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Qualitative analysis on a predator-prey model with Ivlev functional response
Advances in Difference Equations volume 2013, Article number: 164 (2013)
The diffusive predator-prey model with Ivlev functional response is considered under homogeneous Dirichlet boundary conditions. Firstly, we investigate the bifurcation of positive solutions and derive the multiplicity result for γ suitably large. Furthermore, a range of parameters for the uniqueness of positive solutions is described in one dimension. The method we used is based on a comparison principle, Leray-Schauder degree theory, global bifurcation theory and generalized maximum principle.
In this paper, we are concerned with the following reaction-diffusion system:
where Ω is a bounded domain in () with smooth boundary ∂ Ω, u, v represent the population density of prey and predator, respectively. a is the natural growth rate of prey, d is the conversion rate of a consumed prey to a predator, γ is the efficiency of the predator for capturing prey. a, c, d and γ are constants with a, d positive and γ non-negative; c may change sign and indicates the predator has other food sources. This is a prey dependent predator-prey model with the Ivlev-type functional response , which was originally introduced by Ivlev in .
The predator-prey model has long been one of the dominant themes due to its universal existence and importance. Both ecologists and mathematicians are interested in the Ivlev-type predator-prey model; see [2–9] for example. The existence and uniqueness of limit cycle for the Ivlev response predator-prey system were studied in [2, 3]. The conditions for the permanence of the Ivlev system and the existence and stability of a positive periodic solution were investigated in . The dynamical behavior analysis of the Ivlev response predator-prey systems was discussed in [1, 5–7]. To our knowledge, there are few works on such a type of functional response in the reaction-diffusion system. Under Neumann boundary conditions, the spatial pattern formation of the model was carried out by using Hopf bifurcation in . Under Dirichlet boundary conditions, a sufficient and necessary condition for the existence of positive solutions to the model was obtained in .
Now, we introduce some notations and basic facts which will be often used later. Let X be the Banach space
Let be the usual positive cone in X, where ν is the outward unit normal vector on ∂ Ω and . For , let be all eigenvalues of the following problem:
It follows from  that is simple and is strictly increasing in the sense that and implies . When , we denote by for the sake of convenience. Moreover, we denote by (>0) the eigenfunction corresponding to with normalization .
For any , it is well known that the problem
has a unique positive solution which we denote by . It is also known that the mapping is strictly increasing, continuously differentiable in , and that uniformly on as . Moreover, in Ω. Therefore, if , then (1.1) has a semi-trivial solution . Similar results hold with respect to another semi-trivial solution whenever . We extend the definition of by taking if .
This work mainly aims at establishing the existence, multiplicity and uniqueness of positive solutions to (1.1). More precisely, a sufficient and necessary condition for the existence of positive solutions is given when , and when , the multiplicity of positive solutions is obtained under the assumption that γ is suitably large. If γ is suitably small, then we get the uniqueness of positive solutions in one dimension.
The rest of this paper is organized as follows. In Section 2, by calculating the indices of fixed points, we obtain sufficient conditions for the existence of positive solutions to (1.1). In Section 3, by investigating the bifurcation of positive solutions emanating from the semi-trivial solution , we give a sufficient and necessary condition for the existence of positive solutions to (1.1) and establish the multiplicity result of positive solutions when γ is suitably large. In Section 4, assuming that is an interval, we find that (1.1) has at most one positive solution when .
2 The existence of positive solutions
In this section, we establish the existence and nonexistence of positive solutions to (1.1). A necessary condition and a priori estimate are firstly given. The proofs are standard and will be omitted.
Lemma 2.1 If (1.1) has a positive solution, then we have
Lemma 2.2 Assume that is a positive solution of (1.1). Then satisfies
In addition, if .
Next, we set up the fixed point index theory for later use. Let E be a real Banach space and W be a closed convex set of E. For , define and . Let be a compact operator with a fixed point , and denote by L the Fréchet derivative of F at y. Then L maps into itself. We say that L has property α on if there exist and such that .
For an open subset , define , where I is the identity map. If y is an isolated fixed point of F, then the fixed point index of F at y in W is defined by , where is a small open neighborhood of y in W.
Lemma 2.3 (See )
Assume that is invertible on .
If L has property α on , then .
If L does not have property α on , then , where σ is the sum of algebra multiplicities of the eigenvalues of L which are greater than 1.
