The stability and Hopf bifurcation of the diffusive Nicholson’s blowflies model in spatially heterogeneous environment

Abstract

In this paper, we consider the diffusive Nicholson’s blowflies model in spatially heterogeneous environment when the diffusion rate is large. We show that the ratio of the average of the maximum per capita egg production rate to that of the death rate affects the dynamics of the model. The unique positive steady state is locally asymptotically stable if the ratio is less than a critical value. However, when the ratio is greater than the critical value, large time delay can make the unique positive steady state unstable through a Hopf bifurcation. In particular, the first Hopf bifurcation value tends to that of the “average” DDE model when the diffusion rate tends to infinity. Moreover, we show that the direction of the Hopf bifurcation is forward, and the bifurcating periodic solution from the first Hopf bifurcation value is orbitally asymptotically stable, which improves the earlier result by Wei and Li (Nonlinear Anal 60(7):1351–1367, 2005).

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References

  1. 1.

    Busenberg, S., Huang, W.: Stability and Hopf bifurcation for a population delay model with diffusion effects. J. Differ. Equ. 124(1), 80–107 (1996)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Cantrell, R.S., Cosner, C.: Spatial Ecology via Reaction-Diffusion Equations. Wiley Series in Mathematical and Computational Biology. Wiley, Chichester (2003)

    Google Scholar 

  3. 3.

    Cantrell, R.S., Cosner, C., Hutson, V.: Ecological models, permanence and spatial heterogeneity. Rocky Mt. J. Math. 26(1), 1–35 (1996)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Chen, S., Lou, Y., Wei, J.: Hopf bifurcation in a delayed reaction-diffusion-advection population model. J. Differ. Equ. 264(8), 5333–5359 (2018)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Chen, S., Shen, Z., Wei, J.: Hopf bifurcation in a delayed single population model with patch structure. J. Dyn. Differ. Equ. (to appear) (2021)

  6. 6.

    Chen, S., Shi, J.: Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect. J. Differ. Equ. 253(12), 3440–3470 (2012)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Chen, S., Wei, J., Zhang, X.: Bifurcation analysis for a delayed diffusive logistic population model in the advective heterogeneous environment. J. Dyn. Differ. Equ. (2020) (to appear)

  8. 8.

    Chen, S., Yu, J.: Stability and bifurcations in a nonlocal delayed reaction-diffusion population model. J. Differ. Equ. 260(1), 218–240 (2016)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Faria, T.: Normal forms for semilinear functional differential equations in Banach spaces and applications. II. Discrete Contin. Dyn. Syst. 7(1), 155–176 (2001)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Faria, T., Huang, W.: Stability of periodic solutions arising from Hopf bifurcation for a reaction-diffusion equation with time delay. In: Differential Equations and Dynamical Systems (Lisbon, 2000), Volume 31 of Fields Institute Communications, pp. 125–141. American Mathematical Society, Providence (2002)

  11. 11.

    Faria, T., Huang, W., Wu, J.: Smoothness of center manifolds for maps and formal adjoints for semilinear FDEs in general Banach spaces. SIAM J. Math. Anal. 34(1), 173–203 (2002)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Feng, Q., Yan, J.: Global attractivity and oscillation in a kind of Nicholson’s blowflies. J. Biomath. 17(1), 21–26 (2002)

    MathSciNet  Google Scholar 

  13. 13.

    Gourley, S.A., Ruan, S.: Dynamics of the diffusive Nicholson’s blowflies equation with distributed delay. Proc. R. Soc. Edinb. Sect. A 130(6), 1275–1291 (2000)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Guo, S.: Stability and bifurcation in a reaction-diffusion model with nonlocal delay effect. J. Differ. Equ. 259(4), 1409–1448 (2015)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Guo, S.: Spatio-temporal patterns in a diffusive model with non-local delay effect. IMA J. Appl. Math. 82(4), 864–908 (2017)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Guo, S., Ma, L.: Stability and bifurcation in a delayed reaction-diffusion equation with Dirichlet boundary condition. J. Nonlinear Sci. 26(2), 545–580 (2016)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Guo, S., Yan, S.: Hopf bifurcation in a diffusive Lotka–Volterra type system with nonlocal delay effect. J. Differ. Equ. 260(1), 781–817 (2016)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Gurney, W., Blythe, S., Nisbet, R.M.: Nicholson’s blowflies revisited. Nature 287(5777), 17–21 (1980)

    Article  Google Scholar 

  19. 19.

