## Description

Positive dynamical systems comeinto playwhen relevant variables of a systemtake on values which are nonnegative in a natural way. This is the case, for example, in fields as biology, demography and economics, where the levels of populations or prices of goods are positive. Positivity comes in also if the formation of averages by weighted means is relevant since weights, for example probabilities, are not negative. This is the case in quite diverse fields ranging from electrical engineering over physics and computer science to sociology. Thereby averaging takes place with respect to signals

in a sensor network or in a swarm (of birds or robots) or with respect to velocities of particles or the opinions of people. In the fields mentioned the dynamics is often modeled by difference equations which means that time is treated as discrete. Thus, in reality one meets a huge variety of positive dynamical systems in discrete time.

In many cases these systems can be captured by a linearmapping given by a nonnegative matrix. The dynamics (in discrete time) then is given by the powers of the matrix or, equivalently, by the iterates of the linear mapping which maps the positive orthant into itself. A powerful tool then is the Perron–Frobenius Theory of nonnegative matrices (including the asymptotic behavior of powers of those matrices) which has been successful since its inception by O. Perron and G. Frobenius over about hundred years ago. Concerning theory as well as applications there are two insufficient aspects of Perron–Frobenius Theory which later on drove this theory into new directions. The first aspect is that this theory is not just about nonnegative matrices but applies happily also to certain matrices with negative entries. This means that the theory should be understood as dealing with linear selfmappings of convex cones in finite dimensions not just of the standard cone, the positive orthant. The second aspect is that even simple positive dynamical systems are not linear. Thus, what is needed is an extension of classical Perron–Frobenius Theory to nonlinear selfmappings of convex cones in finite dimensions. Moreover, with respect to theory as well as applications, such an extension is needed also in infinite dimensions. Since classical Perron–Frobenius Theory has already so many applications one can imagine the great variety of applications such an extension to nonlinear selfmappings in infinite dimensions will have.