Steady states of a nonlinear elliptic system arising from population dynamics
© Ma et al.; licensee Springer 2014
Received: 24 March 2014
Accepted: 16 July 2014
Published: 24 September 2014
In this paper, we study the existence and multiplicity of coexistence states for an elliptic system modeling two subpopulations of the same species competing for resources. Our results generalize and complement the work of Bouguima et al. (Nonlinear Anal., Real World Appl. 9:1184-1201, 2008). In particular, an open problem proposed by Bouguima et al. is partially solved.
MSC: 34B10, 34B18.
where Ω is a bounded regular domain of (), i.e., is an n-dimensional compact connected smooth submanifold of with boundary ∂ Ω. Throughout the paper, we suppose that a, b, c, d, e, and f are positive functions in for a certain .
System (1.1) arises from population dynamics where it models the steady-state solutions of the corresponding nonlinear evolution problem , where u and v represent, respectively, the concentrations of the adult and juvenile populations. The function a gives the rate at which juveniles become adults and as adults give birth to juveniles, the function b corresponds to the birth rate of the population, e and f reflect the result of harvesting a portion of the population (fishing effort for marine population), c and d measure the competition between u and v. Both populations are living in the same region Ω, and the boundary conditions in (1.1) may be interpreted as the condition that the populations u and v may not stay on ∂ Ω. The Laplacian operator shows the diffusive character of u and v within Ω.
We note that exists if , for ; see Lemma 12 and Corollary 18 in  for the details. In addition, implicit function theorem was adopted in  to discuss the limit case of system (1.1), where the authors supposed that , with sufficiently small and ; the strength of the crowding effect and the competition between u and v are negligible, i.e., , for , and small ; the functions e and f are assumed to have the same effect, that is, with and is small enough. Under the above assumptions, Bouguima et al. obtained the following multiplicity results.
, Theorem 24]
whereandare some functions w.r.t. ε, and, are neighborhoods of 0 and, respectively. The function () appearing in (1.3) is given as in, Lemma 23].
see also , Theorem 2]. However, Bouguima et al. pointed out that the positiveness of the small solution is not obvious and the question remains open. In this paper, we shall give a partial answer to this open problem.
Our main result can be stated as below.
Let, , , and e be positive constants such that. Then the small solutionis positive.
We note that the constants α and β defined in  were wrong, and this mistake is corrected in the statement of Theorem A. Moreover, we would like to point out that under the assumptions (which are similar to the corresponding ones of Theorem A) of Theorem 1.1, the discussions in , Section 5] can also be carried out, and thus the positiveness of remains unclear. However, in our case the positiveness of could be obtained. Consequently, the open problem proposed by Bouguima et al. is partially solved, and it is the first time to obtain the multiplicity of positive solutions for system (1.1). For other related results on this topic, we refer the readers to – and the references therein.
2 Proof of Theorem 1.1
The subsets , , and are open in E.
, Lemma 3.1]
Assume that f and g are continuous functions insatisfying the following conditions:
(H1)andare continuous and locally Lipschitz with respect to.
(H2), at 0, uniformly in.
Supposeand eitheror. Then there exists a neighborhoodof, such that iffor some matrix A and, then eitheror.
Proof of Theorem 1.1
Clearly, the functions and satisfy (H1) and (H2). Consequently, all of the assumptions of Lemma 2.1 are satisfied.
In addition, since is sufficiently small, it follows from Lemma 2.1 that the nontrivial small solution lies in the neighborhood of , and either or .
We claim that .
which together with (2.4) implies , and hence . Consequently, by , Corollary 18], we know that system (1.1) has no positive solutions. But this is impossible since (1.1) admits a positive solution . □
3 Related results
In this section, we shall investigate the effect of death rates and birth rates on the coexistence states. Bouguima et al. established the following existence result.
, Theorem 20]
Assumewith. If, then system (1.1) admits at least one positive solution.
satisfying , in Ω.
which together with (3.1) implies . □
and therefore , which is just the crucial condition used in Theorem B. Although (3.1) seems to be stronger than the assumption of Theorem B, the restrictive condition , is weakened, and Theorem 3.1 is applicable to more general classes of f and e rather than , .
The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC (No. 11361054), SRFDP (No. 20126203110004), Gansu provincial National Science Foundation of China (No. 1208RJZA258).
- Arino O, Montero JA: Optimal control of a nonlinear elliptic population system. Proc. Edinb. Math. Soc. 2000, 116: 225-241. 10.1017/S0013091500020897View ArticleGoogle Scholar
- Canada A, Magal P, Montero JA: Optimal control of harvesting in a nonlinear elliptic system arising from population dynamics. J. Math. Anal. Appl. 2001, 254: 571-586. 10.1006/jmaa.2000.7239MathSciNetView ArticleGoogle Scholar
- Brown KJ, Zhang Y: On a system of reaction-diffusion equations describing a population with two age groups. J. Math. Anal. Appl. 2003, 282: 444-452. 10.1016/S0022-247X(02)00374-8MathSciNetView ArticleGoogle Scholar
- Bouguima SM, Fekih S, Hennaoui W: Spatial structure in a juvenile-adult model. Nonlinear Anal., Real World Appl. 2008, 9: 1184-1201. 10.1016/j.nonrwa.2007.02.011MathSciNetView ArticleGoogle Scholar
- Henaoui O: An elliptic system modeling two subpopulations. Nonlinear Anal., Real World Appl. 2012, 13: 2447-2458. 10.1016/j.nonrwa.2012.01.020MathSciNetView ArticleGoogle Scholar
- Bouguima SM, Mehlia FZ: Asymptotic behavior of an age-structured population model with diffusion. J. Appl. Anal. Comput. 2012, 2: 351-362.MathSciNetGoogle Scholar
- Hei L: Existence and uniqueness of coexistence states for an elliptic system coupled in the linear part. Nonlinear Anal., Real World Appl. 2004, 5: 881-893. 10.1016/j.nonrwa.2004.04.001MathSciNetView ArticleGoogle Scholar
- Hei L, Wu J: Existence and stability of positive solutions for an elliptic cooperative system. Acta Math. Sin. Engl. Ser. 2005, 21: 1113-1120. 10.1007/s10114-004-0467-3MathSciNetView ArticleGoogle Scholar
- Fleckinger J, Gil RPS: Bifurcation for an elliptic system coupled in the linear part. Nonlinear Anal. 1999, 37: 13-30. 10.1016/S0362-546X(98)00138-2MathSciNetView ArticleGoogle Scholar
- Sweers G:Strong positivity in for elliptic systems. Math. Z. 1992, 209: 251-271. 10.1007/BF02570833MathSciNetView ArticleGoogle Scholar
- Kato T: Perturbation Theory for Linear Operators. Springer, Berlin; 1995.Google Scholar
- Wu B, Cui R: Existence, uniqueness and stability of positive solutions to a general sublinear elliptic systems. Bound. Value Probl. 2013, 74: 1-14.MathSciNetGoogle Scholar
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