R E S E A R C H
Open Access
Qualitative analysis on a predator-prey model
with Ivlev functional response
Gaihui Guo
1*, Bingfang Li
2and Xiaolin Lin
1*Correspondence:
1College of Science, Shaanxi
University of Science and Technology, Xi’an, 710021, P.R. China Full list of author information is available at the end of the article
Abstract
The diffusive predator-prey model with Ivlev functional response is considered under homogeneous Dirichlet boundary conditions. Firstly, we investigate the bifurcation of positive solutions and derive the multiplicity result for
γ
suitably large. Furthermore, a range of parameters for the uniqueness of positive solutions is described in one dimension. The method we used is based on a comparison principle, Leray-Schauder degree theory, global bifurcation theory and generalized maximum principle.MSC: 35K57
Keywords: predator-prey model; Ivlev functional response; bifurcation; uniqueness
1 Introduction
In this paper, we are concerned with the following reaction-diffusion system:
⎧ ⎪ ⎨ ⎪ ⎩
–u=u(a–u) –v( –e–γu), x∈, –v=v(c–v+d( –e–γu)), x∈,
u=v= , x∈∂,
(.)
whereis a bounded domain inRN(N≥) with smooth boundary∂,u,vrepresent the population density of prey and predator, respectively.ais the natural growth rate of prey,d
is the conversion rate of a consumed prey to a predator,γ is the efficiency of the predator for capturing prey.a,c,dandγ are constants witha,dpositive andγ non-negative;c
may change sign andc> indicates the predator has other food sources. This is a prey dependent predator-prey model with the Ivlev-type functional response –e–γu, which was originally introduced by Ivlev in [].
The predator-prey model has long been one of the dominant themes due to its universal existence and importance. Both ecologists and mathematicians are interested in the Ivlev-type predator-prey model; see [–] for example. The existence and uniqueness of limit cycle for the Ivlev response predator-prey system were studied in [, ]. The conditions for the permanence of the Ivlev system and the existence and stability of a positive periodic solution were investigated in []. The dynamical behavior analysis of the Ivlev response predator-prey systems was discussed in [, –]. To our knowledge, there are few works on such a type of functional response in the reaction-diffusion system. Under Neumann boundary conditions, the spatial pattern formation of the model was carried out by using Hopf bifurcation in []. Under Dirichlet boundary conditions, a sufficient and necessary condition for the existence of positive solutions to the model was obtained in [].
Now, we introduce some notations and basic facts which will be often used later. LetX
be the Banach space
X=u∈C(¯) :u(x) = ,x∈∂.
LetP={u∈X:u> inand∂νu< on∂}be the usual positive cone inX, whereνis the outward unit normal vector on∂and∂ν=∂/∂ν. Forq(x)∈C(¯), letλ(q) <λ(q)≤
λ(q)≤ · · · be all eigenvalues of the following problem:
–φ+q(x)φ=λφ in, φ= on∂.
It follows from [] thatλ(q) is simple andλ(q) is strictly increasing in the sense that
q≤qandq≡qimpliesλ(q) <λ(q). Whenq(x)≡, we denoteλi() byλifor the sake of convenience. Moreover, we denote by (> ) the eigenfunction corresponding to
λwith normalization = .
For anya>λ, it is well known that the problem
–u= (a–u)u in, u= on∂
has a unique positive solution which we denote byθa. It is also known that the mapping
a→θa is strictly increasing, continuously differentiable in (λ,∞), and thatθa→ uni-formly on¯ asa→λ. Moreover, <θa<ain. Therefore, ifa>λ, then (.) has a semi-trivial solution (θa, ). Similar results hold with respect to another semi-trivial solu-tion (,θc) wheneverc>λ. We extend the definition ofθcby takingθc≡ ifc≤λ.
