Mathematical Problems in Engineering
Volume 2017 (2017), Article ID 4912032, 10 pages
https://doi.org/10.1155/2017/4912032

Research Article

Existence, Multiplicity, and Stability of Positive Solutions of a Predator-Prey Model with Dinosaur Functional Response

Xiaozhou Feng,¹ Kenan Shi,¹ Jianhui Tian,² and Tongqian Zhang3,4

¹Department of Mathematics and Physics, Xi’an Technological University, Xi’an 710032, China
²School of Mechanical Engineering, Xi’an Technological University, Xi’an 710032, China
³College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
⁴State Key Laboratory of Mining Disaster Prevention and Control Co-Founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Xiaozhou Feng; flxzfxz8@163.com

Received 19 April 2017; Accepted 10 July 2017; Published 13 August 2017

Academic Editor: Sebastian Anita

Copyright © 2017 Xiaozhou Feng et al. 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 investigate the property of positive solutions of a predator-prey model with Dinosaur functional response under Dirichlet boundary conditions. Firstly, using the comparison principle and fixed point index theory, the sufficient conditions and necessary conditions on coexistence of positive solutions of a predator-prey model with Dinosaur functional response are established. Secondly, by virtue of bifurcation theory, perturbation theory of eigenvalues, and the fixed point index theory, we establish the bifurcation of positive solutions of the model and obtain the stability and multiplicity of the positive solution under certain conditions. Furthermore, the local uniqueness result is studied when and are small enough. Finally, we investigate the multiplicity, uniqueness, and stability of positive solutions when is sufficiently large.

1. Introduction

The dynamic relationship between predator and their prey is one of dominant themes in ecology and mathematical ecology. Some models have been studied from various views and obtained many good results (see [1–16] and the references therein). In this paper, we are concerned with the predator-prey model with Dinosaur functional response under the homogeneous Dirichlet boundary conditions as follows:

where is a bounded domain in with smooth boundary and , stand for the densities of prey and predators, respectively. , , , , are positive constants. and denote prey intrinsic growth rate and predator intrinsic growth rate and then decrease to zero when , as follows:

The Dinosaur reaction term is an improvement or simplification of the Ivlev-type reaction term, and the change on the species density of prey better than Ivlev-type functional response can be explained. It is easy to see that the Dinosaur reaction term describes prey focus on the fight against predators when the species density of prey is large enough, so as to achieve better defense or disguise itself. To see more biological significance of systems with Ivlev-type functional responses, one can resort to [17–28] and their contents. The research on existence and uniqueness of the limit cycle of a predator-prey model with Ivlev response can be found in [22, 23]. The permanence and existence and stability of positive periodic solutions of the model were studied in [24–26]. The spatial pattern formation of the model was investigated by using Hopf bifurcation in [27, 28]. Some dynamical behavior analysis of the Ivlev response predator-prey systems was established in [17–21]. However, the researches on system (1) are very few. Hence, this paper mainly aims at establishing the existence, bifurcation, and multiplicity of positive solutions on the corresponding elliptic equations to system (1).

The research on the steady-states in reaction-diffusion model is the hot point question [1–4, 9, 10, 13, 16]. In the present paper, we study the steady-state problem corresponding to (1), with the specific form as follows:

Motivated by the papers [1–4], in the present paper, we mainly consider the positive solution of (3). In Section 2, we give some basic results and calculate the fixed point index by the fixed point index theory. In Section 3, we apply the results obtained in Section 2 to study the existence of positive solutions. In Section 4, by virtue of bifurcation theory, perturbation theory of eigenvalues, and the fixed point index theory, we establish the bifurcation of positive solutions of (3) and obtain the stability and multiplicity of the positive solution under certain conditions. Furthermore, the local uniqueness result is studied when and are small enough. Finally, we investigate the multiplicity, uniqueness, and stability of positive solutions when is sufficiently large.

2. Preliminaries

In this section, we mainly calculate the fixed point index by the fixed point index theory, in the following, and set up some definitions which are used in last paper.

Let be a Banach space. is called a wedge if is a closed convex set for all . For any , define

and we always assume that Let be a compact linear operator on We say that has property on if there exist and such that Suppose that is a compact operator for the fixed point , and is Fréchet differentiable at Set as the Fréchet differentiable of at ; hence

Lemma 1 (see [29, 30]). Assuming that is invertible on , then one has

(i) has property on , and then (ii) does not have property on , and then , where is the sum of algebra multiplicities of the eigenvalues of which are greater than one.

Let , denote the principal eigenvalue of the following problem:

and, by , it follows from [3, 4] that is strictly increasing in the case that and implies For convenience we denote by Let be the spectral radius of linear operator on Banach space.

