Existence and stability of positive steady-state solutions for a Lotka-Volterrasystem with intraspecific competition
© Wei et al.; licensee Springer. 2014
Received: 12 February 2014
Accepted: 14 May 2014
Published: 30 May 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.
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.
Then a new structure of coexistence states is obtained and the two species can coexisteven for bigger regions of μ.
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 )
Ifand, thenfor all.
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 )
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).
is Hölder continuous on and .
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.
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.
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).
Ifin, then there existssuch that forsuch that (1.2) has no positive steady-state solutions other than semi-trivial ones for.
- (ii)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.
where and near .
Moreover, we know that the solutions and correspond to the semi-trivial solutions and of (3.1), respectively. Consequently, it remains to solve.
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.
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 (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.
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.
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 .
by the mean value theorem for between μ and .
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.
Finally, based on Lemmas 4.2-4.4, the main result of this section is presented asfollows.
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 .
from Lemma 4.2 and , in . Thus, for large i, we have, which contradicts the assumption.
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 .
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).
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