R E S E A R C H
Open Access
The dynamics of a diffusive logistic model
with nonlocal terms
Xianhua Xie
*and Li Ma
*Correspondence: [email protected]
Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques, College of Mathematics and Computers, Gannan Normal University, Ganzhou, Jiangxi 341000, People’s Republic of China
Abstract
It is well known that the set of positive solutions may contain crucial clues for the stationary patterns. In this paper, we consider a class of diffusive logistic equations with nonlocal terms subject to the Dirichlet boundary condition in a bounded domain. We study the existence of positive solutions under certain conditions on the parameters by using bifurcation theory. Finally, we illustrate the general results by applications to models with one-dimensional spatial domain.
MSC: Primary 35J15; secondary 92B20
Keywords: logistic model; positive steady-state solutions;a prioribounds; bifurcation theory
1 Introduction
Recently, many researchers pay more attention on the studies of reaction-diffusion equa-tions; we refer to, for example, [–]. Ecologically, positive solutions correspond to the existence of steady states of species. It is well known that the set of positive solutions may contain crucial clues for the stationary patterns. From the mathematical viewpoint, it is important to derive some information about the set of positive solutions by means of the coefficients such as the growth rate of the species. Especially, most of the references con-centrated on diffusive models with a single population (see,e.g., [, , ]). One of the most classical diffusive logistic equations is
ut–u=λu( –K(x)up) in,
u= on∂, (.)
which was regarded as a logistic system of individual species in the ecological studies. Here,u(x) is the population density at locationx∈,λ∈R+is the growth rate of the species and is usually deemed to be a variable,Kis a positive function denoting the car-rying capacity, andp> . In (.), we assume thatis surrounded by inhospitable areas, subjected to the homogeneous Dirichlet boundary conditions.
Later, many scientists found that the movement of an individual species is sometimes de-termined by surrounding conditions around the point where the species stays. For exam-ple, we consider movements of animals, where each individual species mutually interacts by seeing, hearing, and smelling around themselves. That is why interaction by chemical
means may take place under certain circumstances. Hence, it seems more realistic to take account of nonlocal effects in the study of species dynamics; see [, , , , , ]. Usually, this nonlocal effect depends on the value of the population aroundx, that is, the crowd-ing effect depends on the series of values ofu. In some special cases, this nonlocal effect also depends on the value in a neighborhoodBr(x) ofx, whereBr(x) represents the ball
centered atxof radiusr> . Along these reasons, system (.) is replaced by the following more general diffusive logistic population models with nonlocal effect:
ut–u=λu( –K(x,y)up(y) dy) in,
u= on∂, (.)
and
ut–u=u(λ–∩B
r(x)K(y)u
p(y) dy) in,
u= on∂, (.)
wherep> , andK:×→RandK:→Rare nonnegative and nontrivial continu-ous functions. Chen and Shi [] considered the dynamical behavior of system (.) when
p= and the kernel functionK(x,y) is a continuous and nonnegative function on× satisfyingK(x,y)u(y) dy> for all positive continuous functionsuon. Applying the implicit function theorem, Chen and Shi [] obtained the existence and uniqueness of a positive steady-state solution of system (.) when <λ–λ, whereλ denotes the
first eigenvalue of the minus Laplacian operator under homogeneous Dirichlet boundary conditions. Some researchers [, , , ] also realized that the kernel functionK(x,y) may have no direct connection with the growth rateλof the species. For example, Allegretto and Nistri [] studied the following model:
–u=u(λ–K(x,y)up(y) dy) in,
u= on, (.)
whereK(x,y) vanishes away from the diagonal domain ofRN×RN. Allegretto and Nistri
[] found that (.) possesses a unique positive solution whenλ>λifK(x,y) =Kδ(|x–y|)
is a mollifier inRN, that is,Kδ(|x–y|)∈C∞
,
RNKδ(|x–y|) dy= for anyxwith
Kδ|x–y|= when|x–y| ≥δ,
andKδ(|x–y|) is bounded away from zero when|x–y|<μ<δ. Later, Corrêaet al.[] proved that (.) possesses a unique positive solution ifK(x,y) is a separable variable, that is, K(x,y) =g(x)h(y), whereh≥,h= , andg(x) > in. Sunet al.[] investigated the existence of positive solutions of system (.) withK(x,y) =K(|x–y|) and= (–, ),
whereK: [, ]→(,∞) is a nondecreasing and piecewise continuous function satisfying
K(y) dy> . Besides, Alveset al.[] also studied the existence of a positive solution of
system (.).
