Abstract
A prey-predator system with the strong Allee effect and generalized Holling type III functional response is presented and discretized. It is shown that the combined influences of Allee effect and step size have an important effect on the dynamics of the system. The existences of Flip and Neimark-Sacker bifurcations and strange attractors and chaotic bands are investigated by using the center manifold theorem and bifurcation theory and some numerical methods.
1. Introduction
Lotka-Volterra model as a paradigm of the fundamental population models has been developed and widely applied in ecological sciences and other fields. Some linear and nonlinear functional responses are employed to describe the phenomena of predation including the Beddington-DeAngelis type [1, 2], Crowley-Martin type [3], and Holling types [4] that are more realistic and proper for modeling most prey-predator interactions [1–8]. Recently, a modified prey-predator model with a generalized Holling response function of type III was addressed [8]. The system exhibits more richer dynamics including the Hopf bifurcation of codimensions 1 and 2, Bogdanov-Takens bifurcation of codimensions 2 and 3.
The researches of the Allee effect on biological population [9–23] imply that the Allee effect probably is a nonignorable factor in ecology, especially for the situation when population density (or size) is low (or small). That fitness of an individual in a small or sparse population decreases as the population density declines leads to Allee effect occurrence. Two most common continuous growth equations to express Allee effect for a single species are given [15–18] which are called multiplicative Allee effect model [15, 16] and the additive Allee effect model [17, 18], respectively. Parameters , , and are the Allee thresholds, where is termed as the strong Allee effect item if and weak Allee effect item if . Mathematically, the systems subjected to the Allee effect can be depicted by the bistability switch, which may lead to more complex dynamics. Thus, the consideration of prey-predator model subjected to an Allee effect is more realistic and has attracted renewed interest, which motivates us to make a modification to the system presented in [8] for further study: where . and denote the prey and predator densities at time , respectively. , , , , , , and are positive constants. is positive or negative. In this paper, we will focus on the case . In particular, and represent the intrinsic growth rate and carrying capacity of prey in the absence of predation, respectively. is so called the strong Allee threshold. The predator consumes the prey with functional response , known as Holling type III response and contributes to its growth rate . is the maximum death rate in the absence of prey. For simplicity, take nondimensional transformations , , to system (2) and rewrite , , as , , ; we have where , , , , and .
Many studies of discrete-time models [19–38] were made and suggested that discrete-time models described by the difference equations are more appropriate than the continuous time models when populations have nonoverlapping generations or the number of populations is small. Moreover, dynamical patterns produced in discrete-time models are much richer than those observed in continuous time models.
Applying the forward Euler scheme, we obtain the discrete-time form of system (3) as follows: where is the step size.
The paper is organized as follows: in Section 2 we show the existence and stability of fixed points of (4). In Section 3 we investigate the existence of flip bifurcation and the Neimark-Sacker bifurcation. In Section 4 we show some numerical simulations to illustrate our main results. Finally, a brief discussion of (4) is given to sum up our analysis.
2. Existence and Stability of Fixed Points
To obtain the fixed points of system (4), it is sufficient to solve the following algebraic equations: After calculating, we have the following statement.
Lemma 1. System (4) always has the trivial fixed point and the boundary fixed points , . If , (4) has a unique positive fixed point , where
To analyze dynamical properties of (4), we need to compute the Jacobian matrix of (4) at any fixed point where The corresponding characteristic polynomial of can be written as where , .
Let and be two roots of equation . Clearly, the local stability of the fixed point is determined by the modules of eigenvalues and ; that is, if and , then is locally asymptotically stable and is termed a sink. If and , then is unstable and is termed a source. If and (or and ), then is termed a saddle. If or , then is termed nonhyperbolic. These can be stated by the following lemma.
Lemma 2 (see [20]). Suppose ; then (i) and if and only if and ;(ii) and if and only if and ;(iii) and if and only if and ;(iv) and if and only if .
A simple computation show that two eigenvalues of are and . Also note that , , , and . Thus, from Lemma 2, we have the following.
Lemma 3. The fixed point of model (4) always exists; moreover, (i) is a source if , ;(ii) is a sink if , ;(iii) is a saddle if ;(iv) is nonhyperbolic if .
Remark 4. Define
From Lemma 3, we see that if or , then one of the two eigenvalues of is and the other is neither nor , which implies that system (4) may undergo the flip bifurcation at the fixed point when parameters vary in a small neighborhood of or .
Similarly, the eigenvalues of Jacobian are and . By Lemma 2, we obtain the following results.
Lemma 5. For , one has the following:(i) is a sink if , ;(ii) is a source if one of the following two cases holds:(1), with ;(2), ;(iii) is a saddle if one of the following three cases holds:(1), with ;(2), ;(3), ;(iv) is nonhyperbolic if one of the following three cases holds:(1);(2) with ;(3).
Remark 6. Lemma 5(iv) implies that there are two parameter surfaces defined, respectively, by
such that one of the eigenvalues of is and the other is neither nor when or . Thus, system (4) may undergo a flip bifurcation when changes in a small neighborhood of or .
After calculating, we also obtain the eigenvalues of the Jacobian are and . Note that so . Obviously, if , then is a source. If , then is a saddle. If , then is nonhyperbolic. By calculating, one can obtain the following.
Lemma 7. The dynamical behaviors of the fixed point can be stated as follows. (i)It is a source if or with .(ii)It is a saddle if .(iii)It is nonhyperbolic if or with .
Now, we discuss local stability of the fixed point . After calculating, we get where and . The discriminant of equation can be calculated . Moreover, and Regarding and as two functions of , then the discriminants of equations and are and , respectively.
Clearly, the sign of is determined by . Therefore we have the following.