Lemma 2 (see [24, 30]). Suppose that , and is large enough such that , Then the following results hold:

(i)(ii)(iii)

Next, we present some a priori estimate by maximum principle which is similar to [3, 4]; its proof will be omitted.

Theorem 3. Any positive solution of (3) has an a priori boundary; that is,

Consider the following equation:

it is easy to see that (7) has unique positive solution denoted by when Hence, if , then (3) has a semitrivial solution ; if , (3) has a semitrivial solution

Now, some notations are introduced as follows:

(1), here (2), here (3)(4)

Define a compact operator by the following form:

where is sufficiently large constant such that It follows that (3) has a positive solution in if and only if has a positive fixed point in Hence, are the only isolated fixed points of if Therefore, are well-defined, and the indices can be calculated, similar to Lemmas  2.3-2.4 in [3]; we can directly give Lemmas 4 and 5 as follows.

Lemma 4. Suppose

(i)(ii)(iii)If , then (iv)If , then

Lemma 5. Suppose (i) If , then ; (ii) if , then

3. Existence of Positive Solutions

In the section, we apply the result of Lemmas 4 and 5 obtained in Section 2 to study the existence and nonexistence of positive solutions of (3).

Theorem 6. (i) If , , then (3) has at least one positive solution.
(ii) If , then (3) has a positive solution if and only if and

Proof. (i) According to Lemmas 4 and 5 and the additivity property of the index, we have

Hence, (3) has at least one positive solution.
(ii) If , then (3) has no solution taking the form Supposing and , note that , and by Lemma 5, we can get the following:

Hence, (3) has at least one positive solution. So the sufficiency is proved. Conversely, suppose that is a positive solution of (3); then , This is because satisfies

It follows that This completes the proof of Theorem 6.

Theorem 7. (i) If , then (3) has no positive solutions; if and , then (3) has no nonnegative nonzero solution.
(ii) If , then the conditions , are necessary for the existence of positive solution of (3). In addition, as , the condition can take place of the necessary condition .
(iii) If , then the conditions are necessary for the existence of positive solutions of (3).

Proof. (i) Suppose that is a positive solution of (3); then satisfies

so we have According to the comparison principle of eigenvalues, we have , a contradiction. Next, we suppose that is a nonnegative nonzero solution of (3). If , , then , a contradiction to the previous proof. Similarly, it is easy to see that , when , , which derives a contradiction.
(ii) Suppose that is a positive solution of (3); it follows that , and then the positive semitrivial solution exists. Thanks to

is a lower solution of (3), and due to the uniqueness of , Thus, since satisfies the problem

it follows from Lemma 2(iii) that

Moreover, as , due to , by the second equation of (3), we get
(iii) Supposing that is a positive solution of (3), by the proof of (ii), exists and Similarly, the given condition can deduce the existence of the positive of (3), Similar to the proof of (i), we also obtain the fact that has the minimum at , or , , which derives the desired result.
This completes the proof of Theorem 7.

Theorem 8. If one of the following conditions holds, then (3) has no positive solutions:

(i) and (ii) and

Proof. (i) Assume that there is a positive solution of (3). Recalling the results of Theorem 3, as and , we get

which is a contradiction to Lemma 2(iii).
(ii) Assume that there is a positive solution of (3). By Theorem 3, if and , we have

which is a contradiction to Lemma 2(iii).
This completes the proof of Theorem 8.

4. Bifurcation and Multiplicity

In this section, taking as a main bifurcation parameter, we shall prove that (3) has at least two positive solutions when the parameters involved in (3) satisfy some ranges. In particular, the uniqueness of positive solutions is established when and are small enough. By the local bifurcation theory [6, 31] or Section  13 in [30], the branch of positive solutions of (3) bifurcates from (or ) when (or ) is established. Now we give the following Lemma where the proof is similar to Theorem  3.8 in [3].

Lemma 9. (i) If and , then (3) has a positive solution which bifurcates from if and only if , and here
(ii) If and , then (3) has a positive solution which bifurcates from if and only if , and here

More precisely, all positive solutions near are defined by

where is the principal eigenfunction of , and it follows from the following problem:

where

Remark 10. In Section 3, by Theorems 6 and 7, we can know the sufficient condition and necessary condition on the existence of positive solutions and find that there exists a gap between and

Next, using the constant as a main bifurcation parameter, we get the following theorem which establishes the multiplicity and stability results of positive solution for (3) in the gap.

Theorem 11. Suppose and . There exists small enough such that the local bifurcation of the positive solution bifurcates from which is nondegenerate and unstable for and . In addition, (3) has at least two positive solutions.

