lε, ∀t>T,  <x<.

Thereuponvobeys ⎧

⎨ ⎩

vtdvxxv(+cvb – (e– u– δ

+a(u–δ))), t>T,  <x<,

vx(t, ) = , v(t,lε)≥, t>T.

Similar to Step , we get

lim inf

t→∞ v(t,x)

c

b eu

+au

– 

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Step . The constructions ofuandv.

For any givenL> ,  <δ,ε, letbe selected by Lemma A. in []. In view of ()

and (), there isT>Tsuch that

u(t,x)u+δ, v(t,x)v–δ, ∀t>T,  <x<,

which implies that

⎧ ⎨ ⎩

utuxxu( –+a(uv–δ+δ)–u), t>T,  <x<,

ux(t, ) = , u(t,lε)≤M, t>T.

Similar to Step , we deduce

lim sup

t→∞ u(t,x)≥ –

v  +au

:=u uniformly on the compact subset of [,∞). ()

Accordingly, for anyL> ,  <δ,ε, letis given by Lemma A. in []. Recalling ()

andh∞=∞, there isT>Tsuch that

u(t,x)u+δ, h(t) >lε, ∀tT,  <x<.

So it follows that ⎧

⎨ ⎩

vtdvxxv(+cvb – (e–+a(uu+δ+δ))), t>T,  <x<,

vx(t, ) = , v(t,lε)≤M, t>T.

Similar to Step  again, it follows that

lim sup

t→∞

v(t,x)≤ c

b eu

+au

– 

:=v

uniformly on the compact subset of [,∞).

Step . Repeating the above procedure, we can find the four sequences{ui},{ui},{vi}and

{vi}satisfying

ui≤lim inf

t→∞ u(t,x)≤lim supt→∞ u(t,x)ui,

vi≤lim inf

t→∞ v(t,x)≤lim supt→∞ v(t,x)vi,

uniformly on the compact subset of [,∞), where

ui=  –

vi  +aui–

, ui=  – vi  +aui–,

vi=

c

b eui

+aui – 

, vi= c

b eui

+aui – 

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It is easy to see that{ui}andviare monotone non-increasing and bounded,{ui}and{vi}

are monotone non-decreasing and bounded, so all of those limits exist, which are denoted byu,u,vandv, respectively. Leti→ ∞, we deduce that

u=  – v

 +au, u=  – v  +au,

v= c

b e+auu – 

, v= c

b e+auu – 

.

()

We now claim thatu=u. In fact, we infer by () that

b c·

uu

[e( +au) –u][e( +au) –u]= (u–u)

 +a(u+u) –a. ()

Ifu=u, () gives that

b c·

[e( +au) –u][e( +au) –u]=  +a(u+u) –a.

Under the condition thatea> , the left side is less than or equal to b

ce, but the right side is

greater than  –a, which leads to a contradiction to condition thatb<ce( –a). Therefore, u=u, andv=vby (). The proof is complete.

4 Numerical analysis

In this section, we conduct numerical analysis to understand the role of generality of the predator in the invasion process. The above analytical analysis shows the existence of global solutions to the system () and further the conditions under which the invader can either spread or vanish as time advances. While these conditions qualitatively show the long-time behaviour of the prey and predator species, they are unfortunately not suffi-ciently clear because it is unknown whenh∞is finite or infinite. Understanding the long-time behaviour is numerically challenging, since the spatial domain is infinitely long. To circumvent this difficulty, we consider a spatial domain of finite range and focus on the in-vasion process of the early stage, which is thought to play a significant role in determining the ultimate fate of the predator species.

To this aim, we restrict the prey species to living in a spatial domain x∈[,L] and L= , and impose an homogeneous Neumann boundary condition atx=L. To min-imise the impact of the right boundary on the potential results, we consider the inva-sion stage fromt=  to t= , at the end of which the invasive species (predator) is far from reaching the right boundary. We employ the numerical scheme in [] to per-form numerical simulation. Moreover, we assume a relatively slow diffusion process by setting d= . to ensure that the predator will not reach the right boundary att= . For the initial condition of the two species, we assume that, prior to the invasion, the prey species is homogeneously distributed in the spatial domain with an equilibrium den-sityu(x, ) = . The predator species starts to invade in a small area with a low density u(x) = .,x∈(,h).