Denote by the spectral radius of a linear operator L.
Lemma 2.4 (See )
Let and let M be a positive constant such that on . Then we have the following conclusions:
Now we introduce the following notations:
, where ;
, where ;
From Lemma 2.2, we see that all the non-negative solutions of (1.1) must be in . For any , define a positive compact operator by
where M is large such that . It follows from the standard elliptic regularity theory that is a completely continuous operator. Observe that (1.1) has a positive solution in W if and only if has a positive fixed point in . If , then , , are the only non-negative fixed points of which are not positive. The corresponding indices in W can be calculated in the following lemmas.
Lemma 2.5 Assume that .
If , then .
If , then .
If , then .
Proof (i) Since has no fixed point on , the degree is well defined. It is easy to see that all fixed points of are in . Therefore, by the homotopy invariance of degree, is independent of τ. Then
Observing that (1.1) has only the trivial solution when , we have
Let . Then
It is easy to see that by Lemma 2.4. This implies that is invertible on and L does not have property α on . By Lemma 2.3, . It follows from (2.1) and (2.2) that .
It is easy to observe that , . Let . Then
Assume that for some . Then
Since , , we have . Thus is invertible on .
Note that . By Lemma 2.4, we know that , and is the principal eigenvalue of the operator with the corresponding eigenfunction . Set . Then and . This shows that L has property α. By Lemma 2.3, .
Let . Then , . Set . Then we have
Assume that for some . Then
If , then . And we further have . Thus is invertible on .
Note that . By Lemma 2.4, we know that is the principal eigenvalue of the operator with the corresponding eigenfunction . Set . Then , and
This shows that L has property α. By Lemma 2.3, .
Since , is invertible on . We claim that L does not have property α on . Note that . By Lemma 2.4, we know that . Suppose that L has property α on . Then there exist and such that . Therefore,
Since , is a principal eigenvalue of the operator , which is contradiction to . Thus L does not have property α on . By Lemma 2.3, , where σ is the sum of the multiplicities of all real eigenvalues of L which are greater than 1.
Assume that is an eigenvalue of L with a corresponding eigenfunction . Then simple calculations yield
If , then from the second equation of (2.3), we obtain
This contradiction shows that . Thus . From the first equation of (2.3), we have
This contradiction shows that L has no eigenvalues being greater than 1. Consequently, . Hence, . □
Similarly, we can obtain the following lemma.
Lemma 2.6 Assume that .
If , then .
If , then .
By the additivity property of the index, the existence of positive solutions to (1.1) is obtained.
Theorem 2.1 (i) If , , then (1.1) has at least a positive solution.
If , , then (1.1) has at least a positive solution.
Proof Argue by contradiction. Suppose that (1.1) has no positive solution.
If and , then by Lemma 2.5 and the additivity property of the index, we have
The contradiction implies that (1.1) has at least a positive solution in .
If and , then by Lemmas 2.5, 2.6 and the additivity property of the index, we have
The contradiction implies that (1.1) has at least a positive solution in . □
3 Bifurcation and multiplicity of positive solutions
In this section, by discussing the bifurcations of positive solutions by using a and c as the main bifurcation parameters, respectively, we establish the multiplicity of positive solutions when γ is suitably large. First, we show that (1.1) has no positive solution when c is sufficiently large.
Lemma 3.1 If (1.1) has a positive solution, then there exists a sufficiently large constant such that .
Proof Suppose that (1.1) has a positive solution . Then by Lemma 2.2, we have
Moreover, since the function is strictly decreasing with respect to , considering the equation of u, we find
Choose c large enough. Then for fixed a, we have , which is a contradiction to (3.1). □
Fixing and taking c as a bifurcation parameter, we shall obtain positive solutions bifurcating from the semi-trivial solution .
Theorem 3.1 Assume that . Let . Then is a bifurcation point of the positive solution to (1.1). Moreover, there exists a constant and a -curve such that
is a positive solution of (1.1) with for each and(3.2)
where is a positive eigenfunction corresponding to with , , , ;
, , and .
Now we state a sufficient condition for the existence of positive solutions as follows.
Theorem 3.2 If the following relationship holds:
then (1.1) has at least a positive solution.
