Complex Boundary Value Problems of Nonlinear Differential Equations: Theory, Computational Methods, and ApplicationsView this Special Issue
Bifurcations of a Ratio-Dependent Holling-Tanner System with Refuge and Constant Harvesting
The bifurcation properties of a predator prey system with refuge and constant harvesting are investigated. The number of the equilibria and the properties of the system will change due to refuge and harvesting, which leads to the occurrence of several kinds bifurcation phenomena, for example, the saddle-node bifurcation, Bogdanov-Takens bifurcation, Hopf bifurcation, backward bifurcation, separatrix connecting a saddle-node and a saddle bifurcation and heteroclinic bifurcation, and so forth. Our main results reveal much richer dynamics of the system compared to the system with no refuge and harvesting.
The Holling-Tanner predator-prey system has attracted much attentions from both theoretical and mathematical biologists, especially, in  the authors considered the ratio-dependent system of the form where and stand for prey and predator population (or densities) at time , respectively. The predator growth is of logistic type with growth rate and carrying capacity in the absence of predation; and stand for the predator capturing rate and half saturation constant, respectively; is the intrinsic growth rate of predator; however, carrying capacity ( is the conversion rate of prey into predators) is the function on the population size of prey. They studied the global properties and the existence and uniqueness of limit cycle for system (1).
Generally speaking, from the views of the optimal management and exploitation of bioeconomic resources, it is necessary and meaningful to consider the refuge or harvesting of populations in some bioeconomic models; one can see [2–11], and the references therein.
In this paper we will analyze the system (1) with refuge and harvesting of the form where , , , , , , , and are positive constants. is a constant number of prey using refuges, and is the rate of prey harvesting.
For simplicity, we first rescale the system (2).
Let , ; system (2) can be written as (still denote , as , )
Next, let , , and , then system (3) takes the form (still denote , , and as , , ) where , , , , and .
From the view of biology, we are only interested in the dynamics of the system (4) in the first quadrant.
The organization of this paper is as follows. In Section 2, we discuss the existence and properties of the equilibria of system (4). In Section 3, all possible bifurcation phenomena of the model in terms of the five parameters are presented, and the numerical simulations about every bifurcation phenomena are exhibited.
2. Qualitative Analysis of Equilibria
To obtain the boundary equilibria the following equation can be obtained Its discriminant is .
Obviously, if and if .
Hence, (5) has two distinct positive solutions , if , a positive solution if , a double solution if , and a solution when and .
One can obtain the positive equilibrium of (4) by solving the equation
We can derive that system (4) has two positive equilibria and if where Moreover, we can show that system (4) just exists one positive equilibrium if and .
The positive equilibrium () of system (4) exists if , and , = + , where , = , = , and .
Summarizing the previous discussion, the number and location of equilibria of system (4) can be described by the following lemmas.
Lemma 1. Let . (i) System (4) has no equilibria when ;(ii) System (4) exist two equilibria and when .
Lemma 2. Let . (i) System (4) has no equilibria when . (ii) System (4) has a unique equilibrium when . (iii) System (4) has two equilibria and when . (iv) System (4) has an equilibrium when . (v) System (4) has two equilibria and when .
Lemma 3. Let and . (i) System (4) has no equilibria when . (ii) System (4) has a unique equilibrium when . (iii) System (4) has two equilibria and when . (iv) System (4) has three equilibria , , and when . (v) System (4) has four equilibria , , , and when . (vi) System (4) has two equilibria and when .(vii) System (4) has two equilibria and when .
Next we discuss the dynamics of system (4) in the neighborhood of each feasible equilibria. Firstly, the Jacobian matrix of system (4) at is It is easy to see that , if exists, is a hyperbolic saddle.
Secondly, the Jacobian matrix of system (4) at is One can see that boundary equilibrium , if exists, is an unstable hyperbolic node.
The Jacobian matrix of system (4) at is Hence, , if exists, is a saddle.
Similarly, we assume boundary equilibrium exists, and the Jacobian matrix of system (4) at is obtained as follows:
Hence, is a saddle node.
The previous discussion can be summarized as follows.
Theorem 4. If the equilibria , , and exist, then and are hyperbolic saddle, and is a hyperbolic unstable node. Moreover, and merge into a saddle node when .
