Abstract
In this paper, we consider two kinds of time-delay slow-fast modified Leslie-Gower models. For the first system, we prove the existence and uniqueness of relaxation oscillation cycle through the geometric singular perturbation theory and entry-exit function. For the second system, we put forward a conjecture that the relaxation oscillation of the system is unique. Numerical simulation also verifies our results for the systems.
1. Introduction
In 2003, Alaoui and Okiye proposed the next modified Leslie–Gower model:where is the density of a prey, is the density of a predator, and the parameters are positive, which play biology roles.
Consider the rescaling,
Substitute (2) into (1), and we have and in terms of and , then we havewhere , and . The parameter can be regard as a small parameter in some special models, such as prey hares and predator lynx.
Consider the case where is small enough, system (3) could be treat as a slow-fast system which has the fast variable and the slow one . Applying the geometric singular perturbation theory, we could study the slow-fast systems [4, 5].
The geometric singular perturbation theory is mathematical rigorous that can analyze dynamics of some slow-fast systems. The study of invariant manifolds in Fenichel’s theory [3] is the basis of geometric singular perturbation theory, and the theory guarantees that any compact normally hyperbolic submanifold of the critical manifold could perturb a locally invariant manifold (0 < ≪ 1), which is O(), close to . In recent years, geometric singular perturbation theory contains a fairly broad class of geometric points used to study the slow-fast systems, see, for example, [6–12]. And, for the exchange lemma, see [13, 14].
Recently, Wang and Zhang proved the existence and uniqueness of relaxation oscillation cycle of the slow-fast modified Leslie-Gower model [15]. Valery et al. studied the global dynamics in the Leslie-Gower model with the Allee Effect [16]. Du et al. considered two delays induced Hopf bifurcation and double Hopf bifurcation in a diffusive Leslie-Gower predator-prey system [17]. Karl et al. discussed the bifurcation of critical sets and relaxation oscillations in singular fast-slow systems [18]. Wang and Zhang considered the stability loss delay and smoothness of the return map in slow-fast systems [19]. Ambrosio et al. addressed the canard phenomenon in a slow-fast modified Leslie-Gower model [20]. Xia et al. discussed relaxation oscillations and the mechanism in a periodically excited vector field with pitchfork-Hopf bifurcation [21]. Atabaigi and Barati studied relaxation oscillations and canard explosion in a predator-prey system of Holling and Leslie types [22]. Ai and Sadhu considered the entry-exit theorem and relaxation oscillations in slow-fast planar systems [23].
The rest of the paper is organized as follows: Section 2 introduces the entry-exit function theory. Section 3 studies the first time-delay slow-fast modified Leslie-Gower model. Section 4 investigates another kind of time-delay slow-fast modified Leslie-Gower model.
2. Entry-Exit Function
We are going to consider a slow-fast vector field in the form ofwhere are state-space variables and the parameter represents the ratio of time scales. We define the functions and satisfying
For system (4), the -axis consists of equilibria when and attracting and repelling (Figure 1(a)). The Figure 1(b) is the case for , since the orbit of system (4) is attracted by the -axis, the orbit starting at with and slightly moves toward the downside at the speed of . After the orbit passes positive -axis, the -axis begins repelling, then the orbit tends to move away from the negative -axis at the point , and then it intersects the line at the point whose -coordinate saying satisfies , where is determined by
(a)
(b)
3. System with Two Positive Time Delays
Consider a system with time delays and , which are sufficiently small constants:
Using the Taylor formula,
Substitute (8) into system (7), it yields
3.1. Equilibria
Lemma 1. For system (9), we have the following result: system (9) and system (3) have the same equilibria. So, that system (9) has equilibria , and .
Moreover, if , system (9) has a unique positive equilibrium with .
3.2. Hopf Bifurcation
We consider the bifurcation of system (9) at the unique positive equilibrium about parameter . For convenience, we denoteso the linearized system of system (9) at the unique positive equilibrium could be written aswith
Then, the characteristic equation of system (11) is
Lemma 2. For equation (13), we have the following result. If , then equation (13) has a pair of pure imaginary roots.
