Abstract and Applied Analysis
Volume 2014 (2014), Article ID 673424, 14 pages
http://dx.doi.org/10.1155/2014/673424

Research Article

Dynamics of a Predator-Prey System with a Mate-Finding Allee Effect

Ruiwen Wu and Xiuxiang Liu

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Received 26 July 2014; Revised 11 September 2014; Accepted 11 September 2014; Published 20 October 2014

Academic Editor: Weinian Zhang

Copyright © 2014 Ruiwen Wu and Xiuxiang Liu. 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.

Abstract.

We consider a ratio-dependent predator-prey system with a mate-finding Allee effect on prey. The stability properties of the equilibria and a complete bifurcation analysis, including the existence of a saddle-node, a Hopf bifurcation, and, a Bogdanov-Takens bifurcations, have been proved theoretically and numerically. The blow-up method has been applied to investigate the structure of a neighborhood of the origin. Our mathematical results show the mate-finding Allee effect can reduce the complexity of system behaviors by making the complicated equilibrium less complicated, and it can be a destabilizing force as well, which makes the system has a high possibility of being threatened with extinction in ecology.

1. Introduction

Just as pointed out by Berryman in [1] that “this dynamical relationship between predators and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance”, both ecologists and mathematicians are interested in the dynamical analysis of predator-prey models. It is well known that the classical Gause type predator-prey system is written by

where and are the densities of prey and predator, respectively, represents the trophic efficiency or the conversion efficiency of predator, and the parameter characterizes the predator natural mortality rate. The functional response , which quantifies the amount of prey consumed per predator per unit time and plays an important role in predator-prey dynamics, is conventionally modeled as prey-dependent, where the prey consumption rate by an average predator is only a function of prey density alone; that is, . Different prey-dependent response types (e.g., the mass-action approach in Lotka-Volterra model and Holling types I–III) have been used to model the predator-prey interactions and get success in describing some ecological communities. When the spatial structure of one or both of the interacting populations is involved, it would be more plausible to take the predator-dependent functional form, where both predator and prey densities affect the response, for example, Hassell and Varley function response [2] and Beddington-DeAngelis functional response functional [3]. In this paper, we are interested in one important form of predator-dependent functional response, called ratio-dependent response; that is, the functional response depends on the term and the corresponding Monod-Holling hyperbolic form is

where is the maximum prey consumption rate and is the predator handling time, which is proposed by Arditi and Ginzburg [4] and studied by many authors; see, for example, Berezovskaya et al. [5, 6], Kuang et al. [7–10], and Zhang et al. [11, 12] in which we know this ratio-dependent functional response provides more reasonable explanations and accurate predictions when community-level situations of food chains and food webs are considered.

Another function in (1) is the per capita growth rate of prey population in the absence of predators. The classical view of popular dynamics is that the major ecological force at work is the release from the constraints of intraspecific competition when a population is small or at low density. The fewer we are, the more we all have and the better welfare be there. In general, takes the form

where and are the intrinsic growth rate and the environment carrying capacity of prey, respectively. However, the ecologist Allee gathered sufficient experimental and observational data to conclude that the evolution of social structures was not only driven by competition, but that cooperation was another, if not the most, fundamental principle in animal species [13]. The dynamical consequences of this importance of animal aggregations directly led to what Odum called in 1953 “the Allee principle,” now known as the Allee effect [14].

There are two conventional types of Allee effects: component Allee effect and demographic Allee effect. The positive relationship between any component of individual fitness and population size can be regarded as a component Allee effect, while a demographic Allee effect is linked to the level of overall individual fitness [15]. Moreover, a demographic Allee effect is strong if there exists an Allee threshold, below which the per capita growth rate becomes negative. The classical and simplest types of demographic Allee effects are considered commonly, which takes the mathematical form of

where denotes the Allee threshold if . Much work has been done on strong Allee effects; see, Hilker et al. [16, 17], Shi et al. [18, 19], and González-Olivares et al. [20, 21] and their citations. A wide range of mechanisms which may result in Allee effects are considered and discovered, from fertilization efficiency in sessile organisms to pollen limitation in plants and to cooperative hunting in animals. So far, one of the most important and probably the most studied mechanisms for Allee effects is called made-finding; that is, individuals in a population fail to find a suitable mate at low density, thus resulting in fewer reproductive outputs and examples include Glanville fritillary butterfly [15], sheep ticks, and whales [15, 22], and such mechanism may cause a mate-finding (component) Allee effect potentially. Mathematically, a mate-finding process can be modeled by a female mating rate which has positive dependence on density, denoted by . The hyperbolic function

where scales the mate-finding Allee effects, is widely used [15, 22–24].