    Győri, I., Trofimchuk, S.I.: On the existence of rapidly oscillatory solutions in the Nicholson blowflies equation. Nonlinear Anal. 48(7, Ser. A: Theory Methods), 1033–1042 (2002)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Hale, J.: Theory of Functional Differential Equations. Applied Mathematical Sciences, vol. 3, 2nd edn. Springer, New York (1977)

    Google Scholar 

  21. 21.

    Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation. London Mathematical Society Lecture Note Series, vol. 41. Cambridge University Press, Cambridge (1981)

    Google Scholar 

  22. 22.

    Hou, X., Duan, L., Huang, Z.: Permanence and periodic solutions for a class of delay Nicholson’s blowflies models. Appl. Math. Model. 37(3), 1537–1544 (2013)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Hu, R., Yuan, Y.: Spatially nonhomogeneous equilibrium in a reaction-diffusion system with distributed delay. J. Differ. Equ. 250(6), 2779–2806 (2011)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Li, W.T., Ruan, S., Wang, Z.C.: On the diffusive Nicholson’s blowflies equation with nonlocal delay. J. Nonlinear Sci. 17(6), 505–525 (2007)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Lin, C.K., Lin, C.T., Lin, Y., Mei, M.: Exponential stability of nonmonotone traveling waves for Nicholson’s blowflies equation. SIAM J. Math. Anal. 46(2), 1053–1084 (2014)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Liu, B.: Global exponential stability of positive periodic solutions for a delayed Nicholson’s blowflies model. J. Math. Anal. Appl. 412(1), 212–221 (2014)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Mei, M., So, J.W.-H., Li, M.Y., Shen, S.S.P.: Asymptotic stability of travelling waves for Nicholson’s blowflies equation with diffusion. Proc. R. Soc. Edinb. Sect. A 134(3), 579–594 (2004)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Shi, Q., Shi, J., Song, Y.: Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity. Discrete Contin. Dyn. Syst. Ser. B 24(2), 467–486 (2019)

    MathSciNet  MATH  Google Scholar 

  29. 29.

    Shi, Q., Song, Y.: Hopf bifurcation and chaos in a delayed Nicholson’s blowflies equation with nonlinear density-dependent mortality rate. Nonlinear Dyn. 84(2), 1021–1032 (2016)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Shu, H., Wang, L., Wu, J.: Global dynamics of Nicholson’s blowflies equation revisited: onset and termination of nonlinear oscillations. J. Differ. Equ. 255(9), 2565–2586 (2013)

    MathSciNet  Article  Google Scholar 

  31. 31.

    So, J.W.-H., Yang, Y.: Dirichlet problem for the diffusive Nicholson’s blowflies equation. J. Differ. Equ. 150(2), 317–348 (1998)

    MathSciNet  Article  Google Scholar 

  32. 32.

    So, J.W.-H., Yu, J.S.: Global attractivity and uniform persistence in Nicholson’s blowflies. Differ. Equ. Dyn. Syst. 2(1), 11–18 (1994)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    So, J.W.-H., Zou, X.: Traveling waves for the diffusive Nicholson’s blowflies equation. Appl. Math. Comput. 122(3), 385–392 (2001)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Su, Y., Wei, J., Shi, J.: Hopf bifurcations in a reaction-diffusion population model with delay effect. J. Differ. Equ. 247(4), 1156–1184 (2009)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Su, Y., Wei, J., Shi, J.: Bifurcation analysis in a delayed diffusive Nicholson’s blowflies equation. Nonlinear Anal. Real World Appl. 11(3), 1692–1703 (2010)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Su, Y., Wei, J., Shi, J.: Hopf bifurcation in a diffusive logistic equation with mixed delayed and instantaneous density dependence. J. Dyn. Differ. Equ. 24(4), 897–925 (2012)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Wei, J., Li, M.Y.: Hopf bifurcation analysis in a delayed Nicholson blowflies equation. Nonlinear Anal. 60(7), 1351–1367 (2005)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Wu, J.: Theory and Applications of Partial Functional-Differential Equations. Applied Mathematical Sciences, vol. 119. Springer, New York (1996)

    Google Scholar 

  39. 39.

    Yan, X.P., Li, W.T.: Stability of bifurcating periodic solutions in a delayed reaction-diffusion population model. Nonlinearity 23(6), 1413–1431 (2010)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Yang, Y., So, J.W.-H.: Dynamics for the Diffusive Nicholson’s Blowies Equation. Number Added, vol. II, pp. 333–352 (1998). Dynamical Systems and Differential Equations, vol. II. Springfield, MO (1996)

  41. 41.