Proof. The condition of equation (13) has a pair of pure imaginary roots:Consider are sufficiently small constants. We can easily get so holds.
According to the equation , we reachTaking into equation (15), then we geti.e.,The proof is completed.
Let be the pair of pure imaginary roots of equation (13) that are satisfying , . Then the following transversality condition holds.
Lemma 3. If holds, then
Proof. The condition of equation (13) has a pair of pure imaginary roots:The proof is completed.
Denote . Combining Lemma 2 and Lemma 3, we conclude the following.
Theorem 1. For system (9), if and , then the equilibrium is a Hopf bifurcation point and a limit cycle occurs.
3.3. Dynamics of Limit Systems
Consider the limit systems of system (9). Setting in (9), we reach the fast subsystem:
Consider the slow time scale and take the singular again, we get the slow subsystem:which is a differential-algebraic equation on the critical set:
We restrict the parameters ,, and to a subset
Under this restriction, and are located in the exclusion part of . Some studies have shown that the branch of has a unique generic fold point . And, the two branches of cross transversally at the transcritical point (Figure 2). Then, the critical set is divided into four parts by points and :where and are normally hyperbolic repelling and and are normally hyperbolic attracting.
In fact, when the parameter is limited to , is on and is on .
3.4. Relaxation Oscillation
Lemma 4. For system (9), there exists a unique such that,
Proof. Considering the system (9), for ,So thatFurthermoreThen, we combine (25) and (27) and conclude that there exist a unique such that,The proof is completed.
Let us define to be the - of the intersection point of and . Then, define a singular slow-fast cycle , which contains two slow segments on from to and on the positive - from to and the two fast connections from to and to , respectively. The next theorem explains the existence and uniqueness of the relaxation oscillation.
Theorem 2. For system (9), restrict the parameters to subset and let be a tubular neighborhood of . Then, for each fixed sufficiently small, system (9) has a unique limit cycle , which is strongly attracting. Moreover, the cycle is the unique limit cycle that converges to in the Hausdorff distance [3] as .
Proof. By Fenichel’s theory, the critical submanifold perturbs to a nearby slow manifold , which is near . By Theorem 2.1 of [9] on the analysis of a jump point, the slow manifold can be continued and passes the fold point , and then it jumps to another attracting branch .
Let be a sufficiently small positive number. Define two vertical sections and as shown in Figure 2:where and are closed intervals centered at and , respectively. Through the flow of system (4), we define the transition map , which is a composition of the next two maps:Then, is given by the composition .
Now, we analyze the properties of these two maps and .(a)Analysis of. Use the same proof as Lemma 4, for each , we can define , with through the following formula:(b)Analysis of. Consider two orbits and starting on , from Fenichel’s theory, and will be attracted to at the exponential rate . By Theorem 2.1 of [9], and pass by the generic fold point contracting exponentially toward each other. Then, they fly to .Therefore, according to the results of (a) and (b), we obtain that the transition map is a contraction at the exponential rate . Then, it follows from the contraction mapping theorem that has a unique fixed point in , which must be stable. This fixed point provides a unique relaxation oscillation cycle of system (9) passing for each .
According to Fenichel’s theory and Theorem 2.1 of [9], we obtain that the relaxation oscillation cycle converges to the slow-fast cycle as in the Hausdorff distance.
So, we can get that the relaxation oscillation cycle is a unique limit cycle of system (9) located in for each .
The theorem is proved.
3.5. Numerical Simulation
First of all, we provide an example to illustrate Theorem 2.
Select the parameter values , and it is easy to state that the system (9) has a unique positive equilibrium point . Numerical simulation shows that it has a unique limit cycle. Figure 3 shows the phase portrait and the time series of the relaxation oscillation cycle.(a)The orbit is shown in red in phase space(b)Time series with the black dashed curve (prey) and red full curve (predator)
(a)
(b)
In addition, we compare the phase portrait and time series of system (9) with system (7). Figure 4 presents the phase portrait of system (9) and system (7). We can see that the two curves are very close to each other. Figure 5 shows the time series of system (9) and system (7).(a)Time series for the relaxation oscillation of the predator with the red dashed curve (system (7)) and the blue full curve (system (9))(b)Time series for the relaxation oscillation of the prey with the blue dashed curve (system (7)) and the red full curve (system (9)). Time series for the relaxation oscillation of prey with the blue dashed curve (system (4)) and the red full curve (system (6)).