Remark 5. Note that if , then , if , then . Thus, the prey species may go extinct as time increases for some initial values when . That is, biological over harvesting occurs.
In the following, we will discuss the properties of interior equilibria of system (4).
2.1. The Properties of Interior Equilibria
The Jacobian matrix of system (4) at is The characteristic equation is , where Denote that The discriminant of is , then has two distinct solutions and denoted by If , it is easy to see that is a node if or , a degenerate node if or , and a focus or a center type nonhyperbolic if .
If is a node if , and a degenerate node if , and a focus or a center-type nonhyperbolic if .
To discuss the stability of , we need to determine the sign of . Define , then .
Clearly, if then for all ; if , then when , when by simple computation, one can obtain , hence .
The Jacobian matrix of system (4) at is Its determinant is .
Through the previous discussion, about the stability of and , we have the following theorem.
Theorem 6. Equilibrium , if exists, must be a hyperbolic saddle. Equilibrium , if exists, may be a node or a focus when . And when , is a stable focus for , a stable degenerate node for , a stable node for , an unstable node for , an unstable degenerate node for , an unstable focus for , and a weak focus or a center for .
The Jacobian matrix of system (4) at is then by the existence condition of , , . Then by taking similar methods used in estimating the properties of , we have the following theorem.
Theorem 7. Let , . Then, (i) assume , then is stable; (ii) assume , then is stable if , is unstable if , and is a weak focus or a center if .
The Jacobian matrix of system (4) at is One can see that , which indicates that is a degenerate singularity and maybe has complicated properties, see the following theorem.
Theorem 8. Let , . Then system (4) has three equilibria, where is a hyperbolic saddle, is a hyperbolic unstable node, and is a degenerate singularity. More precisely, if , then is a saddle node; if , then is a cusp of codimension .
Proof. In order to discuss the properties of system (4) in the neighborhood of the equilibrium , we first take , , then is translated to , and system (4) becomes (still denote , as , )
Clearly, if , then . is a saddle-node. We finish the proof of the part 1°.
When , , which implies that both eigenvalues of the matrix are zero. We rewrite system (20) as where
By introducing variable into previous system and rewriting as for simplicity, then we obtain that where
We take transformation , into (24), then system (24) is transformed to where In order to obtain the canonical normal forms of system (26), we will perform a series of transformations of variables for system (26) in a small neighborhood of .
Firstly, performing the transformation by taking , , then (26) becomes
Secondly, performing the transformation by taking , , then (28) becomes We perform the final transformation of variables by Then, we obtain Note that which indicates that the origin of (31) is a cusp of codimension 2. We complete the proof.
3. Bifurcation Analysis
From previous analysis, we can see the equilibria of system (4) may be hyperbolic or degenerate singularities under appropriate conditions, which indicate that some bifurcations may occur for system (4). It is interesting to investigate what kinds of bifurcations system (4) can undergo with the original parameters varying.
3.1. Hopf Bifurcation
Theorem 6 shows that , if exists, is a weak focus or a center when where .
To determine the direction of Hopf bifurcation and stability of in this case, we need to compute the Liapunov coefficients of the equilibrium . Let and by the variable , . Then we rewrite system (4) (still denote , as , ) as follows: We perform the transformations and rewrite , as , . Then the previous system can be transformed to where the expressions of , , and depend on the parameters , , , , and , and = .
Using the formula, the first Liapunov number is Therefore, there exists a surface () in the parameter space which satisfies
Hence, when the parameter is in (), the equilibrium of system (4) is a weak focus of multiplicity and is unstable (stable) (see ). () is called the subcritical (supercritical) Hopf bifurcation surface of system (4).
From Theorem 6, we know that is a stable focus for and , an unstable focus for and .
(i) System (4) has at least one unstable limit cycle if , , .
(ii) System (4) has at least one stable limit cycle if , , .
Remark 10. When system (4) maybe undergoes degenerate Hopf bifurcation for some parameter values; since the expression of is complicated, we do not discuss this case.
Note that by Theorem 7, if is a weak focus or a center, then we can obtain that its first Lyapunov number is therefore, is a stable weak focus.
By numerical calculation, we give the parameter values , then , and , and the existence condition of subcritical Hopf bifurcation is satisfied. If we keep , , , fixed and choose , then a unstable limit cycle can be shown in Figure 1(a).