    Yi, T., Zou, X.: Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: a non-monotone case. J. Differ. Equ. 245(11), 3376–3388 (2008)

    MathSciNet  Article  Google Scholar 

  42. 42.

    Zhang, J., Peng, Y.: Travelling waves of the diffusive Nicholson’s blowflies equation with strong generic delay kernel and non-local effect. Nonlinear Anal. 68(5), 1263–1270 (2008)

    MathSciNet  Article  Google Scholar 

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Correspondence to Shanshan Chen.

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This research is supported by the National Natural Science Foundation of China (No. 11771109) and Shandong Provincial Natural Science Foundation of China (No. ZR2020YQ01).

A Appendix

A Appendix

The proof of Proposition 3.3

Proof

It follows from Lemmas 2.43.2 and Theorem 2.6 that

$$\begin{aligned} \begin{aligned} \cos 2\theta _0&=\frac{1-c_0^2+2c_0}{(1-c_0)^2},\quad \sin 2\theta _0=-\frac{2{\sqrt{c_0^2 - 2{c_0}} }}{(1 - {c_0})^2},\quad h_0= {\bar{\delta }} \sqrt{c_0^2 - 2{c_0}},\\&\lim _{r \rightarrow 0}r \tau _n=\frac{\theta _0+2n\pi }{h_0},\quad \lim _{r \rightarrow 0} \nu _r \tau _n=\theta _0+2n\pi ,\\ \lim _{r\rightarrow 0 } {E_r}&= \frac{-e^{-2\mathrm{i}\theta _0 }{\overline{p}} f''(c_0)c_0^2}{e^{-2\mathrm{i}\theta _0 }{\overline{p}} f'(c_0)-{\overline{\delta }}-2\mathrm{i} h_0 },\quad \lim _{r\rightarrow 0 } {F_r} = \frac{-{\overline{p}} f''(c_0)c_0^2}{{\overline{p}} f'(c_0)-{\overline{\delta }}}. \end{aligned} \end{aligned}$$
(A1)

Since \(\lim _{r \rightarrow 0 } \psi (x)=\lim _{r \rightarrow 0 } {\overline{\psi }}(x)=c_0\), we see from Eq. (3.8) that

$$\begin{aligned} \begin{aligned} \lim _{r \rightarrow 0 } g_{20}&=\lim _{r \rightarrow 0 }\frac{ r \tau _{n}}{S_{n}(r)} e^{-2 \mathrm{i} \nu _r \tau _n} {\overline{p}} |\Omega | f''(c_0) c_0^3=\lim _{r \rightarrow 0 } g_{11} e^{-2 \mathrm{i} \nu _r \tau _n},\\ \lim _{r \rightarrow 0 } g_{11}&=\lim _{r \rightarrow 0 }\frac{ r \tau _{n}}{S_{n}(r)}{\overline{p}} |\Omega | f''(c_0) c_0^3, \\ \lim _{r \rightarrow 0 } g_{02}&=\lim _{r \rightarrow 0 } \frac{ r \tau _{n}}{S_{n}(r)} e^{2 \mathrm{i} \nu _r \tau _n}{\overline{p}} |\Omega | f''(c_0) c_0^3,\\ \end{aligned} \end{aligned}$$
(A2)

