Summary: Based on the generalized logistic model, we propose and study the global attractivity of the following generalized Lotka-Volterra competition model with distributed delays
\[ \begin{aligned} \dot{x}_1(t) &= r_1 x_1(t) \left (1-\left (\frac{x_1(t)}{K_1}\right )^{\theta_1} - \alpha_{12} \int_{-\infty}^{t} \frac{x_2(s)}{K_1}\,dH_2(t-s) \right ), \\ \dot{x}_2(t) &= r_2 x_2(t)\left (1-\alpha_{21} \int_{-\infty}^{t} \frac{x_1(s)}{K_2} \,dH_1(t-s)-\left (\frac{x_2(t)}{K_2} \right )^{\theta_2} \right), \end{aligned} \]
where all the constants are positive and \(H_i:[0,\infty) \rightarrow \mathbb R\) is a non-increasing function of bounded variation with \(\int_{0}^{\infty}\,dH_i(s) = -1\) \((i = 1, 2)\). Suppose that \(K_{1} > \alpha_{12} K_{2}\) and \(K_{2} > \alpha_{21} K_{1}\). First, we completely describe the structure of equilibria. The system has at least one positive equilibrium. Roughly speaking, the system has a unique positive equilibrium if \(\max \{ \theta_{1} , \theta_{2} \} \geq 1\) and, otherwise, it has at most three positive equilibria. Then, using the iterative method, we show that the system is globally attractive if it has a unique positive equilibrium.