Stability and Hopf bifurcation analysis for a two-enterprise interaction model with delays

New results on competition and cooperation model of two enterprises with multiple delays and feedback controls

“…The research reveals that under fairish conditions, the competition of two species can remain a periodic vibration. The derived results are new and complement the earlier publications (for example, [1][2][3][4][5][6][7][8][9]). In recent years, there have been rare reports on the competition and cooperation model of two enterprises with stochastic perturbation, which might be our future investigation topic.…”

“…where υ i (i = 1, 2) is the time delay in the interior of enterprises and among different enterprises. Li et al [7] considered the stability and bifurcation behavior of the following two-enterprise interaction model with four different delays:…”

Dynamical analysis of a competition and cooperation system with multiple delays

“…As with the calculation of the ODE Hopf bifurcation parameter and as in [28], according to the analysis above and the expressions of g 20 , g 11 , g 02 and g 21 , we can compute the following values: 24) where λ(τ ) = α(τ ) ± iω(τ ) is the characteristic root of (2.3), which is a continuous differentiable family. α ( τ ) and ω ( τ ) can be obtained by taking the derivative of the two sides of (2.3) and taking values at τ .…”

“…Liao [23] assumed τ i (i = 1, 2, 3) = τ and Li [24] considered τ 1 = 0, regarding τ and τ 2 + τ 3 as the bifurcation parameters, respectively. They investigated the existence of the unique positive equilibrium and proved that the Hopf bifurcation can occur as the bifurcation parameter crosses some critical value, and studied the direction of Hopf bifurcation and stability of the periodic solutions.…”

“…In recent decades, scientists have focused on the stability and bifurcation phenomena of the continuous-time autonomous predator-prey system with multiple delays (see, for example, [17][18][19][20][21][22]). In fact, there is a strong relationship between how species co-evolve in nature and how different enterprises co-exist in societal economics, leading to significant research into the delayed competition and cooperation model for business enterprises [23][24][25], which are governed by the following system of ODEs: ⎧ ⎨ ⎩ẋ 1 (t) = r 1 x 1 (t)(1 -…”