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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

x ˙ 1 (t)=x 1 (t)[r 1 -a 11 x 1 (t)-a 13 x 3 (t-τ 3 )],x ˙ 2 (t)=x 2 (t)[r 2 -a 21 x 1 (t-τ 1 )-a 22 x 2 (t)],x ˙ 3 (t)=x 3 (t)[r 3 -a 32 x 2 (t-τ 2 )](1)

with initial conditions

x i (t)=ϕ i (t)0,t[-τ,0),ϕ i (0)>0,i=1,2,3,

here τ=τ 1 +τ 2 +τ 3 . In system (1), x i (t) represents the density of the ith species at time t, respectively, i=1,2,3; τ 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 ith species and r i /a ii is the carrying capacity of the ith species, a 13 ,a 21 and a 32 are competition coefficients.

By choosing τ as a bifurcation parameter, it is shown that system (1) undergoes a Hopf bifurcation at the positive equilibrium as τ 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:
34K60Qualitative investigation and simulation of models
34K18Bifurcation theory of functional differential equations
34K13Periodic solutions of functional differential equations
92D25Population dynamics (general)
34K20Stability theory of functional-differential equations
34K17Transformation and reduction of functional-differential equations and systems; normal forms
34K19Invariant manifolds (functional-differential equations)
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