and

$$\begin{aligned} \begin{aligned} \lim _{r \rightarrow 0 } g_{21}&=\lim _{r \rightarrow 0 } \frac{2 r \tau _{n}}{S_{n}(r)} e^{- \mathrm{i} \nu _r \tau _n} {\overline{p}} |\Omega | f''(c_0) c_0^3 (-\frac{\mathrm{i} g_{11}}{\nu _{r} \tau _{n}} e^{- \mathrm{i} \nu _r \tau _n}+\frac{\mathrm{i} \overline{g}_{11}}{\nu _{r} \tau _{n}} e^{ \mathrm{i} \nu _r \tau _n}+\frac{{F_r}}{c_0})\\&\quad +\lim _{r \rightarrow 0 }\frac{r \tau _{n}}{S_{n}(r)} e^{\mathrm{i}\nu _r \tau _n} {\overline{p}} |\Omega | f''(c_0) c_0^3 (\frac{\mathrm{i} g_{20}}{\nu _{r} \tau _{n}} e^{- \mathrm{i} \nu _r \tau _n}+\frac{\mathrm{i} \overline{g}_{02}}{3 \nu _{r} \tau _{n}} e^{ \mathrm{i} \nu _r \tau _n}+\frac{{E_r}e^{-2 \mathrm{i} \nu _r \tau _n}}{c_0})\\&\quad +\lim _{r \rightarrow 0 }\frac{r \tau _{n}}{S_{n}(r)} e^{-\mathrm{i}\nu _r \tau _n} {\overline{p}} |\Omega | f''(c_0) c_0^3 \\&=\lim _{r \rightarrow 0 }\left\{ \frac{2}{\nu _{r} \tau _{n}}\left[ -\mathrm{i}g_{11}g_{20}+\mathrm{i}|g_{11}|^2\right] +g_{11}e^{-\mathrm{i}\nu _r \tau _n}\frac{2{F_r}}{c_0}\right\} \\&\quad +\lim _{r \rightarrow 0 }\left[ \frac{\mathrm{i}g_{11}g_{20}}{\nu _{r} \tau _{n}}+\frac{\mathrm{i}|g_{02}|^2}{3\nu _{r} \tau _{n}}+g_{11}e^{-\mathrm{i} \nu _r \tau _n}\frac{{E_r}}{c_0}\right] \\&\quad +\lim _{r \rightarrow 0 }\left[ -g_{11}e^{-\mathrm{i} \nu _r \tau _n}c_0+g_{11}e^{-\mathrm{i} \nu _r \tau _n}\frac{c_0}{c_0-2}\right] . \end{aligned} \end{aligned}$$

Therefore,

$$\begin{aligned} \begin{aligned} \lim _{r \rightarrow 0 } {\mathcal {R}}e g_{21}&=\lim _{r \rightarrow 0 } {\mathcal {R}}e \left[ \frac{-\mathrm{i}g_{11}g_{20}}{\nu _{r} \tau _{n}} +g_{11}e^{-\mathrm{i}\nu _r \tau _n}\left( \frac{{E_r}}{c_0}+\frac{2{F_r}}{c_0}+\frac{c_0}{c_0-2}-c_0\right) \right] . \end{aligned} \end{aligned}$$

This, combined with Eq. (3.24), implies that

$$\begin{aligned} \begin{aligned} \lim _{r \rightarrow 0}{\mathcal {R}}e C_{1}(0)&=\lim _{r \rightarrow 0}{\mathcal {R}}e \left[ \frac{\mathrm{i}}{2 \nu _r \tau _{n}}\left( g_{11} g_{20}-2\left| g_{11}\right| ^{2}-\frac{\left| g_{02}\right| ^{2}}{3}\right) +\frac{g_{21}}{2}\right] \\&=\lim _{r \rightarrow 0}{\mathcal {R}}e \left( \frac{\mathrm{i} g_{11} g_{20}}{2 \nu _r \tau _{n}}+\frac{g_{21}}{2} \right) \\&=\lim _{r \rightarrow 0}{\mathcal {R}}e \left[ \frac{\mathrm{i} g_{11} g_{20}}{2 \nu _r \tau _{n}}+ \frac{-\mathrm{i}g_{11}g_{20}}{2\nu _{r} \tau _{n}} +\frac{1}{2}g_{11}e^{-\mathrm{i}\nu _r \tau _n}\left( \frac{{E_r}}{c_0}+\frac{2{F_r}}{c_0}+\frac{c_0}{c_0-2}-c_0\right) \right] \\&=\frac{1}{2}\lim _{r \rightarrow 0} {\mathcal {R}}e\left[ g_{11}e^{-\mathrm{i}\nu _r \tau _n}\left( \frac{{E_r}}{c_0}+\frac{2{F_r}}{c_0}+\frac{c_0}{c_0-2}-c_0\right) \right] \\&=\frac{1}{2}\lim _{r \rightarrow 0}\left[ {\mathcal {R}}e (g_{11}e^{-\mathrm{i} \nu _r \tau _n})({\mathcal {R}}e\frac{{E_r}}{c_0}+\frac{2{F_r}}{c_0}+\frac{c_0}{c_0-2}-c_0)\right. \\&\quad \left. - {\mathcal {I}} m \left( g_{11}e^{-\mathrm{i} \nu _r \tau _n}\right) {\mathcal {I}}m\frac{{E_r}}{c_0} \right] . \end{aligned} \end{aligned}$$
(A3)

