2 Proof of the main result

Hopf bifurcations in Nicholson’s blowfly equation are always supercritical

Istv´ an Bal´ azs

, Gergely R¨ ost

Abstract

We prove that all Hopf bifurcations in the Nicholson’s blowfly equation are su- percritical as we increase the delay. Earlier results treated only the first bifurcation point, and to determine the criticality of the bifurcation, one needed to substitute the parameters into a lengthy formula of the first Lyapunov coefficient. With our result, there is no need for such calculations at any bifurcation point.

Keywords: delay differential equation; Hopf bifurcation; supercritical; normal form

1 Introduction

Nicholson’s blowfly equation

N⁰(t) =−γN(t) +pN(t−τ)e^{−aN(t−τ)} (1) is one of the most studied nonlinear delay differential equations, yet its dynamics is not fully understood. The equation can be interpreted as a population dynamical model with maturation delay and intraspecific competition, with N(t) denoting the population size, and all parameters being positive. There exists a positive equilibrium N^∗ = (1/a) ln(p/γ) of (1) if and only if a >0 andp > γ. These relations are assumed throughout this paper, since we are interested in the bifurcations of periodic orbits from the positive equilibrium.

We can easily see that Nicholson’s equation has Hopf bifurcations at critical delays τ_k with critical eigenvalues ω_k, k ∈ Z. There is a well known method for determining the direction of Hopf bifurcations for this type of equations [3], however, the calculations are rather tedious and rarely followed through completely. In [4], the next theorem was proven.

*MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged Aradi v´ertan´uk tere 1, Szeged, H-6720, Hungary

Department of Mathematics, University of Klagenfurt

Universit¨atsstraße 65-67, Klagenfurt am W¨orthersee, 9020, Austria

„Bolyai Institute, University of Szeged, Aradi v´ertan´uk tere 1, Szeged, H-6720, Hungary

Theorem 1. Let

ReC₁(0) = γ 2 (aN^∗−1)

( τ₀(aN^∗−2)² 3∆²ω₀γ³(aN^∗−1)³

h

2γ γ²−ω₀²

(1 +τ₀γ)²−ω₀²τ₀² +τ₀(1 +τ₀γ) γ⁴+ω₀⁴−6ω²₀γ₀²i

− 2τ₀(aN^∗−2)²

∆aN^∗γ γ+τ₀γ²+ω₀²τ₀ + τ0(aN^∗−3)

∆γ γ+τ₀γ² +ω₀²τ₀

+τ0(aN^∗−2)²

∆

×(ω²τ₀−γ−τ₀γ²)^γ_γ(aN^2aN∗∗^−ω−1)⁰² + (ω₀+ 2ω₀τ₀γ) 2ω_0aN^aN∗−1^∗

(^{γ2aN∗−ω2}0)²

(γ(aN^∗−1))² + 4ω^{2 a2N∗2}

(aN^∗−1)²





 ,

where ∆ = (1 +τ₀γ)²+ω²₀τ₀². Then,

(a) the Hopf bifurcation occurs as τ crosses τ₀ to the right if ReC₁(0) <0, and to the left if ReC₁(0)>0; and

(b) the bifurcating periodic solution is stable if ReC₁(0) <0and unstable ifReC₁(0) >0.

In Section 2, we prove that this long formula for ReC₁(0) is always negative, and this holds not only for the first bifurcation point τ₀, but for all critical parameter values τ_k, k ∈ N0. Here we use the method of [2] (see also [1]) to obtain our main result, which is stated in the following theorem.

Theorem 2. If p > e²γ, equation (1) undergoes a supercritical Hopf bifurcation at N^∗ when τ =τk, for allk ∈N⁰.

2 Proof of the main result

2.1 Preliminary calculations

Let N(t) be an arbitrary solution of equation (1). Setting N(t) = N^∗+ (1/a)x(t), x(t) satisfies

x⁰(t) = −γx(t)−aγN^∗

1−e^−x(t−τ)

+γx(t−τ)e^−x(t−τ). (2) For normalizing the delay, we use the transformation y(s) = x(τ s), and obtain

y⁰(s) =−τ γ y(s) +aN^∗

1−e^−y(s−1)

−y(s−1)e^−y(s−1)

. (3)

Using the Taylor expansion of the exponential function, the linearization of (3) is z⁰(s) =−τ γ z(s) +aN^∗z(s−1)−z(s−1)

. By introducing the new parameter b=aN^∗−1, it can be written as

z⁰(s) = −τ γ z(s) +bz(s−1)

. (4)

Substituting the exponential Ansatz, we find the characteristic equation λ=−τ γ 1 +be^−λ

. (5)

For ω >0,±iω is a pair of complex conjugate roots of (5) if and only if iω =−τ γ 1 +b(cosω−isinω)

. Separating the real and imaginary parts, we obtain

τ γbcosω=−τ γ, (6)

τ γbsinω=ω. (7)

Equation (6) can be simplified to

cosω=−1 b.

