Relaxation Oscillations and Dynamical Properties in Two Time-Delay Slow-Fast Modified Leslie-Gower Models

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.


Introduction
In 2003, Alaoui and Okiye proposed the next modified Leslie-Gower model: where x is the density of a prey, y is the density of a predator, and the parameters r 1 , k 1 , a 1 , b 1 , r 2 , k 2 , and a 2 are positive, which play biology roles. Consider the rescaling, Substitute (2) into (1), and we have x and y in terms of x and y, then we have where a � (a 1 r 2 /a 2 r 1 ), e 1 � (b 1 k 1 /r 1 ), e 2 � (b 1 k 2 /r 1 ), and ε � (r 2 /r 1 ). e parameter ε can be regard as a small parameter in some special models, such as prey hares and predator lynx. Consider the case where ε is small enough, system (3) could be treat as a slow-fast system which has the fast variable x and the slow one y. Applying the geometric singular perturbation theory, we could study the slow-fast systems [4,5]. e geometric singular perturbation theory is mathematical rigorous that can analyze dynamics of some slowfast systems. e 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 So of the critical manifold Co could perturb a locally invariant manifold S ε (0 < ε≪ 1), which is O(ε), close to So. 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][7][8][9][10][11][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 predatorprey system of Holling and Leslie types [22]. Ai and Sadhu considered the entry-exit theorem and relaxation oscillations in slow-fast planar systems [23]. e 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.

Entry-Exit Function
We are going to consider a slow-fast vector field in the form of where (x, y) ∈ R 2 are state-space variables and the parameter ε represents the ratio of time scales. We define the functions f and g satisfying f(0, y, 0) < 0, for y > 0, f(0, y, 0) > 0, for y < 0, For system (4), the y-axis consists of equilibria when ε � 0 and (Figure 1(a)). e Figure 1(b) is the case for ε > 0, since the orbit of system (4) is attracted by the y-axis, the orbit starting at (x 0 , y 0 ) with y 0 > 0 and x 0 > 0 slightly moves toward the downside at the speed of ε. After the orbit passes positive x-axis, the y-axis begins repelling, then the orbit tends to move away from the negative y-axis at the point (0, p 0 (y 0 )), and then it intersects the line x � x 0 at the point whose y-coordinate saying p ε (y 0 ) satisfies lim ε⟶0 p ε (y 0 ) � p 0 (y 0 ), where p 0 (y 0 ) is determined by

System with Two Positive Time Delays
Consider a system with time delays τ 1 and τ 2 , which are sufficiently small constants: Using the Taylor formula, Substitute (8) into system (7), it yields

Hopf Bifurcation.
We consider the bifurcation of system (9) at the unique positive equilibrium E * (x * , y * ) about parameter τ 1 . For convenience, we denote x + e 1 x + e 1 − ayτ 1 , so the linearized system of system (9) at the unique positive equilibrium E * (x * , y * ) could be written as with 2 Complexity en, 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. e condition of equation (13) has a pair of pure imaginary roots: Consider ε, τ 1 , and τ 2 are sufficiently small constants. We can easily get G x > 0, F y < 0, so G x F y < 0 holds.
x + e 1 x + e 1 − ayτ 1 , Consider the slow time scale τ � εt and take the singular again, we get the slow subsystem: which is a differential-algebraic equation on the critical set: We restrict the parameters a, e 1 , and e 2 to a subset Under this restriction, E * (x * , y * ) and E 2 (0, e 2 ) are located in the exclusion part of C 0 . Some studies have shown that the branch (x, y) | y � F(x), x > 0，y > 0 of C 0 has a unique generic fold point D(x M , y M ) � ((1 − e 1 /2), F(1 − e 1 /2)). And, the two branches of C 0 cross transversally at the transcritical point E(x, y) � (0, (e 1 /a)) ( Figure 2). en, the critical set C 0 is divided into four parts by points D and E: where C r 0 and C − 0 are normally hyperbolic repelling and C a 0 and C + 0 are normally hyperbolic attracting. In fact, when the parameter is limited to U, E * (x * , y * ) is on C r 0 and E 2 (0, e 2 ) is on C − 0 .

(25)
So that Furthermore en, we combine (25) and (27) and conclude that there exist a unique y * (e 2 < y * < (e 1 /a)) such that, e proof is completed. Let us define x r to be the x-coordinate of the intersection point of y � y * and C a 0 . en, define a singular slow-fast cycle c 0 , which contains two slow segments on C a 0 from (x r , y * ) to (x M , y M ) and on the positive y-axis from (0, y M ) to (0, y * ) and the two fast connections from (x M , y M ) to (0, y M ) and (0, y * ) to (x r , y * ), respectively. e next theorem explains the existence and uniqueness of the relaxation oscillation. □ Theorem 2. For system (9), restrict the parameters to subset U and let V be a tubular neighborhood of c 0 . en, for each fixed ε > 0 sufficiently small, system (9) has a unique limit cycle c ε ⊂ V, which is strongly attracting. Moreover, the cycle c ε is the unique limit cycle that converges to c 0 in the Hausdorff distance [3] as ε ⟶ 0.
Proof. By Fenichel's theory, the critical submanifold C a 0 perturbs to a nearby slow manifold C a ε , which is O(ε) near C a 0 . By eorem 2.1 of [9] on the analysis of a jump point, the slow manifold C a ε can be continued and passes the fold point D(x M , y M ), and then it jumps to another attracting branch C + 0 . Let x l be a sufficiently small positive number. Define two vertical sections Δ in and Δ out as shown in Figure 2: where I in and I out are closed intervals centered at y M and y out � p 0 (y M ), respectively. rough the flow of system (4), we define the transition map Π: Δ in ⟶ Δ in , which is a composition of the next two maps: en, Π: Δ in ⟶ Δ in is given by the composition Π: � Π 2 ∘ Π 1 .

