DEPENDENCE OF STABILITY OF NICHOLSON’S BLOWFLIES EQUATION WITH MATURATION STAGE ON PARAMETERS∗

The stability of Nicholson’s blowflies equation with maturation stage is investigated by reducing the number of parameters in the original model. We derive the condition on the stability of the positive equilibrium of the model, and discuss the dependence of the stability on the parameters by analyzing geometrically the dependence of real parts of eigenvalues of the characteristic equation with fewer parameters on the parameters. By restoring parameters, the condition on the stability of the positive equilibrium of the original model are formulated explicitly, and the corresponding regions are depicted for some different cases. The obtained result shows that the parameter determining the maximum reproductive success of the population affects only the size of the positive equilibrium, but plays no role in determining its stability.


Introduction
According to data drawn from the experiments on laboratory cultures of the sheep blowfly Lucilia cuprina, Nicholson [14] illustrated in 1953 the oscillatory population fluctuations by using the experimental and mathematical approaches, but the oscillatory behavior was not completely explained. In order to answer the associated questions, in 1980 Gurney et al. [7] formulated the Nicholson's blowflies equation Here N (t) denotes number of mature individuals at time t, the birth of new individuals is assumed to be subject to the Ricker function be −δN in which b is the maximum per capita production rate and 1/δ is the population size at which the population as a whole achieves maximum reproductive success, d is the per capita death rate of mature population, and τ is the constant maturation time of new born individuals. The global attractivity, threshold dynamics and Hopf bifurcation of model (1.1) are further analyzed in [3,8,[10][11][12][13]16]. In model (1.1), the death of the immature population (i.e. in the maturation stage) is neglected. But the death in the stage is objective. Then when the death of the immature population is incorporated into (1.1) and the rate constant is assumed to be µ, Cooke et al. [5] derived the following delay differential equation Model (1.2) has been applied to describe a predator-prey model with juvenile/mature class structure [4], the growth rate of adult female mosquitoes [6], and the change of total population in an SIS epidemic model with standard incidence in the absence of infection [5,17,18]. Dynamical behavior of equation (1.2) is complicated as the positive equilibrium exists, some results have been obtained by many mathematic researchers, but the analysis is far from complete. Shu et al. [15] considered the delay as a bifurcation parameter, examined the onset and termination of Hopf bifurcations of periodic solutions from the positive equilibrium, and showed that the model has only a finite number of Hopf bifurcation values and how branches of Hopf bifurcations are paired. In [19], the necessary and sufficient conditions ensuring the local stability of the positive equilibrium were found by applying the Pontryagin's method, and the effect of parameter values on the local stability of the equilibrium was investigated according to the obtained conditions. Cooke et al. [5] found the existence of stability switch of (1.2) (stable-unstable-stable) of the positive equilibrium as τ increases by using the geometric method. In [1,9], the stability switches with increase of the delay were further investigated, and necessary and sufficient conditions of the local stability are provided by means of the geometric criterion proposed by Beretta and Kuang [2]. Wei and Zou [17] performed a global Hopf bifurcation analysis on (1.2) where b is used as a bifurcation parameter, and showed that model (1.2) actually allows multiple periodic solutions as b increases.
According to the conditions on the local stability of the equilibrium, it is easy to see that, for (1.1) it will no longer be stable once losing stability occurs when the delay increases. Note that ω 0 tends to π/2 as τ → 0, then, when the stability can change with increase of the delay, there must be stability switch for model (1.2), that is, after losing stability, its stability must restor and be preserved until the delay increases to (1/µ) ln(b/d).
In this paper, our study is only concerned with the dependence of stability of the positive equilibrium N * = [ln(b/d) − µτ ]/δ of model (1.2) on all the parameters, so the corresponding discussion is only for the case b > de µτ . To this end, the associated analysis begins from the discussion on the characteristic equation of equation (1.2) at the equilibrium N = N * . It is easy to know that the characteristic equation is For simplicity, denotingλ = λτ andτ = µτ , and then removing the bars, (1.3) reads the following equation where α = ln(b/d) and β = d/µ. Therefore, to discuss the stability of N * is to determine whether the roots of equation (
Then the straight lines β = 0 and α = 2 + τ in the (β, α) plane are the asymptotes of the graph of function α = α k (β). Further, according to the periodicity of cosine function and cotangent function, we know from (2.3) that the larger the subscript k is, the larger the value of β corresponding to the same α is. Therefore, if the curve corresponding to function α = α k (β) is denoted by L k , any two curves L m and L n for m > n (m, n = 0, 1, 2, . . .) do not intersect, and the curve L m is always in the left lower of L n (Fig.  1).
Since ( In order to find the region where all roots of equation (1.4) have negative real part, we have to determine the sign change of real part of root of (1.4) when point (β, α) crosses curves L k . According to the properties of function α = α k (β), this can be carried out by considering the sign of the derivative of real part of the root with respect to β on L k . So substituting ρ = u + iv(v > 0) into (1.4) and separating the real and imaginary parts give Obviously, then, according to the implicit function theorem, (2.4) only determines two functions u = u(β) and v = v(β). So from the implicit differentiation of F (u, v, β, α) = 0 and G(u, v, β, α) = 0 with respect to β, we have
Summarizing the inference and discussion above, we have the following statements with respect to the stability of the positive equilibrium N * .

Conclusion
In Section 2 we have found the parameter separatrix, α = α 0 (β), determining whether (1.4) is stable for the fixed τ . In order to directly determine whether the Equation (3.1) can also be rewritten as   2) is locally asymptotically stable as de µτ < b < b 0 (d), and unstable as b > b 0 (d).
For fixed d and τ , the positive equilibrium of model (1.2) is locally asymptotically stable as de µτ < b < b 1 (µ), and unstable as b > b 1 (µ).
According to Theorem 3.1, the stability regions of model (1.2) for the different cases are showed in Figures 4, 5 and 6, respectively. And the change of the regions with τ is also reflected in the corresponding figures.
At last, it must be pointed out that, with respect to dynamical behavior of model (1.2), the parameter δ determines only the value of the positive equilibrium, but does not affect its stability.