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**Analysis of a delayed two-stage population model with space-limited recruitment.**
*(English)*
Zbl 0847.34076

This paper proposes and analyzes a model of a two-stage population (juvenile and adult) with space-limited recruitment. Real-life examples of such populations are sessile marine populations, like barnacles, that is, populations which, as adults, live with part of their body sealed to the surface of a rock in the sea-bed while the larval and part of the juvenile stages are free-moving. So, the passage to adult stage is subject to the possibility for juveniles to find a space to settle. Short historical remarks made in the introduction of the paper indicate that the first attempt to model this sort of phenomenon is due to Roughgarden et al., who proposed an age-dependent model, with the recruitment in adult stage proportional to the actual free space \(F(t)\). A deficiency in Roughgarden et al.’s model lies in the fact that \(F(t)\) can take negative values. A subsequent model by Bence and Nisbet corrects this deficiency by using the positive part of \(F(t)\). These authors consider a simple model with the density of the total population of adults \(A(t)\) as the only state variable and assume a fixed delay \(\tau\) between settlement and recruitment. The novelty in the present paper is the addition of the juvenile stage in the Bence and Nisbet model, and the consideration of the space occupied by juveniles (not yet recruited), together with the space occupied by adults in the computation of \(F(t)\). Assuming that the density of settling juveniles at time \(t\) is a fixed fraction of the free space \(F(t)\), the model can be reduced to a system of two delay differential equations in \(F(t)\) and \(A(t)\), with delay in \(F\) only, yielding a semigroup in the product \(C ([- \tau,0 ], \mathbb{R})\times \mathbb{R}\). The main mathematical difficulty and interest of the paper lie in the nature of the nonlinearity entailed by the presence of the function \(x\to \max (x,0)\). This difficulty is overcome by a rigorous analysis by which global existence, positivity and boundedness of solutions, and uniform persistence of the adults, are obtained. Global asymptotic stability of the nontrivial steady state is obtained by means of a Lyapunov functional for \(\tau\) large enough (Theorem 5.2), while, using a Razumikhin function, the same conclusion is achieved, independent on the delay, provided that the juvenile and adult mortalities are large enough. Applied to the situation envisaged by Bence and Nisbet, to which the model can be reduced by setting one of the parameters to zero, the above analysis complete the Bence and Nisbet work, which is restricted to the local analysis. Local stability and instability are also analyzed in terms of the characteristic equation near the nontrivial steady state. Two regions of instability are described (Corollaries 4.3 and 4.4). The final discussion proposes to consider the ratio of surface occupancy per individual instead of the settlement rate as an index of stability. It also attracts the attention of the reader to an amazing feature of the type of nonlinearity considered here, that is, the local Hopf bifurcation branch is vertical, all the points of the branch project onto the same value in the parameter space. So, it is not known how the solutions behave asymptotically, when the nontrivial equilibrium is unstable, although numerical results indicate the possibility of a globally asymptotically stable periodic solution.

Reviewer: O.Arino (Pau)

### MSC:

34K20 | Stability theory of functional-differential equations |

34K13 | Periodic solutions to functional-differential equations |

34K18 | Bifurcation theory of functional-differential equations |