This work mainly aims at establishing the existence, multiplicity and uniqueness of posi-tive solutions to (.). More precisely, a sufficient and necessary condition for the existence of positive solutions is given whenc≤λ, and whenc>λ, the multiplicity of positive so-lutions is obtained under the assumption thatγis suitably large. Ifγis suitably small, then we get the uniqueness of positive solutions in one dimension.
The rest of this paper is organized as follows. In Section , by calculating the indices of fixed points, we obtain sufficient conditions for the existence of positive solutions to (.). In Section , by investigating the bifurcation of positive solutions emanating from the semi-trivial solution (θa, ;c), we give a sufficient and necessary condition for the existence of positive solutions to (.) and establish the multiplicity result of positive solutions when
γ is suitably large. In Section , assuming that= (p,q) is an interval, we find that (.) has at most one positive solution whenγ(c+d)≤.
2 The existence of positive solutions
In this section, we establish the existence and nonexistence of positive solutions to (.). A necessary condition anda prioriestimate are firstly given. The proofs are standard and will be omitted.
Lemma . If(.)has a positive solution,then we have
Lemma . Assume that(u,v)is a positive solution of(.).Then(u,v)satisfies
<u<θa<a, <v<θc+d<c+d in.
In addition,v>θcif c>λ.
Next, we set up the fixed point index theory for later use. LetEbe a real Banach space and
Wbe a closed convex set ofE. Fory∈W, defineWy={x∈E:y+γx∈Wfor someγ> } andSy={x∈Wy: –x∈Wy}. LetF:W→W be a compact operator with a fixed point
y∈W, and denote byLthe Fréchet derivative ofFaty. ThenLmapsWyinto itself. We say thatLhas propertyαonWyif there existt∈(, ) andw∈Wy\Sysuch thatw–tLw∈Sy. For an open subsetU⊂W, defineindexW(F,U) =index(F,U,W) =degW(I–F,U, ), whereIis the identity map. Ifyis an isolated fixed point ofF, then the fixed point index ofFatyinWis defined byindexW(F,y) =index(F,U(y),W), whereU(y) is a small open neighborhood ofyinW.
Lemma .(See []) Assume that I–L is invertible on Wy.
(i) IfLhas propertyαonWy,thenindexW(F,y) = .
(ii) IfLdoes not have propertyαonWy,thenindexW(F,y) = (–)σ,whereσis the sum of
algebra multiplicities of the eigenvalues ofLwhich are greater than.
Denote byr(L) the spectral radius of a linear operatorL.
Lemma .(See []) Let q(x)∈C(¯)and let M be a positive constant such that–q(x) +
M> on¯.Then we have the following conclusions:
(i) λ(q(x)) < ⇒r[(–+M)–(–q(x) +M)] > ;
(ii) λ(q(x)) > ⇒r[(–+M)–(–q(x) +M)] < ;
(iii) λ(q(x)) = ⇒r[(–+M)–(–q(x) +M)] = . Now we introduce the following notations:
(i) E:=C(¯)⊕C(¯), whereC(¯) ={u∈C() :u(x) = on∂}; (ii) W:=P⊕P, whereP={u∈C(¯) :u(x) > in};
(iii) D:={(u,v)∈E:u<a,v<c+d}; (iv) D:= (intD)∩W.
From Lemma ., we see that all the non-negative solutions of (.) must be inD. For anyτ∈[, ], define a positive compact operatorAτ :D→Wby
Aτ(u,v) = (–+M)–
τu(a–u) –v( –e–γu) +Mu
τv(c–v+d( –e–γu)) +Mv ,
whereMis large such thatmax{a+ (c+d)γ,d}<M. It follows from the standard elliptic regularity theory that Aτ is a completely continuous operator. Observe that (.) has a
positive solution inW if and only ifA:=A has a positive fixed point inD. Ifa,c>λ, then (, ), (θa, ), (,θc) are the only non-negative fixed points ofAwhich are not positive. The corresponding indices inW can be calculated in the following lemmas.
(i) IndexW(A,D) = .
