Acta Mathematicae Applicatae Sinica, English Series

, Volume 19, Issue 3, pp 491–498

# Global Existence of Positive Periodic Solutions for a Distributed Delay Competition Model

Original papers

## Abstract

By using the continuation theorem of Mawhin’s coincidence degree theory, a sufficient condition is derived for the existence of positive periodic solutions for a distributed delay competition model
\left\{ \begin{aligned} & {u}'{\left( t \right)}{\kern 1pt} = {\kern 1pt} u{\left( t \right)}{\left[ {r_{1} {\left( t \right)} - a_{1} {\left( t \right)}u{\left( t \right)} - b_{1} {\left( t \right)}{\int_{ - T}^0 {L_{1} {\left( s \right)}u{\left( {t + s} \right)}ds - c_{1} {\left( t \right)}{\int_{ - T}^0 {K_{1} {\left( s \right)}v{\left( {t + s} \right)}ds} }} }} \right]}, \\ & {v}'{\left( t \right)} = v{\left( t \right)}{\left[ {r_{2} {\left( t \right)} - a_{2} {\left( t \right)}v{\left( t \right)} - b_{2} {\left( t \right)}{\int_{ - T}^0 {L_{2} {\left( s \right)}v{\left( {t + s} \right)}ds - c_{2} {\left( t \right)}{\int_{ - T}^0 {K_{2} {\left( s \right)}u{\left( {t + s} \right)}ds} }} }} \right]} \\ \end{aligned} \right.
, where r1 and r2 are continuous ω-periodic functions in R+ = [0,∞) with $${\int_0^\omega {r_{i} {\left( t \right)}dt > 0,\;a_{i} ,\;c_{i} {\left( {i = 1,2} \right)}} }$$

are positive continuous ω-periodic functions in R+ = [0,∞), b i (i = 1, 2) is nonnegative continuous ω-periodic function in R+ = [0,∞), ω and T are positive constants, $$K_{i} ,\;L_{i} \in C{\left( {{\left[ { - T,0} \right]},\;(0,\infty )} \right)}$$ and $${\int_{ - T}^0 {K_{i} {\left( s \right)}ds = 1,{\kern 1pt} {\int_{ - T}^0 {L_{i} {\left( s \right)}ds = 1,{\kern 1pt} \;\;i = 1,2} }} }$$ . Some known results are improved and extended.

### Keywords

Global existence positive periodic solution coincidence degree distributed delay model

34K13 34K20

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