Periodic Solution of a Nonautonomous Diffusive Food Chain System of Three Species with Time Delays

Abstract

By using the continuation theorem of coincidence degree theory, the existence of a positive periodic solution for a nonautonomous diffusive food chain system of three species. $$ \left\{ \begin{aligned} & \frac{{dx_{1} {\left( t \right)}}} {{dt}} = x_{1} {\left( t \right)}{\left[ {r_{1} {\left( t \right)} - a_{{11}} {\left( t \right)}x_{1} {\left( t \right)} - a_{{12}} {\left( t \right)}x_{2} {\left( t \right)}} \right]} + D_{1} {\left( t \right)}{\left[ {y{\left( t \right)} - x_{1} {\left( t \right)}} \right]}, \\ & \frac{{dx_{2} {\left( t \right)}}} {{dt}} = x_{2} {\left( t \right)}{\left[ { - r_{2} {\left( t \right)} + a_{{21}} {\left( t \right)}x_{1} {\left( {t - \tau _{1} } \right)} - a_{{22}} {\left( t \right)}x_{2} {\left( t \right)} - a_{{23}} {\left( t \right)}x_{3} {\left( t \right)}} \right]}, \\ & \frac{{dx_{3} {\left( t \right)}}} {{dt}} = x_{3} {\left( t \right)}{\left[ { - r_{3} {\left( t \right)} + a_{{32}} {\left( t \right)}x_{2} {\left( {t - \tau _{2} } \right)} - a_{{33}} {\left( t \right)}x_{3} {\left( t \right)}} \right]}, \\ & \frac{{dy{\left( t \right)}}} {{dt}} = y{\left( t \right)}{\left[ {r_{4} {\left( t \right)} - a_{{44}} {\left( t \right)}y{\left( t \right)}} \right]} + D_{2} {\left( t \right)}{\left[ {x_{1} {\left( t \right)} - y{\left( t \right)}} \right]}, \\ \end{aligned} \right. $$ is established, where r i (t), a ii (t) (i = 1, 2, 3, 4), D i (t) (i = 1, 2), a 12(t), a 21(t), a 23(t) and a 32(t) are all positive periodic continuous functions with period w > 0, τ i (i = 1, 2) are positive constants.