Predator–prey systems with small predator's death rate

The goal of our paper is to study canard relaxation oscillations of predator– prey systems with Holling type II of functional response when the death rate of predator is very small and the conversion rate is uniformly positive. This paper is a natural continuation of [C. Li, H. Zhu, 2013; C. Li, 2016] where both the death rate and the conversion rate are kept very small. We detect all limit periodic sets that can produce the canard relaxation oscillations after perturbations and study their cyclicity by using singular perturbation theory and the family blow-up.


Introduction
The Rosenzweig-MacArthur predator-prey model is typically given by where x ≥ 0 is the population density of prey, y ≥ 0 is the population density of predator and all the parameters are strictly positive. The function P (x) = mx b+x is called the predator response function, δ > 0 represents the death rate of the predator, c > 0 is the rate of conversion of prey to predator and the function x → rx 1 − x K represents the logistic growth model of the prey in the absence of predator. System (1) is a special case of more general predator-prey systems with a Holling type response function P : with the same conditions on x, y and the parameters. The function P in (1) is an increasing function and tends to m > 0 as x → ∞, and it is often called a Michaelis-Menten function or a response function of Holling type II. We call the function P (x) = mx 2 a+x 2 , with m > 0 and a > 0, the response function of Holling type III, and its generalized version is given by P (x) = mx 2 a+bx+x 2 . The Holling type IV response function or the Monod-Haldane function is given by P (x) = response functions of Holling type II and generalized Holling types III and IV, have been studied in [LZ13,Li16], uniformly in (δ, c) → (0, 0). When δ = c = 0, system (2) contains curves of singularities (often called slow curves or critical curves). See e.g. Figure 1. Periodic orbits of (2) that are Hausdorff close to limit periodic sets, at level δ = c = 0, containing pieces of the critical curves and fast orbits are called relaxation oscillations (see e.g. [DR96,DR07]). Thus, [LZ13,Li16] deal with the relaxation oscillations of (2) when (δ, c) ∼ (0, 0), away from the degenerate intersection point (0, br m ), using geometric singular perturbation theory [DR96,KS01,DMD05,DMD08]. The goal of our paper is to study another limiting case observed in [LZ13]: δ → 0 and c is a positive constant. When δ = 0 and c > 0, system (2) contains the critical curve {x = 0} (note that P (0) = 0). Let us focus on system (1) (see Figure 2). All the singularities of the critical curve are normally hyperbolic, except the nilpotent contact point (x, y) = (0, br m ). We distinguish between 3 types of limit periodic sets, at level δ = 0, that can generate limit cycles when δ ∼ 0 and δ > 0 (see also Section 2): (i) the nilpotent contact point (x, y) = (0, br m ), (ii) canard limit cycles consisting of a fast orbit and the part of the critical curve, with a regular slow dynamics, between the α-limit set and the ω-limit set of that fast orbit and (iii) the slow-fast 2-saddle limit periodic set consisting of the hyperbolic saddle (x, y) = (K, 0), the stable manifold of (x, y) = (K, 0), the unstable manifold of (x, y) = (K, 0) and the part of the critical curve between the origin (i.e. the α-limit set of the stable manifold) and the ω-limit set of the unstable manifold. (As we will see in Section 2, the slow dynamics of (1) along the critical curve has a hyperbolic saddle at the origin.) The goal of our paper is to study the cyclicity of each of these limit periodic sets by using geometric singular perturbation theory and the family blow-up at (x, y, δ) = (0, br m , 0). To study the contact point (x, y) = (0, br m ), we use the theory of slow-fast Hopf points of codimension 1 developed in [KS01]. To find the cyclicity of the canard limit cycles, we study zeros of the so-called slowdivergence integral of (1), using results from [DMD05]. The paper [DMD08] is a generalization of [DMD05] allowing zeros in the slow dynamics, away from the contact point. Since the slow-fast 2-saddle limit periodic set contains, beside a zero of the slow dynamics at the origin (x, y) = (0, 0), a (hyperbolic) saddle at (x, y) = (K, 0), away from the critical curve, we cannot use the results of [DMD08] directly. Thus, we have to develop new methods that are suitable for studying the cyclicity of the slow-fast 2-saddle limit periodic set. Clearly, the new techniques can be used not only in the framework of (1) or (2), but also in more general planar slow-fast systems. In the regular case (δ is a positive constant), the system (1) has at most one limit cycle (see [Che81,LC88]). The reason we study limit cycles of (1) uniformly in δ → 0 is twofold. On one hand, we cannot apply the methods of [Che81,LC88] to the singular case δ → 0, and we need to use techniques coming from singular perturbation theory, including the family blow-up. On the other hand, our paper provides a good starting point to get familiar with the methods that can be used not only to study the singular Holling type II case (1) but also the more general system (2), with a response function of Holling type III and IV, in the singular case δ → 0.
In Section 2 we state our main results. Our upper bounds on the number of limit cycles in (1), with δ → 0, are 1 or 2, depending on the region in the parameter space (see Theorem 2.1-Theorem 2.3). We prove our main results in Section 3.