(ii) Ifc=λ,thenindexW(A, (, )) = .
(iii) Ifc>λ(–d( –e–γ θa)),thenindex
W(A, (θa, )) = .
(iv) Ifc<λ(–d( –e–γ θa)),thenindexW(A, (θa, )) = .
Proof (i) SinceAτ has no fixed point onD, the degreedegW(I–Aτ,D, ) is well defined.
It is easy to see that all fixed points ofAτare inD. Therefore, by the homotopy invariance
of degree,degW(I–Aτ,D, ) is independent ofτ. Then
indexW
A,D=deg
W
I–A,D, =degWI–Aτ,D,
=degWI–A,D, . (.)
Observing that (.) has only the trivial solution (, ) whenτ= , we have
degWI–A,D, =indexW
A, (, ). (.)
LetL=A(, ). Then
L= (–+M)–
M
M .
It is easy to see thatr(L) < by Lemma .. This implies thatI–Lis invertible onW(,) andLdoes not have propertyαonW(,). By Lemma .,indexW(A, (, )) = . It follows from (.) and (.) thatindexW(A,D) = .
(ii) It is easy to observe thatW(,)=W,S(,)={(, )}. LetL=A(, ). Then
L= (–+M)–
a+M
c+M .
Assume thatL(ξ,η) = (ξ,η) for some (ξ,η)∈W(,). Then
–ξ=aξ, –η=cη, x∈, ξ=η= , x∈∂.
Sincea=λ,c=λ, we haveξ=η= . ThusI–Lis invertible onW(,).
Note thata>λ. By Lemma ., we know thatra:=r[(–+M)–(a+M)] > , andra is the principal eigenvalue of the operator (–+M)–(a+M) with the corresponding eigenfunctionφ> . Sett= /ra. Thent∈(, ) and (I–tL)(φ, ) = (, )∈S(,). This shows thatLhas propertyα. By Lemma .,indexW(A, (, )) = .
(iii) Let y= (θa, ). Then Wy={(ξ,η)∈E,η≥}, Sy={(ξ, ) :ξ ∈C(¯)}. Set L= A(θ
a, ). Then we have
L= (–+M)–
Assume thatL(ξ,η) = (ξ,η) for some (ξ,η)∈Wy. Then real eigenvalues ofLwhich are greater than .
Assume thatλ> is an eigenvalue ofLwith a corresponding eigenfunction (ξ,η). Then simple calculations yield
This contradiction shows thatLhas no eigenvalues being greater than . Consequently,
Similarly, we can obtain the following lemma.
Lemma . Assume that c>λ.
(i) Ifa>λ(γ θc),thenindexW(A, (,θc)) = .
(ii) Ifa<λ(γ θc),thenindexW(A, (,θc)) = .
By the additivity property of the index, the existence of positive solutions to (.) is ob-tained.
Theorem . (i)If a>λ(γ θc),c>λ,then(.)has at least a positive solution. (ii)If a>λ,λ(–d( –e–γ θa)) <c<λ,then(.)has at least a positive solution.
Proof Argue by contradiction. Suppose that (.) has no positive solution.
(i) Ifa>λandλ(–d( –e–γ θa)) <c<λ, then by Lemma . and the additivity property
The contradiction implies that (.) has at least a positive solution inD.
(ii) Ifa>λ(γ θc) andc>λ, then by Lemmas ., . and the additivity property of the
The contradiction implies that (.) has at least a positive solution inD.
3 Bifurcation and multiplicity of positive solutions
In this section, by discussing the bifurcations of positive solutions by usingaandcas the main bifurcation parameters, respectively, we establish the multiplicity of positive solu-tions whenγ is suitably large. First, we show that (.) has no positive solution whencis sufficiently large.
Lemma . If(.)has a positive solution,then there exists a sufficiently large constant M> such thatλ(–d( –e–γ θa)) <c<M. considering the equation ofu, we find
Fixinga>λand takingcas a bifurcation parameter, we shall obtain positive solutions bifurcating from the semi-trivial solution (θa, ).
