Stable periodic solutions for Nazarenko's equation

In 1976 Nazarenko proposed studying the delay differential equation \begin{document}$ \begin{equation*} \dot{y}(t) = -py(t)+\dfrac{qy(t)}{r+y^{n}(t-\tau)},\qquad t>0, \end{equation*} $\end{document} under the assumptions that \begin{document}$ p,q,r,\tau\in\left(0,\infty\right) $\end{document} , \begin{document}$ n\in\mathbb{N} = \left\{ 1,2,\ldots\right\} $\end{document} and \begin{document}$ q/p>r $\end{document} . We show that if \begin{document}$ \tau $\end{document} or \begin{document}$ n $\end{document} is large enough, then the positive periodic solution oscillating slowly about \begin{document}$ K = \left(q/p-r\right)^{1/n} $\end{document} is unique, and the corresponding periodic orbit is asymptotically stable. We also determine the asymptotic shape of the periodic solution as \begin{document}$ n\rightarrow\infty $\end{document} .


(Communicated by Alfonso Ruiz-Herrera)
Abstract. In 1976 Nazarenko proposed studying the delay differential equationẏ (t) = −py(t) + qy(t) r + y n (t − τ ) , t > 0, under the assumptions that p, q, r, τ ∈ (0, ∞), n ∈ N = {1, 2, . . .} and q/p > r. We show that if τ or n is large enough, then the positive periodic solution oscillating slowly about K = (q/p − r) 1/n is unique, and the corresponding periodic orbit is asymptotically stable. We also determine the asymptotic shape of the periodic solution as n → ∞.
1. Introduction. We consider the scalar delay differential equatioṅ y(t) = −py(t) + qy(t) r + y n (t − τ ) , t > 0, (1.1) under the assumption that p, q, r, τ ∈ (0, ∞) , n ∈ N = {1, 2, . . .} and q p > r. (1.2) This equation was proposed by Nazarenko in 1976 to study the control of a single population of cells [9]. The quantity y(t) is the size of the population at time t. The rate of change y (t) can be given as the difference of the production rate qy(t)/(r + y n (t − τ )) and the destruction rate py(t). We see that the destruction rate at time t depends only on the present state y(t) of the system, while the production rate also depends on the past of y. This is a typical concept in population dynamics; delay appears due to the fact that organisms need time to mature before reproduction. For further population model equations with delay, see e.g., [14]. One of the most widely studied examples is the Mackey-Glass equation: , t > 0. (1.3) In this model the production rate is very similar to the one considered by Nazarenko.
The usual phase space for (1.1) is the Banach space C = C ([−τ, 0] , R) with the supremum norm. A solution of (1.1) is either a continuously differentiable function y : R → R that satisfies (1.1) for all t ∈ R, or a continuous function y : [−τ, ∞) → R that is continuously differentiable for t > 0 and satisfies (1.1) for all t > 0. If for some solution y and t ∈ R, the interval [t − τ, t] is in the domain of y, then the segment y t ∈ C is defined by y t (s) = y (t + s) for −τ ≤ s ≤ 0. To each φ ∈ C there corresponds a solution y φ : [−τ, ∞) → R with y φ 0 = φ. Under condition (1.2), the functions R t → 0 ∈ R and R t → K = (q/p − r) 1/n ∈ R are the only constant solutions, i.e., there exists a unique positive equilibrium besides the trivial one.
Several authors have already examined equation (1.1), see e.g., the works [5,6,17,19]. In this paper we focus on those positive periodic solutions of (1.1) that oscillate slowly about K. A solution y is called slowly oscillatory about K if all zeros of y − K are spaced at distances greater than the delay τ . It is widespread to use the abbreviation SOP for such periodic solutions.