In order to analyze the sign of \(\lim _{r \rightarrow 0}{\mathcal {R}}e C_{1}(0)\), we only need to calculate the signs of \(\lim _{r \rightarrow 0}{\mathcal {R}}e (g_{11}e^{-\mathrm{i} \nu _r \tau _n})\), \(\lim _{r \rightarrow 0}({\mathcal {R}}e\frac{{E_r}}{c_0}+\frac{2{F_r}}{c_0}+\frac{c_0}{c_0-2}-c_0)\), \(\lim _{r \rightarrow 0} {\mathcal {I}} m \left( g_{11}e^{-\mathrm{i} \nu _r \tau _n}\right) \) and \(\lim _{r \rightarrow 0}{\mathcal {I}}m\frac{{E_r}}{c_0} \), respectively. From (2.18), we have

$$\begin{aligned} \lim _{r \rightarrow 0}S_n(r)=\lim _{r \rightarrow 0} (1+r \tau _n f'(c_0)e^{-\mathrm{i} \theta _0}{\overline{p}} )c_0^2 |\Omega |, \end{aligned}$$

then

$$\begin{aligned} \begin{aligned} \lim _{r \rightarrow 0} \frac{1}{S_n(r)}&=\lim _{r \rightarrow 0}\frac{1+r \tau _n f'(c_0)e^{\mathrm{i} \theta _0}{\overline{p}} }{(1+r \tau _n f'(c_0)e^{-\mathrm{i} \theta _0}{\overline{p}} )(1+r \tau _n f'(c_0)e^{\mathrm{i} \theta _0}{\overline{p}} )c_0^2 |\Omega |}\\&=\lim _{r \rightarrow 0} \frac{1+r \tau _n f'(c_0){\overline{p}} \cos \theta _0+\mathrm{i} r \tau _n f'(c_0){\overline{p}}\sin \theta _0}{\left[ 1+2r \tau _n f'(c_0){\overline{p}} \cos \theta _0+(r \tau _n f'(c_0){\overline{p}})^2\right] c_0^2 |\Omega |}. \end{aligned} \end{aligned}$$
(A4)

It follows from (A2) and (A4) that

$$\begin{aligned} \begin{aligned} \lim _{r \rightarrow 0} g_{11}&=\lim _{r \rightarrow 0 }\frac{ r \tau _{n}}{S_{n}(r)}{\overline{p}} |\Omega | f''(c_0) c_0^3 \\&=\lim _{r \rightarrow 0} \frac{ r \tau _{n}{\overline{p}} f''(c_0) c_0\left[ 1+r \tau _n f'(c_0){\overline{p}} \cos \theta _0+\mathrm{i} r \tau _n f'(c_0){\overline{p}}\sin \theta _0\right] }{[1+2r \tau _n f'(c_0){\overline{p}} \cos \theta _0+(r \tau _n f'(c_0){\overline{p}})^2]}. \end{aligned} \end{aligned}$$

Then, together with Eq. (A1), yields

$$\begin{aligned} \begin{aligned} \lim _{r \rightarrow 0} {\mathcal {R}}e \left( g_{11}e^{-\mathrm{i} \theta _0}\right)&=\lim _{r \rightarrow 0} \left\{ {\mathcal {R}}e g_{11} \cos \theta _0+ {\mathcal {I}} m g_{11} \sin \theta _0\right\} \\&=\lim _{r \rightarrow 0} \frac{ r \tau _{n}{\overline{p}} f''(c_0) c_0\left( 1+r \tau _n f'(c_0){\overline{p}} \cos \theta _0\right) \cos \theta _0}{[1+2r \tau _n f'(c_0){\overline{p}} \cos \theta _0+(r \tau _n f'(c_0){\overline{p}})^2]}\\&\quad +\lim _{r \rightarrow 0} \frac{ r \tau _{n}{\overline{p}} f''(c_0) c_0\left( r \tau _n f'(c_0){\overline{p}}\sin \theta _0\right) \sin \theta _0}{[1+2r \tau _n f'(c_0){\overline{p}} \cos \theta _0+(r \tau _n f'(c_0){\overline{p}})^2]}\\&=\lim _{r \rightarrow 0} \frac{r \tau _{n}{\overline{p}} f''(c_0) c_0 \cos \theta _0+ \left( r \tau _{n}{\overline{p}}\right) ^2 f'(c_0) f''(c_0) c_0}{[1+2r \tau _n f'(c_0){\overline{p}} \cos \theta _0+(r \tau _n f'(c_0){\overline{p}})^2]}\\&=\frac{(\theta _0+2n\pi ) \sqrt{c_0^2 - 2{c_0}}\frac{1}{1-c_0}+(\theta _0+2n\pi )^2(1-c_0)}{1+2\frac{(\theta _0+2n\pi )}{\sqrt{c_0^2 - 2{c_0}}}+\left[ \frac{(\theta _0+2n\pi )(1-c_0)}{\sqrt{c_0^2 - 2{c_0}}}\right] ^2}, \end{aligned} \end{aligned}$$
(A5)