] and references therein. However, the discussion of the dynamical behavior of two inter-acting species in the presence of nonlocal term effects is more difficult than those models without nonlocal term effects. Lately, Guo and Yan [] employ Lyapunov-Schmidt reduc-tion to investigate the existence of the positive solureduc-tion of the following model:
⎧ ⎪ ⎨ ⎪ ⎩
–u=λu( –A(x,y)u(y) dy–
A(x,y)v(y) dy) in,
–v=λv( –A(x,y)u(y) dy–
A(x,y)v(y) dy) in,
u=v= on,
(.)
whereu(x) andv(x) are the population densities at locationx,λ> is a scaling constant, andis a connected bounded open domain inRN (N≥) with a smooth boundary∂.
The kernel functionsAij(i,j= , ) describe the dispersal behaviors of the populations. A natural problem is whether (.) has a positive solution forλnot only near to but also far away fromλ. Moreover, it is very interesting to investigate the following more general
population model with nonlocal delay effect:
⎧ ⎪ ⎨ ⎪ ⎩
–u=u(λ–A(x,y)up(y) dy–
A(x,y)vq(y) dy) in,
–v=v(λ–A(x,y)up(y) dy–
A(x,y)v
q(y) dy) in,
u=v= on,
(.)
where⊂RN(N≥) is a bounded domain with a smooth boundary∂,p,qare positive constants, andAij∈L∞(×,R),i,j= , . Hereu(x) andv(x) can be interpreted as the densities of prey and predator populations at a spatial positionx∈, and the parameter λis a positive real number representing the growth rate of the prey and predator.
The purpose of this paper is to find sufficient conditions ensuring the existence of a positive solution for allλ>λ. Our main approach is global bifurcation theory, which is
different from the method adopted in []. Moreover, we also obtain the stability of the positive solution by analyzing the distribution of the eigenvalues, which was not consid-ered by Alveset al.[]. Throughout this paper, we impose the following assumptions on the dispersal kernel functionsAij(x,y),i,j= , .
(C) TandTare positive on the spaceC+()×C+()in the sense that
Ti(C+()×C+())⊂C+()×C+()\ {(, )},i= , , whereC+()represents the space of positive continuous functions, and
Tj(u,v) =
Aj(·,y)up(y) +Aj(·,y)vq(y)
dy, j= , .
(C) Ifu,vare measurable and satisfy
×[A(x,y)|uu(y)|p+A(x,y)|v(y)|
q
up](uu(x))dydx= ,
×[A(x,y)|u(y)|
p
vq +A(x,y)|vv(y)|q](vv(x))dydx= ,
thenu=v= a.e. in. Here we setuu = foru= and denote by · the usual norm inH
(), that is,
u=u
H ()=
where the spaceH
() ={U∈H()|U(x) = ,∀x∈∂}andHk()(k≥) is
the Sobolev space ofL-functionsf onwith derivativesdnf
dxn (n= , , . . . ,k)
belonging toL().
Our main results are stated as follows.
Theorem . Suppose that Aij(i,j= , )satisfy(C)and(C).Then problem(.)has a
positive solution if and only ifλ>λ.
In view of Theorem ., we see that system (.) withAij(x,y) =Aij(x,y)χBr(x)(i,j= , )
has a positive solution if and only ifλ>λ, whereAij(x,y) (i,j= , ) are positive functions
on×. IfAij(i,j= , ) do not satisfy assumption (C), thenAij(i,j= , ) may vanish in some neighborhood of the diagonal of×(see Section ). In this case, Theorem . is inapplicable. However, we are also able to investigate the existence and nonexistence of positive solutions for some value ofλunder the following assumptions on the dispersal kernel functionsAij(x,y),i,j= , .
