We consider a generalization of the well-known nonlinear Nicholson blowflies model with stochastic perturbations. Stability in probability of the positive equilibrium of the considered equation is studied. Two types of stability conditions: delay-dependent and delay-independent conditions are obtained, using the method of Lyapunov functionals and the method of linear matrix inequalities. The obtained results are illustrated by numerical simulations by means of some examples. The results are new, and complement the existing ones.

We introduce a multi-species chemotaxis type system admitting an arbitrarily large number of population species, all of which are attracted versus repelled by a single chemical substance. The production versus destruction rates of the chemotactic substance by the species is described by a probability measure. For such a model, we investigate the variational structures, in particular, we prove the existence of Lyapunov functionals, we establish duality properties as well as a logarithmic Hardy–Littlewood–Sobolev type inequality for the associated free energy. The latter inequality provides the optimal critical value for the conserved total population mass.

In this paper, we consider diffusion, reaction and dissolution of mobile and immobile chemical species present in a porous medium. Inflow–outflow boundary conditions are considered at the outer boundary and the reactions amongst the species are assumed to be reversible which yield highly nonlinear reaction rate terms. The dissolution of immobile species takes place on the surfaces of the solid parts. Modelling of these processes leads to a system of coupled semilinear partial differential equations together with a system of ordinary differential equations (ODEs) with multi-valued right-hand sides. We prove the global existence of a unique positive weak solution of this model using a regularization technique, Schaefer's fixed point theorem and Lyapunov type arguments.

In this paper we study asymptotic behaviour of distributed parameter systems governedby partial differential equations (abbreviated to PDE). We first review some recently developed resultson the stability analysis of PDE systems by Lyapunov's second method. On constructing Lyapunov functionalswe prove next an asymptotic exponential stability result for a class of symmetric hyperbolic PDEsystems. Then we apply the result to establish exponential stability of various chemical engineeringprocesses and, in particular, exponential stability of heat exchangers. Through concrete examples weshow how Lyapunov's second method may be extended to stability analysis of nonlinear hyperbolic PDE.Meanwhile we explain how the method is adapted to the framework of Banach spaces Lp , $1<p\leq \infty$ .