Abstract
A class of Lotka-Volterra mutualistic system with time delays of benefit and feedback delays is introduced. By analyzing the associated characteristic equation, the local stability of the positive equilibrium and existence of Hopf bifurcation are obtained under all possible combinations of two or three delays selecting from multiple delays. Not only explicit formulas to determine the properties of the Hopf bifurcation are shown by using the normal form method and center manifold theorem, but also the global continuation of Hopf bifurcation is investigated by applying a global Hopf bifurcation result due to Wu (1998). Numerical simulations are given to support the theoretical results.
Highlights
In recent years, population models have merited a great deal of attention due to their theoretical and practical significance since the pioneering theoretical works by Lotka [1] and Volterra [2]; see [3,4,5,6] and references therein
He and Gopalsamy [17] considered a Lotka-Volterra mutualistic system (4) with delay, in which all delays take the same value; Meng and Wei only considered a mutualistic system (5) with the same value of feedback delay for each population in [18] while we introduce feedback delays of two populations to grow of the species in system (6) and take time delay of each species benefit into account
By theory analysis and simulations, we find that time delays of the benefit τ2 and τ3 can not affect the stability of the system (6) when we only choose different values of τ2, τ3 with τ1 = τ4 = 0
Summary
Population models have merited a great deal of attention due to their theoretical and practical significance since the pioneering theoretical works by Lotka [1] and Volterra [2]; see [3,4,5,6] and references therein. Song et al [4] analyzed the stability of the interior positive equilibrium and the existence of local Hopf bifurcation for system (1) with τ11 = τ22 = 0 by taking the delay τ = τ12 + τ21 as the bifurcation parameter. Ẏ (t) = y (t) [−r2 + a21x (t) − a22y (t − τ)] , where τ denotes the feedback time delay of prey species to the growth of species itself, and for the predator They found that the unique positive equilibrium of system (3) will no longer be absolutely stable and the switches from stability to instability to stability disappear as the feedback time delay increases monotonously from zero, which had been obtained for system (3) by Song and Wei [12]. A brief discussion is given in the last section
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