If we restrict our examinations only to positive solutions, then we can apply the transformation x = log y − log K. Thereby we obtain the equation (1.4) where the feedback function f ∈ C 1 (R, R) is defined as for all x ∈ R, (1.5) see Fig. 1. Note that f (0) = 0. Condition (1.2) implies that f is strictly increasing, hence we are in the so called "negative feedback" case. Solutions and segments of solutions of (1.4) are defined analogously as for equation (1.1). Note that the positive equilibrium of (1.1) given by K is transformed into the trivial equilibrium of (1.4). In accordance, an SOP solution of (1.4) is a periodic solution that have zeros spaced at distances greater than τ . We know from Theorem 7.1 of Mallet-Paret and Sell in [8] that if T 0 < T 1 < T 2 are three consecutive zeros of an SOP solution of (1.4), then T 2 − T 0 is its minimal period.
Nussbaum verified the global existence of SOP solutions for equations of the form (1.4) and for a wide class of feedback functions containing (1.5), see [11] and also [10]. His proof applies the Browder ejective fixed point principle. By [10,11], equation (1.4) has at least one SOP solution for Nussbaum also established results on the uniqueness of the SOP solution (up to translation of time) in [13]. However, paper [13] demands f to be odd, thus it cannot be applied for (1.5). Paper [2] of Cao, a second result on uniqueness, requires h(x) = xf (x)/f (x) < 1 to be monotone decreasing in x ∈ (0, b) and monotone increasing in x ∈ (−a, 0) with some a > 0 and b > 0. One can easily check that this concavity condition does not necessarily hold in our case either. We need to choose a different approach to guarantee the uniqueness of the SOP solution. The monotonicity of f is not sufficient: Cao proved the existence of a monotone f in [1] such that equation (1.4) has at least two distinct SOP solutions.
The stability of the SOP solutions is another central question. A well-known result is due to Kaplan and Yorke [3]: under certain restrictions on f , if the SOP solution is unique, then it is orbitally asymptotically stable. The region of attraction consists of the segments of all eventually slowly oscillatory solutions.
For a more detailed summary on SOP solutions of equation (1.4), we refer to the work [4] of Kennedy and Stumpf.
Song, Wei and Han studied the equation in the form (1.1), and they showed that a series of Hopf bifurcations takes place at the positive equilibrium as τ passes through the critical values see [17]. They gave explicit formulae to determine the stability, direction and the period of the bifurcating periodic solutions. Then they verified the global existence of the bifurcating periodic solutions by applying the global Hopf bifurcation theory in [20]. They showed that equation (1.1) has at least k periodic solutions if τ > τ k , k ≥ 1. Song, Wei and Han could not decide whether equation (1.1) has a periodic solution if τ ∈ (τ 0 , τ 1 ). As we have mentioned above, Nussbaum solved this problem in [11]. Our work is motivated by the fact that Song and his coauthors could not determine the stability of the periodic orbits for τ far away from the local Hopf bifurcation values. Uniqueness of the SOP solution has not been studied either.
The following theorems are the main results of this paper.
Theorem 1.1. Set p, q, r and n as in (1.2). (i) If τ > 0 is large enough, then equation (1.1) has a unique positive periodic solutionȳ : R → R oscillating slowly about K. The corresponding periodic orbit is asymptotically stable, and it attracts the set φ : y φ (t) > 0 for t ≥ −τ, y φ t − K has at most one sign change for large t .
Uniqueness of the periodic solution is always meant up to time translation. If we fix p, q, r and τ , we can determine the asymptotic shape of the periodic solution as n → ∞. Theorem 1.2. Set p, q, r and τ such that (1.2) and τ min{p, q/r − p} > 8 hold.
(i) Theorem 1.1.(i) is true for all sufficiently large n.