Proof Fixing and taking c as the main bifurcation parameter, we can obtain a supercritical bifurcating branch of positive solutions to (1.1), which emanates from the semi-trivial solution at the value of . The existence was given in Theorem 3.1. It suffices to show the bifurcation direction. To this end, substitute given by (3.2) into the second equation of (1.1), divide by s, differentiate with respect to s and set , which leads to
Now, multiply by and integrate over Ω to get
Hence, we have
Noting that , we obtain , which shows that the bifurcating branch is supercritical.
In the following, we shall investigate the global structure of bifurcation solutions given by Theorem 3.1 and then the relationship in (3.2) can be obtained. By the global bifurcation theorem in , we can extend the local bifurcation positive solution to the global one. One can see [15, 16] for similar arguments. From Lemma 3.1, we know that the parameter c is bounded. And from a priori estimates given by Lemma 2.2, it follows that and are also bounded. Hence, we claim that the continuum of positive solutions bifurcating from cannot remain in the interior of , which implies that there must exist (), at which one of the components of the continuum of positive solutions vanishes. Let be a strictly increasing sequence converging to for which (1.1) has at least a positive solution such that or as . If we denote by the limit of as , then we have the following three cases:
Since is non-degenerate, (iii) is excluded. Set . Then satisfies
Letting , we have
It follows from that , which contradicts the global bifurcation theorem. (ii) is also excluded. Hence, we know that (i) holds true. Set . Then satisfies
Letting , we have
It follows from that . Now, we know that is uniquely determined by . Observe that implies that . Hence, if and , then (1.1) has at least a positive solution. □
Remark 3.1 The condition (3.3) is better than those obtained in Theorem 2.1. In particular, (3.3) includes the case . Note that Theorem 2.1 tells us nothing when . Since is not invertible on when , Lemma 2.3 is not satisfied and so we cannot use it to get the index of the fixed point .
Fix and take c as the bifurcation parameter, then by the proof of Theorem 3.2, we can obtain a supercritical bifurcating branch from the point . Moreover, resorting to the global bifurcation theory, we get the maximal continuum of positive solutions, which tells us the range of the parameter c is , where is determined uniquely by . Hence, a sufficient and necessary condition for the existence of positive solutions can be stated in the following remark.
Remark 3.2 Assume . Then (1.1) has a positive solution if and only if , where is determined uniquely by .
In the case , Theorem 3.2 tells us that if , then (1.1) has at least a positive solution. A natural question is whether (1.1) has a positive solution if . In fact, making use of bifurcation theory and degree theory, we can solve this problem and further establish the multiplicity of positive solutions to (1.1) when γ is suitably large. We first take a as a bifurcation parameter and give the bifurcation solutions emanating from the semi-trivial solution .
Theorem 3.3 Assume . Then is a bifurcation point of the positive solution to (1.1). Moreover, there exist a constant and a -curve such that
is a positive solution of (1.1) with for each and
where is the positive eigenfunction corresponding to with , , , ;
, , and ;
has the derivative , where is defined by(3.4)
Using the above theorem, we can obtain the following multiplicity result of positive solutions to (1.1).
Theorem 3.4 Assume that . Let . If , then there exists a constant such that (1.1) has at least two positive solutions for each and has at least one positive solution for .
Proof It follows from Theorem 3.2 that (1.1) has at least one positive solutions for each . So we only have to show that (1.1) has at least two positive solutions for each and has at least one positive solution for .
Let . Note that the direction of the bifurcation of (1.1) emanating from the semi-trivial solution is determined by the sign of given in (3.4). If , then we see that and the bifurcation is subcritical. So there exists a positive constant such that for each , (1.1) has a unique positive solution . To prove the existence of a positive solution when and two positive solutions when , it suffices to show that for each , (1.1) has a positive solution in .
where , , W and M are defined in Section 2. It follows from standard elliptic regularity theory that is a completely continuous operator. Obviously, (1.1) has non-negative solutions if and only if the operator has fixed points in . It is easy to check that has no fixed point on for . Hence, . In particular, . For small , has only three fixed points , and in . It is well known that is linearly stable while and are unstable, which implies and . Hence, we have
On the other hand, has only two non-negative fixed points and which are not positive in . By Lemma 2.5, both and are zero. Thus one can assert that has a fixed point in other than and , which shows that (1.1) has a positive solution in . This completes the proof. □
4 The uniqueness of positive solutions
The main result in this section is the following.
Theorem 4.1 Assume for some real numbers . If , then for every satisfying (3.3), the following boundary value problem:
has exactly one positive solution.