In our paper, we aim to study a predator-prey system which is subject to the ratio-dependent functional response and a mate-finding Allee effect, and we believe it will give people a better understanding of Allee effects in predator-prey systems. More precisely, we consider the following system:

with the initial conditions , . By introducing the dimensionless variables given by , , and , then the system (6) becomes

where , , , , and (relative strength of the mate-finding Allee effect). Now we have rescaled the system from (6) to (7) by reducing the number of parameters from seven to five. For the sake of consistency, we make the same dimensionless transformation of as that in the system studied by Pavlová et al. [25]. Note that , so

.

We assume that holds in our paper and all the other parameters are positive; that is, .

We organize the paper as follows. In Section 2, we show the system is dissipative and the stability and bifurcation analysis of complicated equilibrium , predator-free equilibria, and positive equilibria are given. We discuss our findings and summarize our conclusions in Section 3, and numerical simulations are also carried out to support our findings.

2. Stability and Bifurcation Analysis

2.1. Dissipativetity

Note that is not defined at , and we can redefine the derivative as follows:

Finally we define . Clearly, with the extended definition, there globally exists a unique solution of system (7) for any given nonnegative initial condition. Obviously, is an invariant set. The following lemma shows that system (7) is dissipative.

Lemma 1. Let be a solution of system (7); then, one has

Proof. Let . Differentiating , one has

Thus, we have and system (7) is dissipative. This completes the proof.

2.2. Complicated Equilibrium and Its Stability

Since the Jacobian matrix cannot be evaluated at (noting that and are not differentiable at ), the classical stability anlysis methods are not applied. We apply the algorithm presented in [6] to investigate the structure of a neighborhood of and show that system (7) has a stable for all system parameters. Such result is obtained numerically in Figures 1, 2, 4, and 6(a).

(a) ,

(b) ,

(c) ,

Figure 1: Phase portraits of system (7) if and for , , , , and .

(a) A saddle and a stable ()

(b) A saddle and a stable surrounded by a homoclinic loop ()

(c) A saddle and an unstable ()

Figure 2: Phase portraits of system (7) if and for , , , and .

For the sake of simplicity in calculation, we focus on the following system which is equivalent to system (7):

Definition 2. A vector field is nondegenerate if it satisfies the following.

(A1)Polynomials and have no common factors of the form , where , , and at least one of the constants , is nonzero.(A2)Polynomial has no factors of the form , where , and .

By the time scale change , the system (11) takes the form

Let

and then one has

It is easy to see that the vector field of (12) is nondegenerate. After applying the blow-up transformations with

and the time change , one has

where

Thus, if , has only one nonnegative root , and consequently, system (16) has one equilibrium . has two nonnegative roots: and if , and thus system (16) has two equilibria and on -axis. We state the following lemma.

Lemma 3. We have the following results.

(1)If and , is a single stable node.(2)If and , is a single saddle.(3)If and , is a stable node and is a saddle.(4)If and , is a saddle and is a stable node.

Proof. We only verify (3) since the other three results can be shown in a similar way. It is easy to see that

If and , that is, , then , , and

Based on [6, Proposition 3], we can conclude that that is a stable node and is a saddle, respectively.

We obtain the following system in a similar way by making transformations with

and ; then, we obtain

where

Thus, if , system (21) has only one nonnegative equilibrium (0,0). The system (21) has two equilibria and on the -axis if . However, corresponds to ; thus, we do not need to study it again. is a new equilibrium and it does not exist in the -plane. Note that and . Again, based on [6, Proposition 3], then comes the following conclusion.

Lemma 4. We say that

(1)if , is a saddle;(2)if , is a stable node.

According to Lemmas 3 and 4, it then follows from [6] that the following result holds for the origin of system (7), which is equivalent to system (11).

Theorem 5. If , then there is at least one attracting parabolic sector in the neighborhood of in the first quadrant, and no hyperbolic or elliptic sector can be found; that is, the complicated equilibrium is stable.