(ii) Define v : R → R as the ω-periodic extension of the piecewise linear function where ω is given by (1.6). Let η 1 > 0 and η 2 > 0 be arbitrary. If n is large enough, then there exists T ∈ R for theω-periodic SOP solutionȳ, such that |ω − ω| < η 1 , and The proofs of these theorems are similar, and they are organized as follows. Throughout the paper we examine equation (1.1) in the form (1.4)-(1.5). First we calculate an SOP solution v for the "limit equation" v (t) = −g(v(t − τ )), where g : R → R is a piecewise constant function chosen so that (1.5) is close to g outside a neighborhood of 0. Then we consider (1.5) as a perturbation of g and follow the technique used by Walther in [18] (for a slightly different class of equations) to obtain information about the solutions of equation (1.4). We show the existence of a convex closed subset A(β) ⊆ C such that all solutions of (1.4) with initial segments in A(β) return to A(β). Thereby a return map P : A(β) → A(β) can be introduced. Next we explicitly evaluate a Lipschitz constant L(P ) for P . If τ or n is large enough, then L(P ) < 1, i.e., P is a contraction. The unique fixed point of P is the initial segment of an SOP solution. Besides this, we need the results of paper [12] of Nussbaum to show that all SOP solutions have segments in A(β), and hence the SOP solution is unique. The rest of the theorems will follow easily. In particular, stability comes from the work [3] of Kaplan and Yorke.
We give full proofs only when q pr and pr q − pr (1.7) are not integers. In this case we determine the sizes of τ and n exactly. We indicate the necessary modifications when either q/(pr) or pr/(q − pr) is an integer.
The particular form of f is actually not important. It is possible to show that equation (1.4) admits an SOP solution if the feedback function f is Lipschitzcontinuous, there are constants A > 0, B > 0 and small β > 0 such that |f (x) + A| is small for x < −β and |f (x) − B| is small for x > β, furthermore, the Lipschitz constant for f restricted to the interval (−∞, −β] ∪ [β, ∞) is also sufficiently small. The method of Walther in [18] works for all such nonlinearities. If, in addition, f (0) = 0, f is continuously differentiable and f (x) > 0 for all real x, then one can prove uniqueness and stability using papers [12] of Nussbaum and [3] of Kaplan and Yorke.
Let us also mention that the Schwarzian derivative of (1.5) equals −n 2 /2 at all x ∈ R. Hence Proposition 2 of Liz and Röst in [7] gives bounds for the global attractor: If τ f (0) > 3/2, then for all positive solutions y of (1.1). If τ f (0) ≤ 3/2, then Proposition 2 of [7] states that all solutions of (1.4) converge to 0 (and hence all positive solutions of (1.1) converge to K) as t → ∞. This result improves the well-known fact that the trivial equilibrium of (1.4) (and hence the positive equilibrium of (1.1)) is locally asymptotically stable whenever τ f (0) < π/2. Note that if p, q, r are fixed according to (1.2), then f (x) converges to p − q/r = −A as nx → −∞ and f (x) tends to p = B as nx → ∞. Therefore we examine the Given any φ ∈ C, a solution v φ : [−τ, ∞) → R of (2.1) is an absolutely continuous function such that v φ | [−τ,0] = φ and the integral equation is satisfied for t > 0. Similarly, an absolutely continuous function v : R → R is a solution of (2.1), if the integral equation is satisfied for all t ∈ R. In this section we evaluate an SOP solution for (2.1).
where σ = (1 + B/A)τ is the first positive zero, and ω = (2 + A/B + B/A)τ is the second positive zero and the minimal period of v.
This function has zero at σ From this formula we see that ω can be defined as ,ω] to R ω-periodically, then we get a periodic solution of (2.1) on R.

Preliminary estimates. For
This is true because In this section we study the solution x = x φ of (1.4) when f ∈ N (A, B, β, ε) and φ ∈ A(β). Our main goal is to show that if φ ∈ A(β), then there exist q > 0 and q > 0 such that Define the integer N by (N − 1)τ < σ ≤ N τ , where σ is the first positive zero of the periodic function v given by Proposition 1. As σ = (1 + B/A)τ , we get that and Then and It is also clear from the choice of φ that x (t) < 0 for t ∈ (0, τ ). Proof of (3.2) for k = 2. Condition (C.1) guarantees that δ < τ , and therefore v(δ) = −Bδ by Proposition 1. Then, by (3.3) and by the definition of δ, By the monotonicity of x on [0, τ ], we obtain that This observation together with (3.1) gives that for t ∈ [τ, 2τ ], We have verified (3.2) for k = 2.
