Monotone systems revisited

There are some topics in mathematics and physics which are a lasting source of dissatisfaction for me since I feel that I have not properly understood them despite having made considerable efforts to do so. In the case of physics the reason is often that the physicists who understand the subject are not able to explain it in a way which provides what a mathematician sees as a comprehensible account. In mathematics the problem is a different one. Mathematicians frequently have a tendency (often justified) to discuss things on a level which is as general as possible. This leads to theorems which are loaded down with detail and where the many technical conditions make it difficult to see the wood for the trees. When confronted with such things I sometimes feel exhausted and give up. I prefer an account which builds up ideas step by step from simple beginnings. Here I return to a subject which I have written about more than once in this blog before but where the sense of dissatisfaction remains. I hope to reduce it here.

I start with a system of ordinary differential equations $\dot x_i=f_i(x)$. It should be defined on the $n$-dimensional Euclidean space or on one of its orthants. (An orthant is the subset of Euclidean space defined by making a choice of the signs of its components. It generalises a quadrant in the two-dimensional case.) The system is said to be cooperative if $\frac{\partial f_i}{\partial x_j}>0$ for all $i\ne j$. The name comes from the fact that the equations for the population dynamics of a set of species has this property if each species benefits the others. Suppose we now have two solutions $x$ and $\bar x$ of the system and that $x_i(t_0)\le\bar x_i(t_0)$ for all $i$ at some time time $t_0$. We may abbreviate this relation by $x(t_0)\le\bar x(t_0)$. Here we see a partial order on Euclidean space defined by the ordering of the components. A theorem of Müller and Kamke says that if the initial data for two solutions of a cooperative system at time $t_0$ satisfies this relation then $x(t)\le\bar x(t)$ for all $t\ge t_0$. Another way of saying this is that the time-$t$ flow of the system is preserves the partial order. A system of ODE with this property is called monotone. Thus the Müller-Kamke theorem says that a cooperative system is monotone.

The differential condition for monotonicity can be integrated. If $x$ and $\bar x$ are two points in Euclidean space with $x_i=\bar x_i$ for a certain $i$ and $x_j\le\bar x_j$ for $j\ne i$ then $f_i(x)\le f_i(\bar x)$. To see this we join $x$ to $\bar x$ by a piecewise linear curve where the coordinates other than the $i$th are increased successively from $x_j$ to $\bar x_j$. On each segment of this curve the value of $f_i$ does not decrease, as a consequence of the fundamental theorem of calculus. Hence its value at the end of the entire path is at least as big as its value at the beginning. We now want to prove that a certain inequality holds at all times $t\ge t_0$. In order to do this we would like to consider the first time $t_*>t_0$ where the inequality fails and get a contradiction. Unfortunately there might be no such time – in principle the condition might fail immediately. To get around this we deform the system for the solution $\bar x$ to $\frac{d\bar x_i}{dt}=f_i(\bar x)+\epsilon$. If we can prove the result for the deformed system the result for the initial system follows by continuous dependence of the solution on $\epsilon$. For the deformed system let $t_*$ be the supremum of the times where the desired inequality holds. If the inequality does not hold globally then the system is still defined at $t=t_*$. For $t=t_*$ we have $x_i=\bar x_i$ for some $i$ and we can assume w.l.o.g. that $x_j<\bar x_j$ for some $j$ since otherwise the two solutions would be equal and the result trivial. The integrated form of the cooperativity condition implies that at $t_*$ the right hand side of the evolution equation for $\bar x_i-x_i$ is positive. On the other hand the fact that it just reached zero coming from positive values implies that the right hand side of the evolution equation is non-positive and we get a contradiction.

A key source of information about monotone dynamical systems is the book of Hal Smith with this title. I have repeatedly looked at this book but always got bogged down quite quickly. Now I realise that for my purposes it would have been much better if I had started with chapter 3. The Müller-Kamke theorem is discussed in section 3.1. The range of application of this theorem can be extended considerably by the following trick, discussed in section 3.5. Suppose that we define $y_i=(-1)^{m_i}x_i$ where each of the $m_i$ are zero or one. This transforms the signs of $Df$ in a certain way and so cooperativity of the system for $y$ corresponds to a certain sign pattern for the entries of $Df$. A first important condition is that each off-diagonal element of $Df(x)$ should be either non-negative or non-positive. Next, the sign of $\frac{\partial f_i}{\partial x_j}\frac{\partial f_j}{\partial x_i}$ is not changed be the transformation and must thus be non-negative. In the context of population models this can be interpreted as saying that there is no pair of species which are in a predator-prey relationship. Given that these two conditions are satisfied we consider a labelled graph where the nodes are the numbers from $1$ to $n$ and there is an edge between two nodes if at least one of the corresponding partial derivatives is non-zero at some point. The edge is then labelled with the sign of this non-zero value. A loop in the graph can be assigned the sign which is the product of those of its edges. It turns out that a system can be transformed to a cooperative system in the way indicated if and only if the graph contains no negative loops. I will call a system of this type ‘cooperative up to sign reversal’. The system can be transformed by a permutation of the variables into one where $Df$ has diagonal blocks with non-negative entries and off-diagonal elements with non-positive entries.

If all elements of $Df$ are required to be non-positive we get the class of competitive systems. It should be noted that being competitive leads to less restrictions on the dynamics of a system (towards the future) than being cooperative. We can define a class of systems which are competitive up to sign reversal. An example of such a system is the basic model of virus dynamics. In that system the unknowns are the populations of uninfected cells $x$, infected cells $y$ and virus particles $v$. The transformation $y\mapsto -y$ makes it into a competitive system. In various models of virus dynamics including the immune response the target cells of the virus and the immune cells are in a predator-prey relationship and so these systems can be neither cooperative up to sign or competitive up to sign.

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