As ω >0, (7) implies sinω > 0. Then we have the critical imaginary parts ω_k= arccos

−1 b

+ 2kπ, k ∈N⁰. The inequality b >1 implies the estimates −1<−¹_b <0 and

π

2 <arccos

−1 b

< π. (8)

From sinω_k >0, we get sinω_k = sin

arccos

−1 b

+ 2kπ

= sin

arccos

−1 b

= r

1− 1 b². Thus, the critical parameter values are

τ_k = ω_k γbsinωk

= arccos −¹_b

+ 2kπ γb

q 1− _b¹2

= arccos −¹_b

+ 2kπ γ√

b²−1 , k ∈N0.

For checking the transversality condition, we differentiate the characteristic equation (5) with respect to the parameter τ:

λ⁰ =−γ 1 +be^−λ

+τ γbe^−λλ⁰, and express the derivative

λ⁰ =−γ 1 +be^−λ 1−τ γbe^−λ .

Substituting −τ γbe^−λ =λ+τ γ and −γ(1 +be^−λ) = λ/τ from (5), we can see that

λ⁰ = λ

τ(1 +λ+τ γ).

Considering λ in the form µ+iω, whereµ, ω ∈R, and taking the real part, we get

µ⁰ = Re iω

τ(1 +µ+iω+τ γ) = Re iω(1 +µ+τ γ−iω)

τ((1 +µ+τ γ)²+ω²) = ω²

τ((1 +µ+τ γ)²+ω²) >0.

As the real parts of the eigenvalues are strictly increasing in the parameterτ, the transver- sality condition holds, and we have Hopf bifurcations at critical values τ_k,k ∈Z, ifb >1.

The calculation in this section is equivalent with Section 2 of [4], however, our notations will be more convenient for us in the sequel.

2.2 Directions of the Hopf bifurcations

We follow the argument of [2], and apply it to equation (3). We denote the difference between the parameter and the critical value by α=τ−τ_k, and use the notationy_s(u) = y(s+u), −1 ≤ u ≤ 0 for solutions segments, as usual. Let L and F be defined by the relation

L(α)y_s+F(y_s, α) = (τ_k+α) −γy(s)−aγN^∗

1−e^−y(s−1)

+γy(s−1)e^−y(s−1) , where L(α) is a linear operator, F(0,0) = 0 and D1F(0,0) = 0. Then we have

L(α)ϕ=−(τ_k+α)γ ϕ(0) +bϕ(−1) and

F(ϕ, α) =−(τk+α)γ (b+ 1) 1−e^−ϕ(−1)

−ϕ(−1)e^−ϕ(−1)−bϕ(−1)

. (9)

For L₀ =L(0) we get

L₀(ϕ) =−τ_kγ(ϕ(0) +bϕ(−1)).

By substitution,

L₀(1) =−τ_kγ(1 +b), L₀ θe^iωkθ

=τ_kγbe^−iωk, L₀ e^2iωkθ

=−τ_kγ 1 +be^−2iωk follows. Expanding (9) into a Taylor series with respect to the first variable and substi- tuting α= 0, we have

F(ϕ,0) =τ_kγb−1

2 ϕ²(−1)−τ_kγb−2

6 ϕ³(−1) +h.o.t.