Proposition 3. In addition to the assumptions of the previous proposition, suppose that x (t) < 0 for t ∈ (0, τ ), (3.8) x (t) > 0 for t ∈ (τ + δ, N τ ), (3.9) and for the unique solution q = q(φ) of the equation where σ is the smallest positive zero of v.
Exchanging the role of A and B in the proofs of Propositions 2 and 3, we get the desired estimates for |x −ṽ|,x,x and |q − (ω − σ)|.
is negative for any β > 0 and ε > 0. In consequence, estimate (3.11) in the proof of Proposition 3 is not true in this case, that is, we cannot guarantee inequality (3.7) (which is a key property in proving the existence of q with x q ∈ −A(β)).
Similarly, (C.6) in Proposition 5 cannot be satisfied with any β > 0 and ε > 0 if A/B is an integer.
Next we discuss how Proposition 3 or Proposition 5 should be modified in case B/A or A/B is an integer. Remark 1. One can change Proposition 3 as follows in case B/A is an integer.
The proof of (3.6) is independent of the value of c 3 , so it is correct even if B/A is an integer. First, inequality (3.6) has be extended for a larger interval. Assume that It is clear that T < 1. Then estimate (3.2) and the definition of v yield that for This result and (3.6) together give that By (3.2), the right hand side is not greater than β + (N + T )ετ + (A + B)δ. It is clear from N τ = σ and T < 1 that we need to consider the second line of the definition of v in Proposition 1 to evaluate v((N + T )τ ): We see from the last two results that one can achieve if β > 0 and ε > 0 are small enough.
Results (3.20) and (3.21) guarantee the existence of q. It is now easy to modify the rest of Proposition 3.
Proposition 5 has to be altered in a similar fashion if A/B is an integer. The subsequent proofs of the paper also need to be slightly changed if either B/A or A/B is an integer. We omit the details. In this section F denotes the semiflow corresponding to (1.4): Then subset F (τ + δ, A(β)) ⊂ C consists of the τ + δ-segments of those solutions that have initial segments in A(β): . We introduce the map where ψ = F (τ + δ, φ). In other words, if ψ ∈ F (τ + δ, A(β)), then s(ψ) is the time in (0, (N − 1)τ − δ) for which x ψ s(ψ) ∈ −A(β). Proposition 3 guarantees that s is well-defined.
Consider the map Our next goal is to determine Lipschitz constants for these maps. Proposition 6. τ L β is a Lipschitz constant for F τ , and 1 + δL(f ) is a Lipschitz constant for F δ .
Next, consider This proves the Lipschitz constant for F δ .
In order to determine Lipschitz constants for the maps s and F s , we estimate Proposition 7. If φ,φ ∈ F (τ + δ, A(β)), then for the solutions x = x φ andx = xφ we have max Proof. The assertion is clearly true if N = 2, so we may suppose that N > 2. We verify that max for all k ∈ {0, 1, ..., N − 2} by induction on k. Estimate (4.1) obviously holds for k = 0. We need to show that it holds for k + 1 provided it true for some k ∈ {0, 1, ..., N − 3}. It follows from (3.6) that Now we are in position to evaluate Lipschitz constants for both s and F s .
We obtain the following corollary.
is a Lipschitz constant for R.

Now consider the map
whereq is introduced in Proposition 5.

Proposition 10. The constant
The proof of this proposition is analogous to the reasoning above, thus we leave it to the reader. One needs to use Proposition 5.
As a consequence we can state the following.

5.
On the ranges of the SOP solutions. In this section we show that if τ is large enough and β is small enough, then any SOP solution x : R → R of (1.4) has segments in A(β).
We define g : R → R such that and g(0) = 0. By Lemma 1 of [12], g is continuous and nondecreasing on R, and there exists d > 0 such that |g(x)| ≥ d|x| for |x| ≤ 1 and |g(x)| ≥ d for |x| ≥ 1. We may choose d as follows.
