Global Positive Periodic Solutions of Generalized n -Species Competition Systems with Multiple Delays and Impulses

Abstract

By applying the fixed point theorem in a cone of Banach space, we obtain an easily verifiable necessary and sufficient condition for the existence of positive periodic solutions of two kinds of generalized $n$-species competition systems with multiple delays and impulses as follows: $x_i^{\prime }(t)=x_i(t)[a_i(t)-b_i(t)x_i(t)-{\sum }_{j=\mathrm{1}}^n{c}_{ij}(t)x_i(t-{\tau }_{ij}(t))-{\sum }_{j=\mathrm{1}}^n{d}_{ij}(t)x_j(t-{\gamma }_{ij}(t))-{\sum }_{j=\mathrm{1}}^n{e}_{ij}(t){\int }_{-{\sigma }_{ij}}^{\mathrm{0}}\mathrm{‍}{f}_{ij}(s)x_j(t+s)ds],\text{\hspace{0.17em}a}\text{.}\text{e}\text{.},\mathrm{\hspace{0.17em}t}>\mathrm{0},t\ne t_k,\mathrm{\hspace{0.17em}k}\in Z_{+},\mathrm{\hspace{0.17em}i}=\mathrm{1,2},\dots ,n;{\mathrm{\hspace{0.17em}x}}_i(t_k^{+})-x_i(t_k^{-})={\theta }_{ik}{x}_i(t_k),\mathrm{\hspace{0.17em}i}=\mathrm{1,2},\dots ,n,\mathrm{\hspace{0.17em}k}\in Z_{+};\text{and}{x}_i^{\prime }(t)=x_i(t)[a_i(t)-b_i(t)x_i(t)+{\sum }_{j=\mathrm{1}}^n{c}_{ij}(t)x_i(t-{\tau }_{ij}(t))-{\sum }_{j=\mathrm{1}}^n{d}_{ij}(t)x_j(t-{\gamma }_{ij}(t))-{\sum }_{j=\mathrm{1}}^n{e}_{ij}(t){\int }_{-{\sigma }_{ij}}^{\mathrm{0}}\mathrm{‍}{f}_{ij}(s)x_j(t+s)ds],\text{\hspace{0.17em}a}\text{.}\text{e}\text{.},\mathrm{\hspace{0.17em}t}>\mathrm{0},\mathrm{\hspace{0.17em}t}\ne t_k,\mathrm{\hspace{0.17em}k}\in Z_{+},\mathrm{\hspace{0.17em}i}=\mathrm{1,2},\dots ,n;\mathrm{}{\mathrm{\hspace{0.17em}x}}_i(t_k^{+})-x_i(t_k^{-})={\theta }_{ik}{x}_i(t_k),\mathrm{\hspace{0.17em}i}=\mathrm{1,2},\dots ,n,\mathrm{\hspace{0.17em}k}\in Z_{+}.$ It improves and generalizes a series of the well-known sufficiency theorems in the literature about the problems mentioned previously.