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Dynamics of a competitive Lotka-Volterra system with three delays. (English) Zbl 1238.34148
The authors study the following three-species Lotka-Volterra type competition system with three discrete time delays \aligned \dot x_1(t)=&x_1(t)[r_1-a_{11}x_1(t)-a_{13}x_3(t-\tau_3)],\\ \dot x_2(t)=&x_2(t)[r_2-a_{21}x_1(t-\tau_1)-a_{22}x_2(t)],\\ \dot x_3(t)=&x_3(t)[r_3-a_{32}x_2(t-\tau_2)] \endaligned\tag1 with initial conditions $$x_i(t)=\phi_i(t)\geq 0, \ t\in [-\tau, 0), \ \phi_i(0)>0, \ i=1,2,3,$$ here $\tau=\tau_1+\tau_2+\tau_3$. In system (1), $x_i(t)$ represents the density of the $i$th species at time $t$, respectively, $i=1,2,3$; $\tau_i$ is the feedback time delay of the species $x_i (i=1,2,3)$ to the growth of the species itself; $r_i$ is the intrinsic growth rate of the $i$th species and $r_i/a_{ii}$ is the carrying capacity of the $i$th species, $a_{13}, a_{21}$ and $a_{32}$ are competition coefficients. By choosing $\tau$ as a bifurcation parameter, it is shown that system (1) undergoes a Hopf bifurcation at the positive equilibrium as $\tau$ crosses some critical values. The formulae determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained by using the normal form theory and center manifold theorem.

##### MSC:
 34K60 Qualitative investigation and simulation of models 34K18 Bifurcation theory of functional differential equations 34K13 Periodic solutions of functional differential equations 92D25 Population dynamics (general) 34K20 Stability theory of functional-differential equations 34K17 Transformation and reduction of functional-differential equations and systems; normal forms 34K19 Invariant manifolds (functional-differential equations)
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