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Journal of Function Spaces
Volume 2019, Article ID 2193670, 9 pages
https://doi.org/10.1155/2019/2193670
Research Article

Fixed Point Theory and Positive Solutions for a Ratio-Dependent Elliptic System

School of Mathematic Sciences, Qufu Normal University, Qufu Shandong, 273165, China

Correspondence should be addressed to Aixia Qian; nc.ca.ssma@aixiaq

Received 3 July 2018; Accepted 26 November 2018; Published 1 January 2019

Academic Editor: Yong H. Wu

Copyright © 2019 Jingmei Liu and Aixia Qian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider a ratio-dependent predator-prey model under zero Dirichlet boundary condition. By using topological degree theory and fixed index theory, we study the necessary and sufficient conditions for the existence of positive solutions. Then we present the asymptotic behavior analysis of positive solutions, by bifurcation theory and energy estimates.

1. Introduction and Preliminaries

There is growing results about elliptic system, which comes from biological and physiological evidence, such as [19]. A more suitable general predator-prey model should be based on the so-called ratio-dependent theory, which asserts that the per capita predator growth rate should be a function of the ratio of prey to predator abundance. In recent years, such model has been studied extensively, and many important phenomena have been observed (see [46, 1012] and references therein).

In this paper, we consider the following semilinear elliptic system with ratio-dependent function response and Dirichlet boundary condition:Here is a bounded domain in with smooth boundary . In the biological model, the two unknown functions represent the spatial distribution density of the prey and predator. is termed the birth rate of the prey, while is the death rate of the predator and is a positive constant sometimes referred to as the conversion rate. The Holling-Tanner interaction term represents the rate at which the prey is consumed by the predator with .

We remark that problem (1) with homogeneous Neumann boundary condition was discussed in [6]. We will consider what is more general in this paper, where the parameters are positive and . What is more, since (1) comes from the biological module, our results and methods are different from those in [1316] and references therein, which are also the Dirichlet boundary problems.

We say that a solution of (1) is positive solution if both for all and for all ; i.e., is also a coexistence state of (1).

Now, we give some notations, definitions and well-known facts which will be used in the sequel.

For each , let be the principal eigenvalue ofAs is well known, the principal eigenvalue is given by the following variational characterizationWe denote by ; some useful properties can be seen [6, Proposition A.1] or [17, Lemma 5.2].

For , let be a sufficiently large constant such that for all . Define a compact linear operator by , where is the unique solution of the following problem:Denote be the spectral radius of . Then the relationship between and can be given as [2, Proposition 1].

Theorem 1 (see [2]). ;
;
.
From Theorem 1, we see that it is crucial to determine the eigenvalue . The following theorem is established by [2] (see also Theorem 2.4 [1]).

Theorem 2 (see [1, 2]). Let and in with on . Then we have(a)If , then ;(b)If , then ;(c)If , then .Some concepts of cone, total wedge, topological degree and fixed point index in a cone can be seen in [1821] and so on.

Let be a Banach space and be a total wedge. Let be a compact operator with a fixed point and let be a relatively open subset of such that has no fixed point on the boundary of . We denote by the degree of in relative to and by the fixed point index of at relative to .

Theorem 3 (see [21, 22]). Assume that is a total wedge, and let be a compact operator with a fixed point and it is Frechlet differentiable at . Let be the Frechlet derivative of at . Then maps into itself. Moreover, if is invertible on , then the following results hold:(i)If has property on , then ;(ii)If does not has property on , then , where is the sum of multiplicities of all eigenvalues of , which is greater than .

2. Existence of Positive Solutions of (1)

In the sequel, for simplicity of notation and more transparent analysis, we redefineIt is easy to see that, for ,while Under the definition of , it is obvious that (1) has a trivial solution and only one semitrivial solution (if ), where is the unique positive solution of

Lemma 4. If model (1) has a positive solution, then and .

Proof. Assume is a positive solution of (1), and thenIt follows from the property of the principal eigenvalue that and , that is, .
On the other hand, since satisfiesone hasSo .

Remark 5. Above lemma is about the necessary condition for (1) to have positive solutions. Next we show that and are also the sufficient conditions for the existence of a positive solution of (1). We will use fixed point index theory.

