Periodicity in a Nonlinear Predator-prey System with State Dependent Delays

Abstract

With the help of a continuation theorem based on Gaines and Mawhin’s coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of the following nonlinear state dependent delays predator-prey system

$$ \left\{ {\begin{array}{*{20}c} {\frac{{dN_{1} {\left( t \right)}}} {{dt}} = N_{1} {\left( t \right)}\left[ {b_{1} {\left( t \right)} - {\sum\limits_{i = 1}^n {a_{i} {\left( t \right)}{\left( {N_{1} {\left( {t - \tau _{i} {\left( {t,N_{1} {\left( t \right)},N_{2} {\left( t \right)}} \right)}} \right)}} \right)}^{{\alpha _{i} }} } }} \right.} \\ {\;\left. { - {\sum\limits_{j = 1}^m {c_{j} {\left( t \right)}{\left( {N_{2} {\left( {t - \sigma _{j} {\left( {t,N_{1} {\left( t \right)},N_{2} {\left( t \right)}} \right)}} \right)}} \right)}^{{\beta _{j} }} } }} \right]} \\ {\frac{{dN_{2} {\left( t \right)}}} {{dt}} = N_{2} {\left( t \right)}{\left[ { - b_{2} {\left( t \right)} + {\sum\limits_{i = 1}^n {d_{i} {\left( t \right)}{\left( {N_{1} {\left( {t - \rho _{i} {\left( {t,N_{1} {\left( t \right)},N_{2} {\left( t \right)}} \right)}} \right)}} \right)}^{{\gamma _{i} }} } }} \right]},} \\ \end{array} } \right. $$

, where a _i (t), c _j (t), d _i (t) are continuous positive periodic functions with periodic ω > 0, b ₁(t), b ₂(t) are continuous periodic functions with periodic ω and ${\int_o^\omega {b_{i} {\left( t \right)}dt > 0,\tau _{i} \sigma _{j} ,\rho _{i} {\left( {i = 1,2,...,n,j = 1,2,...,m} \right)}} }$ are positive constants.