Theorem . Assume that a>λ.Let˜c=λ(–d( –e–γ θa)).Then(θ
a, ;c˜)is a bifurcation
point of the positive solution to(.).Moreover,there exists a constantδ> and a C-curve (u(s),v(s);c(s)) : (,δ)→X×X×R such that
(i) (u(s),v(s))is a positive solution of(.)withc=c(s)for eachs∈(,δ]and
u(s) =θa+s
˜
φ+φ(s), v(s) =sψ˜ +ψ(s), (.)
whereψ˜ is a positive eigenfunction corresponding to˜cwithψ˜dx= ,
˜
φ= (+a– θa)–(( –e–γ θa)ψ˜) < ,(φ,ψ)∈Z,Z⊕span{(φ˜,ψ˜)}=X×X;
(ii) c() =˜c,(u(),v()) = (θa, ),andφ() =ψ() = .
By the classic Crandall-Rabinowitz bifurcation theorem in [], one can obtain Theo-rem . easily. So the proof is omitted here. One can refer to [, ] for similar arguments.
Now we state a sufficient condition for the existence of positive solutions as follows.
Theorem . If the following relationship holds:
a>λ(γ θc) and c>λ
–d –e–γ θa, (.)
then(.)has at least a positive solution.
Proof Fixinga>λ and takingcas the main bifurcation parameter, we can obtain a su-percritical bifurcating branch of positive solutions to (.), which emanates from the semi-trivial solution (θa, ) at the value ofc˜=λ(–d( –e–γ θa)). The existence was given in Theo-rem .. It suffices to show the bifurcation direction. To this end, substitute (u(s),v(s);c(s)) given by (.) into the second equation of (.), divide bys, differentiate with respect tos
and sets= , which leads to
–ψ() =c˜+d –e–γ θaψ() +c() –ψ˜ +dγe–γ θaψ˜ψ˜. Now, multiply byψ˜ and integrate overto get
–
ψ()ψ˜dx=
˜
ψc˜+d –e–γ θaψ() dx+
c() –ψ˜ +dγe–γ θaψ˜dx.
Hence, we have
c() =
˜
ψdx–dγ
e–γ θaφ˜ψ˜dx.
Noting thatφ˜< , we obtainc() > , which shows that the bifurcating branch is super-critical.
can see [, ] for similar arguments. From Lemma ., we know that the parametercis bounded. And froma prioriestimates given by Lemma ., it follows thatu∞andv∞
are also bounded. Hence, we claim that the continuum of positive solutions bifurcating from (θa, ;c˜) cannot remain in the interior ofP×P, which implies that there must exist
ˆ
c(>˜c), at which one of the components of the continuum of positive solutions vanishes. Letcnbe a strictly increasing sequence converging toˆc∈(˜c,∞) for which (.) has at least a positive solution (un,vn) such thatun→ orvn→ asn→ ∞. If we denote by (uˆ,vˆ) the limit of (un,vn) asn→ ∞, then we have the following three cases:
(i) (uˆ,vˆ) = (,θˆc), (ii) (uˆ,vˆ) = (θa, ), (iii) (uˆ,vˆ) = (, ).
Since (, ) is non-degenerate, (iii) is excluded. Setv˜n=vn/vn∞. Thenv˜nsatisfies –v˜n=
cn–vn+d
–e–γunv˜
n, ˜vn∞= in, v˜n= on∂. Lettingn→ ∞, we have
–v˜=cˆ+d –e–γ θav˜, ˜v
∞= in, v˜= on∂.
It follows fromv˜∈P–{}thatcˆ=λ(–d( –e–γ θa)) =c˜, which contradicts the global
bifur-cation theorem. (ii) is also excluded. Hence, we know that (i) holds true. Setu˜n=un/un∞. Thenu˜nsatisfies
–u˜n=
a–un–vn
–e–γun un
˜
un, ˜un∞= in, u˜n= on∂.