Statement of results
We state our main results in Section 2.3-Section 2.5, clearly distinguishing between three different types of limit periodic sets defined in Section 2.2. Instead of working with the original predator-prey model (1), it is more convenient to work with a polynomial normal form (6) obtained in Section 2.1.
Using a time rescaling, i.e. multiplication by b + x > 0, system (3) becomes a polynomial vector field: When δ = 0, (4) becomes and we call (5) the fast subsystem of (4). The critical curve {x = 0} of (5) is normally attracting when y > b and normally repelling when 0 ≤ y < b. When y = b, we deal with a nilpotent contact point. See Figure 2. One of the reasons we work with the polynomial system (4), instead of (3), is we want to apply a family blow-up of (4) to the point (x, y, δ) = (0, b, 0) (see Section 2.2). After translationȳ = y − b, we may also assume that the nilpotent contact point is situated at (x, y) = (0, 0). We get where x ≥ 0 and y ≥ −b (we denoteȳ again by y).
If we apply the coordinate change ( ,δ, t) → (− , −δ, −t) to (11), then we obtain the blown-up vector field in the phase directional chart {ȳ = −1} (see (8)). In the charts {x = 1} and {x = −1}, we find no extra singularities. See  Thus, the passage along the critical curve {x = 0}, with y ≥ −b, satisfies the assumptions of [DMD08]. Now we define the canard limit cycle Γ (b,c) y , y ∈]−b, 0[, consisting of a regular orbit of the fast subsystem of (6) and the part of the critical curve between the α-limit set (0, y) and the ω-limit set (0, F (y)) of that regular orbit, where the fast relation function F depends on (b, c) and F (y) > 0. Since the slow dynamics (7) is regular on the segment [y, F (y)] for each y ∈] − b, 0[, we can use the so-called slow divergence integral of (6) along [y, F (y)] to study limit cycles of (6) Hausdorff close to Γ (b,c) y (see [DMD08] or [DMD05]). The slow divergence integral is defined as: where we use the fact that −y is the divergence of (6), with δ = 0, along the critical curve {x = 0} and dτ = dy −b(y+b) (see (7)). Following [DMD05] or [DMD08], the cyclicity of Γ −b the slow-fast 2-saddle limit periodic set in (6), at level δ = 0, consisting of the horizontal orbit y = −b connecting the hyperbolic saddle (x, y) = (0, −b) of the slow dynamics with the hyperbolic saddle (x, y) = (1, −b), the unstable manifold of (x, y) = (1, −b), the ω-limit set (x, y) = (0, F (−b)) of the unstable manifold and the critical curve contains the singularity of the slow dynamics, we cannot use (12) to study the cyclicity of Γ [DMD08]). See Sections 2.3 and 3.1 for more details.
Remark 1. Let us explain why the slow-fast 2-saddle limit periodic set Γ . Since the vector field (6), with δ = 0, points inwards the (positively invariant) set D := {0 ≤ x ≤ 1, y ≥ −b} along the line {x = 1, y > −b}, the unstable manifold enters the set D and stays in D. For the system (6), with δ = 0, the parabola y = (1 − b)x − x 2 and the critical curve x = 0 represent the x-nullcline (see Figure 4). To show that F (−b) < ∞ for b > 0, we have to study (6) near y = +∞. After a coordinate change (x, y) = (X, 1 Y ), with Y ∼ 0 and Y > 0, and multiplication by Y , system (6), for δ = 0, becomes where X ≥ 0 and Y ≥ 0. Singularities of (13) are given by {X = 0}; all the singularities are semi-hyperbolic and attracting. Notice also that {Y = 0} is the stable manifold of the singular point (X, Y ) = (0, 0) and that the (local) stable manifold of the singular point (X, As explained in Section 1 and Section 2.2, the limit periodic set Γ (b,c) −b contains one "singular" hyperbolic saddle and one "regular" hyperbolic saddle. Using [Mou90], [DMD08] and the fact that the connection between the two saddles is unbroken, we show that the singular hyperbolic saddle is dominant, and we obtain the following cyclicity result: Figure 4: Nullclines and direction field for (6) with δ = 0.
such that system (6) has at most one limit cycle in V, for each value (δ, b, c) ∈ [0, δ 0 ] × W. If the limit cycle exists, it is hyperbolic and repelling.
Theorem 2.1 will be proved in Section 3.1.

