# Steady states of a nonlinear elliptic system arising from population dynamics

- Ruyun Ma
^{1}Email author, - Ruipeng Chen
^{1}and - Yanqiong Lu
^{1}

**2014**:185

https://doi.org/10.1186/s13661-014-0185-4

© Ma et al.; licensee Springer 2014

**Received: **24 March 2014

**Accepted: **16 July 2014

**Published: **24 September 2014

## Abstract

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.

## Keywords

## 1 Introduction

where Ω is a bounded regular domain of ${\mathbb{R}}^{n}$ ($n\ge 1$), *i.e.*, $\overline{\mathrm{\Omega}}$ is an *n*-dimensional compact connected smooth submanifold of ${\mathbb{R}}^{n}$ with boundary *∂* Ω. Throughout the paper, we suppose that *a*, *b*, *c*, *d*, *e*, and *f* are positive functions in ${C}^{\alpha}(\overline{\mathrm{\Omega}})$ for a certain $\alpha \in (0,1)$.

System (1.1) arises from population dynamics where it models the steady-state solutions of the corresponding nonlinear evolution problem [1], 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.

*F*the nonlinear term such that $F(U)(x)=\left(\begin{array}{c}c(x)u(u+v)\\ d(x)v(u+v)\end{array}\right)$. In the following, for $q\in {L}^{\mathrm{\infty}}(\mathrm{\Omega})$, we denote by $\underline{q}$ and $\overline{q}$ the essential infimum and supremum of

*q*, respectively. Let ${\rho}_{1}(q)$ be the principal eigenvalue of the linear problem

**n**is the outward unit normal vector on

*∂*Ω; see [3], Theorem 3.2]. Recently, Bouguima

*et al.*[4] investigated system (1.1) and showed that it has a positive solution if and only if ${\lambda}_{1}(\mathfrak{L}-A(x))<0$ via fixed point theory, where ${\lambda}_{1}(\mathfrak{L}-A(x))$ is the principal eigenvalue of the linear eigenvalue problem

We note that ${\lambda}_{1}(\mathfrak{L}-A(x))$ exists if $a(x)>0$, $b(x)>0$ for $x\in \mathrm{\Omega}$; see Lemma 12 and Corollary 18 in [4] for the details. In addition, implicit function theorem was adopted in [4] to discuss the limit case of system (1.1), where the authors supposed that $a(x)={\rho}_{1}(e)+\epsilon {a}_{1}(x)$, $b(x)={\rho}_{1}(e)+\epsilon {b}_{1}(x)$ with $\epsilon >0$ sufficiently small and ${a}_{1},{b}_{1}\in {L}_{+}^{\mathrm{\infty}}(\mathrm{\Omega})$; the strength of the crowding effect and the competition between *u* and *v* are negligible, *i.e.*, $c(x)=\epsilon {c}_{1}(x)$, $d(x)=\epsilon {d}_{1}(x)$ for ${c}_{1},{d}_{1}\in {L}_{+}^{\mathrm{\infty}}(\mathrm{\Omega})$, and small $\epsilon >0$; the functions *e* and *f* are assumed to have the same effect, that is, $f(x)=e(x)+\epsilon {f}_{1}(x)$ with ${f}_{1}\in {L}_{+}^{\mathrm{\infty}}(\mathrm{\Omega})$ and $\epsilon >0$ is small enough. Under the above assumptions, Bouguima *et al.* obtained the following multiplicity results.

### Theorem A

[4], Theorem 24]

*Assume that*

*Then there exists*${\epsilon}_{0}>0$,

*such that system*(1.1)

*has two nontrivial solutions of the form*

*where*${S}^{1}:(-{\epsilon}_{0},{\epsilon}_{0})\to {V}_{0}$*and*${S}^{2}:(-{\epsilon}_{0},{\epsilon}_{0})\to {V}_{\ast}$*are some functions w*.*r*.*t*. *ε*, *and*${V}_{0}$, ${V}_{\ast}$*are neighborhoods of* 0 *and*$\frac{\alpha}{\beta}$, *respectively*. *The function*$U({S}^{i}(\epsilon ),\epsilon )$ ($i=1,2$) *appearing in* (1.3) *is given as in*[4], *Lemma * 23].

see also [4], Theorem 2]. However, Bouguima *et al.* pointed out that the positiveness of the small solution ${\xi}_{1}$ 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.

### Theorem 1.1

*Let*${a}_{1}$, ${b}_{1}$, ${f}_{1}$, *and* *e* *be positive constants such that*${a}_{1}+{b}_{1}-{f}_{1}>0$. *Then the small solution*${\xi}_{1}$*is positive*.

