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

Periodicity in a Nonlinear Predator-prey System with State Dependent Delays 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

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

, Volume 21 (1) – Jan 1, 2005

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

Acta Mathematicae Applicatae Sinica, English Series Vol. 21, No. 1 (2005) 49–60 Periodicity in a Nonlinear Predator-prey System with State Dependent Delays Feng-de Chen, Jin-lin Shi School of Mathematics and Computer, Fuzhou University, Fuzhou 350002, China (E-mail: fdchen@fzu.edu.cn; fdchen@263.net) 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 dN (t) α ⎪ 1 i = N (t) b (t) − a (t) N (t − τ (t, N (t),N (t))) 1 1 i 1 i 1 2 dt i=1 − c (t) N t − σ (t, N (t),N (t)) , j 2 j 1 2 j=1 dN (t) γ ⎪ 2 i = N (t) − b (t)+ d (t) N (t − ρ (t, N (t),N (t))) , ⎩ 2 2 i 1 i 1 2 dt i=1 where a (t),c (t),d (t) are continuous positive periodic functions with periodic ω> 0, b (t),b (t) are continuous i j i 1 2 periodic functions with periodic ω and b (t)dt > 0.τ ,σ ,ρ (i =1, 2,·· · ,n, j =1, 2,··· ,m) are continuous i i j i and ω-periodic with respect to their ﬁrst arguments, respectively. α ,β ,γ (i =1, 2,··· ,n, j =1, 2,·· · ,m) i j i are positive constants. Keywords periodic solutions; nonlinear; delay; predator-prey; coincidence degree 2000 MR Subject Classiﬁcation 34K15; 34C25; 92D25 1 Introduction [13] [17] As was pointed out by Freedman and Wu and Kuang , it would be of interest to study the global existence of periodic solutions for systems with periodic delays, representing predator- prey or competition systems. Also, at present, there are only a few papers dealing with the existence of periodic solutions of state dependent delay diﬀerential equation (see for instance [10] [1–13,15–20] and the references therein). G. H. Fan...

/lp/springer_journal/periodicity-in-a-nonlinear-predator-prey-system-with-state-dependent-fw87FizO0x
Publisher
Springer-Verlag
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/s10255-005-0214-2
Publisher site
See Article on Publisher Site

### 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.

### Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Jan 1, 2005

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