and

$$\begin{aligned} \begin{aligned} \lim _{r \rightarrow 0} {\mathcal {I}}m \left( g_{11}e^{-\mathrm{i} \theta _0}\right)&=\lim _{r \rightarrow 0}\left\{ -{\mathcal {R}}e g_{11} \sin \theta _0+ {\mathcal {I}} m g_{11} \cos \theta _0\right\} \\&=-\lim _{r \rightarrow 0} \frac{ r \tau _{n}{\overline{p}} f''(c_0) c_0\left( 1+r \tau _n f'(c_0){\overline{p}} \cos \theta _0\right) \sin \theta _0}{[1+2r \tau _n f'(c_0){\overline{p}} \cos \theta _0+(r \tau _n f'(c_0){\overline{p}})^2]}\\&\quad +\lim _{r \rightarrow 0} \frac{ r \tau _{n}{\overline{p}} f''(c_0) c_0\left( r \tau _n f'(c_0){\overline{p}}\sin \theta _0\right) \cos \theta _0}{[1+2r \tau _n f'(c_0){\overline{p}} \cos \theta _0+(r \tau _n f'(c_0){\overline{p}})^2]}\\&=\lim _{r \rightarrow 0} \frac{- r \tau _{n}{\overline{p}} f''(c_0) c_0\sin \theta _0}{[1+2r \tau _n f'(c_0){\overline{p}} \cos \theta _0+(r \tau _n f'(c_0){\overline{p}})^2]}\\&=\frac{(\theta _0+2n\pi )(c_0 - 2)c_0\frac{1}{1-c_0}}{1+2\frac{(\theta _0+2n\pi )}{\sqrt{c_0^2 - 2{c_0}}}+\left[ \frac{(\theta _0+2n\pi )(1-c_0)}{\sqrt{c_0^2 - 2{c_0}}}\right] ^2}. \end{aligned} \end{aligned}$$
(A6)

Note that

$$\begin{aligned} \begin{aligned} \lim _{r\rightarrow 0 } \frac{{E_r}}{c_0}&= \frac{-e^{-2\mathrm{i}\theta _0 }(c_0-2)c_0}{e^{-2\mathrm{i}\theta _0 }(1-c_0)-1-2\mathrm{i} \sqrt{c_0^2 - 2{c_0}}}\\&=(-\cos 2\theta _0+\mathrm{i}\sin 2\theta _0) (c_0-2)c_0\\&\quad \times \frac{[(1-c_0)\cos 2\theta _0-1]+ \mathrm{i}[(1-c_0)\sin 2\theta _0+2\sqrt{c_0^2 - 2{c_0}}]}{((1-c_0)\cos 2\theta _0-1)^2+ ((1-c_0)\sin 2\theta _0+2\sqrt{c_0^2 - 2{c_0}})^2}. \end{aligned} \end{aligned}$$

Therefore, we obtain

(A7)

and

$$\begin{aligned} \begin{aligned} \lim _{r\rightarrow 0 } {\mathcal {I}}m \frac{{E_r}}{c_0}&=\frac{\left[ -\sin 2\theta _0-2\sqrt{c_0^2 - 2{c_0}}\cos 2\theta _0\right] (c_0-2)c_0}{((1-c_0)\cos 2\theta _0-1)^2+ ((1-c_0)\sin 2\theta _0+2\sqrt{c_0^2 - 2{c_0}})^2}\\&=\frac{\frac{2\sqrt{c_0^2 - 2{c_0}}}{(1-c_0)^2}(c_0^2-2c_0)^2}{((1-c_0)\cos 2\theta _0-1)^2+ ((1-c_0)\sin 2\theta _0+2\sqrt{c_0^2 - 2{c_0}})^2}\\&=\frac{2(c_0^2-2c_0)^{\frac{5}{2}}}{5c_0^4-14c_0^3+9c_0^2}. \end{aligned} \end{aligned}$$
(A8)

From (A1), one also have

$$\begin{aligned} \begin{aligned} \lim _{r\rightarrow 0 }\frac{2{F_r}}{c_0}&=\frac{-2{\overline{p}} f''(c_0)c_0}{{\overline{p}} f'(c_0)-{\overline{\delta }}}=\frac{-2(c_0-2)c_0}{-c_0}=2(c_0-2)>0. \end{aligned} \end{aligned}$$
(A9)