Proof. Firstly, we prove that any positive solutions bifurcated from are nondegenerate and unstable. To complete this, we need only show that there exists a sufficiently small such that, for , any positive solution of (3) is nondegenerate and the linearized eigenvalue problem

has a unique eigenvalue such that with multiplicity one. Suppose that , , are the sequences which converge to as . Thanks to , there exists the sequence such that as . Set as a solution of (3). So linearized problem (20) can be denoted by

Notice that, as , problem (21) converges to the following problem:

and since , (22) has as a simple eigenvalue with corresponding eigenfunction . Moreover, all the other eigenvalues are positive and stand apart from . Thus, it follows from perturbation theory [32] that problem (21) has a unique eigenvalue which is near to zero for large . In particular, all the other eigenvalues of problem (21) have positive real parts and stand apart from . Note that is simple real eigenvalue which converges to zero and we can denote the corresponding eigenfunction such that

Now we prove that for large in following. Multiplying the first equation of (21) with and integrating on , we get

Meanwhile, multiplying the first equation of (3) with by and integrating, we obtain

Thanks to , adopting variational principle, by the above equation, we have

According to (24) and (26), we have

Taking into (27) and so dividing by and taking the limit, we obtain

recalling that , it is easy to see that for large . This proves our claim.
Next, in order to prove the existence of at least two positive solutions, we may use reduction to absurdity and suppose that (3) has a unique coexistence state ; it follows from local bifurcation theory that solutions must be positive solutions bifurcated from near ; moreover, is nondegenerate and the corresponding linearized eigenvalue problem has a unique eigenvalue such that with multiplicity one. Thanking to the above facts, it is easy to see that is invertible and does not have property on , and it follows from Lemma 1(ii) that . Thus, applying Lemmas 4-5 and the additivity property of the index, we obtain

which derives a contradiction. The proof is completed.

Remark 12. Theorem 6 gives the multiplicity of positive solutions when . Note that since with for , there exists . Remark 10 shows that there is no positive solution of (1) if . Hence, by establishing a bifurcation sketch map as in Figure 1, we can easily get the fact that there must be at least two coexistence states for and some .

Figure 1: Bifurcation sketch map.

Theorem 13. Suppose that , , and . If

where

then (3) has a unique positive solution.

Proof. We suppose that and are two positive solutions of (3). Due to and , it follows from the comparison argument for elliptic problem that we have the following results:

Setting ; , according to system (3) and the differential mean value theorem, there exists such that and satisfy

In view of the fact that is a solution of (3), 0 is the principal eigenvalue for the following two eigenvalue problems:

Hence, applying Rayleigh’s formula for the principal eigenvalue, it follows that

Multiplying two equations of (33) by and , respectively, and integrating on , we get the fact that

Thanks to , thus

Hence, the integral in (36) has a nonnegative value, and then which show the desired results.

5. Stability and Multiplicity of Positive Solutions

In this section, the multiplicity, stability, and some uniqueness of positive solutions of (3) are considered by using as a parameter. Next, some lemmas to obtain the main results of this section are given. These lemmas will show that there exist some upper and lower solutions which do not depend on and indicate the nondegeneracy at any solution of (3) with certain hypothesis. In the following, an asymptotic result is given firstly.

Lemma 14. Suppose that , . For any given small , there exists such that for , (3) has at least one positive solution which satisfies

Proof. Set , . It is easy to see that functions and satisfy the Lipschitz continuous in . If we can show that and are the upper and lower solutions of (3), respectively, then (3) has at least one positive solution with . Thanks to Chapter  10 Comparison Theorems and Monotonicity Methods in [30] for nonquasimonotone functions, it suffices to require that the following hold:

The first and fourth inequalities in (39) are obvious. Next we check the second and third of (21), by the direct calculation as follows:

Therefore, for , the inequalities hold when

Lemma 15. Suppose that , .

(i)The positive solution of (3) meets (38) and

(ii)There exists sufficiently large such that, for , the positive solution of (3) which meets (38) is nondegenerate and linearly stable.

Proof. (i) As , it is clear to see that the operator defined in Section 2 converges to the following operator:

therefore,
(ii) Suppose that the result does not hold. Then we can obtain ; there exists with and with such that

where is a positive solution of (1) as which meets (38). By multiplying and to two equations of (3), respectively, and integrating on , we get

where and are the complex conjugates of and , respectively. By the above equation, it is easy to see that and are bounded. Then we can suppose that and Re. Since and are also bounded, so let , . Taking the limit on (44), we get

It follows from (i) that must be real number with . If then is an eigenvalue of the following problem:

Due to , , which is a contradiction, so we get . Similarly, we also can prove , a contradiction to ; the proof is complete.

