Bistability and Oscillatory Behaviours of Cyclic Feedback Loops

Abstract

In this paper, we study the stability of an Ordinary Differential Equation (ODE) usually referred to as Cyclic Feedback Loop, which typically models a biological network of $d$ molecules where each molecule regulates its successor in a cycle ($A_{1}\rightarrow A_{2}\rightarrow \cdots \rightarrow A_{d-1} \rightarrow A_{d} \rightarrow A_{1}$). Regulations, which can be either positive or negative, are modelled by increasing or decreasing functions. We make an analysis of this model for a wide range of functions (including affine and Hill functions) by determining the parameters for which bistability and oscillatory behaviours arise. These results encompass previous theoretical studies of gene regulatory networks, which are particular cases of this model.

1 Introduction

We aim to characterise the stability of the ODE system

$$\begin{aligned} \textstyle\begin{cases} \dot{x}_{1}=\alpha _{1} f_{1}(x_{d})-x_{1}, \\ \dot{x}_{2}=\alpha _{2} f_{2}(x_{1})-x_{2}, \\ \vdots \\ \dot{x}_{d}=\alpha _{d} f_{d}(x_{d-1})-x_{d}, \end{cases}\displaystyle \end{aligned}$$

(1)

where $f_{1},\ldots, f_{d} \in \mathcal{C}^{1}(\mathbb{R}_{+}, \mathbb{R}_{+})$ (with $\mathbb{R}_{+}=[0, +\infty )$) are non-negative functions, at least one of them is bounded, one is positive and $\alpha _{1},\ldots, \alpha _{d}$ are positive parameters.

Throughout this paper, we use the convention $x_{0}=x_{d}$, which allows us to write (1) under the compacted form

$$ \forall i\in \{1,\ldots, d\}, \quad \dot{x}_{i}=\alpha _{i}f_{i}(x_{i-1})-x_{i}. $$

This model is a generalisation of a gene regulatory network initially proposed by Goodwin [1, 2], usually referred to as Cyclic feedback loop. Such systems represent interactions between genes, mRNAs, enzymes and proteins called repressors which have the ability to inhibit the expression of some genes. In system (1), $x_{1}, \ldots, x_{d}$ represent the concentration of each of the molecules involved in the network (denoted $A_{1}$,…, $A_{n}$), and $f_{1},\ldots, f_{d}$ the regulation between them. The system is assumed to be cyclic ($A_{i}$ regulates $A_{i+1}$ and only $A_{i+1}$, as illustrated by Fig. 1), and each regulation can be positive ($f_{i}$ increasing) or negative ($f_{i}$ decreasing). The relevance of these cyclic models has been established in [3] and [4] where some theoretical predictions (oscillatory phenomena and bistability) have been observed experimentally. This highlights the importance of understanding the dynamical behaviour of such systems i.e. determining the number of stable equilibrium points and their basins of attraction, as well as the possible existence of periodic solutions or chaotic behaviours.

Fig. 1

Schematic representation of the cyclic feedback loop (1): The blue circles represent the molecules of the network, and the arrows between them regulation, which can be positive ($f_{i}$ increasing) or negative ($f_{i}$ decreasing)

Equations similar to System (1) have been, for some specific choices of $f_{i}$, the subject of several theoretical studies [5–10]. In these papers, restrictions on the functions $f_{i}$ were notably imposed by the necessity to compute the values of the equilibrium points of the system, which is intricate when more than two functions are not affine, and are not identical. In the present paper, we follow a method initiated by Cherry and Adler [11] allowing to avoid explicitly computing the equilibrium points. In the two dimensional case, which writes

$$\begin{aligned} \textstyle\begin{cases} \dot{x}_{1}=\alpha _{1}f_{1}(x_{2})-x_{1}, \\ \dot{x}_{2}=\alpha _{2}f_{2}(x_{1})-x_{2}, \end{cases}\displaystyle \end{aligned}$$

(2)

the following results have already been established:

• 1. If $f_{1}$ and $f_{2}$ are both increasing or both decreasing, and if

  $$\begin{aligned} \underset{x>0}{\sup}\left \lvert \frac{xf_{1}'(x)}{f_{1}(x)}\right \rvert \underset{x>0}{\sup}\left \lvert \frac{xf_{2}'(x)}{f_{2}(x)} \right \rvert >1, \end{aligned}$$

  (3)

  then there exist values of $(\alpha _{1}, \alpha _{2})\in {\mathbb{R}_{+}^{*}}^{2}$ such that system (2) is multistable i.e. there exist at least two equilibrium points which are asymptotically stable [11].

• 2. If $f_{1}$ and $f_{2}$ are both increasing or both decreasing, and if¹

  $$ \frac{1}{\sqrt{\lvert f_{1}'\rvert}} \quad \text{and} \quad \frac{1}{\sqrt{\lvert f_{2}'\rvert}} \quad \text{are strictly convex}, $$

  (4)

  then system (2) is either monostable or bistable, i.e. there exist exactly one or exactly two equilibrium points which are asymptotically stable. Moreover, it is possible to determine, up to a set of measure zero, the set of parameters for $(\alpha _{1}, \alpha _{2})$ for which the system is monostable and the set of parameters for which it is bistable [12].