The proof will be finished in several steps. The technique we used here can be found in the papers [17, 18]. The basic ingredient is non-degeneration of positive solutions, which is summarized as the following lemma.
Lemma 4.1 Let be an arbitrary positive solution of (4.1). Then the linearized problem of (4.1) at ,
has only the trivial solution . In other words, any positive solution is non-degenerate.
Proof Since is a positive solution of (4.1), by the Krein-Rutman theorem, we have
The linearized problem (4.2) can be written as
From the monotonicity of , it follows that
If , then we claim that
Now we shall prove that (4.4) is true. Let . It suffices to show . Obviously, and . Reminding given by Lemma 2.2, we get and thus . This shows that (4.4) holds true.
Define the operators and by
So (4.2) can be written as
By (4.3) and (4.5), and have inverses, say , , which are compact and order-preserving, i.e., for . Now we shall show that the only solution to (4.6) is , which completes the proof of this lemma. To this end, we argue by contradiction, assuming that there exists a solution to (4.2). From (4.6), it follows that
Since the right-hand side of (4.7) defines a compact strongly order-preserving operator, we find that ϕ must change sign in . Similarly, ψ must change sign in . Moreover, ϕ and ψ cannot vanish on an interval of positive length by the maximum principle, i.e., the zeros of ϕ, ψ are isolated each from the others. In fact, if ϕ vanishes on an interval, then at the boundary of such an interval where or , which contradicts the maximum principle. Thus there exists a partition of , say
and we can choose ϕ such that
Here we claim that
Since the principal eigenvalues of and on any subinterval of are strictly positive, the generalized maximum principle will be used to show this claim. By hypothesis, , and . Thus
We claim that . In fact, if , then by the generalized maximum principle, we have , . Thus
Therefore , , which contradicts (4.8). So .
Again by hypothesis, , and . Thus
We claim that . In fact, if , then by the generalized maximum principle, we have , . Thus
Therefore , , which contradicts (4.8). So . Arguing recursively, we show (4.9) holds.
According to the parity of n, either
is satisfied. Assume (4.10) holds. Then
Since and , by the generalized maximum principle, we have , . Thus
Therefore , , which contradicts (4.8). Similarly, if (4.11) holds, we also get a contradiction. The proof of Lemma 4.1 is complete. □
Applying the implicit function theorem, we can show that if (4.1) has exactly one positive solution, which in addition is non-degenerate, then the following problem:
has also exactly one positive solution provided ϵ is small enough. The proof is omitted.
Lemma 4.2 Suppose that (3.3) is satisfied and (4.1) has exactly one positive solution , which is non-degenerate. Then there exists such that for every the problem (4.12) has exactly one positive solution . Moreover, and the mapping , from a neighborhood of in R to , belongs at least to the class .
Proof of Theorem 4.1 Consider the set
Since (4.1) with is uncoupled, if it has a positive solution, then it has exactly one. Thus , i.e., Γ is not empty. By Lemma 4.2, we know that Γ is open in . Now we shall show that Γ is closed in . Hence , which completes the proof. To show this, consider a sequence in Γ satisfying as . Since the mapping is increasing in γ, both (3.3) and are satisfied for and , . Let be the unique positive solution of (4.1) with . By passing to a subsequence if necessary, converges to as . From the proof of Theorem 3.2, it follows that is in the interior of . Since (3.3) is satisfied for , we know that is a positive solution of (4.1) with . In fact, the problem (4.1) with has exactly one positive solution. Otherwise the application of the implicit function theorem would imply that (4.1) with has at least two positive solutions for sufficiently large n, contradicting the fact that . Hence and . This finishes the proof. □
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The work is supported by the Natural Science Foundation of China (Nos. 11271236, 11001160), the Natural Science Basic Research Plan in Shaanxi Province of China (No. 2011JQ1015) and the Foundation of Shaanxi Educational Committee of China (No. 12JK0856).
The authors declare that they have no competing interests.
GG performed the theory analysis and carried out some computations. BL participated in the sequence alignment and also undertook some computations. XL participated in the design of the study. All authors read and approved the final manuscript.
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Guo, G., Li, B. & Lin, X. Qualitative analysis on a predator-prey model with Ivlev functional response. Adv Differ Equ 2013, 164 (2013). https://doi.org/10.1186/1687-1847-2013-164
- predator-prey model
- Ivlev functional response