##
**Delayed responses and stability in two-species systems.**
*(English)*
Zbl 0552.92016

At first the model
\[
(1)\quad \dot x_ 1(t)=b_ 1(x_ 1(t))-m_ 1(x_ 1(t),x_ 2(t))
\]

\[ \dot x_ 2(t)=b_ 2(x_ 2(t))-m_ 2(x_ 1(t),x_ 2(t)) \] is investigated, where \(b_ i\) and \(m_ i\) 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 \[ (2)\quad \dot x_ 1(t)=b_ 1(x_ 1(t-\tau_{11}))-m_ 1(x_ 1(t),x_ 2(t-\tau_{12})) \]

\[ \dot x_ 2(t)=b_ 2(x_ 2(t- \tau_{21}))-m_ 2(x_ 1(t-\tau_{22}),x_ 2(t)) \] have the same qualitative properties what regards the domain of solutions, nonnegativity and asymptotic stability if the additional conditions \[ (3)\quad \partial_ im_ i>\partial_ ib_ i+\partial_ im_{3- i}\quad (i=1,2) \] at the steady state hold whatever be the time delays \(\tau_{ij}=0\). The meaning of (3) is that the intraspecific self- regulating feedback effects \(\partial_ im_ i\) are higher than its own positive feedback \(\partial_ ib_ i\) together with its negative (competitive) effects on its competitor \(\partial_ jm_ i.\)

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.

\[ \dot x_ 2(t)=b_ 2(x_ 2(t))-m_ 2(x_ 1(t),x_ 2(t)) \] is investigated, where \(b_ i\) and \(m_ i\) 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 \[ (2)\quad \dot x_ 1(t)=b_ 1(x_ 1(t-\tau_{11}))-m_ 1(x_ 1(t),x_ 2(t-\tau_{12})) \]

\[ \dot x_ 2(t)=b_ 2(x_ 2(t- \tau_{21}))-m_ 2(x_ 1(t-\tau_{22}),x_ 2(t)) \] have the same qualitative properties what regards the domain of solutions, nonnegativity and asymptotic stability if the additional conditions \[ (3)\quad \partial_ im_ i>\partial_ ib_ i+\partial_ im_{3- i}\quad (i=1,2) \] at the steady state hold whatever be the time delays \(\tau_{ij}=0\). The meaning of (3) is that the intraspecific self- regulating feedback effects \(\partial_ im_ i\) are higher than its own positive feedback \(\partial_ ib_ i\) together with its negative (competitive) effects on its competitor \(\partial_ jm_ i.\)

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

### MSC:

92D40 | Ecology |

34K20 | Stability theory of functional-differential equations |

92D25 | Population dynamics (general) |