Remark 6. If , the neighborhood of in the first quadrant has a parabolic (attracting) sector, but if , there are two parabolic (attracting) sectors; in fact, we can regard those two sectors as a whole attracting one. If , , are also complicated equilibria and needed to apply the blow-up transformation again. Now we can see that the relative strength of mate-finding Allee effect greatly changes the structure of the neighborhood of in our system (7), because and always stay negative and and cannot be unstable; thus, the diversity of phase portraits around of ratio-dependent prey-predator systems decreases. More importantly, the system becomes fragile and even has a higher risk of extinction due to mate-finding Allee effect.

2.3. Existence and Stability of Predator-Free Equilibria

In order to analyze the predator-free equilibria of the system (7), we set

We can notice that, for , there exist two boundary equilibria and if and ; that is,

 ,

where . Let , be the roots of the quadratic equation ; that is,

and they appear in pairs. Besides, there is no equilibrium on -axis.

The Jacobian matrix evaluated at , is given by

and one eigenvalue is , , and the second eigenvalue is , . Note that the sign of , depends on , . Using the expression of , , from (24), we can have

Under the restriction , it follows that

Hence is always positive, while is negative. By using the facts above, we have the following theorem.

Theorem 7. Assume () holds. Then

(1)if , is a saddle and is a stable node (Figure 1(c)),(2)if , is an unstable node and is a saddle (Figure 2).

Remark 8. If , system (7) has a unique predator-free boundary equilibrium point (also known as the saddle-node equilibrium), where

Figure 1(b) shows the instantaneous equilibrium in the case of and the behaviors of trajectories are divided by the stable manifold of (the green curve).

2.4. Positive Equilibria and Their Stability and Bifurcation Results

2.4.1. Existence and Stability of Positive Equilibria

The interior (positive) equilibria can be evaluated by the intersections of the zero isoclines

in the first quadrant. The predator zero isocline, with the slope , is a straight line passing through the origin, and it lies in the first quadrant if the following restriction holds

 .

Besides, we can define

Note that under the restriction . And for , we also set

If and , that is,

 ,

where , there are two positive equilibria and , which also appear simultaneously, with the coordinates

Note that , it suggests that if holds, then holds; that is, if there exists any interior equilibrium in the first quadrant, then two predator-free equilibria will be detected. The Jacobian matrix , evaluated at an interior equilibrium is given by

Similarly, we have and . Moreover,

and it is interesting to find that is an increasing function of . In fact, we can have

and we can define . Then, it follows that

since

Moreover, we can have the following conclusion.

Theorem 9. Assume that and hold. Then one has the following.

(1) is a saddle.(2)If , is stable.(3)If , can be stable or unstable depending on (Figure 2).

2.4.2. Hopf Bifurcation

Here assume that and hold, and we consider as the bifurcation parameter to discuss the Hopf bifurcation. It is easy to know that the Hopf bifurcation can occur only at as is always a saddle point. Then we try to show the existence of such that and . From the expression of and some numerical attempts, we find that other parameters , , , and still have an important impact on the qualitative properties of the Hopf bifurcation though only is considered as the bifurcation parameter. The following assumption is used to guarantee the exsitence of the Hopf bifurcation with bifurcation parameter :

 , .

Assumption implies that and , and we have known that is a continuous increasing function of in ; thus, there exists such that . Furthermore, the transversality condition is satisfied. Using the facts above, system (7) can undergo a Hopf bifurcation at for if holds.

Now we try to discuss the stability of the limit cycle of system (7) as a Hopf bifurcation occurs by computing the first Lyapunov coefficient [26] at the . Transformations and are used to translate the equilibrium of the system (7) to the origin. Then we get

where

and , are power series in the powers of satisfying . The Lyapunov number (as defined in [26]) is given by

where . Since the expression Lyapunov number for is very complex, we fail to discuss the sign of precisely though a subcritical Hopf bifurcation has been found numerically with the Lyapunov number (Figure 3). Obviously, the assumption is satisfied in Figure 3. We need to require (green line) and (blue line) to guarantee the existence of ; that is, and hold (Figure 3(a)). is a continuous increasing function of . If , system (7) has a stable (Figure 2(a)). As increases, a homoclinic loop is created by joining the stable and unstable manifolds of the saddle at some () with a stable inside (Figure 2(b)). Then as gets larger, an unstable limit cycle appears and coexists with a stable (Figure 4). However, the limit cycle begins to shrink with the increasing . The shrinking unstable limit cycle (Figures 3(b) and 3(c)) disappears as passes through is also the root of equation , . Then , becomes unstable (Figure 2(c)).