Lemma 6. Assuming , then any positive solution of (1) has a priori bounds

Proof. Since part 1 is a simple consequence of a standard comparison argument, we omit it. For part 2, by a direct calculation, we get thatand henceAs , we haveTherefore . Otherwise, if , then , which implies , a contradiction. Consequently is necessary for the existence of a positive solution; in such casethen

Remark 7. Above result shows that the coexistence states of (1) have a priori bounds as soon as varies in compact subinterval of .
Now we introduce the following notations:where
From Lemma 6, nonnegative solutions of (1) must be in . Define a positive and compact operator bywhere is s sufficiently large number such that and for .

Remark 8. Note that (1) is equivalent to by elliptic regularity, and therefore in order to show the existence of positive solutions of (1), it suffices to prove has nontrivial fixed points in .

We next compute the fixed point index of the trivial and semitrivial solution of (1). We have shown that (1) has a trivial solution and a semitrivial solution since and . Moreover the following lemma holds.

Lemma 9. Assuming that and , then(i);(ii);(iii) if ; and if .

Proof. (i) For each , we define a positive and compact operator byThen has no fixed point on and . Thus is well defined for all . By the homotopy invariance of Laray-Schauder degree and the only fixed point of in , we obtainSetAssume that for some . It is easy to see . Thus is invertible on . Since , we see by Theorem 1, this implies that does not have property on . Thus by Theorem 3 (ii), .
(ii) Observe that . Let and thenAssuming that for some , thenSince and , we see that . Thus is invertible on . By Theorem 1, we see and is the principle eigenvalue of the operator with a corresponding eigenfunction in . Set and then , butThis shows that has property , and thus by Theorem 3 (i).
(iii) The second part is a straightforward consequence of Lemma 4 and (i)(ii). In fact, from Lemma 4, the nonnegative solutions of (1) are and if , so by the properties of topological degree, we haveCombining (i) and (ii) above and (26), we have .
Next, give the proof of the first part. Observe . Let , and thenAssume that for some ; i.e., satisfiesTaking , it follows from the second equation of (28) and Theorem 2 that , if , which contradicts . So . Then we get from the first equation of (28)If , then by Theorem 2. On the other hand, , and we get a contradiction. Therefore and is invertible on .
We claim that has property on . In fact, setSince , by Theorem 1 (ii), is an eigenvalue of with a corresponding eigenfunction . Since , we see . Set , and thenThis proves that has property . Therefore .

Lemma 10. If and , then (1) has a positive solution.

Proof. By Lemma 9, we haveHence (1) has at least a positive solution.

From Lemmas 4 and 10, we get the following.

Lemma 11. Model (1) admits a positive solution if and only if and .

3. Structure of Solutions with as a Bifurcation Parameter

In this section we shall regard as a bifurcation parameter and suppose that all other constants are fixed. For all values of , we have the branch of zero solutions of (1) . Suppose and , and then (1) also has the branch of semitrivial solutions . We next give some results about the bifurcation from .

Lemma 12. (i) The trivial steady state is locally asymptotically stable if and , while it is unstable if or .
(ii) Assume that . Then the semitrivial solution steady state is locally asymptotically stable if , while it is unstable if .

Proof. From the linearization principle eigenvalue problemThe stability of is determined by the problemSince (34) is not completely coupled, we only need to consider the following two eigenvalue problems:andThen an eigenvalues of (34) must be an eigenvalues of (35) or (36) (see [11, 12] and references therein). Denote the principle eigenvalue of (35) and (36) by and , respectively. ThenCombining above results, one can see that if , then all eigenvalues of (34) are positive, and thus is locally asymptotically stable. On the other hand, if , then (34) has a negative eigenvalue, which implies the instability of .
Similarly, combining with the definition of , we can get (i).

First we shall obtain a local result on bifurcation from , as the results of [23].

Lemma 13. Let be fixed. Then is a bifurcation value of (1) where positive solutions bifurcate from the line of semi-trivial solutions . The set of positive solutions to (1) near is a smooth curvesuch that , , and . Moreoverand is the unique bifurcation value for which positive solutions bifurcate form .

Proof. We apply the similar proof of Lemma 12 [4]. By changing the variables , defineA simple calculation implies thatBy letting , we can find that only when , does have a solution with . Thus is the only bifurcation point along , where positive solution of (1) bifurcates. At , it is easy to verify that the kernelwhere satisfiesSince , we can choose , and thenThe range of the operator is given bywhich is of codimension one andsince . Thus we can conclude that the set of positive solutions to (1) near is . Moreoverwhere is the linear functional defined by . Therefore the bifurcation at is always subcritical.
Suppose that there is a sequence of coexistence states of (1) with for some . Then from the -equation of (1), we find for every or equivalentlyBy the compactness of , it is easy to see that along some subsequence, relabeled by , we havefor some with . Thus passing the limit in (49), we find thatTherefore , which concludes the proof.