(C) There arer> andmconnected open sets,, . . . ,m⊂such that
i∩j=∅,i=j, andAij(x,y) > for all(x,y)∈×satisfyingx∈/
m
j=jand |x–y|<r,i,j= , .
In view of [], we know that if⊂, thenλ()≥λ(). Moreover, the inequality
is strict as soon as\contains a set of positive capacity (since the first eigenfunction
cannot vanish on such a set). Hence, we have the following result.
Theorem . Suppose that Aij(i,j= , )satisfy(C)and(C).Then,problem(.)has a positive solution whenλ<λ<min{λ(), . . . ,λ(m)},whereλ(i)denotes the principal eigenvalue of the minus Laplacian operator iniunder homogeneous Dirichlet boundary conditions,i= , , . . . ,m.Moreover,the solution is stable.
The remaining parts of the paper are structured in the following way. In Section , we employ the global bifurcation theory to obtain the existence and stability of positive so-lutions of (.) under conditions (C) and (C). Section is devoted to the case whereAij
(i,j= , ) satisfy condition (C). Section is devoted to the application of our theoretical results to some one-dimensional models.
2 Proof of Theorem 1.1
In this section, we introduce some basic results. First, consider the functionsφpij,ω:→R
(i,j= , ) given by
φijp,ω(x) =
Aij(x,y)ω(y)pdy, x∈.
IfAij(i,j= , ) andωare bounded, thenφijp,ω (i,j= , ) are well defined. Moreover, we
have the following observations:
and
φijp:L∞()→L∞(),
φijp(u) =φpij,u
(i,j= , ) are uniformly continuous inL∞(). (.)
Using these notations, it is easy to observe that (u,v) is a positive solution of (.) if and only if (u,v) is a positive solution of
First, we show the nonexistence of a positive solution of (.) for smallλ.
Lemma . Suppose that Aij(i,j= , )satisfy(C)and(C).Then system(.)withλ<λ
has no positive solutions.
Proof We prove this lemma by contradiction. Assume that (.) withλ≤λhas a positive
solution (u∗,v∗). Then we have
equations of (.) is less than and that each of the right-hand sides of the two equations is greater than , which is a contradiction. So system (.) has no positive solution for
λ≤λ.
In the following section, we intend to prove the existence of a positive solution for (.) by using the classical bifurcation result of Rabinowitz []. To this end, we recall that there existsc∞=c∞() > such that, for eachf ∈L∞(), there exists a uniqueω∈C()
sat-isfying
–ω=f(x), x∈,
ω= , x∈∂, (.)
andωC()≤c∞f∞. Thus, the solution operatorS:C()→C() can be given by
SU=ω ⇔
–ω=U, x∈, ω= , x∈∂.
Obviously,Sis well defined, linear, and satisfies
SUC()≤c∞UC(), ∀U∈C().
Moreover, by the Schauder imbedding theorem,S:C()→C() is a compact operator.
In view of the spectrum ofS, it is easy to see that
σ(S) =λ–j |λjis an eigenvalue of the minus Laplacian operator
.
On the other hand, define the nonlinear operatorF:C()→C() as
FV=ω ⇔
–ω+VV= , x∈,
ω= , x∈∂,
whereV= (u,v)T∈C() and
V=
φp,u+φq,v φ
p,u+φq,v
.
Obviously,Fis continuous and satisfies
FVC()≤c∞V∞VC(), ∀V∈C(),
Using again the Schauder imbedding theorem, we see thatF:C()→C() is compact.
Furthermore, note that
FVC()≤ FVC().
Then we have
VFVC()
C()
≤FVC()
VC() ≤
from which it follows that
lim V→
FV VC()
= , i.e., FV=oVC()
. (.)
Obviously,V= (u,v)Tsolves (.) if and only if
V=G(λ,V)λSV+FV.
