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

Global Existence of Positive Periodic Solutions for a Distributed Delay Competition Model 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 r 1 and r 2 are continuous ω-periodic functions in R + = [0,∞) with $$ {\int_0^\omega {r_{i} {\left( t \right)}dt > 0,\;a_{i} ,\;c_{i} {\left( {i = 1,2} \right)}} } $$ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

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

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

Acta Mathematicae Applicatae Sinica, English Series Vol. 19, No. 3 (2003) 491–498 Global Existence of Positive Periodic Solutions for a Distributed Delay Competition Model 1,2 3 Xian-yi Li ,De-ming Zhu 1,3 Department of Mathematics, East China Normal University, Shanghai 200062, China Department of Mathematics and Physics, Nanhua University, Hengyang 421001, China ( E-mail: dmzhu@math.ecnu.edu.cn) 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 0 0 u (t)= u(t) r (t) − a (t)u(t) − b (t) L (s)u(t + s)ds − c (t) K (s)v(t + s)ds , 1 1 1 1 1 1 −T −T 0 0 v (t)= v(t) r (t) − a (t)v(t) − b (t) L (s)v(t + s)ds − c (t) K (s)u(t + s)ds , 2 2 2 2 2 2 −T −T where r and r are continuous ω-periodic functions in R =[0,∞)with r (t)dt > 0, a ,c (i =1, 2) are 1 2 + i i i positive continuous ω-periodic functions in R =[0,∞),b (i =1, 2) is nonnegative continuous ω-periodic + i function in R =[0,∞), ω and T are positive constants, K ,L ∈ C([−T, 0], (0,∞)) and K (s)ds =1, + i i i −T L (s)ds =1,i =1, 2. Some known results are improved and extended. −T Keywords Global existence, positive periodic solution, coincidence degree, distributed delay model 2000 MR Subject Classification 34K13, 34K20 1 Introduction Consider the following competition model  u (t)= u(t) r − a u(t) − c K (s)v(t + s)ds , 1 1 1 1 −T (1)  v (t)= v(t) r − a v(t) − c K (s)u(t + s)ds , 2 2 2 2 −T where r ,a ,c (i=1, 2) and T are positive constants, K ∈ C [−T, 0], (0,∞) and K (s)ds i i i i i −T [2] =1, i =1, 2. Gopalsamy studied the stability of the equilibrium of system (1). For the ecological significance of system (1), one can refer to [2, 3] and the...
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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2003 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
D.O.I.
10.1007/s10255-003-0125-z
Publisher site
See Article on Publisher Site

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 r 1 and r 2 are continuous ω-periodic functions in R + = [0,∞) with $$ {\int_0^\omega {r_{i} {\left( t \right)}dt > 0,\;a_{i} ,\;c_{i} {\left( {i = 1,2} \right)}} } $$

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Mar 3, 2017

References

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