Persistence and permanence of dynamical systems

In mathematical models for chemical or biological systems the unknowns, which might be concentrations or populations, are naturally positive quantities. It is of interest to know whether in the course of time these quantities may approach zero in some sense. For example in the case of ecological models this is related to the question whether some species might become extinct. The notions of persistence and permanence are attempts to formulate precise concepts which could be helpful in discussing these questions. Unfortunately the use of these terms in the literature is not consistent.

I first consider the case of a dynamical system with continuous time defined by a system of ODE \dot x=f(x). The system is defined on the closure \bar E of a set E which is either the positive orthant in Euclidean space (the subset where all coordinates are non-negative) or the intersection of that set with an affine subspace. The second case is included to allow the imposition of fixed values of some conserved quantities. Let \partial E be the boundary of E, where in the second case we mean the boundary when E is considered as a subset of the affine subspace. Suppose that for any x_0\in \bar E there is a unique solution x(t) with x(0)=x_0. If x_0\in E let c_1(x_0)=\liminf d(x(t),\partial E), where x(t) is the solution with initial datum x_0 and d is the Euclidean distance. Let c_2(x_0)=\limsup d(x(t),\partial E). A first attempt to define persistence (PD1) says that it means that c_1(x_0)>0 for all x_0\in E. Similarly, a first attempt to define uniform persistence (PD2) is to say that it means that there is some m>0 such that c_1(x_0)\ge m for all x_0\in E. The system may be called weakly persistent (PD3) if c_2(x_0)>0 for all x_0\in E but this concept will not be considered further in what follows. A source of difficulties in dealing with definitions of this type is that there can be a mixing between what happens at zero and what happens at infinity. In a system where all solutions are bounded PD1 is equivalent to the condition that no positive solution has an \omega-limit point in \partial E. Given the kind of examples I am interested in I prefer to only define persistence for systems where all solutions are bounded and then use the definition formulated in terms of \omega-limit points. In that context it is equivalent to PD1. The term permanence is often used instead of uniform persistence. I prefer the former term and I prefer to define it only for systems where the solutions are uniformly bounded at late times, i.e. there exists a constant M such that all components of all solutions are bounded by M for times greater than some t_0, where t_0 might depend on the solution considered. Then permanence is equivalent to the condition that there is a compact set K\subset E such that for any positive solution x(t)\in K for t\ge t_0. A criterion for permanence which is sometimes useful will now be stated without proof. If a system whose solutions are uniformly bounded at infinity has the property that \overline{\omega(E)}\cap\partial E=\emptyset then it is permanent. Here \omega(E) is the \omega-limit set of E, i.e. the union of the \omega-limit sets of solutions starting at points of E. If there is a point in \overline{\omega(E)}\cap\partial E there is a solution through that point on an interval (-\epsilon,\epsilon) which is non-negative. For some systems points like this can be ruled out directly and this gives a way of proving that they are permanent.

Let \phi:[0,\infty)\times \bar E\to \bar E be the flow of the dynamical system, i.e. \phi(t,x_0) is the value at time t of the solution with x(0)=x_0. The time-one map F of the system is defined as F(x)=\phi(1,x). Its powers define a discrete semi-dynamical system with flow \psi(n,x)=F^n(x)=\phi(n,x), where n is a non-negative integer. For a discrete system of this type there is an obvious way to define \omega-limit points, persistence and permanence in analogy to what was done in the case with continuous time. For the last two we make suitable boundedness assumptions, as before. It is then clear that if the system of ODE is persistent or permanent its time-one map has the corresponding property. What is more interesting is that the converse statements also hold. Suppose the the time-one map is persistent. For a given E let K be the closure of the set of points F^n(x). This is a compact set and because of persistence it lies in E. Let K'=\phi (K\times [0,1]). This is also a compact subset of E. If t is sufficiently large then \phi([t],x)\in K where [t] is the largest integer smaller that t. This implies that \phi(t,x)\in K' and proves that the system of ODE is persistent. A similar argument shows that if the time-one map is permanent the system of ODE is permanent. We only have to start with the compact set K provided by the permanence of the time-one map.

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