When taking , then , , and which satisfy the existence condition of supercritical Hopf bifurcation. Furthermore, we choose ; according to Theorem 9, there exists a stable limit cycle, which can be shown in Figure 1(b).
3.2. Backward Bifurcation
Lemmas 2–3 and Theorems 6–8 illustrate that when the parameter varies in the range of , system (4) just has only one positive equilibrium which is stable. However, when varies in the range , system (4) has two distinct positive equilibria and , where is a stable node or focus and is a saddle. Furthermore, when , system (4) has unique positive equilibrium . The previous discussion indicates the possibility of a backward bifurcation, which can be summarized as follows.
Theorem 11. Let , . Then system (4) has a unique positive equilibrium when , has two distinct positive equilibria and when , where is a stable node and is a saddle, and has one positive equilibrium or when . Therefore, system (4) undergoes a backward bifurcation when .
We give a numerical example in Figure 2 which displays that system (4) has a backward bifurcation at .
3.3. Saddle-Node Bifurcations
From Lemmas 2–3 and Theorem 4, we see that when , , and degenerate into a saddle-node . This indicates that there is a saddle node bifurcation surface which takes the form
Similarly, from Lemma 3 and the part of Theorem 8, we know that when and , in , system (4) admits the double point . And is a saddle node if .
One also can see that when the parameter varies in the range of , system (4) has two distinct positive equilibria and . From Theorem 6, we know that may be a stable node, or a focus, and is a saddle. These imply that system (4) undergoes another saddle-node bifurcation of codimension 1. That is, there is a second saddle-node bifurcation surface which is defined by
3.4. Bogdanov-Takens Bifurcation
From the part of Theorem 8, we can see that system (4) exists a cusp of codimension 2, which implies that there may exist the Bogdanov-Takens bifurcation in system (4). Clearly, there exists a parameter space such that system (4) has a cusp of codimension 2 when .
To show that system (4) undergoes the Bogdanov-Takens bifurcation we choose and as bifurcation, parameters. We need to find the universal unfolding of .
Let , and consider the following unfold system where and are small parameters and vary in the neighborhood of the origin.
Translating to by the transformation and . Then system (43) is rewritten as where and are smooth functions of , at least of the third order. And
Taking the change of variables , and rewriting , as , , we obtain where with , .
Taking , substituting in system (46), and rewriting as , we get that where is a smooth function of , , and at least of order three. When , ,
Next, let , , and into (48) and rewriting , , and as , , and yields where is a smooth function of , , and at least of order three and when , . Let , , , and rewrite as . Then system (50) becomes where is a smooth function of , , and at least of order three and .
Then system (4) exists the following bifurcation curves in a small neighborhood of the origin in the plane.
Theorem 12. Let , , . Then system (43) admits the following bifurcations: (i) there exists a saddle node bifurcation curve ; (ii) there is a Hopf bifurcation curve ; (iii) there is a homoclinic bifurcation curve .
The biological interpretation for the Bogdanov-Takens bifurcation is that if the harvesting rate and the prey refuge value satisfy , , and , then the predator and prey coexist in the form of a positive equilibrium or a periodic orbit for different initial values, respectively. And there exist other values of parameters, such that the predator and prey coexist in the form of a positive equilibrium for all initial values lying inside the homoclinic loop, and the predator and prey coexist in the form of a periodic orbit with infinite period for all initial values on the homoclinic loop. By choosing , , , , and , the numerical simulations for the Bogdanov-Takens bifurcation in Theorem 12 can be shown in Figures 3, 4 and 5.
3.5. Separatrix Connecting a Saddle-Node and a Saddle Bifurcation and Heteroclinic Bifurcation
From Theorem 8 and Lemma 3, when , , , there may exist a separatrix connecting the saddle-node and the saddle . When , , the saddle node separates into the hyperbolic node and the hyperbolic saddle , which implies that system (4) undergoes a separatrix connecting a saddle node and a saddle bifurcation. Furthermore, the heteroclinic bifurcation may occur if there exists a heteroclinic orbit connecting the separatrix of saddle and saddle .
This paper is supported by NSFC (11226142), Foundation of Henan Educational Committee (2012A110012), Foundation of Henan Normal University (2011QK04, 2012PL03), Natural Science Foundation of Shanghai (12ZR1421600), and Shanghai Municipal Educational Committee (10YZ74).
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