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Figure 1 The bigger predator’s attack rate.Graphical illustration of the invasion process of the generalist predator into the native prey species. The left two panels describe the spatio-temporal distribution of population density while the right-most panel shows population biomass. The bold red curve in the middle panel indicates the free boundary. Parameter values area= 3,b= 4,c= 1,e= 1,h0= 1,μ= 0.8.

Figure 2 The smaller predator’s attack rate.Graphical illustration of the invasion process of the generalist predator into the native prey species. The left two panels describe the spatio-temporal distribution of population density while the right-most panel shows population biomass. The bold red curve in the middle panel indicates the free boundary. Parameter values area= 3,b= 3,c= 1,e= 1,h0= 1,μ= 0.8.

the predator does not grow. After this transient period, population biomass of the preda-tor starts to increase and the domain starts to grow linearly. Decreasing the generality of the predator by changingb=  tob= , we found that invasion is unsuccessful (Figure ). This contrasting result shows that generalist predator is more likely to invade successfully. We tried longer simulation time but the conclusion remains unaltered. Numerical simu-lations also show that the size of the initial spatial domain of the predator has a significant impact on invasion. Figure  shows that increasing the size of the initial invading domain (i.e.,h= .) facilitates invasion and Figure  shows: the larger the better.

5 Conclusion

In summary, we consider an invasion scenario of a generalist predator into prey, which is formulated by a predator-prey model with free boundary. In our model (), the so-called free boundaryx=h(t) characterises the change of the expanding front for the predator v, as well as the long-time behaviours of the predatorvand preyuare focussed on. We conclude that the predator will gradually vanish if the limit of front functionh(t) is finite, while the predator will spread if the limit is infinite under some assumptions, and it fur-ther will stabilise to an equilibrium. These analytical findings indicate that the change of invasion region to predator can determine whether it successfully invade or not.

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Figure 3 The smaller size of the initial invading domain.Graphical illustration of the invasion process of the generalist predator into the native prey species. The left two panels describe the spatio-temporal distribution of population density while the right-most panel shows population biomass. The bold red curve in the middle panel indicates the free boundary. Parameter values area= 3,b= 3,c= 1,e= 1,h0= 1.2,

μ= 0.8.

Figure 4 The larger size of the initial invading domain.Graphical illustration of the invasion process of the generalist predator into the native prey species. The left two panels describe the spatio-temporal distribution of population density while the right-most panel shows population biomass. The bold red curve in the middle panel indicates the free boundary. Parameter values area= 3,b= 3,c= 1,e= 1,h0= 2,μ= 0.8.

a specialist predator. There is a threshold size of the initial spatial domain for a successful invasion below which invasion can fail. Moreover, we also noticed that there is a time period during which invasive specie increases in density slowly. The existence of such a time period implies that invasion can possibly be unsuccessful if we take into account stochastic effects. We conclude that the more general the invasive species, the more likely to be successful the invasion is.

The free boundary in model () describes the one-dimensional environment. With re-gard to realities of situation, however, we realize the two or three-dimensional case to more match the reality. Naturally, there will be more challenges in mathematical analy-sis and numerical simulation for multi-dimensional free boundary, and we will pay more attention to these questions in future work.

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Acknowledgements

Zhi Ling and Lai Zhang gratefully acknowledge the financial support by the PRC Grant NSFC 11571301.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Author details

1School of Mathematical Science, Yangzhou University, Si Wangting Road, Yangzhou, 225002, China.2College of Mathematics and Systems Science, Shandong University of Science and Technology, Qian Wangang Road, Qingdao, 24105, China.3Department of Mathematics, Jiangsu University, ZhenJiang, 212013, China.4Department of Mathematics, Anhui Normal University, Wuhu, 241000, China. 5Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, 208016, India.

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Received: 25 May 2017 Accepted: 11 September 2017

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Figure

Figure 1 The bigger predator’s attack rate. Graphical illustration of the invasion process of the generalistpredator into the native prey species
Figure 3 The smaller size of the initial invading domain. Graphical illustration of the invasion process ofthe generalist predator into the native prey species

References

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