Global Hopf bifurcation of a population model with stage structure and strong Allee effect
Abstract: 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 ca… Show more
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Cited by 4 publications
References 19 publications
Stable Periodic Orbits for Delay Differential Equations with Unimodal Feedback
We consider delay differential equations of the form $$ y^{\prime }(t)=-ay(t)+bf(y(t-1)) $$ y ′ ( t ) = - a y ( t ) + b f ( y ( t - 1 ) ) with positive parameters a, b and a unimodal $$f:[0,\infty )\rightarrow [0,1]$$ f : [ 0 , ∞ ) → [ 0 , 1 ] . It is assumed that the nonlinear f is close to a function $$g:[0,\infty )\rightarrow [0,1]$$ g : [ 0 , ∞ ) → [ 0 , 1 ] with $$g(\xi )=0$$ g ( ξ ) = 0 for all $$\xi >1$$ ξ > 1 . The fact $$g(\xi )=0$$ g ( ξ ) = 0 for all $$\xi >1$$ ξ > 1 allows to construct stable periodic orbits for the equation $$x^{\prime }(t)=-cx(t)+dg(x(t-1))$$ x ′ ( t ) = - c x ( t ) + d g ( x ( t - 1 ) ) with some parameters $$d>c>0$$ d > c > 0 . Then it is shown that the equation $$ y^{\prime }(t)=-ay(t)+bf(y(t-1)) $$ y ′ ( t ) = - a y ( t ) + b f ( y ( t - 1 ) ) also has a stable periodic orbit provided a, b, f are sufficiently close to c, d, g in a certain sense. The examples include $$f(\xi )=\frac{\xi ^k}{1+\xi ^n}$$ f ( ξ ) = ξ k 1 + ξ n for parameters $$k>0$$ k > 0 and $$n>0$$ n > 0 together with the discontinuous $$g(\xi )=\xi ^k$$ g ( ξ ) = ξ k for $$\xi \in [0,1)$$ ξ ∈ [ 0 , 1 ) , and $$g(\xi )=0$$ g ( ξ ) = 0 for $$\xi >1$$ ξ > 1 . The case $$k=1$$ k = 1 is the famous Mackey–Glass equation, the case $$k>1$$ k > 1 appears in population models with Allee effect, and the case $$k\in (0,1)$$ k ∈ ( 0 , 1 ) arises in some economic growth models. The obtained stable periodic orbits may have complicated structures.
Stable Periodic Orbits for Delay Differential Equations with Unimodal Feedback
We consider delay differential equations of the form $$ y^{\prime }(t)=-ay(t)+bf(y(t-1)) $$ y ′ ( t ) = - a y ( t ) + b f ( y ( t - 1 ) ) with positive parameters a, b and a unimodal $$f:[0,\infty )\rightarrow [0,1]$$ f : [ 0 , ∞ ) → [ 0 , 1 ] . It is assumed that the nonlinear f is close to a function $$g:[0,\infty )\rightarrow [0,1]$$ g : [ 0 , ∞ ) → [ 0 , 1 ] with $$g(\xi )=0$$ g ( ξ ) = 0 for all $$\xi >1$$ ξ > 1 . The fact $$g(\xi )=0$$ g ( ξ ) = 0 for all $$\xi >1$$ ξ > 1 allows to construct stable periodic orbits for the equation $$x^{\prime }(t)=-cx(t)+dg(x(t-1))$$ x ′ ( t ) = - c x ( t ) + d g ( x ( t - 1 ) ) with some parameters $$d>c>0$$ d > c > 0 . Then it is shown that the equation $$ y^{\prime }(t)=-ay(t)+bf(y(t-1)) $$ y ′ ( t ) = - a y ( t ) + b f ( y ( t - 1 ) ) also has a stable periodic orbit provided a, b, f are sufficiently close to c, d, g in a certain sense. The examples include $$f(\xi )=\frac{\xi ^k}{1+\xi ^n}$$ f ( ξ ) = ξ k 1 + ξ n for parameters $$k>0$$ k > 0 and $$n>0$$ n > 0 together with the discontinuous $$g(\xi )=\xi ^k$$ g ( ξ ) = ξ k for $$\xi \in [0,1)$$ ξ ∈ [ 0 , 1 ) , and $$g(\xi )=0$$ g ( ξ ) = 0 for $$\xi >1$$ ξ > 1 . The case $$k=1$$ k = 1 is the famous Mackey–Glass equation, the case $$k>1$$ k > 1 appears in population models with Allee effect, and the case $$k\in (0,1)$$ k ∈ ( 0 , 1 ) arises in some economic growth models. The obtained stable periodic orbits may have complicated structures.
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