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

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


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][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and the references therein).In this paper, we are concerned with the predatorprey model with Dinosaur functional response under the homogeneous Dirichlet boundary conditions as follows: where Ω is a bounded domain in   ( ≥ 1) with smooth boundary Ω and , V 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 ℎ(1 −  − ) 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][18][19][20][21][22][23][24][25][26][27][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][25][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][18][19][20][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][2][3][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  > 0 is sufficiently large.

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  ≥ 0. For any  ∈ , define and we always assume that  =  − .Let  :   →   be a compact linear operator on .We say that  has property  on   if there exist  ∈ (0, 1) and  ∈   \   such that − ∈   .Suppose that  :  →  is a compact operator for the fixed point  0 ∈ , and  is Fréchet differentiable at  0 .Set  =   ( 0 ) as the Fréchet differentiable of  at ; hence  :  → .
(ii)  does not have property  on , and then   (,  0 ) = (−1)  , where  is the sum of algebra multiplicities of the eigenvalues of  which are greater than one.

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).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.
Proof.(i) Suppose that (, V) is a positive solution of (3); then (, V) satisfies so we have  =  1 ( + V − ).According to the comparison principle of eigenvalues, we have  >  1 , a contradiction.Next, we suppose that (, V) is a nonnegative nonzero solution of (3).If  ̸ ≡ 0, V ≡ 0, then  >  1 , a contradiction to the previous proof.Similarly, it is easy to see that  >  1 , when  ≡ 0, V ̸ ≡ 0, which derives a contradiction.(ii) Suppose that (, V) is a positive solution of (3); it follows that  >  1 , and then the positive semitrivial solution   exists.Thanks to is a lower solution of (3), and due to the uniqueness of   ,  <   .Thus, since V satisfies the problem Moreover, as  < 1/, due to  <   ≤  < 1/, by the second equation of (3), we get  >  1 (−   −  ).
(iii) Supposing that (, V) is a positive solution of (3), by the proof of (ii),   exists and  ≤   .Similarly, the given condition  >  1 can deduce the existence of the positive   of (3),   ≤ V. Similar to the proof of (i), we also obtain the fact that  =  1 ( + V − ) has the minimum at  = 0, V =   or  =   , V =   , 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  ≤ .

Mathematical Problems in Engineering
Proof.(i) Assume that there is a positive solution (, V) of (3).
Recalling the results of Theorem 3, as  ≥   and  ≤ , we get which is a contradiction to Lemma 2(iii).

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 (0,   , ) (or (  , 0, )) when  >  More precisely, all positive solutions near ã fl  1 (  ) are defined by where Φ is the principal eigenfunction of ã, and it follows from the following problem: where Ψ = (−Δ +  − 2  ) −1 (  Φ).
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  >  1 (  ) and  >  1 (  ) −  .
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  >  1 and ∫ Ω Φ 3 (1 −   ) < 0. There exists small enough  > 0 such that the local bifurcation of the positive solution ((), V()) bifurcates from (0,   , ã) which is nondegenerate and unstable for  ∈ (ã − , ã) and 0 <  ≪ 1.

In addition, (3) has at least two positive solutions.
Proof.Firstly, we prove that any positive solutions bifurcated from (0,   ) are nondegenerate and unstable.To complete this, we need only show that there exists a sufficiently small  > 0 such that, for  ∈ (ã − , ã), any positive solution ((), V()) of ( 3 has a unique eigenvalue μ such that Re( μ) < 0 with multiplicity one.Suppose that {  > 0}, {  }, {  > 0} are the sequences which converge to 0 as  → ∞.Thanks to  = ã +  1  + ( 2 ), there exists the sequence {  } such that   ∈ (ã−  , ã) as  → ∞.Set (  , V  ) as a solution of (3).So linearized problem (20) can be denoted by and since  =  1 (  ) <  1 (2  ), (22) has 0 as a simple eigenvalue with corresponding eigenfunction (Φ, 0).Moreover, all the other eigenvalues are positive and stand apart from 0. 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 0. Note that   is simple real eigenvalue which converges to zero and we can denote the corresponding eigenfunction (  ,   ) such that lim →∞ (  ,   ) → (Φ, 0) . ( Now we prove that   < 0 for large  in following.Multiplying the first equation of ( 21) with Φ and integrating on Ω, we get Meanwhile, multiplying the first equation of ( 3) with (, V) = (  , V  ) by   and integrating, we obtain Thanks to   =   Φ + ( 2 ), adopting variational principle, by the above equation, we have According to ( 24) and ( 26), we have Taking ((), V()) = (Φ + ( 2 ),   + Ψ + ( 2 )) into (27) and so dividing by   and taking the limit, we obtain lim recalling that ∫ Ω Φ 3 (1 −   ) < 0, it is easy to see that  < 0 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 (û, V); it follows from local bifurcation theory that solutions must be positive solutions bifurcated from (0,   ) near ã; moreover, (û, V) is nondegenerate and the corresponding linearized eigenvalue problem has a unique eigenvalue μ such that Re( μ) < 0 with multiplicity one.Thanking to the above facts, it is easy to see that  −   (û, V) is invertible and does not have property  on , and it follows from Lemma 1(ii) that index  (, (û, V)) = (−1) 1 = −1.Thus, applying Lemmas 4-5 and the additivity property of the index, we obtain which derives a contradiction.The proof is completed.
Proof.We suppose that ( 1 , V 1 ) and ( 2 , V 2 ) are two positive solutions of (3).Due to  >  1 and  >  1 + ( + /()), it follows from the comparison argument for elliptic problem that we have the following results: Setting  fl  1 −  2 ;  fl V 1 − V 2 , according to system (3) and the differential mean value theorem, there exists  1 ,  2 ∈ [ 1 ,  2 ] such that  and  satisfy In view of the fact that ( 1 , V 1 ) 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 Ω, we get the fact that Hence, the integral in (36) has a nonnegative value, and then ,  ≡ 0 which show the desired results.

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.
Proof.(i) As  → ∞, it is clear to see that the operator (, V) defined in Section 2 converges to the following operator: therefore, (, V) → (  ,   ).
(i)  ≥ ω; that is, ω is a lower solution for .
(ii) A positive solution of (3) converges to (  ,   ) as  → ∞.Meanwhile, the positive solution is nondegenerate and linearly stable.