Complexity
Now, we analyze the properties of these two maps Π 1 and Π 2 .
(a) Analysis of Π 1 . Use the same proof as Lemma 4, for each y 0 ∈ I in , we can define p 0 (y 0 ), with e 2 < p 0 (y 0 ) < (e 1 /a) through the following formula: (b) Analysis of Π 2 . Consider two orbits c 1 ε and c 2 ε starting on Δ out , from Fenichel's theory, c 1 ε and c 2 ε will be attracted to C a ε at the exponential rate O(e − 1/ε ). By eorem 2.1 of [9], c 1 ε and c 2 ε pass by the generic fold point D(x M , y M ) contracting exponentially toward each other. en, they fly to Δ in . erefore, according to the results of (a) and (b), we obtain that the transition map Π: Δ in ⟶ Δ in is a contraction at the exponential rate O(e − 1/ε ). en, it follows from the contraction mapping theorem that Π has a unique fixed point in Δ in , which must be stable. is fixed point provides a unique relaxation oscillation cycle c ε ⊂ V of system (9) passing Δ in for each 0 < ε ≪ 1.
According to Fenichel's theory and eorem 2.1 of [9], we obtain that the relaxation oscillation cycle c ε converges to the slow-fast cycle c 0 as ε ⟶ 0 in the Hausdorff distance.
So, we can get that the relaxation oscillation cycle c ε is a unique limit cycle of system (9) located in V for each 0 < ε ≪ 1.
e theorem is proved.
First of all, we provide an example to illustrate eorem 2. Select the parameter values (a, e 1 , e 2 , ε, τ 1 , τ 2 ) � (1, 0.2, 0.022, 0.01, 0.2, 0.2), and it is easy to state that the system (9) has a unique positive equilibrium point E * (0.33, 0.36). 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) e orbit is shown in red in phase space (b) Time series with the black dashed curve (prey) and red full curve (predator) 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)).

System with Two Time Delays
In this section, we consider time delays τ 1 and τ 2 in system (3), then we have where τ 1 stands for delayed maturation of the predator and τ 2 stands for the time needed to digest the prey.
In order to get the conditions for Hopf bifurcation, we reach the linearized system of system (32) at the equilibrium E * (x * , y * ): where y C (x r , y out ) Figure 2: Illustration for the relaxation oscillations of system (9).
e critical manifold C 0 contains repelling part C r 0 and attracting part C a 0 . e dashed black line is the predator isocline, the red dot is the fold point D(x M , y M ), the solid blue line is the slow-fast cycle c 0 , the double and single arrows are fast and slow flows, and the solid red is the attracting relaxation oscillation c ε .

Analysis of Limit Systems.
In order to get rid of the delay τ 1 and τ 2 , we denote x � x 1 , x(t − τ 2 ) � x 2 , y � y 1 y(t − τ 1 ) � y 2 . en, we get i.e., When τ � 0 , system (50) goes to Consider the case where ε is small enough, system (51) could be viewed as a slow-fast system which has the fast variable X � x 1 x 2 and the slow variable Y � y 1 en, we arrive at the critical set: Since en, the eigenvalues of matrix (55) are (56) us, we reach the attractive part and the repulsive part of C 1 0 are, respectively, (57) e attractive part and the repulsive part of C 2 0 are, respectively, e attractive part and the repulsive part of C 3 0 are, respectively, and the attractive part and the repulsive part of C 4 0 are, respectively, For system (32), we choose the parameter values (a, e 1 , e 2 , ε) � (0.805, 0.2, 0.022, 0.01), then we get (x * , y * ) � (0.422, 0.444). A series of calculations gives us that τ 00 ≈ 11.50. us, we choose (τ 1 , τ 2 ) � (3, 1), and then the equilibrium E * (x * , y * ) of system (32) is asymptotically stable ( Figure 6). Besides, we choose (τ 1 , τ 2 ) � (3,13) and get the periodic solutions of system (32) (Figure 7).

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 τ � τ 1 + τ 2 crosses some critical values τ 0j , the Hopf bifurcation occurs. Meanwhile, we put forward a conjecture that the relaxation oscillation cycle of the system is unique when τ � 0. 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.