Remark 14. Above results together with Lemma 4 show that the bifurcation of nonnegative solutions from at must be to the left, while Lemma 4 also shows that the branch of nontrivial solutions cannot extend too far to the left.
We now investigate the global nature of the above curve of nontrivial nonnegative solution in the plane, i.e., in , to show that hypotheses of Th [24] are satisfied.
Writing and , it is easy to check that is a nonnegative solution of (1) if and only if and satisfiesRewrite (52) asDefine byThen is a linear compact operator and the Frechlet derivative . Let and then with and if and only if is a nonnegative solution of (1).
We must calculate the index when is close to . This index is equal to , where is the sum of the algebraic multiplicities of eigenvalue of .
Suppose that is an eigenvalue of . Then there exists a nonzero function such thati.e., is an eigenvalue of (55). Conversely, if and is an eigenvalue of (55) with corresponding eigenfunction , then is an eigenfunction of corresponding to the eigenvalue , where is the unique solution ofSince all eigenvalues of are positive and , it follows that is invertible. The eigenvalues of (55) form an increasing sequence and is a continuous increasing function. Thus is an eigenvalue of if and only if for some . Clearly .
(i) Suppose and is an eigenvalue of . Then is an eigenvalue of (55) and the least eigenvalue of , i.e., . But , a contradiction. Hence has no eigenvalues and so .
(ii) Now suppose , i.e., . Since is increasing with , there exists a unique ( say) such that . Since , it follows that for and . Thus is the only eigenvalue of which is greater than . We now to show that is a simple eigenvalue of . The discussion above shows that , where is the principal eigenfunction corresponding to the eigenvalue ofand . Thus . Suppose that . Then there exists such thatMultiplying by and integrating over shows that , which is impossible. Hence and so is a simple eigenvalue of . Thus whenever .
Therefore Theorem 3.2 [24] can be applied to .
LetBy arguments similar to those used in Section 4 of [24], it can be proved that there exists a continuum in emanating from such that(i)if , then ;(ii)if and , then is a solution of (1).(iii)close to the bifurcation point , consists of the points on the curve given by Lemma 13.Clearly in a neighbourhood of the bifurcation point .

Theorem 15. Assume (i)If , then , i.e., ;(ii) is unbounded in .

Proof. (i) Suppose that contains a point which lies outside of . Then there exists a point which is the limit of a sequence of points in . As , either or .
(1) Suppose . Then for and either for some or for some . It follows thatwhere is a constant chosen sufficient large so that the term in the square bracket is positive for all . It follows from the strong maximum principle that . A similar argument shows that if , then . Thus or .
(2) Suppose and . Then , and so lies on the trivial branch of solutions . The only trivial nonnegative solutions which are close to lie on the semitrivial , and so there cannot exist a sequence in converging to . Therefore it is impossible that both and are identically zero.
(3) Suppose , then and we find from the first equation of (1)Then by theory and bootstrapping arguments, such that
andwhich is impossible, since .
(4) Suppose that . Then and there bifurcate from nontrivial, nonnegative solutions. While we have shown that is the unique bifurcation value for which positive solutions bifurcate from , then . That is impossible.
Therefore, if , then .
(ii) must satisfy one of the three alternatives discussed before Theorem 15. Because of (i) above, contains no pair of points of the form and and cannot join up with another bifurcation point of the form on . Hence joins to ; i.e., is unbounded.
Based on above preparations, we have the bifurcation results as follows.

Theorem 16. AssumeThen there exists an unbounded component of the set of positive solutions of (1) such that and if . i.e., , where stands for the projection operator into the component of the term.

4. Asymptotic Behavior Analysis of Positive Solutions of (1)

In this section, we give the sketch of asymptotic behavior for positive solution as the parameter .

Suppose (63) and let be a coexistence of (1). Then, since , there exists a constant such that . Hence owing to Lemma 6, there exists a constant such thatNote that

Theorem 17. Suppose . Let be a sequence such thatFor each , let be a coexistence of (1); thenandwhere is the unique positive solution of

Proof. By the global bifurcation theory or Theorem 15, we get . From the second equation of (1)Since in , by using regularity theory and Schauder bootstrapping technique (see [17] Chapter 5), in such that . In fact, is the principal eigenfunction corresponding to . This implies that for any compact subset of , uniformly in .
On the other hand, from the first equation of (1)Similar to arguments above, in , such that on