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Delayed responses and stability in two-species systems. (English) Zbl 0552.92016
At first the model $$ (1)\quad \dot x\sb 1(t)=b\sb 1(x\sb 1(t))-m\sb 1(x\sb 1(t),x\sb 2(t)) $$ $$ \dot x\sb 2(t)=b\sb 2(x\sb 2(t))-m\sb 2(x\sb 1(t),x\sb 2(t)) $$ is investigated, where $b\sb i$ and $m\sb i$ fulfil standard minimal assumptions used in all ecological models. It is shown that the first quadrant is an invariant set of (1). A set of sufficient conditions is given for the coexistence of the two competing species. The main result of the paper is that the solutions to the equations $$ (2)\quad \dot x\sb 1(t)=b\sb 1(x\sb 1(t-\tau\sb{11}))-m\sb 1(x\sb 1(t),x\sb 2(t-\tau\sb{12})) $$ $$ \dot x\sb 2(t)=b\sb 2(x\sb 2(t- \tau\sb{21}))-m\sb 2(x\sb 1(t-\tau\sb{22}),x\sb 2(t)) $$ have the same qualitative properties what regards the domain of solutions, nonnegativity and asymptotic stability if the additional conditions $$ (3)\quad \partial\sb im\sb i>\partial\sb ib\sb i+\partial\sb im\sb{3- i}\quad (i=1,2) $$ at the steady state hold whatever be the time delays $\tau\sb{ij}=0$. The meaning of (3) is that the intraspecific self- regulating feedback effects $\partial\sb im\sb i$ are higher than its own positive feedback $\partial\sb ib\sb i$ together with its negative (competitive) effects on its competitor $\partial\sb jm\sb i.$ Analogous questions are investigated for another pair of equations, too. An estimation of the decay rate (rate of convergence to the asymptotically stable steady state) is given including the case when the systems are linear. Ecological and biological implications of the results are discussed in comparison to a wide selection from the literature.
Reviewer: J.Tóth

34K20Stability theory of functional-differential equations
92D25Population dynamics (general)
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