Lettingn→ ∞, we have
–u˜= (a–γ θˆc)u˜, ˜u∞= in, u˜= on∂.
It follows fromu˜∈P–{}thata=λ(γ θcˆ). Now, we know thatcˆis uniquely determined
bya=λ(γ θc). Observe thatc<cˆimplies thata>λ(γ θc). Hence, ifa>λ(γ θc) andc>c˜,
then (.) has at least a positive solution.
Remark . The condition (.) is better than those obtained in Theorem .. In particu-lar, (.) includes the casec=λ. Note that Theorem . tells us nothing whenc=λ. Since
I–Lis not invertible onW(,)whenc=λ, Lemma . is not satisfied and so we cannot use it to get the index of the fixed point (, ).
Fixa>λ and takecas the bifurcation parameter, then by the proof of Theorem ., we can obtain a supercritical bifurcating branch from the point (λ(–d( –e–γ θa));θ
a, ). Moreover, resorting to the global bifurcation theory, we get the maximal continuum of positive solutions, which tells us the range of the parametercisλ(–d( –e–γ θa)) <c<cˆ, wherecˆis determined uniquely bya=λ(γ θˆc). Hence, a sufficient and necessary condition for the existence of positive solutions can be stated in the following remark.
Remark . Assumea>λ. Then (.) has a positive solution if and only ifλ(–d( –
e–γ θa)) <c<ˆc, whereˆcis determined uniquely bya=λ(γ θ
ˆ
In the casec>λ, Theorem . tells us that ifa>λ(γ θc), then (.) has at least a positive solution. A natural question is whether (.) has a positive solution ifa∈(λ,λ(γ θc)]. In fact, making use of bifurcation theory and degree theory, we can solve this problem and further establish the multiplicity of positive solutions to (.) whenγ is suitably large. We first takeaas a bifurcation parameter and give the bifurcation solutions emanating from the semi-trivial solution (,θc).
Theorem . Assume c>λ.Then(,θc;λ(γ θc))is a bifurcation point of the positive
solu-tion to(.).Moreover,there exist a constantδ> and a C-curve(a(s);U(s),V(s)) : (,δ)→
R×X×X such that
(i) (U(s),V(s))is a positive solution of(.)witha=a(s)for eachs∈(,δ]and
U(s) =s˜ + (s), V(s) =θc+s
˜
+(s),
where ˜ is the positive eigenfunction corresponding toλ(γ θc)with
˜
dx= ,
˜
=dγ(––c+ θc)–(θc˜) > ,( ,)∈Z,Z⊕span{(˜,˜)}=X×X;
(ii) a() =λ(γ θc),(U(),V()) = (,θc),and () =() = ;
(iii) a(s)has the derivativea() =ρ(γ),whereρ(γ)is defined by
ρ(γ) =γ
˜
˜dx– γ
θc˜dx. (.)
Using the above theorem, we can obtain the following multiplicity result of positive so-lutions to (.).
Theorem . Assume that c>λ.Letγ=
˜ ˜dx/
θc˜dx.Ifγ >γ,then there exists a constant a∗ ∈(λ,λ(γ θc))such that(.)has at least two positive solutions for
each a∈(a∗,λ(γ θc))and has at least one positive solution for a≥λ(γ θc).
Proof It follows from Theorem . that (.) has at least one positive solutions for each
a>λ(γ θc). So we only have to show that (.) has at least two positive solutions for each
a∈(a∗,λ(γ θc)) and has at least one positive solution fora=λ(γ θc).
LetD ={(u,v)∈W:(u,v) – (,θc)E<}. Note that the direction of the bifurcation of (.) emanating from the semi-trivial solution (,θc) is determined by the sign ofρ(γ) given in (.). Ifγ >γ, then we see thatρ(γ) < and the bifurcation is subcritical. So there exists a positive constanta∗∈(λ,λ(γ θc)) such that for eacha∈(a∗,λ(γ θc)), (.) has a unique positive solution (ua,va)∈D. To prove the existence of a positive solution
whena=λ(γ θc) and two positive solutions whena∈(a∗,λ(γ θc)), it suffices to show that for eacha∈(a∗,λ(γ θc)], (.) has a positive solution inD\D.
