Permanence for two-species Lotka-Volterra systems with delays
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Department of Mathematics, Sichuan Normal University, Chengdu 610068
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Department of Mathematics, Wenzhou University, Wenzhou, 325035
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Received:
01 January 2005
Accepted:
29 June 2018
Published:
01 November 2005
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MSC :
92D30.
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The permanence of the following Lotka-Volterra system with time delays
$\dot{x}_ 1(t) = x_1(t)[r_1 - a_1x_1(t) + a_11x_1(t - \tau_11) + a_12x_2(t - \tau_12)]$,
$\dot{x}_ 2(t) = x_2(t)[r_2 - a_2x_2(t) + a_21x_1(t - \tau_21) + a_22x_2(t - \tau_22)]$,
is considered. With intraspecific competition, it is proved that in competitive case, the system is permanent if and only if the interaction matrix of the system satisfies condition (C1) and in cooperative case it is proved that condition (C2) is sufficient for the permanence of the system.
Citation: Suqing Lin, Zhengyi Lu. Permanence for two-species Lotka-Volterra systems with delays[J]. Mathematical Biosciences and Engineering, 2006, 3(1): 137-144. doi: 10.3934/mbe.2006.3.137
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Abstract
The permanence of the following Lotka-Volterra system with time delays
$\dot{x}_ 1(t) = x_1(t)[r_1 - a_1x_1(t) + a_11x_1(t - \tau_11) + a_12x_2(t - \tau_12)]$,
$\dot{x}_ 2(t) = x_2(t)[r_2 - a_2x_2(t) + a_21x_1(t - \tau_21) + a_22x_2(t - \tau_22)]$,
is considered. With intraspecific competition, it is proved that in competitive case, the system is permanent if and only if the interaction matrix of the system satisfies condition (C1) and in cooperative case it is proved that condition (C2) is sufficient for the permanence of the system.
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