Next, by the condition , one can easily get the fact that ; then there exists a sufficiently small such that , and then we can know that the problem

has a unique positive solution by the result of (7).

Lemma 16. Suppose that and . Then, for some , there exists a sufficiently large such that the following result holds when .

(i); that is, is a lower solution for .(ii)A positive solution of (3) converges to as . Meanwhile, the positive solution is nondegenerate and linearly stable.

Proof. (i) According to Theorem 3 and the comparison argument, we show ; then there exists a sufficiently large such that with ; hence

So the proof is complete.
(ii) As , is a lower solution for which does not depend on ; the proof of (ii) is similar to Lemma 15, so we omit it.

Lemma 17. Suppose that and ; here . Then there exists a sufficiently large such that ; the following claims hold:

(i).(ii)The positive solution of (3) is nondegenerate and linearly stable.

Proof. (i) Applying the comparison argument for elliptic problem, we obtain , and so as is large enough we get

which proves claim (i).
(ii) Let , where . Thanks to , where , then . Due to , hence . Since , so .
Set ; then we get

According to standard elliptic regularity theory and Sobolev embedding theorems, we may suppose that . Taking the limit, the above equation satisfies

Hence, must be the principal eigenfunction corresponding to denoted by and .
In the following, we prove that linearized eigenvalue problem (44) has no eigenvalue as . Taking , problem (44) converges to the following:

Obviously, for (53), is a simple eigenvalue with its corresponding eigenfunction . Applying the method of the proof in Theorem 11, it suffices to prove for . Take an eigenfunction such that .
Applying the integral equation on which can be established by multiplying to the second equation of (44) and integrating on , we get

Multiplying to the second equation of (3) and using variational principle on , we can deduce the following integral equation:

By substituting into the above two equations and taking the limit, we can get

which get the desired result.

Theorem 18. (i) If , , then there exists a sufficiently large such that (3) has at least two positive solutions for .
(ii) If , , then there exists a sufficiently large such that (3) has a unique positive solution for .
(iii) If and , where , then there exists a sufficiently large such that (3) has a unique positive solution for .

Proof. (i) Suppose the result is false; we can assume that (3) has a unique positive solution ; then must satisfy (38). As is large enough, by Lemma 14, it is easy to see that is invertible and has no property on . Furthermore, has no real eigenvalue larger than or equal to one. It follows from Lemma 1(ii) that . Applying Lemmas 4-5 and the additivity property of the index, we get

Which derive a contradiction, so the result of (i) is established.
(ii) Suppose that (3) has many positive solutions with the hypothesis of (ii), thanks to the compactness of the operator which has at most finitely many positive fixed points in the given region which are denoted by

Similar to the proof of (i), it is easy to see that , . Finally, applying Lemmas 16(ii) and 1(ii), along with the additivity property of the index, we get

Thus, the uniqueness is proved.
(iii) By , we can get . The proof of (iii) is similar to (ii); just the semitrivial solution does not exist in this case. So we omit it.

Theorem 19. If the assumption of Theorem 18(i) holds, then as , either or , where is a positive solution of

Proof. In the event that , it is easy to see that holds. Hence, we only investigate the case of . Let , it follows that

Taking , the solution of the above problem converges to a nonzero solution of the following problem:

Obviously, the result holds.

Remark 20. By Theorem 19, we can understand that the exact number of positive solutions is dependent on the multiplicity of positive solutions with the same hypothesis in Theorem 18(i).

6. Conclusion

The positive solution of a predator-prey model with Dinosaur functional response under Dirichlet boundary conditions is investigated. Firstly, using the comparison principle and fixed point index theory, the sufficient conditions and necessary conditions on coexistence of positive solutions on the predator-prey model with Dinosaur functional response are established. See Theorems 6–8. Secondly, by virtue of bifurcation theory, perturbation theory of eigenvalues, and the fixed point index theory, we establish the bifurcation of positive solutions to (3) and obtain the stability and multiplicity of the positive solution under certain conditions. See Theorem 11. Furthermore, the local uniqueness result is studied when and are small. See Theorem 13. Finally, we investigate the multiplicity, uniqueness, and stability of positive solutions when is large. See Lemmas 15–17 and Theorems 18-19.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The work was partially supported by the National Natural Science Foundation of China (nos. 61102144, 11371230), the Natural Science Basic Research Plan in Shaanxi Province of China (no. 2016JM6041), Shandong Provincial Natural Science Foundation of China (no. ZR2015AQ001), Research Funds for Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources by Shandong Province and SDUST (2014TDJH102), and the president of the Xi’an Technological University Foundation (XAGDXJJ1423).

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