The coefficients of the right-hand side of F(y₁e^iωkθ+y₂e^−iωkθ+y₃·1 +y₄e^2iωkθ,0)

=B_(2,0,0,0)y²₁ +B_(1,1,0,0)y₁y₂+B_(1,0,1,0)y₁y₃+B_(0,1,0,1)y₂y₄+B_(2,1,0,0)y₁²y₂+. . . (10) are

B_2,0,0,0 =τ_kγb−1

2 e^−2iωk, B_1,1,0,0 =τ_kγ(b−1), B_1,0,1,0 =τ_kγ(b−1)e^−iωk, B_0,1,0,1 =τ_kγ(b−1)e^−iωk, B_2,1,0,0 =−τ_kγb−2

2 e^−iωk. The direction of the bifurcation is determined by the sign of

K = Re

1 1−L₀(θe^iωkθ)

B_(2,1,0,0)−B_(1,1,0,0)B_(1,0,1,0)

L₀(1) + B_(2,0,0,0)B_(0,1,0,1) 2iω_k−L₀(e^2iωk)

. (11) We use the notation a ∼ b for real numbers a and b having the same sign. Substituting all terms into K, we have

K ∼ 2b⁵ −12b⁴ + 23b³−23b²+ 4b+ 4√ b²−1 +b² −6b⁴+ 19b³−27b² + 10b+ 2

arccos

−1 b

+ 2kπ

=:K∗

(12)

The polynomial p₁(b) := −6b⁴+ 19b³−27b²+ 10b+ 2 is the solution of the 4th order IVP (initial value problem)

p₁(1) =−2, dp1

db (1) =−11, d²p1

db² (1) =−12, d³p1

db³ (1) =−30, d⁴p1

db⁴ (b) = −144,

so p₁(b) is negative for all b > 1. In (12), the coefficient of 2kπ is also negative for all b >1. From (8), for k ∈N⁰, we conclude

K∗ < 2b⁵−12b⁴+ 23b³−23b²+ 4b+ 4√

b²−1 +b²p₁(b)π 2. This expression is negative for all b >1 if

2b⁵−12b⁴+ 23b³−23b²+ 4b+ 4√

b²−1<−b²p₁(b)π 2. As the terms √

b²−1 and −b²p₁(b) are positive, the last inequality holds if p₂(b) := 2b⁵−12b⁴+ 23b³−23b²+ 4b+ 42

b² −1

−b⁴p₁(b)²π² 4 <0.

The polynomial p2(b) is the unique solution of the 12th order IVP p₂(1) =−π² <0, d⁷p2

db⁷ (1),= 534240−3064320π² <0, dp₂

db (1),= 8−15π² <0, d⁸p₂

db⁸ (1),=−120960−18103680π² <0, d²p₂

db² (1),= 184− 345π²

2 <0, d⁹p₂

db⁹ (1),=−13063680−98340480π² <0, d³p2

db³ (1),= 2004−1518π² <0, d¹⁰p2

db¹⁰ (1),=−116121600−501681600π² <0, d⁴p₂

db⁴ (1),= 10776−10968π² <0, d¹¹p₂

db¹¹ (1),=−2035756800π² <0, d⁵p₂

db⁵ (1),= 55200−72240π² <0, d¹²p₂

db¹² (b),= 1916006400−4311014400π² <0, d⁶p₂

db⁶ (1),= 249120−474840π² <0,

sop₂(b) is negative for all b >1. Hence, we conclude thatK <0 for allb >1 andk ∈N0, by [2], the theorem holds.

We have proven that all Hopf bifurcations for the Nicholson’s blowfly equation are su- percritical, hence there is no need to calculate the complicated first Lyapunov coefficients in the future. Our theorem significantly improves Theorem 2 of [4], where only the first bifurcation point was studied, and even for that a lengthy formula needed to be checked for any particular parameter combination to ensure supercriticality.

Acknowledgments

IB was supported by NKFI K129322 and EFOP-3.6.2-16-2017-00015. GR was supported by NKFI FK124016 and TUDFO/47138-1/2019-ITM.

References

[1] I. Bal´azs and G. R¨ost. “Hopf bifurcation for Wright-type delay differential equations:

The simplest formula, period estimates, and the absence of folds”. In: Communica- tions in Nonlinear Science and Numerical Simulation 84 (2020).

[2] T. Faria and L. T. Magalh˜aes. “Normal Forms for Retarded Functional Differential Equations with Parameters and Applications to Hopf Bifurcation”. In: Journal of Differential Equations 122.2 (1995), pp. 201–224.

[3] B. D. Hassard, N. D. Kazarinoff, and Y-H. Wan. Theory and Applications of Hopf Bifurcation. Cambridge University Press, 1981.

[4] J. Wei and M. Y. Li. “Hopf bifurcation analysis in a delayed Nicholson blowflies equation”. In: Nonlinear Analysis 60.7 (2005), pp. 1351–1367.