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Qualitative Analysis for a Predator Prey System with Holling Type III Functional Response and Prey Refuge
A predator prey system with Holling III functional response and constant prey refuge is considered. By using the Dulac criterion, we discuss the global stability of the positive equilibrium of the system. By transforming the system to a Liénard system, the conditions for the existence of exactly one limit cycle for the system are given. Some numerical simulations are presented.
Recently, the qualitative analysis of predator prey systems with Holling II or III types functional response and prey refuge has been done by several papers, see [1–5]. Their main objective is to discuss under what conditions the positive equilibrium of the corresponding system is stable or unstable and the existence of exactly one limit cycles. In general, the prey refuge has two types, one is the so-called constant proportion prey refuge: , where , the other type is called constant prey refuge: .
In , the authors considered the following system with a constant proportion prey refuge: where and denote the prey and predator density, respectively, at time , the parameters are positive constants, and their biological meanings can be seen in . The main result is that when system (1.1) admits only one limit cycle which is globally asymptotically stable.
In paper , the authors only gave the local stability analysis to the following system with a constant prey refuge: In this paper, we will research under what conditions that the positive equilibrium is globally asymptotically stable and the existence of exactly one stable limit cycle of system (1.2). For ecological reason, we only consider system (1.2) in or .
It easy to obtain the following lemma.
Lemma 1.1. Any solution of system (1.2) with initial condition is positive and bounded for all .
2. Basic Results
It is easy to obtain the following lemma.
Lemma 2.1. Let hold. Further assume that and , . Then is locally asymptotically stable, if any of and holds. When is unstable, furthermore, is a saddle point.
About the properties of the positive equilibrium, we have the following theorem.
Theorem 2.2. Assume . Then(I) is locally asymptotically stable for if holds.(II) is locally asymptotically stable for and is locally unstable for if holds, where (III)system (2.1) undergoes Hopf bifurcation at if holds.
Proof. The Jacobian matrix of system (2.1) at is
where . Then , where , the discriminant of is . Hence, the equation has two roots and , where .
Note that and implies . Consider Then (I)If holds, then holds for . Considering (H2) and , for , , which implies is locally asymptotically stable.(II)If holds, then , for , since , by , we obtain . Together with (H2), for , which means is locally asymptotically stable. On the other hand, for is locally unstable.(III)We have these satisfy Liu’s Hopf bifurcation criterion (see , page 255); hence, the Hopf bifurcation occurs at . This ends the proof.
3. Global Stability of the Positive Equilibrium
Theorem 3.1. If is locally stable. Further assume that , then the positive equilibrium of system (2.1) is globally asymptotically stable.
Proof. Take the Dulac function , for system (2.1) we have
If for .
On the other hand, there exist The equation has two roots .
Case 1. If , then for , ; for , . Hence, is the least value of the function . If , it has for all , then is increasing for , notice that . Therefore, for . Since, for , system (2.1) does not exist limit cycle.
Case 2. If , then , for , hence, for is increasing. Evidently, , then there exists such that , where , hence, when , when . We know that takes the least value at , that is, . According to , for we obtain , where .
To prove for , it suffices to prove for . Clearly, takes the least value at , and is strictly decreasing at the interval . Hence, for holds. Since . Therefore, for holds if holds, then for holds.
In sum, if one of the following three conditions holds (1) ; (2) ; (3) , , the function does not change the sign for , then system (2.1) does not exist limit cycle. It is easy to see that the conditions , and are equal to . The proof is completed.
4. Existence and Uniqueness of Limit Cycle
Theorem 4.1. If holds, for system (2.1) admits at least one limit cycle in .
Proof. We construct a Bendixson loop which includes of system (2.1). Let be a length of the line be a length of line . Define where . The orbit of system (4.1) with initial value intersects with the line and the intersection point , we obtain the orbit arc . Let be a length of line be a length of line . Because is a length of orbit line of system (2.1) and , , the orbits of system (2.1) tend to the interior of the Bendixson loop from the outer of , and , by comparing system (2.1) to system (4.1): and . Then the orbits of system (2.1) tend to the interior of the Bendixson loop from the outer of . On the other hand, under the condition of Theorem 4.1, is unstable, by Poincaré-Bendixson Theorem, system (2.1) admits at least one limit cycle in the region . This ends the proof.
Lemma 4.2 (see ). Let , be continuously differentiable functions on the open interval , and be continuously differentiable functions on in such that(1), (2)having a unique , such that for and ,(3) for ,then system (4.1) has at most one limit cycle.
Theorem 4.3. If holds, for system (2.1) exists exactly one limit cycle which is globally asymptotically stable in .
Proof. Let , still denote , as , then system (2.1) becomes
the positive equilibrium changes .
Let , then transform to the origin , still denote , as yield where .
Clearly, . It is easy to see that the conditions (1) and (2) of Lemma 4.2 for are satisfied. Consider Note that by the assumption of Theorem 4.3, is unstable equilibrium and then . Consider where where
Then, we have where
By a simple computation, we obtain It is easy to verify that and has two roots and defined by, respectively, Obviously, . Therefore, for and for which indicates that is the minimum point of the function when . Substituting into , we obtain It is easy to see that if , then , which implies for all . That is, the function is a strictly increasing function for .
Note that for and . It follows from (4.6) that
Hence, there exists a point , such that , that is, This, together with the monotonicity of when , we may conclude that for and for . Therefore, is the minimum point of the function for .
Together with (4.16), we obtain It follows from (4.6), we have . This indicates for all .
Then all the conditions of Lemma 4.2 are satisfied, considering Theorem 4.1, we obtain the conclusion of this theorem. The proof is completed.
5. Numerical Simulations
Take , , , , , and . Then , and . One can see a Hopf bifurcation occurring at and the bifurcated periodic solution is stable in Figure 1.
This work is partially supported by the National Natural Science Foundation of China (11226142), Foundation of Henan Educational Committee (2012A110012), Youth Science Foundation of Henan Normal University (2011QK04), Natural Science Foundation of Shanghai (no. 12ZR1421600), and Shanghai Municipal Educational Committee (no. 10YZ74).
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Copyright © 2012 Xia Liu and Yepeng Xing. 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.