Abstract
The qualitative properties of a nonautonomous competitive Lotka-Volterra system with infinite delays are studied.By using a result of matrix theory and the fluctuation lemma, we establish a series of easily verifiable algebraic conditions on the coefficients and the kernel, which are sufficient to ensure the survival and the extinction of a determined number of species. The surviving part is stabilized around a globally stable critical point of a subsystem of the system under study. These conditions also guarantee the asymptotic behavior of the system.
Highlights
This article deals with nonautonomous competitive systems of integro-differential equations with infinite delays
This work is included within the area of mathematical biology, in population dynamics, for we study non-autonomous competitive LotkaVolterra systems of n-species with infinite delays
X∗ = col(x∗1, . . . , x∗r) is a positive equilibrium point that attracts all solutions of the system (3) with initial condition (4)
Summary
This article deals with nonautonomous competitive systems of integro-differential equations with infinite delays,. This work is included within the area of mathematical biology, in population dynamics, for we study non-autonomous competitive LotkaVolterra systems of n-species with infinite delays. These models with delay are called hereditary models with memory. The system (1) describes the competition between n species, where xi(t) denotes the population density of the i-th species at time t It is well known, by the fundamental theory of systems of functional differential equations [15], that (1) has only one solution x(t) = col(x1(t), . The proof is analogous to the proof of the Proposition 2.3 and 2.4 in [10]
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