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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)
##### Keywords:
Lotka-Volterra type infinite delay system
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##### References:
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