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Existence and stability of positive steady-state solutions for a Lotka-Volterrasystem with intraspecific competition
Advances in Difference Equations volume 2014, Article number: 159 (2014)
In this paper, we investigate the existence and stability of the positivesteady-state solutions for a Lotka-Volterra system with intraspecific competition byusing the Lyapunov-Schmidt reduction technique. To do this, we must firstly obtainthe semi-trivial steady states as their base, which extend the method used in theprevious studies. Our results show that the two competition species withintraspecific competition can coexist for bigger regions of the diffusion rateμ and also complete the existing works.
The maintenance of biodiversity has received increasing attention from ecologists andmathematicians. Moreover, resource competition is thought as an important factor indriving evolutionary diversification, in which intraspecific competition for resourcesplays a major role; see [1–4]. The classical Lotka-Volterra competition system [5–8] gives a better description of the population competition, whose dynamicalbehaviors have been studied extensively; see [9–16] and the references therein. In this paper a competitive Lotka-Volterradiffusion model of two slightly different species is studied, namely, the two speciesare identical except for their intraspecific competition rates. For this system, oncethe diffusion is involved biologically, it would be very interesting to find out whetherand when the two species competing for the same limited resources to survive cancoexist.
To begin with, we present the classical Lotka-Volterra system with spatiallyinhomogeneous terms as follows
where Ω is a bounded domain in with smooth boundary ∂ Ω,μ is the diffusion rate, and denote the densities of two competing species, and represents the intrinsic growth rate of species.
The system above shows that the two species have the same diffusion rates and intrinsicgrowth rates. However, the two species with slight difference, such as the originalspecies and the mutant, are common in biology; they correspond to the perturbed systemsof (1.1), see [17–21] for example, and are originally introduced in . The two species with different diffusion rates discussed in  correspond to the system
where . It is shown from (θ given in the next section) as that the slower diffuser can drive the faster one toextinction, and thus the two species cannot coexist. In , it is further stated that the slower diffuser may fail with a time periodicfunction instead of . In , the model with small variations of intrinsic growth rates is given by
It is shown that the stability of the two competing species varies in a complicated wayas μ increases, which implies that the two species only coexist for smallregions of μ if they could. For taking the form in , the authors further illustrate that the mutant can always survive andinvade, but the original species can only coexist in some cases. Furthermore, in  the system with different interspecific competition rates is given as
Then a new structure of coexistence states is obtained and the two species can coexisteven for bigger regions of μ.
Motivated by the studies above, in this paper, we continue the analytic works foranother perturbation system
where τ is a small positive constant (i.e.,) and , are smooth functions, which indicates that the two speciesonly have different intraspecific competition rates. Our main purpose is to study theexistence and stability of the positive steady-state solutions (that is, the coexistencestates) of (1.2) by using the Lyapunov-Schmidt reduction technique, the implicitfunction theorem combined with finite covering theorem and the perturbation theory forcompleting the previous studies. To do this, however, we must firstly get thesemi-trivial steady-state solutions of (1.2) according to , which is different from the corresponding results already known, and extendthe method used in the existing works. The main results we obtained show that the twocompetition species with different intraspecific competition can also coexist for biggerregions of the diffusion rate μ, but it is also important to caution thatthe conclusions for stability obtained in this paper are contrary to the ones analyzedin .
The rest of this paper is organized as follows. In Section 2, from  we firstly give the semi-trivial steady-state solutions of (1.2), and thenanalyze the stability of the solutions. In Section 3, on the basis of thesemi-trivial solutions obtained in Section 2, we investigate the existence andnonexistence of positive steady-state solutions of (1.2) by using the Lyapunov-Schmidtreduction technique, the implicit function theorem and the finite covering theorem. InSection 4, combining stability theory with perturbation theory, we further discussthe stability of positive steady-state solutions in detail.
2 Existence and stability of semi-trivial steady-state solutions
In this section we firstly establish the existence of semi-trivial steady-statesolutions of (1.2) by , whose proof is omitted. Then we simply analyze the stability of thesemi-trivial solutions.
Lemma 2.1 (see )
Suppose that λ is a positive parameter and thateitheron an open subset of ∂ Ω, orthe boundary condition is a Dirichlet condition on partof ∂ Ω, orholds. The principaleigenvalueof
is positive if and only if, whereis the positive principal eigenvalue of
Ifand, thenfor all.
Consider the following system:
where , , , , is Lipschitz in u and is a measurable function inx which is bounded if u is restricted to a bounded set and.
Lemma 2.2 (see )
Suppose thatwithof classin u andin x, and thereexistssuch thatfor. If the principaleigenvalueis positive in the problem
then (2.1) has a minimal positive equilibrium, and all solutions to (2.1) that areinitially positive on an open subset of Ω are eventually bounded byorbits which increase towardas.
