Nonlinear impulse evolution systems and applications to population models (Q1353749)

This article deals with applications to population theory of some results about existence and stability of solutions (in particular, bounded and periodic) of the following impulse differential system: \[ {\textstyle {\frac{dx}{dt}}}+Ax= f(t,x)\qquad (t\in\mathbb{R}\setminus J), \] \[ x(t+0)-x(t-0)= g(t,x(t- 0))\qquad (t\in J), \] where \(J\) is some discrete set, \(A\) a sectorial linear operator in a Banach space \(X\), \(f(t,x)\) and \(g(t,x)\) smooth functions defined on \((\mathbb{R}\setminus J)\times X\) and \(J\times X\), respectively, and taking their values in \(X\); these results were obtained by the author and S. I. Trofimchuk earlier. The basic applications are related with the partial differential equation of type \[ {\textstyle {\frac{\partial u}{\partial t}}}= D\Delta u(t,x)+\varphi(t,x,u(t,x)) \qquad (t\neq t_j) \] (under standard boundary value conditions), which is perturbated with impulse members \[ u(t_j+0,x)- u(t_j-0,x)= g_j(u(t_j-0,x)). \] These systems can be treated as mathematical models for the growth and diffusion of a population subject under abrupt changes (such as disasters, harvesting, heavy immigrations, etc.).