Examining the second derivative of f , one can check that f (x) > 0 for x ∈ (−∞, x * ) and f (x) < 0 for x ∈ (x * , ∞), where So f is strictly concave up on (−∞, x * ], strictly concave down on [x * , ∞) and x * is the unique inflection point of f . If f is concave down on [0, 1], i.e., x * ≤ 0, then -as the graph of f is above the straight line joining (0, 0) and (1, f (1)) -we see that Now suppose that x * > 0, i.e., x is concave up on [0, x * ]. Then, on the interval (0, x * ], the graph of f is above the tangent line at x = 0: If x * ≥ 1 or the estimate f (x) > f (0)x holds for all x ∈ (0, 1], then we have verified assertion (i) for all x ∈ [0, 1].
Following the proof of Lemma 10 in [12], we get the subsequent estimates forx 0 andx 1 .
As we are going to apply estimate (24) of [12] in the forthcoming proof, let us note that the second line of (24) contains a typo. With the notations of [12], the correct form of this estimate is the following: if 1 ≤ z ≤ 3/2, then There is also a mistype in Lemma 9 of [12]: estimate (33) holds if z 1 ≥ 3/2.
Proof. We use the results of [12] with parameters α = τ , ε = 1/2 and x 0 =x 0 , x 1 =x 1 . Constants c and k of [12] both equal 1 in our case. We need to consider three cases according to the sizes of z and z 1 . Case max{z 1 , z} ≤ 3/2: By Lemma 6 of [12], From the second line of (23) in Lemma 7 of [12] we see that Case z 1 ≥ 3/2: As τ d > 1, we get from the first line of estimate (33) in [12] that and from the last line of (33) in [12] that Here we also used the fact that f is an increasing function. Case z ≥ 3/2: Using estimate (34) of [12], the inequality τ d > 1 and the monotonicty of f , we deduce that Summing up the above estimates and using that g is monotone increasing, we obtain thatx 0 ≥ min τ d, and Recall that constant d is chosen so that |g(x)| ≥ min{d, d|x|} for all x ∈ R. By applying this estimate in (5.7), we conclude that Estimate (5.8) and inequality |g(x)| ≥ min{d, d|x|} now imply that It is analogous to show that Since τ d > 4, we must havex 0 ≥ τ d/2 and |x 1 | ≥ τ d/2.
Now we are able to determine a positive lower bound forx on a specific interval of length 1.
Proposition 15. If τ d > 4 and B is an upper bound for f , then for each SOP solutionx : R → R of (5.1), one can give an interval I of length 1 such that Proof. Actually we prove that for any γ ∈ (0, 1), We show this again by applying the results of [12]. Note that as d ≤ f (1)/2 and B is an upper bound for f , we have d ≤ B/2 and thus 0 < γd/(2B) < 1/4.
Observe that τ d(1 − γ)/2 is decreasing in γ, and γτ d 2 /(4B − 2γd) is increasing in γ. The lower bound given forx in (5.10) is maximal if we choose γ such that the two terms are equal, i.e., if We obtain the statement of the proposition with this choice of γ.
The main result of this section follows easily from the last proposition. Proof. Proposition 15 guarantees that for any SOP solution x : R → R of (1.4), there exists an interval J of length τ such that x(t) ≥ β for t ∈ J. Let q * ≥ sup J be minimal with x(q * ) = β. It is clear that q * exists (as x is continuous and has arbitrary large zeros), and then x q * ∈ A(β).
6. Proofs of the main theorems. The main theorems follow from the partial results of the previous sections.
Proof of Theorem 1.1. Consider (1.4)-(1.5), where p, q, r, n are fixed according to (1.2). We prove that for all sufficiently large τ > 0, equation (1.4) has a unique SOP solutionx : R → R. The corresponding periodic orbit is asymptotically stable and its region of attraction is φ : x φ t has at most one sign change for sufficiently large t .