Then we see from (A7) and (A9) that

$$\begin{aligned} \begin{aligned}&\lim _{r \rightarrow 0}({\mathcal {R}}e\frac{{E_r}}{c_0}+\frac{2{F_r}}{c_0}+ \frac{c_0}{c_0-2}-c_0)\\&\quad =\frac{\left[ (c_0-1)+\frac{1+3c_0^2-6c_0}{(1-c_0)^2}\right] (c_0-2)c_0}{((1-c_0)\cos 2\theta _0-1)^2+ ((1-c_0)\sin 2\theta _0+2\sqrt{c_0^2 - 2{c_0}})^2}+(c_0-4)+\frac{c_0}{c_0-2}\\&\quad =\frac{(c_0^3-3c_0)(c_0-2)^2c_0+(c_0-4)(5c_0^4-14c_0^3+9c_0^2)(c_0-2)+c_0(5c_0^4-14c_0^3+9c_0^2)}{(5c_0^4-14c_0^3+9c_0^2)(c_0-2)}\\&\quad =\frac{(c_0^2-3)(c_0-2)^2c_0^2+(5c_0^4-14c_0^3+9c_0^2)(c_0^2-5c_0+8)}{(5c_0^4-14c_0^3+9c_0^2)(c_0-2)}>0. \end{aligned} \end{aligned}$$
(A10)

It follows from (A5), (A6), (A8) and (A10) that

$$\begin{aligned} \begin{aligned}&\lim _{r \rightarrow 0}{\mathcal {R}}e C_{1}(0)=\lim _{r \rightarrow 0} {\mathcal {R}}e\left[ g_{11}e^{-\mathrm{i}\nu _r \tau _n}\left( \frac{{E_r}}{c_0}+\frac{2{F_r}}{c_0}+\frac{c_0}{c_0-2}-c_0\right) \right] \\&=\lim _{r \rightarrow 0}\left[ {\mathcal {R}}e (g_{11}e^{-\mathrm{i} \nu _r \tau _n})({\mathcal {R}}e\frac{{E_r}}{c_0}+\frac{2{F_r}}{c_0}+\frac{c_0}{c_0-2}-c_0)- {\mathcal {I}} m \left( g_{11}e^{-\mathrm{i} \nu _r \tau _n}\right) {\mathcal {I}}m\frac{{E_r}}{c_0} \right] \\&=\frac{(\theta _0+2n\pi ) \sqrt{c_0^2 - 2{c_0}}\frac{1}{1-c_0}+(\theta _0+2n\pi )^2(1-c_0)}{1+2\frac{(\theta _0+2n\pi )}{\sqrt{c_0^2 - 2{c_0}}}+\left[ \frac{(\theta _0+2n\pi )(1-c_0)}{\sqrt{c_0^2 - 2{c_0}}}\right] ^2}\\&\quad \times \frac{(c_0^2-3)(c_0-2)^2c_0^2+(5c_0^4-14c_0^3+9c_0^2)(c_0^2-5c_0+8)}{(5c_0^4-14c_0^3+9c_0^2)(c_0-2)}\\&\quad -\frac{(\theta _0+2n\pi )(c_0 - 2)c_0\frac{1}{1-c_0}}{1+2\frac{(\theta _0+2n\pi )}{\sqrt{c_0^2 - 2{c_0}}}+\left[ \frac{(\theta _0+2n\pi )(1-c_0)}{\sqrt{c_0^2 - 2{c_0}}}\right] ^2} \times \frac{2(c_0^2-2c_0)^{\frac{5}{2}}(c_0-2)}{(5c_0^4-14c_0^3+9c_0^2)(c_0-2)}. \end{aligned} \end{aligned}$$
(A11)

For simplicity, we only calculate the numerator of (A11):

