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Convergence in Lotka-Volterra-type delay systems without instantaneous feedbacks. (English) Zbl 0795.34067
The authors consider the Lotka-Volterra type infinite delay system \[ u_ i'(t)= b_ i (u_ i(t)) G_ i(u_ t(\cdot)), \qquad i=1,\dots,n, \] where \(G_ i(u_ t(\cdot))= r_ i-a_ i \int_{\tau_ i}^ 0 u_ i(t+ \theta) d\mu_ i(\theta)+ \sum_{j=1}^ n \int_ \infty^ 0 u_ j(t+\theta)d\mu_{ij}(\theta)\), and \(u_ t(\theta)= u(t+\theta)\) for \(\theta\leq 0\), \(u(t)= (u_ 1(t),\dots, u_ n(t))\), in the following hypotheses (for \(i=1,\dots,n\), \(j=1,\dots,n)\):
\((\text{H}_ 1)\) \(b_ i(0)=0\), \(b_ i(\cdot)\) is continuously differentiable, \(b_ i'(\cdot)>0\),
\((\text{H}_ 2)\) \(r_ i\), \(a_ i\) and \(\tau_ i\) are constants, \(a_ i,\tau_ i>0\),
\((\text{H}_ 3)\) \(\mu_ i(\theta)\) are nondecreasing, \(\mu_ i(0)- \mu_ i(-\tau_ i) =1\),
\((\text{H}_ 4)\) \(\mu_{ij}(\theta)\) are bounded real-valued Borel measures on \((\infty,0]\) with total variation \(| \mu_{ij}|\).
The solutions are sought in the Banach space \((Bc)\) of bounded continuous functions that map \((-\infty,0]\) into \(\mathbb{R}^ n\), with the uniform norm. The initial condition
\((\text{H}_ 5)\) \(u_ 0(s)= \varphi(s)\geq 0\), for \(s\leq 0\), \(\varphi\in Bc\), \(\varphi(0)>0\) (\(\varphi(0)\) means \(\varphi_ i(0)>0\), \(i=1,\dots,n\)), is also assumed.
If, in addition to \(\text{H}_ 1 \text{-H}_ 5\), some assumptions are satisfied, then \(\lim u(\varphi)(t)= \rho\), where \(\rho= (\rho_ 1,\dots, \rho_ n)\in \mathbb{R}^ n\) is the unique saturated equilibrium of the system \((\rho_ i\geq 0\), \(G_ i(\rho)\leq 0\) and \(\rho_ i G_ i(\rho)=0)\).
Analogous results are established for a finite delay version of the considered system.

MSC:
34K20 Stability theory of functional-differential equations
93D15 Stabilization of systems by feedback
92D25 Population dynamics (general)
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