zbMATH — the first resource for mathematics

Convergence in Lotka-Volterra type diffusive delay systems without dominating instantaneous negative feedbacks. (English) Zbl 0807.35149
The authors study a system of diffusive Lotka-Volterra functional differential equations of the following form: \[ {\delta\over\delta t}u_ i (x,t) = d_ i\;\Delta u_ i(x,t) + b_ i u_ i (x,t)G_ i \bigl( u_ t(x, \cdot) \bigr) \quad \text{on} \quad \Omega \times (0,\infty), \] \(\delta/ \delta v\) \(u_ i (x,t) = 0\) on \(\delta \Omega \times (0,\infty)\), \(u_ i(x,\theta) = \zeta_ i(x,\theta)\) on \(\Omega \times (-\infty,0]\), where \(i=1, \dots,n\), \(u=(u_ 1, \dots, u_ n)\), \(x=(x_ 1, \dots,x_ n)\), and \(\Omega\) is a bounded region in \(\mathbb{R}^ n\). The functionals \(G_ i\) have the form \[ G_ i \bigl( u_ t(x, \cdot) \bigr) = r_ i - a_ i \int^ 0_{-\tau_ i} u_{it} (x,\theta) d \mu_ i (\theta) + \sum^ n_{j=1} \int^ 0_ \infty u_{jt} (x,\theta) d \mu_{ij} (\theta), \] where \(u_ t(x,\theta) = u(x,t+ \theta)\) for \(\theta \in (-\infty,0]\). The authors assume the existence of a unique spatially homogeneous steady state and prove that it is globally asymptotically stable if the feedbacks of the intraspecific competitions are dominant and the delays involved are sufficiently small. Applications are made to a diffusive Lotka-Volterra competition model and to a diffusive Lotka-Volterra food chain model.

35R10 Functional partial differential equations
92D25 Population dynamics (general)
35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI