Article Contents
Article Contents

# Global Hopf bifurcation of a population model with stage structure and strong Allee effect

• * Corresponding author: Junjie Wei
This research is supported by National Natural Science Foundation of China (No. 11371111)
• This paper is devoted to the study of a single-species population model with stage structure and strong Allee effect. By taking $τ$ as a bifurcation parameter, we study the Hopf bifurcation and global existence of periodic solutions using Wu's theory on global Hopf bifurcation for FDEs and the Bendixson criterion for higher dimensional ODEs proposed by Li and Muldowney. Some numerical simulations are presented to illustrate our analytic results using MATLAB and DDE-BIFTOOL. In addition, interesting phenomenon can be observed such as two kinds of bistability.

Mathematics Subject Classification: Primary: 34C23, 34D05; Secondary: 34K28.

 Citation:

• Figure 1.  The growth rate per capita $r(y)=\frac{y^{n-1}}{1+y^m}-0.15$ in system $\dot y=\frac{y^n}{1+y^m}-0.15y=r(y)y$

Figure 2.  Figures of $\bar{y}$ and equilibria $y_1$, $y_2$ with parameters given in (18)

Figure 3.  Graphs of $S_n(\tau)$ on $\left[0, \tau^{1}\right)$ with parameters given in (18)

Figure 4.  $y_1\approx0.6986$ is unstable, and sustained oscillation occurs when $\tau\in[0, \tau^0)$, where $0 < \tau=8 < \tau^0\approx22.2$, and the initial condition is $\varphi=0.8$ for $t\in[-\tau, 0]$

Figure 5.  $y_2$ is asymptotically stable when $\tau\in[0, \tau_0)\cup\left(\tau_1, \tau^0\right)$, and the initial condition is $\varphi=1.1$ for $t\in[-\tau, 0]$

Figure 6.  $y_2\approx1.0962$ is unstable, and sustained oscillation occurs when $\tau\in(\tau_0, \tau_1)$, where $2.8\approx\tau_0 < \tau=10 < \tau_1\approx14.8$, and the initial condition is $\varphi=1.1$ for $t\in[-\tau, 0]$

Figure 7.  $\tau$, h) plane, where $h=\sqrt{2}D-\alpha |b'| e^{-\delta\tau}$

Figure 8.  Hopf bifurcation branch on the ($\tau$, d) plane, where $d=\max y(t)-\min y(t)$

Figure 9.  Stability of equilibria $0, ~y_1, ~y_2$ and periodic solutions bifurcated from $y_2$

Table 1.  List of quantities of periodic solution bifurcating from $y_2$ under (18)

 $\delta$ $\text{Re}(c_1(0))$ $\mu_2$ $\beta_2$ $~~\tau_0\approx 2.8$ $>0$ $-37.3115 < 0$ $>0$ $< 0~~$ $~~\tau_1\approx14.6$ $< 0$ $-72.7255 < 0$ $< 0$ $< 0~~$
•  W. C. Allee , Animal aggregations: A study in general sociology, The Quarterly Review of Biology, 2 (1927) , 367-398.  doi: 10.1086/394281. E. Beretta  and  Y. Kuang , Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM Journal on Mathematical Analysis, 33 (2002) , 1144-1165.  doi: 10.1137/S0036141000376086. W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations Heath Boston, 1965. K. Engelborghs , T. Luzyanina  and  D. Roose , Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, ACM Transactions on Mathematical Software, 28 (2002) , 1-21.  doi: 10.1145/513001.513002. K. Engelborghs, T. Luzyanina and G. Samaey, DDE-BIFTOOL v. 2.00: a Matlab Package for Bifurcation Analysis of Delay Differential Equations, Ph. D thesis, Katholieke Universiteit Leuven, 2001. D. Fan , L. Hong  and  J. Wei , Hopf bifurcation analysis in synaptically coupled HR neurons with two time delays, Nonlinear Dynamics, 62 (2010) , 305-319.  doi: 10.1007/s11071-010-9718-2. J. Hale, Theory of Functional Differential Equations Springer-Verlag, New York, 1977. J. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University, Cambridge, 1981. J. Jiang and J. Shi, Bistability Dynamics in Some Structured Ecological Models, in: Spatial Ecology, in: Chapman and Hall/CRC Mathematical and Computational Biology, CRC press, Boca Raton, 2009. L. Junges  and  J. A. C. Gallas , Intricate routes to chaos in the {M}ackey-{G}lass delayed feedback system, Physics Letters A, 376 (2012) , 2109-2116.  doi: 10.1016/j.physleta.2012.05.022. M. Y. Li  and  J. S. Muldowney , On bendixson's criterion, Journal of Differential Equations, 106 (1993) , 27-39.  doi: 10.1006/jdeq.1993.1097. M. Y. Li  and  J. S. Muldowney , On R.A. Smith's autonomous convergence theorem, Journal of Mathematics, 25 (1995) , 365-379.  doi: 10.1216/rmjm/1181072289. M. C. Mackey  and  L. Glass , Oscillation and chaos in physiological control systems, Science, 197 (1977) , 287-289.  doi: 10.1126/science.267326. A. Y. Morozov , M. Banerjee  and  S. V. Petrovskii , Long-term transients and complex dynamics of a stage-structured population with time delay and the Allee effect, Journal of Theoretical Biology, 396 (2016) , 116-124.  doi: 10.1016/j.jtbi.2016.02.016. J. S. Muldowney , Compound matrices and ordinary differential equations, Rocky Mountain Journal of Mathematics, 20 (1990) , 857-872.  doi: 10.1216/rmjm/1181073047. Y. Qu  and  J. Wei , Bifurcation analysis in a time-delay model for prey-predator growth with stage-structure, Nonlinear Dynamics, 49 (2007) , 285-294.  doi: 10.1007/s11071-006-9133-x. Y. Qu , J. Wei  and  S. Ruan , Stability and bifurcation analysis in hematopoietic stem cell dynamics with multiple delays, Physica D: Nonlinear Phenomena, 239 (2010) , 2011-2024.  doi: 10.1016/j.physd.2010.07.013. S. Ruan  and  J. Wei , On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynamics of Continuous, Discrete and Impulsive Systems, 10 (2003) , 863-874. H. Shu , L. Wang  and  J. Wu , Global dynamics of Nicholson's blowflies equation revisited: Onset and termination of nonlinear oscillations, Journal of Differential Equations, 255 (2013) , 2565-2586.  doi: 10.1016/j.jde.2013.06.020. J. Wang , J. Shi  and  J. Wei , Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, Journal of Differential Equations, 251 (2011) , 1276-1304.  doi: 10.1016/j.jde.2011.03.004. J. Wei , Bifurcation analysis in a scalar delay differential equation, Nonlinearity, 20 (2007) , 2483-2498.  doi: 10.1088/0951-7715/20/11/002. J. Wei  and  D. Fan , Hopf bifurcation analysis in a Mackey-Glass system, International Journal of Bifurcation and Chaos, 17 (2007) , 2149-2157.  doi: 10.1142/S0218127407018282. J. Wu , Symmetric functional differential equations and neural networks with memory, Transactions of the American Mathematical Society, 350 (1998) , 4799-4838.  doi: 10.1090/S0002-9947-98-02083-2.

Figures(9)

Tables(1)