DefineFτ :D→Wby
Fτ(u,v) = (–+M)–
u(a–u) –τv( –e–γ θa) +Mu v(c–v+τd( –e–γ θa)) +Mv ,
where τ ∈[, ],D,W andMare defined in Section . It follows from standard ellip-tic regularity theory that Fτ is a completely continuous operator. Obviously, (.) has
check thatFτ has no fixed point on∂(D\D) forτ∈[, ]. Hence,indexW(Fτ,D\D)≡
The main result in this section is the following.
Theorem . Assume= (p,q)for some real numbers p<q.Ifγ(c+d)≤,then for
every(a,c)satisfying(.),the following boundary value problem:
⎧
The proof will be finished in several steps. The technique we used here can be found in the papers [, ]. The basic ingredient is non-degeneration of positive solutions, which is summarized as the following lemma.
has only the trivial solution(, ).In other words,any positive solution is non-degenerate.
Proof Since (u∗,v∗) is a positive solution of (.), by the Krein-Rutman theorem, we have
The linearized problem (.) can be written as
From the monotonicity ofλ(·), it follows that
λ
v∗–d –e–γu∗–c>λ
v∗–d –e–γu∗–c= . (.)
Ifγ(c+d)≤, then we claim that
u∗+γv∗e–γu∗>u∗+v∗( –e –γu∗)
u∗ . (.)
Thus
λ
u∗+γv∗e–γu∗–a>λ
u∗+v∗( –e
–γu∗)
u∗ –a
= . (.)
Now we shall prove that (.) is true. Leth(u∗) =u∗+γu∗v∗e–γu∗–v∗( –e–γu∗). It suffices
to showh(u∗) > . Obviously,h() = andh(u∗) =u∗( –γv∗e–γu∗). Remindingv∗<c+d
given by Lemma ., we geth(u∗) > and thush(u∗) > . This shows that (.) holds true.
Define the operatorsLandLby
Lφ= –ψ+ (u∗+γv∗e–γu∗–a)φ, φ∈X,
Lψ= –ψ+ (v∗–d( –e–γu∗) –c)ψ, ψ∈X.
So (.) can be written as
Lφ= –
–e–γu∗ψ, Lψ=dγv∗e–γu∗φ. (.)
By (.) and (.), L andL have inverses, sayL– ,L– , which are compact and order-preserving,i.e.,L–i (P–{})⊂intPfori= , . Now we shall show that the only solution to (.) is (, ), which completes the proof of this lemma. To this end, we argue by contra-diction, assuming that there exists a solution (φ,ψ)= (, ) to (.). From (.), it follows that
–φ=L– –e–γu∗L– dγv∗e–γu∗φ. (.)
Since the right-hand side of (.) defines a compact strongly order-preserving operator, we find thatφmust change sign in (p,q). Similarly,ψmust change sign in (p,q). Moreover,
φ andψ cannot vanish on an interval of positive length by the maximum principle,i.e., the zeros ofφ,ψ are isolated each from the others. In fact, ifφ vanishes on an interval, thenφ= at the boundary of such an interval whereφ> orφ< , which contradicts the maximum principle. Thus there exists a partition of (p,q), say
Q={p=x<x<x<· · ·<xn–<xn=q}, and we can chooseφsuch that
φ(x) > , x∈(xk,xk+),k≥, k+ ≤n,
φ(x) < , x∈(xk–,xk),k≥, k≤n,
φ(x) = , x=xk, ≤k≤n.