Cyclicity of Γ
To study zeros of the slow divergence integral (12), first we have to find properties of the fast relation function F on ] − b, 0[ by using a first integral of (6), with δ = 0. When c = 2b + 2, we find the first integral by using Darboux theory of integrability, and we get the following cyclicity result: The following statements are true: 1. If 0 < b 0 < 1, then there exist δ 0 > 0, a neighborhood W of (b 0 , c 0 ) in the (b, c)-space such that the set ∪ y∈C Γ (b0,c0) y can produce at most two limit cycles in (6), for each value (δ, b, c) ∈ [0, δ 0 ] × W.

If
can produce at most one limit cycle in (6), for each value (δ, b, c) ∈ [0, δ 0 ] × W. If the limit cycle exists, then it is hyperbolic and repelling.
Theorem 2.2 will be proved in Section 3.2. The case c = 2b + 2 is a topic of further studies.

Cyclicity of the contact point
To study the number of limit cycles close to a contact point, we typically blow up the contact point and detect all possible limit periodic sets on the blow-up locus that can produce limit cycles after perturbation: a center, closed orbits surrounding the center and a singular cycle consisting of two semi-hyperbolic singularities on the equator of the blow-up locus and the regular orbits that are heteroclinic to them (see Figure 3). When b = 1, we can bring our system (6), near the contact point at the origin, to a normal form for a slow-fast Hopf point of codimension 1 studied in e.g. [KS01], and we obtain at most one (hyperbolic) limit cycle.
Theorem 2.3 will be proved in Section 3.3.
We define a difference map near the slow-fast 2-saddle limit periodic set Γ where ∆ + (resp. ∆ − ) is a transition map of (6) in forward time (resp. in backward time) from S to T (see Figure 5). The horizontal section S is parametrized by x (we give a precise definition of S later in this section) and the section T ⊂ {ȳ = 0} is defined in the family directional chart of (8) and parametrized byx ∼ 0, where (x,ȳ) are the coordinates of (9). Thus, T is transverse to the connection {x = 0}, on the blow-up locus, between the attracting part of the critical curve and the repelling part of the critical curve (see Figure 3). It is clear that limit cycles of (6), close to Γ (b0,c0) −b0 in the Hausdorff sense, are given as zeros of the difference map ∆. In the rest of this section, we show that ∆ has at most one zero in x (counting multiplicity), for each fixedδ > 0,δ ∼ 0 and (b, c) ∼ (b 0 , c 0 ). −b , can be studied as zeros of a difference map ∆ := ∆ + − ∆ − where ∆ + (resp. ∆ − = ∆ 2 • ∆ 1 ) is a transition map defined by following the trajectories of (6) in forward time (resp. in backward time) from the section S to the section T .
First, we study the transition map ∆ − . We split up ∆ − into two parts (see Figure 5): 1. The Dulac map ∆ 1 near the hyperbolic saddle (x, y) = (1, −b) defined by following the orbits of (6) in backward time from S to S 1 . We study the transition ∆ 1 by using results from e.g. [Mou90].
2. The transition map ∆ 2 defined by following the trajectories of (6) in backward time from S 1 to T . This transition map includes the passage near the critical curve with a hyperbolic saddle in the slow dynamics and it has been studied in detail in [DMD08] or [HDMD13].

Proof of Theorem 2.2
Let b 0 be an arbitrary positive constant and c 0 = 2b 0 + 2. We suppose that C ⊂] − b 0 , 0[ is an arbitrary compact set. We study zeros of the slow divergence integral I(y, b 0 , c 0 ), given in (12), w.r.t. y ∈ C, depending on the constant b 0 .