Lemma 2.3 (see )
Suppose that the hypotheses of Lemma 2.2 are satisfied andthatwithstrictly decreasingin u for. Then the minimal positiveequilibriumis the only positive equilibrium for (2.1).
For the further discussions, we make the following assumption on .
is Hölder continuous on and .
Then, from Lemmas 2.1-2.3, it is well known that
has only a positive solution denoted by when the condition (H) is valid, and it is also easilyfound that
respectively have unique positive equilibriums and , denoted by and , for (H) and . Hence, we can find the following result.
Theorem 2.1 Assume that (H) holds and. Then (1.2) has semi-trivialsteady-state solutionsand.
To discuss the stability of semi-trivial steady-state solutions and , we need to consider the eigenvalue problem
corresponding to the solution . Due to [23, 24], one can show that (2.3) has a principal eigenvalue , which is real, algebraically simple, and all othereigenvalues have their real parts less than . Therefore, the stability of is decided by the principal eigenvalue of the problem
Similarly, the stability of is dependent on the principal eigenvalue of the problem
Theorem 2.2 Suppose that the hypotheses of Theorem 2.1 arevalid. If (), thenis asymptotically stable (unstable).If (), thenis asymptotically stable (unstable).
Remark 2.1 In this section, we only give a simple conclusion for the stability ofsemi-trivial steady-state solutions of (1.2), whose detailed proofs are analogous toSection 3 of .
3 Existence and nonexistence of positive steady-state solutions
The research on the steady-state solutions in a competition-diffusion system is always ahot issue. In this section we establish the existence and nonexistence of positivesteady states of (1.2). For this purpose, we must make use of the semi-trivial solutionsobtained in Section 2 for the following results.
To discuss the steady states of (1.2), we deal with the elliptic system corresponding to(1.2), which takes the form as follows
For convenience of the following discussions, we denote
and give the Sobolev spaces
where such that .
According to (2.2), it is obvious that (3.1) has nontrivial nonnegative solutions for . Then for we will look for the solutions of (3.1) near, namely, the steady states of (1.2).
Theorem 3.1 Suppose thatandhave no common roots, andsetandare the consecutive and simple rootsof.
Ifin, then there existssuch that forsuch that (1.2) has no positive steady-state solutions other than semi-trivial ones for.
Ifin, then for any sufficiently smalland, (1.2) has the positive steady-state solutionssatisfyingin the neighborhood ofbesides semi-trivial ones. Here(3.2)
with, and. Moreover, the positive steady-statesolution branchesconnect with the semi-trivialonesand, and the smooth functionsandare defined onsuch thatand.
Proof For , our purpose is to find the positive solutions of (3.1)near . Following the Lyapunov-Schmidt reduction technique , we know that on the basis of the spaces defined above. Then we can set
that is to say, the solution form we will look for is given by
where and near .
Substituting the expression above into (3.1), we have
and the map defined above is a smooth function. Obviously, on thebasis of the semi-trivial steady states obtained in Section 2, we have
Let . Then
which is denoted by L, and L is a Fredholm operator of index zerosince X is compactly imbedded in Y. It is easy to check that is in the kernel of L. Due to, we know that 0 is a simple eigenvalue of L,which leads to
Define the projection operator as
It is easily found that
which results in
According to the Lyapunov-Schmidt reduction technique, the equivalent expression of(3.3) is
We know that L is an isomorphic mapping from to Y, then, by applying the implicit theorem tothe second equation of (3.7), it can be solved to get a unique solution near . Furthermore, combining the finite covering theorem, thereexists such that
Thus, the solvability of (3.3) is converted to that satisfies the first equation of (3.7), that is,
Combining with (3.4), we have
On the basis of the definition of P, we can get a smooth function satisfying
Hence we need to solve . From (3.8) and (3.9), we can find
and then there exists a smooth function such that
Moreover, we know that the solutions and correspond to the semi-trivial solutions and of (3.1), respectively. Consequently, it remains to solve.
From (3.11), it follows that
Differentiate (3.10) with respect to τ, set and combine with (3.6) and (3.8), which leads to
From (3.3), we obtain
and then it follows from (3.5) that
Hence, according to (3.12)-(3.14), we have
For in , for example, we let , and then we obtain when and . By using the finite covering theorem, there exists taken smaller if necessary such that has no solution for . This results in statement (i) of Theorem 3.1.
For in , we can see that with and for and since and have no common roots. Combining the implicit functiontheorem with the finite covering theorem, we have that for chosen yet smaller if necessary and, the equation has only solution given by the smooth function with , which shows that (3.3) has solutions with . Due to and given in (3.9), we see
and then denote and by and , respectively. From (3.8), it follows that and . Therefore, for , the equation has solutions given by , and , which shows that (1.2) has not only semi-trivialsteady-state solution branches, but also positive branches given by (3.2) which meet thesemi-trivial ones whose form is discussed below.