(a) Bifurcation structure

(b) , the unstable limit cycle when a subcritical Hopf bifurcation occurs

(c) , the trajectory around the unstable limit cycle when a subcritical Hopf bifurcation occurs

Figure 3: Bifurcation structure of for Hopf bifurcation and phase portraits of system (7) for , , , and .

(a) Inside the limit cycle

(b) Outside the limit cycle

Figure 4: An unstable limit cycle coexists with a stable , for , , , , and .

2.4.3. Saddle-Node Bifurcation

We know that when , , , , and satisfy and the following equation

where , then there is only one interior equilibrium of system (7), whose coordinates are given by

And the Jacobian matrix evaluated at is

Next we choose the relative strength of mate-finding Allee effect as the bifurcation parameter; then, we have the following theorem.

Theorem 10. Assume () holds. For and , a saddle-node bifurcation occurs at the unique positive equilibrium of system (7).

Proof. First, we have ; therefore, has an eigenvalue , and if , then is simple. Let and be the eigenvectors corresponding to the for and , respectively. Then we obtain

From the expressions for and , we get

where

Thus by Sotomayor’s theorem [26], system (7) undergoes a saddle-node bifurcation at as passes if .

A numerical example has been carried out to demonstrate a saddle-node bifurcation of system (7) (Figure 5). The green curves represent the smaller unstable equilibria. The stability of the larger equilibria changes from stable (blue curves) to unstable (black curves) as passes through .

(a) Prey population scenario

(b) Predator population scenario

Figure 5: A saddle-node bifurcation of system (7) at (mark as SN), for , , , and . A Hopf bifurcation also occurs.

(a) Bifurcation structure

(b) A cusp

Figure 6: Bifurcation structure of system (7) in with a Bogdanov-Takens bifurcation and phase portraits for , , , , and .

2.4.4. Bogdanov-Takens Bifurcation

Let us focus on again. After knowing that a saddle-node bifurcation can occur at if and , we go on to consider the case of , , and . If and , then the Jacobian matrix of has . When , it is easily shown that if then , and hence we choose the relative mate-finding strength and the predator growing ability as two bifurcation parameters. Then the following statement holds.

Theorem 11. Let () hold. The system (7) undergoes a Bogdanov-Takens bifurcation around the equilibrium point when and .

Proof. Following the ideas in [27, 28], we consider the neighborhood of ; that is, and , where , , are sufficient small, and system (7) becomes

We make , for translating to the origin , and we obtain

where , are polynomial functions of with satisfying , and and are functions satisfying .
Then we apply the affine transformation

to obtain

where and are polynomial functions with satisfying and satisfying , respectively. , , are functions in their variables with satisfying .
Furthermore, if , we can see that

(BT0)

(BT1),(BT2),(BT3)the map

 is regular at

According to [27, 28], the system (7) undergoes a Bogdanov-Takens bifurcation around the equilibrium point when and . This completes the proof.

Remark 12. Furthermore, the quantity [27, 28] which determines the structure of the bifurcation is given by

The bifurcation diagram of system (7) in is presented in Figure 6(a). There are no positive equilibria of system (7) in Domain 1. In the domain bounded by Limit point cycle (green) and Hopf curve (blue), there exist a saddle and an unstable . In the domain bounded by Hopf curve (blue) and Homoclinic curve (dot and magenta), there is a saddle and a stable surrounded by an unstable limit cycle. In Domain 2, system (7) has a saddle and a stable . At BT point, is the unique positive equilibrium of system (7) in the first quadrant, and the local dynamics near is characterized by two trajectories (green curves, Figure 6(b)), which meet at , but with opposite time orientation, and also behave like a cusp-like configuration.

3. Discussion

In this paper, we have discussed a ratio-dependent predator-prey model with a (component) mate-finding Allee effect on prey. Allee effects and ratio-dependent functional response are very popular among theoretical ecologists as they can display realistic and complicated dynamical behaviors, which provide better explanations for the ecological observations. And the hyperbolic function that we concerned for mate-finding Allee effects is considered as the most appropriate analytical form from ecological point of view.