In view of [], consideringE=C(), we have the following result.
Theorem . Let E be a Banach space.Suppose that F satisfies(.),S is a compact linear operator,andλ–∈σ(S)with odd algebraic multiplicity.Let
=(λ,V)∈R×E:V=λSV+FV,V= ,
and letCbe a closed connected component ofthat contains(λ, ).Then either () Cis unbounded inR×E,or else
() there existsλ˜=λsuch that(λ˜, )∈Candλ˜–∈σ(S).
Remark . Becauseλis the principle eigenvalue of the eigenvalue problem (.) with
the associated eigenfunctionψ> onand its multiplicity is simple, by the global
bi-furcation theorem there exists a closed connected componentC containing (λ, ) and
satisfying () or () for solutions to (.).
In order to prove the existence of positive solutions of (.) withλ>λ, it follows from
Lemma . and Theorem . that it only suffices to prove that conclusion () of Theo-rem . holds and thatV is bounded whenλ>λ.
Lemma . There exists ε> such that if (λ,V) = (λ,u,v)∈C with λ–λ <ε and
VC()<εwhere u= and v= ,then u and v have definite signals,that is, (i) u(x) > andv(x) > for allx∈,or
(ii) u(x) > andv(x) < for allx∈,or
(iii) u(x) < andv(x) > for allx∈,or
(iv) u(x) < andv(x) < for allx∈.
Proof TakeVn= (un,vn)∈C() andλn→λasn→ ∞such thatVnC(×)→ as n→ ∞and
Vn=G(λn,Vn).
Letw
n=unuCn() andw
n=vnvCn(). Then we have
⎧ ⎪ ⎨ ⎪ ⎩
–w
n+φp,unw
n+φq,vnw
n=λnwn in,
–wn+φp,unwn+φq,vnwn=λnwn in,
w
n=wn= on∂.
It follows from (.) thatφ
integrating on, we have
allx∈because the other three cases can be discussed analogously. Note thatwandw
are theC()-limits ofwnandwn, respectively. Then, on,win> andwin> fornlarge enough. Thus, the signs ofunandvnare the same as those ofw
nandwnfornlarge enough.
This completes the proof.
It is easy to check that if (λ,u,v)∈, then the pairs (λ, –u,v), (λ,u, –v), and (λ, –u, –v) are also in. In what follows, we decomposeCintoC=C+,+∪C+,–∪C–,+∪C–,–, where
C+,+=(λ,u,v)∈C:u(x)≥,v(x)≥,∀x∈;
C–,+=(λ,u,v)∈C:u(x)≤,v(x)≥,∀x∈;
C–,–=(λ,u,v)∈C:u(x)≤,v(x)≤,∀x∈.
The following lemma tells the fact that system (.) satisfies Theorem .().
Lemma . Each ofC+,+,C–,+,C–,+,andC–,–is unbounded.
Proof It is easy to see that if one ofC+,+,C–,+,C–,+, andC–,–is unbounded then the others
are also unbounded. Therefore, it suffices to show thatC+,+is unbounded. Suppose that
C+,+is bounded. ThenCis also bounded. In view of the global bifurcation theorem
(Theo-rem .),Csatisfies conclusion () of Theorem ., that is,Ccontains (λ˜, , ), whereλ˜=λ
andλ˜–∈σ(S).
We take{(λn,un,vn)}∞n=⊆C+,+ such thatunvn= and (un,vn) =G(λn,un,vn) and such
thatλn→ ˜λ,unC()→, andunC()→ asn→ ∞. Letting
wn= un
unC()
, wn= vn
vnC()
and using similar arguments as in the proof of Lemma ., we get that (w
n,wn) converges
to (w,w) inC()×C() asn→ ∞, which is a nonzero solution pair of the eigenvalue
problem
⎧ ⎪ ⎨ ⎪ ⎩
–w=λ˜w in,
–w=λ˜w in,
w=w= on∂,
which implies that bothwandware eigenfunctions associated withλ˜. Sinceλ˜=λ, both
wandwchange signs in. Thus, fornlarge enough,w
nandwnchange signs, and hence
the same results hold forun=unC()wnandvn=vnC()wn. However, this contradicts
the assumption that (λn,un,vn)∈C+,+. This completes the proof.