Clearly, and correspond to the semi-trivial stationary solution of(1.2). For , there is either or due to that and have no common roots. Without loss of generality, we take, and then . Moreover, since is the simple root of . Thus, by the implicit function theorem, we can obtain aunique solution of defined by a smooth function for with small enough and , which implies that the corresponding semi-trivialsteady-state solution of (1.2) can be described by . Similarly, for the case , we can get only the solution of still given by , which corresponds to the semi-trivial steady-statesolution of (1.2). Moreover, for , similar results hold true, and the unique solution of or is denoted by with . Thus the relevant solution of (1.2) is given by or . □
Remark 3.1 The proof of Theorem 3.1 shows the existence of loops or branchesof positive steady states just as , whose details are omitted here.
4 Stability of positive steady-state solutions
In this section, we analyze the stability of positive steady states of (1.2). For , we know , and then the principal eigenvalue of the correspondingeigenvalue problem (2.3) is because of , which shows that all other eigenvalues have negative realparts. Furthermore, for , since all other eigenvalues also have negative real partsby the perturbation theory , the stability of is determined by the principal eigenvalue near 0 of (2.3), that is,
Firstly, for , we can set
in (4.1) with smooth functions and . Furthermore, the emphasis is on the sign of. Hence, for small τ, we divide thediscussions of the sign of into three cases, that is, μ close to, μ close to and μ bounded away from and . To do this, we must first prove the following lemmas.
Lemma 4.1 For, the principal eigenvalueof (4.1) satisfies
Proof Multiply the first equation of (4.1) by v and integrate overΩ to get
In the same way, multiplying the second equation of (4.1) by u and integratingby parts, we can obtain
From (4.4) and (4.5), it suffices to show that (4.3) is valid. □
Furthermore, on the basis of Lemma 4.1, we can discuss the sign of.
Lemma 4.2 For, , we have the following conclusion:
Proof It is obvious that and as from (3.2) and (4.2). Then, for , it follows that
Thus, we can obtain (4.6) from (4.3), (4.7) and (4.8). □
For the discussions below, without loss of generality, we can assume the case (the case is similar), and then and .
Lemma 4.3 Assume that. Then we have the following conclusion:
Proof According to Theorem 3.1, we know that is the bifurcation point of the positive solutions to(1.2). Then and the eigenfunction satisfies
Multiplying the equation above by and integrating by parts, we can get
and then combining with it follows that , where
in (4.3). Thus it leads to
by the mean value theorem for between μ and .
For , we know that , , , as and . Then, differentiating (4.10) with respect toμ, we obtain
as and . From (4.3), (4.11) and (4.12), we can get the relation(4.9). □
In the same way, for μ close to , the sign of is decided by the result below, whose details are omittedhere.
Lemma 4.4 Assume that. Then we have the following conclusion:
Finally, based on Lemmas 4.2-4.4, the main result of this section is presented asfollows.
Theorem 4.1 Suppose thatandhave no common roots, are two consecutive and simple rootsofandin. Then, forand, we have:
Ifandin, then the positive steady statesof (1.2) are asymptotically stable;
Ifandin, thenis unstable.
Proof Now, we only consider the statement (i) (the statement (ii) can be provedsimilarly). The key point in the proof of (i) is to obtain by contradiction. Suppose that there exist and such that for . By passing to the limit, we have, and as with .
For , we know that
from Lemma 4.2 and , in . Thus, for large i, we have, which contradicts the assumption.
For or , we still only consider the case , and then the case can be treated similarly. In this case, we may suppose (if , then it is analogous). It is clear that since in and is the simple root of . Therefore, we see
which also leads to for large i combining with and . The contradiction completes the proof. □
Remark 4.1 Theorem 4.1 is contrary to the stability results in . But it is shown that the two competition species with differentintraspecific competition rates can also coexist for bigger regions of μ,which is similar to .
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The authors would like to express sincere thanks to the anonymous referee for his/hercarefully reading the manuscript and valuable comments on this paper. The work issupported by the Education Committee Foundation of Shaanxi Province, and the NationalNatural Science Foundation of China (No. 11271236).
The authors declare that they have no competing interests.
MW performed the theory analysis and carried out the computations. JC participated inthe design of the study and helped to draft the manuscript. LQ conceived of the studyand participated in some computations. All authors have read and approved the finalmanuscript.
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Wei, M., Chang, J. & Qi, L. Existence and stability of positive steady-state solutions for a Lotka-Volterrasystem with intraspecific competition. Adv Differ Equ 2014, 159 (2014). https://doi.org/10.1186/1687-1847-2014-159
- steady-state solutions
- Lyapunov-Schmidt reduction technique