$$\begin{aligned} \begin{aligned}&\left[ (\theta _0+2n\pi ) \sqrt{c_0^2 - 2{c_0}}\frac{1}{1-c_0}+(\theta _0+2n\pi )^2(1-c_0)\right] \\&\quad \times \left[ (c_0^2-3)(c_0-2)^2c_0^2+(5c_0^4-14c_0^3+9c_0^2)(c_0^2-5c_0+8)\right] \\&\quad -(\theta _0+2n\pi )(c_0 - 2)c_0\frac{1}{1-c_0}\times 2(c_0^2-2c_0)^{\frac{5}{2}}(c_0-2)\\&=\frac{2(c_0^2-2c_0)^{\frac{7}{2}}(c_0-2)(\theta _0+2n\pi )}{c_0-1}\\&\quad -\left[ (c_0^2-3)(c_0-2)^2c_0^2+(5c_0^4-14c_0^3+9c_0^2)(c_0^2-5c_0+8)\right] \\&\quad \times \left[ (c_0-1)(\theta _0+2n\pi )^2+\frac{\sqrt{c_0^2 - 2{c_0}}(\theta _0+2n\pi )}{c_0-1}\right] \\&<\frac{2(c_0^2-2c_0)^{\frac{7}{2}}(c_0-2)(\theta _0+2n\pi )}{c_0-1}\\&\quad -\left[ (c_0^2-3)(c_0-2)^2c_0^2+(5c_0^4-14c_0^3+9c_0^2)(c_0^2-5c_0+8)\right] (c_0-1)(\theta _0+2n\pi )\\&\quad -\left[ (c_0^2-3)(c_0-2)^2c_0^2+(5c_0^4-14c_0^3+9c_0^2)(c_0^2-5c_0+8)\right] \frac{\sqrt{c_0^2 - 2{c_0}}(\theta _0+2n\pi )}{c_0-1}\\&=(\theta _0+2n\pi )\left\{ \frac{2(c_0^2-2c_0)^{\frac{7}{2}}(c_0-2)}{c_0-1}\right. \\&\quad \left. -\left[ (c_0^2-3)(c_0-2)^2c_0^2+(5c_0^4-14c_0^3+9c_0^2)(c_0^2-5c_0+8)\right] (c_0-1)\right\} \\&\quad -\left[ (c_0^2-3)(c_0-2)^2c_0^2+(5c_0^4-14c_0^3+9c_0^2)(c_0^2-5c_0+8)\right] \frac{\sqrt{c_0^2 - 2{c_0}}(\theta _0+2n\pi )}{c_0-1}. \end{aligned} \end{aligned}$$
(A12)

Let

$$\begin{aligned} \begin{aligned} A&=\frac{2(c_0^2-2c_0)^{\frac{7}{2}}(c_0-2)}{c_0-1}\\&\quad -\left[ (c_0^2-3)(c_0-2)^2c_0^2+(5c_0^4-14c_0^3+9c_0^2)(c_0^2-5c_0+8)\right] (c_0-1),\\ B&=-\left[ (c_0^2-3)(c_0-2)^2c_0^2+(5c_0^4-14c_0^3+9c_0^2)(c_0^2-5c_0+8)\right] . \end{aligned} \end{aligned}$$

Then, when \(c_0>2\), we have

$$\begin{aligned} \begin{aligned} A&\le \left\{ 2c_0^2(c_0-2)(c_0-1)^3(c_0-2)^2\right. \\&\quad \left. -c_0^2(c_0-1)^2[(c_0^2-3)(c_0-2)^2+(c_0-1)(2c_0-4)(c_0^2-5c_0+8)]\right\} \frac{1}{c_0-1}\\&=\frac{2c_0^2(c_0-2)(c_0-1)^2(c_0-1)(c_0-2)^2-c_0^2(c_0-1)^2(c_0-2)[3c_0^3-14c_0^2+23c_0-10]}{c_0-1}\\&=\frac{c_0^2(c_0-2)(c_0-1)^2[-c_0^3+4c_0^2-7c_0+2]}{c_0-1}\\&=\frac{c_0^2(c_0-2)(c_0-1)^2[-c_0(c_0-2)^2-3c_0+2]}{c_0-1}\\&<0, \end{aligned} \end{aligned}$$
(A13)

and

$$\begin{aligned} \begin{aligned} B=-\left\{ (c_0^2-3)(c_0-2)^2c_0^2+(c_0-1)(5c_0-9)c_0^2\left[ (c_0-\frac{5}{2})^2+\frac{7}{4}\right] \right\} <0. \end{aligned} \end{aligned}$$
(A14)

Summarizing the (A11), (A12), (A13) and (A14), we have \(\lim _{r \rightarrow 0}{\mathcal {R}}e C_{1}(0)<0\). \(\square \)

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Huang, D., Chen, S. The stability and Hopf bifurcation of the diffusive Nicholson’s blowflies model in spatially heterogeneous environment. Z. Angew. Math. Phys. 72, 41 (2021). https://doi.org/10.1007/s00033-021-01473-2

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Keywords

  • Hopf bifurcation
  • Delay
  • Diffusion
  • Heterogeneous environment

Mathematics Subject Classification

  • 35R10
  • 37G15
  • 37N25
  • 92D25