(a)
(b)
4. System with Two Time Delays
In this section, we consider time delays and in system (3), then we havewhere stands for delayed maturation of the predator and stands for the time needed to digest the prey.
4.1. Equilibria and Hopf Bifurcation
System (32) has the same equilibria with system (9). So, system (32) has a unique positive equilibrium .
In order to get the conditions for Hopf bifurcation, we reach the linearized system of system (32) at the equilibrium :where
Substituting into (34) yieldsi.e.,
Denote , the associated characteristic equation of system (33) can be obtained as follows:
Lemma 5. For equation (37), if , then equation (37) ha no pairs of pure imaginary roots. If , then equation (37) has a pair of pure imaginary roots.
Proof. Let be the pair of pure imaginary roots of equation (37), then we take into equation (37) with , it yieldsSeparating the real and imaginary parts of equation (38), then we getSo thatAccording to , we havewithSinceWe can conclude that if , that, which implies that Equation (41) has no positive roots, so equation (37) has no pairs of pure imaginary roots. Besides, if , then and equation (41) has a positive root:which implies equation (37) has a pair of pure imaginary roots whenThe proof is completed.
Denote be the pair of pure imaginary roots of equation (37) that are satisfying , .
Lemma 6. (transversality condition) If holds, then , .
Proof. Differentiating both sides of equation (37) with respect to , we reachthenSoTo sum up, we get .
The proof is completed.
Theorem 3. For system (32), if , , system (32) undergoes a Hopf bifurcation and are the Hopf bifurcation values.
Proof. If , then the equation (37) has a pair of purely imaginary roots which satisfy the transversality condition. Thus system (32) undergoes a Hopf bifurcation and are the Hopf bifurcation values.
The theorem is proved.
Remark 1. If , then is locally asymptotically stable and if , then is unstable.
4.2. Analysis of Limit Systems
In order to get rid of the delay and , we denote , , . Then, we geti.e.,
When , system (50) goes to
Consider the case where is small enough, system (51) could be viewed as a slow-fast system which has the fast variable and the slow variable . We have
Then, we arrive at the critical set:with
SinceThen, the eigenvalues of matrix (55) are
Thus, we reach the attractive part and the repulsive part of are, respectively,
The attractive part and the repulsive part of are, respectively,
The attractive part and the repulsive part of are, respectively,and the attractive part and the repulsive part of are, respectively,
4.3. Conjecture of the System
In this section, we combine the result of Subsection 4.1 and Subsection 4.2 and propose a conjecture.
Conjecture 1. If and , then the equilibrium is a Hopf bifurcation point of system (51) and a periodic solution is bifurcated from . Moreover, the relaxation oscillation cycle of the system (51) is unique.
4.4. Numerical Simulation
We show some numerical simulations in this part to verify our theoretical results.
For system (32), we choose the parameter values , then we get . A series of calculations gives us that . Thus, we choose , and then the equilibrium of system (32) is asymptotically stable (Figure 6). Besides, we choose and get the periodic solutions of system (32) (Figure 7).
For system (51), we choose the parameter values . Figure 8 shows the phase portrait and the time series of the relaxation oscillation cycle.
(a)
(b)
5. Conclusion
In this paper, we mainly study two kinds of time-delay slow-fast modified Leslie-Gower models. For the first system, through the geometric singular perturbation theory and entry-exit function, we prove the existence and uniqueness of relaxation oscillation cycle. For another system, when crosses some critical values , the Hopf bifurcation occurs. Meanwhile, we put forward a conjecture that the relaxation oscillation cycle of the system is unique when . Numerical simulation verified our theoretical results and indicated that our method is effective. Our results show that delay affects the stability of the positive equilibrium and produces more complex dynamics than the model without delay.[1, 2]
Data Availability
All data, models, and code generated or used during the study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through grant no.11572288 and the Natural Science Foundation of Zhejiang through grant no.LY20A020003.