Analysis of the limit cycle properties of a fast – slow predator – prey system

We consider fast–slow planar systems of predator-prey models with the prey growing much faster than the predator. By using basic differential and integral calculus, Lyapunov functions and phase plane analysis, other than the geometric singular perturbation theory, we derive that the limit cycle exhibits the temporal pattern of a stable relaxation oscillator as a parameter tends to 0, such result shows the coexistence of the predator and the prey with quite diversified time response, which typically happens when the prey population grows much faster than those of predator.


Introduction
The existence and uniqueness of the limit cycle of the biological mathematical model, which is not only of important theoretical meaning but also has certain application background, has attracted the interest of scholars for a long time.For the research of limit cycle, Cheng [2] proved the first uniqueness result, and more general uniqueness results for limit cycle in Rosenzweig-MacArthur predator-prey systems have been reached comprehensively [5,6,9,10].
It is increasingly recognized that changes in ecologically important species' traits can occur fast enough to affect interspecific interactions while they are taking place [1], which suggests that these two means of adaptive change, although both occurring at ecological rates, do not have the same effects on community dynamics.Cortez and Ellner [3] considered a modified Rosenzweig-MacArthur model, which implies that the prey responds to ecological conditions faster than the predator does: Corresponding author.Email: jinfengwangmath@163.com where x is prey population, y is predator population, r is the exponential growth rate of the prey without density limitation, K is the prey carrying capacity, a is the encounter rate, h is the handling time, and d is the per capita death rate of the predator, ε is a small positive number that represents the difference in timescales between the prey and predator species, and the prey population has fast dynamics, that is, the prey grows much faster than the predator, which also makes (1.1) a fast-slow system.The numerical simulation results in [3] show that rapidly induced defences tend to stabilize community dynamics and that some behaviors observed in rapidly evolving systems cannot be produced by phenotypic plasticity.It is well known that many predator-prey systems with oscillatory behavior possess a unique limit cycle which is globally asymptotically stable.Our goal in the paper is to give the rigorous mathematical verification for the changes of the limit cycle as ε tends to 0, we find that the limit cycle exhibits the temporal pattern of a relaxation oscillator as ε tends to 0, which coincides with the numerical results in [3] completely.
It is noticed that Hsu and Shi [4] studied relaxation oscillation profile of the limit cycle when d → 0 or a → 0 for the Rosenzweig-MacArthur predator-prey system [7]: where a, m, d > 0. In the same way, Wang et al. [8] studied the more general class of predatorprey system, here we follow their methods to get the relaxation oscillation profile of limit cycle when ε → 0. The phase portrait analysis for certain parameters can show that a prey-only or coexistence equilibrium is globally stable; and there exists a periodic solution for other parameters and the periodic solution is the only globally stable limit cycle.The predator population y(t) is near 0 for a long time when ε −1 is small for relaxation oscillation profile.
Our result is strictly proved by using basic differential and integral calculus, a Lyapunov function, and phase plane analysis.
We review some known results regarding the dynamics of system (1.1) in Section 2.1, and we prove our main results in Section 2.2 for the case ε → 0. We will use δ i and C i , (i ∈ N), to denote a variety of positive constants, and in Section 2.2 these constants are independent of ε.

Asymptotic behavior of limit cycle for ε small
After transformation τ = εt, the topologically equivalent form of the system (1.1) is (2.1) So we focus on the system (2.1) in the followings to investigate the profile of limit cycle as ε → 0.

Existence of limit cycle
It is obvious to obtain the Jacobian matrix of (2.1) at (λ, y λ ): . Then we can get the following basic dynamical behavior of (2.1): 1. λ = rhK−1 2h is the unique Hopf bifurcation value where a locally stable Hopf bifurcation point emerges, and a periodic solution of small amplitude appears near (λ, y λ ); 2h , (λ, y λ ) is unstable, and (2.1) has a unique limit cycle which is globally asymptotically orbital stable [4].

