Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous negative feedbacks.

*(English)*Zbl 1146.34053The paper deals with linear functional differential equations
\[
x_{i}^{\prime}(t)=-\left[ b_{i}x_{i}(t)+\sum_{j=1}^{n}l_{ij}\int_{-\tau} ^{0}x_{j}(t+\theta)\,d\eta_{ij}(\theta)\right] ,\qquad i=1,2,\ldots,n, \tag{3}
\]
and multiple species Lotka-Volterra type models
\[
x_{i}^{\prime}(t)=r_{i}(t)x_{i}(t)\left[ 1-b_{i}x_{i}(t)-\sum_{j=1}^{n} l_{ij}\int_{-\tau}^{0}x_{j}(t+\theta)\,d\eta_{ij}(\theta)\right] ,\qquad i=1,2,\ldots,n, \tag{4}
\]
where \(b_{i},\) \(l_{ij}\in\mathbb{R},\) \(\tau>0,\) \(r_{i}\) are positive continuous functions and \(\eta_{ij}:[-\tau,0]\rightarrow\mathbb{R}\) are normalized functions of bounded variation. If delays are sufficiently small, they can be neglected, and solutions of functional differential equations behave mainly like those of ordinary differential equations.

In fact, for a system ({4}) without nondelayed intraspecific competition (\(b_{i}=0\)), attractivity of the positive equilibrium \(x^{\ast},\) if exists, is traditionally investigated by imposing restrictions on the size of delay in intraspecific terms. However, such assumptions may not be realistic in applied problems, and the authors assume that the intraspecific competition terms \(b_{i}x_{i}(t)\) without delay dominate, in certain sense, intraspecific competition with delay and interspecific interactions.

A key problem is to establish sufficient conditions that insure diagonal dominance of the instantaneous negative feedbacks over the total competition matrix, in which case the stability of ({4}) follows irrespective of the choice of delay functions \(\eta_{ij}.\) First a criterion for the exponential stability of the linear functional differential equations ({3}) extending known results for linear equations with discrete delays is obtained. It is shown that, for \(b_{i}>0,\) the criterion is optimal and provides necessary and sufficient conditions for the stability of ({3}) in terms of the coefficients \(b_{i}\) and \(l_{ij}\) independently of the delay functions \(\eta_{ij}.\) Then new sufficient conditions for the boundedness of solutions and for the global asymptotic stability of the positive equilibrium \(x^{\ast}\) (if exists) are obtained. These also improve results reported in the literature, and even sharper conditions are derived for the system ({4}) with nondecreasing delay functions \(\eta_{ij}.\) Remarkably, the methods used in the paper do not involve Lyapunov functionals and apply to more general systems.

In fact, for a system ({4}) without nondelayed intraspecific competition (\(b_{i}=0\)), attractivity of the positive equilibrium \(x^{\ast},\) if exists, is traditionally investigated by imposing restrictions on the size of delay in intraspecific terms. However, such assumptions may not be realistic in applied problems, and the authors assume that the intraspecific competition terms \(b_{i}x_{i}(t)\) without delay dominate, in certain sense, intraspecific competition with delay and interspecific interactions.

A key problem is to establish sufficient conditions that insure diagonal dominance of the instantaneous negative feedbacks over the total competition matrix, in which case the stability of ({4}) follows irrespective of the choice of delay functions \(\eta_{ij}.\) First a criterion for the exponential stability of the linear functional differential equations ({3}) extending known results for linear equations with discrete delays is obtained. It is shown that, for \(b_{i}>0,\) the criterion is optimal and provides necessary and sufficient conditions for the stability of ({3}) in terms of the coefficients \(b_{i}\) and \(l_{ij}\) independently of the delay functions \(\eta_{ij}.\) Then new sufficient conditions for the boundedness of solutions and for the global asymptotic stability of the positive equilibrium \(x^{\ast}\) (if exists) are obtained. These also improve results reported in the literature, and even sharper conditions are derived for the system ({4}) with nondecreasing delay functions \(\eta_{ij}.\) Remarkably, the methods used in the paper do not involve Lyapunov functionals and apply to more general systems.

Reviewer: Yuri V. Rogovchenko (Kalmar)

##### MSC:

34K20 | Stability theory of functional-differential equations |

34K60 | Qualitative investigation and simulation of models involving functional-differential equations |

92D25 | Population dynamics (general) |

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\textit{T. Faria} and \textit{J. J. Oliveira}, J. Differ. Equations 244, No. 5, 1049--1079 (2008; Zbl 1146.34053)

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