Here we claim that
ψ(xk) > , ψ(xk+) < , xk,xk+∈Q–{p,q}. (.) Since the principal eigenvalues ofLandLon any subinterval of (p,q) are strictly positive, the generalized maximum principle will be used to show this claim. By hypothesis,φ(x) > ,x∈(x,x) andφ(x) =φ(x) = . Thus
Lψ(x) =dγv∗e–γu∗φ(x) > , x∈(x,x).
We claim thatψ(x) < . In fact, ifψ(x)≥, then by the generalized maximum principle, we haveψ(x) > ,x∈(x,x). Thus
Lφ(x) = –
–e–γu∗ψ(x) < , x∈(x,x).
Thereforeφ(x) < ,x∈(x,x), which contradicts (.). Soψ(x) < . Again by hypothesis,φ(x) < ,x∈(x,x) andφ(x) =φ(x) = . Thus
Lψ(x) =dγv∗e–γu∗φ(x) < , x∈(x,x).
We claim thatψ(x) > . In fact, ifψ(x)≤, then by the generalized maximum principle, we haveψ(x) < ,x∈(x,x). Thus
Lφ(x) = –
–e–γu∗ψ(x) > , x∈(x,x).
Thereforeφ(x) > ,x∈(x,x), which contradicts (.). Soψ(x) > . Arguing recursively, we show (.) holds.
According to the parity ofn, either
φ(x) > , x∈(xk,q), ψ(xk) > (.)
or
φ(x) < , x∈(xk+,q), ψ(xk+) < (.)
is satisfied. Assume (.) holds. Then
Lψ(x) =dγv∗e–γu∗φ(x) > , x∈(xk,q).
Sinceψ(xk) > andv(q) = , by the generalized maximum principle, we haveψ(x) > ,
x∈(xk,q). Thus
Lφ(x) = –
–e–γu∗ψ(x) < , x∈(x
k,q).
Thereforeφ(x) < ,x∈(xk,q), which contradicts (.). Similarly, if (.) holds, we also
Applying the implicit function theorem, we can show that if (.) has exactly one positive solution, which in addition is non-degenerate, then the following problem:
⎧ ⎪ ⎨ ⎪ ⎩
–u=u(a–u) –v( –e–(γ+)u), x∈(p,q), –v= (c–v+d( –e–(γ+)u))v, x∈(p,q),
u(p) =u(q) =v(p) =v(q) =
(.)
has also exactly one positive solution providedis small enough. The proof is omitted.
Lemma . Suppose that (.) is satisfied and (.) has exactly one positive solution
(u∗,v∗),which is non-degenerate.Then there exists=(a,c,d,γ) > such that for
ev-ery∈(–,)the problem(.)has exactly one positive solution(u(),v()).Moreover, (u(),v()) = (u∗,v∗)and the mapping→(u(),v()),from a neighborhood of= in R to X,belongs at least to the class C.
Proof of Theorem. Consider the set
:=γˆ∈[,γ] : (.) withγ=βhas a unique positive solution,∀β∈[,γˆ].
Since (.) withγ = is uncoupled, if it has a positive solution, then it has exactly one. Thus ∈,i.e., is not empty. By Lemma ., we know that is open in [,γ]. Now we shall show thatis closed in [,γ]. Hence= [,γ], which completes the proof. To show this, consider a sequence{γn}n≥ insatisfyingγn→ ˜γ ≤γ asn→ ∞. Since the mappingγ →λ(γ θc) is increasing inγ, both (.) andγ(c+d)≤ are satisfied for
γ =γ˜ andγ =γn,n≥. Let (un,vn) be the unique positive solution of (.) withγ =γn. By passing to a subsequence if necessary, (un,vn) converges to (u˜,v˜) asn→ ∞. From the proof of Theorem ., it follows that (u˜,v˜) is in the interior ofP×P. Since (.) is satisfied forγ =γ˜, we know that (u˜,v˜) is a positive solution of (.) withγ =γ˜. In fact, the problem (.) withγ =γ˜has exactly one positive solution. Otherwise the application of the implicit function theorem would imply that (.) withγ=γnhas at least two positive solutions for sufficiently largen, contradicting the fact thatγn∈. Henceγ˜ ∈ and= [,γ]. This
finishes the proof.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
GG performed the theory analysis and carried out some computations. BL participated in the sequence alignment and also undertook some computations. XL participated in the design of the study. All authors read and approved the final manuscript.