Phase analysis of limit cycle
In this section, we always assume that a, d > 0, rhK > 1, and λ = d a−hd satisfies 0 < λ < rhK−1 2h , then (2.1) has a unique limit cycle from Section 2.1.Now we define Firstly, we construct an invariant region where the limit cycle is located in the first quadrant.For this reason, an estimate of the unstable manifold U = {(x 1 (t), y 1 (t)) : t ∈ R} is given at the saddle point (rK, 0).From the comparison principle, it satisfies 0 < x 1 (t) < rK for all t ∈ R; U is above the isocline y 0 (x) = (1+hx)(rK−x) aK when λ < x < rK.Because it is monotone for λ < x < rK, denote this part by {(x, y 1 (x)) : λ ≤ x ≤ rK} with y 1 (rK) = 0. Furthermore, we denote Then we can estimate the unstable manifold U at the saddle point (rK, 0) by y 2 (x) and y 3 (x).
Proof.From the two equations in system (2.1), we have Since the unstable manifold satisfies 0 < x 1 (t) < rK for all t ∈ R, then along U, we have Integrating along the portion of U from x = rK to some x < λ, we obtain For the upper bound we notice that the tangent line of the unstable manifold is y = ε + (1+rhK)(r−εd) arK (rK − x), which is below y = y 2 (x).So we just have to prove that the vector field ( f (x, y), g(x, y)) points towards the region below the line y = y 2 (x) when (x, y) = (x, y 2 (x)) and λ < x < rK, which is equivalent to .
That proves that the upper bound y 1 (x) ≤ y 2 (x).
From Lemma 2.1, the unstable manifold achieves its maximum y-value when x = λ, and the maximum value y * can be estimated as From the phase portrait, the limit cycle is below the unstable manifold U, then we have the following upper bound for the location of the limit cycle.
Then the orbit of the limit cycle By constructing a more precise region R 2 ⊂ R 1 containing Σ, we prove that for a sub-region where where g(x) = ax 1+hx .Note that (rK − 1/h − λ, y λ ) is the reflection of (λ, y λ ) with respect to the line x = rhK−1 2h .
Lemma 2.3.Let R 3 be defined as in (2.7).Then R 3 is a bounded convex subset of R 2 + containing (λ, y λ ), and Σ R and is strictly increasing in (λ, ∞); in the same way, because W 2 (y) = 1 − (y λ /y), W 2 (y) is strictly decreasing in [0, y λ ) and is strictly increasing in (y λ , ∞).Thus W(x, y) reaches the global minimum at the unique critical point (λ, y λ ), and every level curve of W(x, y) is a bounded closed curve.The level curves have convex boundary since W 1 and W 2 are both convex one-variable functions.For R 3 be defined as in (2.7), (rK y λ )}, the vector field ( f (x, y), g(x, y)) points outwards.Thus from the properties of periodic orbit, Σ R 3 = ∅.
To obtain the global asymptotical behavior of the limit cycle Σ, we divide the orbit with four reference points (see Figure 2.1): .1: Illustration of the phase portrait (not up to scale) and the limit cycle in the proof.The isoclines are the thin solid curves: x = 0, y = 0, x = λ and the parabola y = y 0 (x); the limit cycle is the thick solid curve Let T = T(λ) be the period of Σ.Then T = T 1 + T 2 + T 3 + T 4 , where T i is the time taken from Q i to Q i+1 (with Q 5 = Q 1 ).We also assume that x(0) = λ and y(0) = y λ,+ , i.e. the orbit starts from the highest point of y(t).Now we obtain our main result.Theorem 2.4.Let Σ = {(x(t), y(t)) : t ∈ R} be the orbit of the unique periodic solution of (2.1) when 0 < λ < rhK−1 2h .Assume that rhK > 1 and a > d > 0 are fixed, the extremal points of Σ are defined as in (2.10) and Q i , T i (i = 1, 2, 3, 4) and the period T are defined as above.When ε > 0 is sufficiently small, then there exist constants Moreover, for ε > 0 is sufficiently small, there exists some C i > 0, such that (2.14) Proof.We prove the theorem in several steps.
Step 6: We show that there exist constants y 1 , y 2 > 0 such that y λ,+ < y 1 and y 2 < y λ,− .From Lemma 2.1 and (2.6), we obtain the estimate of upper bound of y λ,+ by letting y 1 = ε ε + 1+rhK arK .For the estimate of y 2 > 0, we notice that any solution orbit satisfies > y λ,− > y 2 .The estimates for T i for i = 2, 4 are clear from Step 3 and Step 5.This completes the proof.

Conclusion
The periodic solution of the system (2.1) is unique thus exists a globally stable limit cycle.It is assumed that the prey population has fast dynamics in this paper and it studies the asymptotic behavior of the limit cycle of (2.1) when ε tends to zero.The predator population y(t) is near 0 for a long time when ε −1 is small for relaxation oscillation profile (see Figure 2.2).We show that the period T of Σ tends to ∞ as ε → 0, see Theorem 2.4 for a more mathematical description.
the boundary of the invariant region R 3 : y = y 4 (x) is the outer boundary (together with x = 0 and y = 0); y = y 5 (x) and y = y 6 (x) are the upper and lower portions of inner boundary respectively; the line λ * = rK − 1/h − λ is the reflection of x = λ with respect to λ = (rhK − 1)/2h.(This graph is essentially from[4].)