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Global stability for infinite-delay, dispersive Lotka-Volterra systems: Weakly interacting populations in nearly identical patches. (English) Zbl 0731.92029

The authors consider the system of equations

$\left(1\right)\phantom{\rule{1.em}{0ex}}\left(d{u}_{j}^{i}/dt\right)\left(t\right)={b}_{j}^{i}{u}_{j}^{i}\left(t\right)\left[{r}_{j}^{i}-\sum _{k=1}^{m}{\int }_{-\infty }^{0}{u}_{j}^{k}\left(t+s\right)d{v}_{j}^{ik}\left(s\right)\right]+$
$+\sum _{\ell =1}^{n}{d}_{\ell j}^{i}\left({u}_{\ell }^{i}-{u}_{j}^{i}\right),\phantom{\rule{1.em}{0ex}}{u}_{j}^{i}\left(s\right)={\phi }_{j}^{i}\left(s\right)\ge 0,\phantom{\rule{1.em}{0ex}}-\infty

where ${u}_{j}^{i}$ represents the density of species i in habitat j, $1\le i\le m$, $1\le j\le n$; and ${v}_{j}^{ik}$ are bounded, real-valued Borel measures on (-$\infty ,0\right]$ with total variation $|{v}_{j}^{ik}|$ satisfying

${\int }_{-\infty }^{0}{e}^{-{\gamma }_{0}s}d|{v}_{j}^{ik}|\left(s\right)<\infty$

for some positive number ${\gamma }_{0}$. The dispersal coefficients ${d}_{\ell j}^{i}\ge 0$, ${d}_{\ell j}^{i}\ne {d}_{j\ell }^{i}$ in general, may be functions or functionals of u for one result, but are assumed constant for the main result. The ${b}_{j}^{i}>0$ are not necessarily nearly independent of j.

It is proved that there exists a globally stable equilibrium solution provided that (a) intraspecific competition is strong relative to interspecific coupling (weakly interacting species), (b) the immediate (undelayed) deleterious effect of intraspecific competition on a species growth rate dominates the corresponding delayed effect, and (c) the habitats are nearly identical. The assumption of nearly identical habitats takes the form

${r}_{j}^{i}={r}^{i}+{\Delta }{r}_{j}^{i},\phantom{\rule{1.em}{0ex}}{v}_{j}^{ik}={v}^{ik}+{\Delta }{v}_{j}^{ik},$

where ${\Delta }{r}_{j}^{i}$ and $|{\Delta }{v}_{j}^{ik}|\left(-\infty ,0\right]$ are small and ${v}^{ik}$ are bounded Borel measures on (-$\infty ,0\right]·$

This work is distinguished from previous work principally by the generality of the unbounded delays allowed and by the consideration of multiple, not necessarily identical, habitats. The mathematical techniques are adaptions of those used by R. H. Martin and H. L. Smith [Convergence in Lotka-Volterra systems with diffusion and delay, Proc. Workshop Diff. Eq. Appl., Retzhof/Austria 1989, Lect. Notes Pure Appl. Math., Marcel Dekker, New York].

##### MSC:
 92D40 Ecology 34K20 Stability theory of functional-differential equations 34K30 Functional-differential equations in abstract spaces 35R10 Partial functional-differential equations 34K99 Functional-differential equations
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