Lemma 8. Assume , then (i) is a source if one of the following three cases holds:(1);(2), , ;(3), , ;(ii) is a sink if one of the following two cases holds:(1), , ;(2), , ;(iii) is a saddle if and , ;(iv) is nonhyperbolic if one of the following three cases holds:(1), , ;(2), , ;(3), , .
Proof. That always holds. Set and are eigenvalues of matrix . We will consider two cases to prove Lemma 8.
Case 1. Assume , then and which yield and .
Case 2. Assume . One can see that if , then , ; moreover if and only if and . By comparing and , we obtain that if , then and which lead to the fact that is a source. If the and which lead to the fact that is a sink. If the and which lead to and , that is, is nonhyperbolic.
In Case 2, the fact that the assumption leads to that the equation has two distinct positive roots and so that when or , when . Also note that if , if . Thus, we derive that and if , and if , if or . Following above discussion and Lemma 2, the proof of Lemma 8 is complete.
From above analysis and Lemma 8, we have two parameter surfaces:
From Lemma 8, we can see that if , then two eigenvalues of are a pair of conjugate complex numbers and satisfy . Thus, when vary in a small neighborhood of model (4) undergoes a Neimark-Sacker bifurcation from the fixed point . Similarly, if , then one of two eigenvalues of is and the other is neither nor ; that is, when vary in a small neighborhood of , a flip bifurcation from the fixed point can occur.
3. Analysis of Bifurcation
On the basis of the analysis in Section 2, by choosing parameter as a bifurcation parameter, we mainly investigate the flip bifurcation and Neimark-Sacker bifurcation of positive fixed point by using the center manifold theorem and bifurcation theory of [39–41] in this section.
Giving a perturbation () to parameter we consider a perturbation of model (4) as follows:
Let be a feasible fixed point of model (4). We translate to the origin by using transformations , . Then we have (rewrite , as , ) Expanding model (16) as Taylor series at to the second order, then we have where
The flip bifurcation of model (4) at will be investigated firstly when varies in the small neighborhood of . Similar arguments can be applied to the other cases , , , , and .
Let arbitrarily. From the discussion in Section 2, we can see model (4) has a unique positive fixed point ; its eigenvalues are and . In this case, , , and . Also note that and so . Moreover, there is a transformation so that model (17) becomes where
By the center manifold theorem [41], we can get the approximate representation of the center manifold of model (20) at the fixed point in a small neighborhood of as follows: where Furthermore, on the center manifold , we have Therefore, the map which is model (4) restricted to the center manifold takes the form where In order to undergo a flip bifurcation for map (25), we require that two discriminatory quantities and are not zero, where From above analysis and the theorem in [40], we have the following.
Theorem 9. If , then model (4) undergoes a flip bifurcation at equilibrium when the parameter varies in the small neighborhood of . Moreover, if (resp. ), then the period points that bifurcate from are stable (resp., unstable).
Next, we discuss the Neimark-Sacker bifurcation of when parameters vary in the small neighborhood of . Taking (for simplicity, denoting by ) and choosing as a perturbation of bifurcation parameter, a perturbed form of model (4) is obtained as follows: where which is a small perturbation.
We transform the unique positive fixed point into the origin by using , : where The characteristic equation associated with linearization system of model (29) at is where When parameters and vary in a small neighborhood of , the roots of the characteristic equation (31) are a pair of complex conjugate numbers and denoted by One can see that
On the other hand, it is required that , when , which is equivalent to . Also note that and when . Thus, , which indicates . We only need to require that ; that is, which implies that defined by (33) do not lie in the intersection of the unit circle with the coordinate axes when .
Next we study the normal form of model (29) when .
Let , , and and perform the transformations , to (29); we can obtain where
Let By [39], we know that if the following discriminatory quantity holds, then model (38) undergoes a Neimark-Sacker bifurcation. where Form above analysis and the theorem in [39], we have the following.
Theorem 10. Let and (41) hold. Then system (4) undergoes Neimark-Sacker bifurcation at the positive fixed point when the parameter varies in the small neighborhood of . Moreover, if (resp., ), then an attracting (resp., repelling) invariant closed curve bifurcates from the fixed point for (resp., ).
4. Numerical Simulations
To confirm the above theoretical analysis, in this section we present the bifurcation diagrams and phase portraits for system (4). is chosen as the bifurcation parameter to verify the existence of flip bifurcation and Neimark-Sacker bifurcation.
Firstly, let , , , , , , and . One can verify that all conditions in are satisfied. That is, if varies in a small neighborhood of () and the other parameters are kept fixed, then system (4) undergoes a Neimark-Sacker bifurcation, which is illustrated in Figure 1.
(a)
(b)
(c)
(d)
Let , , , , , and . By , we obtain ; that is, when , which leads to and . Moreover, is stable when and loses its stability when reaches and passes , which leads to a cascade of the period doubling with increasing (see Figure 2).
(a)
(b)
5. Conclusion
In this paper, a discrete-time prey-predator model with generalized Holling type III functional response and Allee effect has been presented. The stability of fixed points and the existence of flip bifurcation and Neimark-Sacker bifurcation are investigated. Our analysis shows that the strong Allee effect can change topological structure and dynamics of the system. The combined influence of Allee effect and the other parameters induce system (4) to yield more rich dynamical behaviors including chaotic bands and chaotic attractors which do not exist in the continuous case. These results reported in this paper could be very useful for the biologists who devote to study the discrete-time prey-predator models.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This paper is supported by NSFC (11226142), Foundation of Henan Educational Committee (2012A110012), Foundation of Henan Normal University (2011QK04, 2012PL03), and Scientific Research Foundation for Ph.D. of Henan Normal University (no. 1001).