In addition, we show that ifω denotes the minimal period ofx, thenω/ω tends to 1, where ω is defined by (1.6). Theorem 1.1 will follow by settingȳ = Kex. Set A = q/r − p > 0, B = p > 0, N = 1 + B/A andÑ = 1 + A/B . We consider only the case when Existence of an SOP solutionx. Recall c i , i ∈ {1, ..., 6}, from (C.1)-(C.6). Using the definitions of δ andδ, we can write c i in the form c i = a i τ + b i β for all i ∈ {1, ..., 6}, where a i = 0 and b i = 0 are functions of A, B and ε. We emphasize that a i and b i are independent of τ and β. Fix ε > 0 such that Inequalities (6.2) guarantee that the minimum on the right hand side is positive, so this choice of ε is possible. One can easily check that for such ε, the coefficient a i is positive for all i ∈ {1, ..., 6}. In consequence, if τ is an arbitrary positive number and β = ατ , where then c i is positive for all i ∈ {1, ..., 6}, that is, (C.1)-(C.6) are satisfied. In the following, we fix ε as above, and use β = ατ with α set as above.
As nonlinearity f defined in (1.5) is strictly increasing with lim x→−∞ f (x) = −A and lim x→∞ f (x) = B, it is clear that f ∈ N (A, B, ατ, ε) if

This inequality holds if
In addition, recall that f admits a unique inflection point x * ∈ R (given in (5.3)), f is strictly increasing on (−∞, x * ] and strictly decreasing on [x * , ∞). Hence f is Lipschitz continuous with Lipschitz constant In consequence, we can use the results of Sections 3 and 4 for τ ≥ τ 1 . We conclude that is a Lipschitz constant for the Poincaré map P . If τ ≥ τ 2 = x * /α, then ατ ≥ x * . Since f is decreasing on [x * , ∞), we see that L ατ can be chosen as This formula shows that lim τ →∞ τ k L ατ = 0 for any positive integer k. Similarly, lim τ →∞ τ k L −ατ = 0 for any positive integer k.
As L(f ), N,Ñ are independent of τ , and δ,δ are linear functions of β = ατ , we obtain that lim τ →∞ L(P ) = 0. Therefore there exists τ 3 ≥ max{τ 1 , τ 2 } such that L(P ) < 1 for τ > τ 3 , and hence P is a contraction on A(ατ ). The unique fixed point of P in A(ατ ) is the initial segment of a periodic solutionx. It is clear from the construction thatx is an SOP solution.
Uniqueness. We may assume that the parameter α was fixed so small above that α ≤ ( √ B 2 + d 2 − B)/2. If τ d > 4, where d is set in Proposition 12, then Corollary 2 gives that all SOP solutions of (1.4) have segments in A(ατ ). Hence all SOP solutions arise as fixed points of P in A(ατ ). The uniqueness of the fixed point of P yields the uniqueness of the SOP solution for τ > max{τ 3 , 4/d}.
Stability. Kaplan and Yorke proved that the uniqueness of the SOP orbit gives its asymptotic stability if τ > π/(2f (0)), see Theorem 2.1 and Remark 2.5 of [3]. Note that our previous assumption τ > 4/d and the definition of d together guarantee that τ > π/(2f (0)). The region of attraction is also determined in [3].
Minimal period. The statement regarding the limit of the minimal period ofx follows at once from Theorem 1 of [12].
One can modify the proof of Theorem 1.1 to cover the case when either A/B or B/A is an integer using Remark 1. Table 1 presents some examples when Theorem 1.1 is true. Now let η 1 > 0 and η 2 > 0 be arbitrary. Choose β ∈ (0, β] and ε ∈ (0, ε] so that By the monotonicity of f , this yields that there exists n 1 = n 1 (β , ε ) > n 0 so that f ∈ N (A, B, β , ε ) for all n > n 1 . In addition, as the initial function of the SOP solutionx (corresponding to any n > n 1 ) belongs to A(β), andx is a continuous function oscillating about 0, there exists T = T (β ) such thatx T ∈ A(β ). Therefore we can indeed apply Propositions 2, 3, 4 and 5 for any n > n 1 with β and ε chosen above. It comes from the construction that the minimal period ofx isω = q +q. Condition (6.7) ensures that we can apply estimates (3.10) and (3.19) from Propositions 3 and 5. These results together with assumption (6.8) imply that for n > n 1 , |ω − ω| = |q +q − ω| ≤ |q − σ| + |q − (ω − σ)| < η 1 .