## Monotone dynamical systems

In previous posts I have written a little about monotone dynamical systems, a class of systems which in some sense have simpler dynamical properties than general dynamical systems. Unfortunately this subject was always accompanied by some confusion in my mind. This results from the necessity of a certain type of bookkeeping which I was never really able to get straight. Now I think my understanding of the topic has improved and I want to fix this knowledge here. There are two things which have led to this improvement. One is that I read an expository article by Eduardo Sontag which discusses monotone systems in the context of biochemical networks (Systems and Synthetic Biology 1, 59). The other is that I had the chance to talk to David Angeli who patiently answered some of my elementary questions as well as providing other insights. In what follows I only discuss continuous dynamical systems. Information on the corresponding theory in the case of discrete dynamical systems can be found in the paper of Sontag.

Consider a dynamical system $\dot x=f(x)$ defined on an open subset of $R^n$. The system is called monotone if $\frac{\partial f_i}{\partial x_j}\ge 0$ for all $i\ne j$. This is a rather restrictive definition – we will see alternative possibilities later – but I want to start in a simple context. There is a theorem of Müller and Kamke which says that if two solutions $x$ and $y$ of a monotone system satisfy $x_i(0)\le y_i(0)$ for all $i$ then they satisfy $x_i(t)\le y_i(t)$ for all $i$ and all $t\ge 0$. This can be equivalently expressed as the fact that for each $t\ge 0$ the time $t$ flow of the dynamical system preserves the partial order defined by the condition that $x_i\le y_i$ for all $i$. This can be further reexpressed as the condition that $y-x$ belongs to the positive convex cone in $R^n$ defined by the conditions that the values of all Cartesian coordinates are non-negative. This shows the way to more general definitions of monotone flows on vector spaces, possibly infinite dimensional. These definitions may be useful for the study of certain PDE such as reaction-diffusion equations. The starting point is the choice of a suitable cone. This direction will not be followed further here except to consider some other simple cones in $R^n$.

A monotone system in the sense defined above is also sometimes called cooperative. The name comes from population models where the species are beneficial to each other. Changing the sign in the defining conditions leads to the class of competitive systems. These can be transformed into cooperative systems by changing the direction of time. However for a given choice of time direction the competitive systems need not have the pleasant properties of cooperative systems. Another simple type of coordinate transformation is to reverse the signs of some of the coordinates $x_i$. When can this be used to transform a given system into a monotone one?. Two necessary conditions are that each partial derivative of a component of $f$ must have a (non-strict) sign which is independent of $x$ and that the derivatives are symmetric under interchange of their indices. What remains is a condition which can be expressed in terms of the so-called species graph. This has one node for each variable $x_i$ and an arrow from node $i$ to node $j$ if $\frac{\partial f_j}{\partial x_i}$ is not identically zero. If the derivative is positive the arrow bears a positive sign and if it is negative a negative sign. Alternatively, the arrows with positive sign have a normal arrowhead while those with negative sign have a blunt end. In this way the system gives rise to a labelled oriented graph. To each (not necessarily oriented) path in the graph we associate a sign which is the product of the signs of the individual edges composing the path. The graph is said to be consistent if signs can be associated to the vertices in such a way that the sign of an edge is always the product of the signs of its endpoints. This is equivalent to the condition that every closed loop in the graph has a positive sign. In other words, every feedback in the system is positive. Given that the other two necessary conditions are satisfied the condition of consistency characterizes those networks which can be transformed by changes of sign of the $x_i$ to a monotone system. A transformation of this type can also be thought of as replacing the positive orthant by another orthant as the cone defining the partial order.

Next I consider some examples. Every one-dimensional system is monotone. In a two-dimensional system we can have the sign patterns $(+,+)$, $(-,-)$ and $(+,-)$. In the first case the system is monotone. In the second case it is not but can be made so by reversing the sign of one of the coordinates. This is the case of a two-dimensional competitive system. In the third case the system cannot be made monotone. A three-dimensional competitive system cannot be made monotone. The species graph contains a negative loop. Higher dimensional competitive systems are no better since their graphs all contain copies of that negative loop.

A general message in Sontag’s paper is that consistent systems tend to be particularly robust to various types of disturbances. Large biochemical networks are in general not consistent in this sense but they are close to being consistent in the sense that removing a few edges from the network make them consistent. This also means that they can be thought of as a few consistent subsystems joined together. Since biological systems need robustness this suggests a topological property which biochemical networks should have compared to random networks. Sontag presents an example where this has been observed in the transcription network of yeast.

A more sophisticated method which can often be used to obtain monotone systems from systems of chemical reactions by a change of variables has been discussed in a previous post. The advantage of this is that together with other conditions it can be use to show that generic solutions, or sometimes even all solutions, of the original system converge to stationary solutions.