3.1. The Less Complicated Equilibrium

We know that ratio-dependent predator-prey systems with the logistic growth of prey exhibit very complicated dynamic behaviors around the origin [5, 6, 8, 9]; for instance, the origin behaves like a stable node and an unstable saddle at the same time [5]. However, such dynamic properties can not be observed in our system, because is always asymptotically stable, which implies that any trajectory, starting from a certain neighborhood of , goes towards the origin for all system parameters. In other words, the complicated equilibrium becomes less complicated because of the mate-finding Allee effect on prey, which provides insights into pest control by introducing a mate-finding Allee effect to reduce the complexity around the origin.

We have constructed diagrams with and without a mate-finding Allee effect in the same ratio-dependent predator-prey system for better understanding and comparison. It is easy to see that the origin can be a unstable node and a saddle at the same time in a ratio-dependent predator-prey system where the mate-finding Allee effect is absent in Figure 7(b). It is interesting to note that, in both cases, we set . Generally, the predator growing ability is assumed to be less than the consumption ability in predator-prey systems. Hence describes the case that predators can reply on other resources, but the prey is still the limiting factor [5, 8]. So we can see obviously how the mate-finding Allee effect in the prey population influences the dynamics of the ratio-dependent predator-prey systems: it increases the extinction risk of both prey and predators, even in the case that prey population is not the only resources of the predator population.

(a)

(b)

Figure 7: The same ratio-dependent predator-prey system with and without a mate-finding Allee effect for , , , and .

3.2. The Impacts of

In order to understand the influence of the relative strength of the mate-finding Allee effect on the ratio-dependent prey-predator system, we consider as the bifurcation parameter and find that system (7) can exhibit a saddle-node, a subcritical Hopf, and a co-2 Bogdanov-Takens bifurcations with the other bifurcation parameter .

It is easy to see that if , system (7) has no interior equilibria or other boundary equilibria except ; then, is globally asymptotically stable. In other words, if the mate-finding Allee effect on the prey population is very strong, any trajectory converges to as tends to infinity; that is, both the prey and predators become extinct regardless of their initial density, because the prey population has great difficulty in finding mates at low density and suffers a heavy loss from predation at the same time; thus, such situation leads to a continued decline in predator population which is illustrated in Figure 1(a). Furthermore, different from general predator-prey systems with logistic growth on prey population [5, 6] or those subject to the demographic Allee effects on prey [29], the existence of predator-free boundary equilibria becomes conditional in prey-predator systems where the prey population suffers from a mate-finding Allee effect, and if such equilibria exist, they appear simultaneously with an unstable smaller equilibrium .

If , , there are two predator-free equilibria (an unstable node and a saddle whose stable manifolds lie on -axis) and two positive equilibria: (always a saddle) and of system (7) simultaneously. In some cases, changes from stable to unstable as the mate-finding Allee effect strength gets stronger. It is interesting that, following from Theorem 5, there do not exist any global asymptotically stable positive equilibria for any , , , , and , which means that a mate-finding Allee effect on prey increases the risk of extinction for system (7). In particular, there can exist a stable surrounded by an unstable limit cycle for some ; then, the other trajectories outside the unstable limit cycle are attracted to except the stable manifolds of , which indicates that the mate-finding Allee effect on prey population may destabilize the system.

Though system (7) may experience unstable oscillation for some certain , it still needs extra assumption for other system parameters apart from those basic assumptions and , which indicates actually that the impacts are limit and system (7) itself is highly sensitive to its initial condition and present states. Harder to encounter a receptive mate at low density leads to fewer reproductive output directly, however, we take measures to mitigate the negative effects of mate-finding Allee effects by changing other system parameters.

We also leave the co-3 bifurcation problem for future discussion. Another important and interesting issue that requires further exploration is a more sophisticated proof of the existence and stability of the limit cycles. It has been shown that, with different mathematical forms of the Allee effect in a predator-prey system, the number of limit cycles changes [20]. Recently, uniqueness of limit cycle in a Gause-type predator-prey system with an Allee effect has been proved by Olivares et al. [21]. We believe their work on the form of demographic Allee effects will be helpful for future studies on those with component Allee effects.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are very thankful to Dr. Ludĕk Berec for his nice suggestions and the two anonymous reviewers for their careful reading and thoughtful comments. The work of is supported in part by the NSF of Guangdong Province (S2012010010034).

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