Author details
1College of Science, Shaanxi University of Science and Technology, Xi’an, 710021, P.R. China.2Department of Basic
Course, Shaanxi Railway Institute, Weinan, 714000, P.R. China.
Acknowledgements
The work is supported by the Natural Science Foundation of China (Nos. 11271236, 11001160), the Natural Science Basic Research Plan in Shaanxi Province of China (No. 2011JQ1015) and the Foundation of Shaanxi Educational Committee of China (No. 12JK0856).
References
1. Ivlev, V: Experimental Ecology of the Feeding Fishes. Yale University Press, New Haven (1961)
2. Kooij, RE, Zegling, A: A predator-prey model with Ivlev’s functional response. J. Math. Anal. Appl.198, 473-489 (1996) 3. Sugie, J: Two-parameter bifurcation in a predator-prey system of Ivlev type. J. Math. Anal. Appl.217, 349-371 (1998) 4. Baek, HK, Kim, SD, Kim, P: Permanence and stability of an Ivlev-type predator-prey system with impulsive control
strategies. Math. Comput. Model.50, 1385-1393 (2009)
5. Wang, HL, Wang, WM: The dynamical complexity of a Ivlev-type prey-predator system with impulsive effect. Chaos Solitons Fractals38, 1168-1176 (2008)
6. Xiao, HB: Global analysis of Ivlev’s type predator-prey dynamic systems. Appl. Math. Mech.38, 461-470 (2007) 7. Ling, L, Wang, WM: Dynamics of a Ivlev-type predator-prey system with constant rate harvesting. Chaos Solitons
Fractals41, 2139-2153 (2009)
8. Wang, WM, Zhang, L, Wang, HL, et al.: Pattern formation of a predator-prey system with Ivlev-type functional response. Ecol. Model.221, 131-140 (2010)
9. Jia, YF: A sufficient and necessary condition for the existence of positive solutions for a prey-predator system with Ivlev-type functional response. Appl. Math. Lett.24, 1084-1088 (2011)
10. Smoller, J: Shock Waves and Reaction-Diffusion Equations. Springer, New York (1983)
11. Dancer, EN: On the indices of fixed points of mappings in cones and applications. J. Math. Anal. Appl.91, 131-151 (1983)
12. Cassanova, CS: Existence and structure of the set of positive solutions of a general class of sublinear elliptic non-classical mixed boundary value problems. Nonlinear Anal. TMA49(3), 361-430 (2002)
13. Crandall, MG, Rabinowitz, PH: Bifurcation from simple eigenvalues. J. Funct. Anal.8(2), 321-340 (1971)
14. Yamada, Y: Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions. SIAM J. Math. Anal.21, 327-345 (1990)
15. Guo, GH, Wu, JH: Multiplicity and uniqueness of positive solutions for a predator-prey model with B-D functional response. Nonlinear Anal. TMA72, 1632-1646 (2010)
16. Blat, J, Brown, KJ: Global bifurcation of positive solutions in some systems of elliptic equations. SIAM J. Math. Anal.17, 1339-1353 (1986)
17. López-Gómez, J, Pardo, R: Existence and uniqueness of coexistence states for the predator-prey Lotka-Volterra model with diffusion on intervals. Differ. Integral Equ.6, 1025-1031 (1993)
18. Casal, A, Eilbeck, JC, López-Gómez, J: Existence and uniqueness of coexistence states for a predator-prey model with diffusion. Differ. Integral Equ.7, 411-439 (1994)
doi:10.1186/1687-1847-2013-164