Now, we shall prove that system (.) satisfies Theorem .(). It suffices to show that the connected componentC+,+intersects any set of the form{λ} ×H
()×H() forλ>λ.
Lemma . Suppose that Aij(i,j= , )satisfy conditions(C)and(C).For any> ,
there exists a constant r> such thatuC()≤r andvC()≤r whenever(λ,u,v)∈C+,+ andλ≤.
Proof First, we denote by · the usual norm inH
(), that is,
u=u H()=
|∇u|dx.
Indeed, if this were not true, there would exist{(λn,un,vn)}∞n=⊂[,]×H()×H()
such that
Case . un → ∞,vn → ∞andλn→λ∈asn→ ∞, and(un,vn) =G(λn,un,vn).
We only discuss Case because the other cases can be dealt with analogously. Let
wn= un respectively, equations (.) reduce to
⎧
Passing to the limit in these equalities and using the Fatou lemma, we have
Then we have
3 Proof of Theorem 2.1
We observe that ifAij(i,j= , ) do not satisfy condition (C), that is:
Proof In view of equation (.), we have
We conclude that|Dε∩Br|= for allε> andr> , whereBr(x)Bris the ball centered atxof radiusr> . Hence,|Dε|= for allε> . This completes the proof.
Hereafter, we proceed as in the proof of Lemma .. Our goal is also to derive an es-timate ofa prioribounds for (λ,u,v)∈C+,+, whereλ∈ ˜= [λ
Proof Indeed, arguing by contradiction, if this were not true, then there would exist
{(λn,un,vn)}∞n=⊂ ˜×H()×H() such that one of the following three cases holds:
. un → ∞,λn→λ∈ ˜asn→ ∞,vn ≤l, and(un,vn) =G(λn,un,vn); . vn → ∞,λn→λ∈ ˜asn→ ∞,un ≤l, and(un,vn) =G(λn,un,vn); . un → ∞,vn → ∞,λn→λ∈ ˜asn→ ∞, and(un,vn) =G(λn,un,vn).
We just discuss the case because the other two cases can be dealt with analogously. Let
ϕn= un
Using similar arguments as in the proof of Lemma ., we haveϕ= , which implies that at least one ofϕandϕis not equal to . Without loss of generality, assume thatϕ= .
4 Examples
In this section, we consider system (.) with= (,π) to check the validity of the main results obtained in Sections and . Notice thatλσn=n(n∈N) are the eigenvalues of
the linear eigenvalue problemu+λu= withu() =u(π) = andsinnxis the eigenfunc-tion associated with the eigenvaluen,n∈N. In particular,σ= andsinx> on (,π).
In what follows, we consider the following example:
Ai,j(x,y) =
if (x,y)∈[(,π)×(π,π)]∪[(π,π)×(,π)],
otherwise
for alli,j= , . Obviously,Aij(i,j= , ) vanish on the diagonal. It follows from Theorem . that there exists a positive solution for all λ> . In fact, we see that system (.) with = (,π) and givenAij(i,j= , ) has a positive solutionu=χsinxandv=χsinx, where χandχare positive constants satisfying
λ= (χ)p
π
sinpxdx+ (χ)q
π
sinqxdx+ .
Competing interests
The authors declare that they have no competing interests regarding the publication of this paper.
Authors’ contributions
Both authors, XHX and LM, contributed substantially to this paper, participated in drafting and checking the manuscript, and have approved the version to be published.
Acknowledgements
Special thanks to the anonymous referees for very useful suggestions. The research has been supported by the Natural Science Foundation of China (11361004). Xian-Hua Xie is supported by the Bidding Project of Gannan Normal University (16zb01). Both authors are supported by the Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques of Gannan Normal University.
Received: 25 November 2016 Accepted: 26 December 2016
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