A discrete analogue of an integrodifferential equation
Consider the integrodifferential equation \[ dx(t)/dt=x(t)\bigl[a-x(t)-b\int_{-\infty}^t K(t-s)x(s)ds\bigr], \quad t>0, \tag{1} \] under additional conditions and appropriate initial condition, which models the dynamics of a single species with negative feedback and accumulation of toxic effects in the environment. The authors propose the delay difference equation \[ x(n+1)=\alpha x(n)/\bigl[1+x(n)+b\mu\sum_{j=1}^\infty j^m\sigma^j x(n-j)\bigr], \quad m,n=0,1,2,\ldots, \] under additional conditions and appropriate initial condition, but without any new discretization parameter. It is shown, by means of the discrete ``linear chain trick'', that equation (2) has a globally asymptotically stable positive equilibrium under the same conditions under which equation (1) is globally asymptotically stable.
scientific article; zbMATH DE number 1391501