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On bifurcation of a stage-structured single-species model with harvest

Honghua Bin, Yuying Liu, Junjie Wei

TL;DR

We study a stage-structured Nicholson-type population model with constant harvest and maturation delay. Existence of positive equilibria is obtained via the Lambert $W$ function, and a harvest threshold $h^*$ is derived so that two equilibria exist for $h\in[0,h^*)$, coalescing at $h^*$, and none for $h>h^*$. Local stability is analyzed through the characteristic equation, and Hopf bifurcations are shown to occur as the delay $\tau$ varies, with the critical delays identified by a finite set $\tau_n\in J$. By applying center-manifold reduction and the Balázs–Rost normal-form theory, the direction of the Hopf bifurcations is determined (forward at the first, backward at the last) and the bifurcating periodic solutions are shown to be asymptotically stable for small harvest. Numerical simulations with representative parameter values corroborate the analytical predictions and illustrate harvest- and delay-induced oscillations.

Abstract

This paper investigates the dynamics of the Nicholson's blowffies equation with stage structure and harvest. By employing the property of Lambert W function, the existence of positive equilibria is obtained. With aid of the distribution of the eigenvalues in the characteristic equation, the local stability of the equilibria and the existence of Hopf bifurcation of the singlespecies model are obtained. Furthermore, by applying the results due to Balazs I., Rost G. (Internat. J. Bifur. Chaos 31(2021):2150071), when the harvest rate is sufffciently small, the direction of the Hopf bifurcations at the ffrst and last bifurcation values are forward and backward, respectively, and the bifurcating periodic solutions are all asymptotically stable. Finally, Numerical simulations are conducted to validate the theoretical conclusions. These results can be seen as the complement of the works of Shu et al. (J. Differential Equations 255 (2013) 2565).

On bifurcation of a stage-structured single-species model with harvest

TL;DR

We study a stage-structured Nicholson-type population model with constant harvest and maturation delay. Existence of positive equilibria is obtained via the Lambert function, and a harvest threshold is derived so that two equilibria exist for , coalescing at , and none for . Local stability is analyzed through the characteristic equation, and Hopf bifurcations are shown to occur as the delay varies, with the critical delays identified by a finite set . By applying center-manifold reduction and the Balázs–Rost normal-form theory, the direction of the Hopf bifurcations is determined (forward at the first, backward at the last) and the bifurcating periodic solutions are shown to be asymptotically stable for small harvest. Numerical simulations with representative parameter values corroborate the analytical predictions and illustrate harvest- and delay-induced oscillations.

Abstract

This paper investigates the dynamics of the Nicholson's blowffies equation with stage structure and harvest. By employing the property of Lambert W function, the existence of positive equilibria is obtained. With aid of the distribution of the eigenvalues in the characteristic equation, the local stability of the equilibria and the existence of Hopf bifurcation of the singlespecies model are obtained. Furthermore, by applying the results due to Balazs I., Rost G. (Internat. J. Bifur. Chaos 31(2021):2150071), when the harvest rate is sufffciently small, the direction of the Hopf bifurcations at the ffrst and last bifurcation values are forward and backward, respectively, and the bifurcating periodic solutions are all asymptotically stable. Finally, Numerical simulations are conducted to validate the theoretical conclusions. These results can be seen as the complement of the works of Shu et al. (J. Differential Equations 255 (2013) 2565).

Paper Structure

This paper contains 7 sections, 9 theorems, 51 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Existence of equilibrium
  3. Stability and Bifurcation Analysis
  4. Stability of equilibria and existence of Hopf bifurcation
  5. The direction and stability of the Hopf bifurcation
  6. Numerical simulations
  7. Conclusion

Key Result

Theorem 1

If $(H_1)$ is satisfied, then there exists $h^*>0$ such that (i) two positive equilibria exist in system (m) for $h<h^*$, (ii) a unique positive equilibrium exists in system (m) for $h=h^*$, (iii) no positive equilibrium exists in system (m) for $h>h^*$, where and $W(\cdot)$ is Lambert W function.

Figures (5)

  • Figure 1: $(a)$ Two positive equilibria in system (\ref{['m']}), $(b)$ a unique positive equilibrium in system (\ref{['m']}), $(c)$ no positive equilibrium in system (\ref{['m']}).
  • Figure 2: The diagram of equilibria in system (\ref{['m']}) for $h>0$.
  • Figure 3: (a) The equilibria of system (\ref{['m']}) with the varying of $\tau$. (b) The function curve of $I(\tau)$ on $(0,14.81)$.
  • Figure 4: The graphs of $S_m(\tau)$ on $\mathcal{I}$.
  • Figure 5: The solution $M(t)$ of system (\ref{['m']}) with different values of $\tau$. (a) $\tau=4$, (b) $\tau=4.7$, (c) $\tau=11$ and (d) $\tau=11.8$. The initial value is $M_0(t)=10+\cos(2\pi t/\tau)$.

Theorems & Definitions (25)

  • Claim 1
  • proof
  • Claim 2
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 15 more