### Remark 1.1

We note that the constants *α* and *β* defined in [4] 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 [4], Section 5] can also be carried out, and thus the positiveness of ${\xi}_{1}$ remains unclear. However, in our case the positiveness of ${\xi}_{1}$ could be obtained. Consequently, the open problem proposed by Bouguima *et al.*[4] 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 [6]–[8] and the references therein.

## 2 Proof of Theorem 1.1

The subsets ${K}^{+}$, ${K}^{-}$, and $K={K}^{+}\cup {K}^{-}$ are open in *E*.

### Lemma 2.1

[9], Lemma 3.1]

*Assume that* *f* *and* *g* *are continuous functions in*$\mathrm{\Omega}\times {\mathbb{R}}^{2}$*satisfying the following conditions*:

(H1)$f(x,r)$*and*$g(x,r)$*are continuous and locally Lipschitz with respect to*$r=({r}_{1},{r}_{2})\in {\mathbb{R}}^{2}$.

(H2)$f(\cdot ,r)=o(|r|)$, $g(\cdot ,r)=o(|r|)$*at* 0, *uniformly in*$x\in \overline{\mathrm{\Omega}}$.

*Suppose*$\sigma ({\tilde{A}}_{0})\cap \sigma (-\mathrm{\Delta})=\{{\lambda}_{1}\}$*and either*${\tilde{b}}_{0}({\lambda}_{1}-{\tilde{a}}_{0})>0$*or*${\tilde{c}}_{0}({\lambda}_{1}-{\tilde{d}}_{0})>0$. *Then there exists a neighborhood*${\mathcal{U}}_{0}$*of*$({\tilde{A}}_{0},(0,0))$, *such that if*$(A,(u,v))\in {\mathcal{U}}_{0}\cap \mathcal{S}$*for some matrix* *A* *and*$(u,v)\ne (0,0)$, *then either*$u,v\in {K}^{+}$*or*$u,v\in {K}^{-}$.

### Proof of Theorem 1.1

Clearly, the functions $f(x,u,v):=-\epsilon {c}_{1}(x)u(u+v)$ and $g(x,u,v):=-\epsilon {d}_{1}(x)v(u+v)$ satisfy (H1) and (H2). Consequently, all of the assumptions of Lemma 2.1 are satisfied.

*i.e.*, the nontrivial solution $({A}_{\epsilon},({u}_{\epsilon},{v}_{\epsilon}))$ with ${u}_{\epsilon}$ and ${v}_{\epsilon}$ small, where

In addition, since $\epsilon >0$ is sufficiently small, it follows from Lemma 2.1 that the nontrivial small solution $({A}_{\epsilon},({u}_{\epsilon},{v}_{\epsilon}))$ lies in the neighborhood ${\mathcal{U}}_{0}$ of $({A}_{0},(0,0))$, and either ${u}_{\epsilon},{v}_{\epsilon}\in {K}^{+}$ or ${u}_{\epsilon},{v}_{\epsilon}\in {K}^{-}$.

We claim that ${u}_{\epsilon},{v}_{\epsilon}\in {K}^{+}$.

which together with (2.4) implies ${\lambda}_{1}(\mathfrak{L}-{A}_{\epsilon}^{T})>0$, and hence ${\lambda}_{1}(\mathfrak{L}-{A}_{\epsilon})>0$. Consequently, by [4], Corollary 18], we know that system (1.1) has no positive solutions. But this is impossible since (1.1) admits a positive solution ${\xi}_{2}$. □

## 3 Related results

In this section, we shall investigate the effect of death rates and birth rates on the coexistence states. Bouguima *et al.*[4] established the following existence result.

### Theorem B

[4], Theorem 20]

*Assume*$f=e+k$*with*$k\in \mathbb{R}$. *If*$\underline{a}\underline{b}>{\rho}_{1}(e){\rho}_{1}(f)$, *then system* (1.1) *admits at least one positive solution*.

*a*,

*b*,

*c*,

*d*,

*e*, and

*f*are ${C}^{\alpha}(\overline{\mathrm{\Omega}})$ continuous positive functions, system (1.2) has a principal eigenvalue ${\lambda}_{1}(\mathfrak{L}-A(x))$, which corresponds to a unique (up to a constant multiple) eigenfunction

satisfying ${u}_{0}>0$, ${v}_{0}>0$ in Ω.

### Theorem 3.1

*System*(1.1)

*has at least one positive solution provided that*

### Proof

which together with (3.1) implies ${\lambda}_{1}(\mathfrak{L}-A(x))<0$. □

### Remark 3.1

and therefore $\underline{a}\underline{b}>{\rho}_{1}(e){\rho}_{1}(f)$, 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 $f=e+k$, $k\in \mathbb{R}$ is weakened, and Theorem 3.1 is applicable to more general classes of *f* and *e* rather than $f=e+k$, $k\in \mathbb{R}$.

## Declarations

### Acknowledgements

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).

## Authors’ Affiliations

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