• 3. All the solutions to system (2) converge (even without assuming any monotonicity).

This last result is a direct application of the Poincaré-Bendixson theorem and the Dulac-Bendixson theorem [13].

A natural question at this stage is which of these properties generalise to higher dimensions ($d\geq 3$). A major result was achieved by Mallet-Paret and Smith in [14], showing that the Poincaré-Bendixson theorem can be adapted to monotone feedback systems, including (1). Furthermore, a series of results of Hirsch [15–17] generalised and summarised in [9], has shown that in the case where an even number of functions $f_{i}$ is decreasing, the solutions of (1) converge to an equilibrium point for almost every initial condition. It is well-known that this property does not hold when the number of decreasing functions is odd (in which case the system is often called ‘repressilator’), and that there can there exist stable orbits [6, 18].

In this paper, we prove the following result, which is a generalisation of our previous paper on the two-dimensional case [12]:

Theorem 1

Let us assume that $f_{1},\ldots, f_{d}\in \mathcal{C}^{3}(\mathbb{R}_{+}, \mathbb{R}_{+}^{*})$ are monotone, non-negative, and that (at least) one of them is bounded. Moreover, let us assume that the functions $\frac{1}{\sqrt{\lvert f'_{1} \rvert }}, \ldots, \frac{1}{\sqrt{\lvert f'_{d} \rvert }}$ are defined and convex, and that (at least) one of them is strictly convex. Lastly, let us denote by $n$ the number of these functions which are decreasing, and let

$$ D:=\prod \limits _{k=1}^{d}{\underset{x>0}{\sup}\left \lvert \frac{xf_{k}'(x)}{f_{k}(x)}\right \rvert}\in (0, +\infty ]. $$

• 1. If $n$ is even, then

  • (i) If $D<1$, then for any $\alpha \in \left (\mathbb{R}_{+}^{*}\right )^{d}$ system (1) has a unique equilibrium point which is globally asymptotically stable.

  • (ii) If $D>1$, then there exists a non-empty set $A_{\textrm{bis}}\subset \left (\mathbb{R}_{+}^{*}\right )^{d}$ such that

    • If $\alpha \in A_{\textrm{bis}}$, then system (1) has exactly two asymptotically stable equilibria, and the union of their basins of attraction is a dense open subset of $\mathbb{R}_{+}^{d}$, with a complement of Lebesgue measure zero.

    • If $\alpha \in \overline{A_{\textrm{bis}}}^{C}$, then system (1) has a unique equilibrium point which is globally asymptotically stable.

  • (iii) If $d\geq 5$ and $D>\frac{1}{\cos \left ( \frac{2 \pi}{d}\right )^{d}}$, then there exists a non-empty set $A_{\textrm{per}}\subset A_{\textrm{bis}}$ such that if $\alpha \in A_{\textrm{per}}$, then system (1) has periodic solutions.

• 2. If $n$ is odd, then system (1) has a unique equilibrium point. Moreover, if $d\geq 3$, then

  • (i) If $D<\frac{1}{\cos \left (\frac{\pi}{d}\right )^{d}}$, then for any $\alpha \in \left (\mathbb{R}_{+}^{*}\right )^{d}$, this equilibrium point is asymptotically stable, and all the solutions of (1) either converge to this point or to a periodic orbit.

  • (ii) If $D>\frac{1}{\cos \left (\frac{\pi}{d}\right )^{d}}$, then there exists $A_{\textrm{unst}}\subset \left (\mathbb{R}_{+}^{*} \right )^{d}$ a non-empty set such that

    • If $\alpha \in A_{\textrm{unst}}$, then this equilibrium point is unstable, and there exists a finite number of periodic solutions, among which at least one is asymptotically stable. Moreover, the set of initial conditions for which the solution converges to a periodic solution is a dense open subset of $\mathbb{R}_{+}^{d}$, and its complement, which is the set of initial conditions for which the solution converges to the equilibrium point, has Lebesgue measure zero.

    • If $\alpha \in \overline{A_{\textrm{unst}}}^{C}$, this equilibrium point is asymptotically stable, and all the solutions of (1) either converge to this point or to a periodic orbit.

In each of these cases, the sets $A_{\textrm{bis}}$, $A_{\textrm{per}}$, $A_{\textrm{unst}}$ can be explicitly expressed, as we will show in Sects. 3 and 4. Moreover, note that any Hill function (even shifted), i.e. function of the form $x \mapsto \frac{1+\lambda x^{r}}{1+x^{r}}$, with $\lambda \in \mathbb{R}_{+} \backslash \{1\}$ and $r\geq 1$, as well as linear functions, satisfy the convexity hypothesis of this theorem, which means that this result encompasses the other theoretical studies mentioned above [5–10].

It is worth noting that, when $n$ is even, the set of initial conditions for which the solution converges to a periodic orbit has Lebesgue measure zero, and can therefore hardly be reached numerically. Nevertheless, we highlight this result since, up to our knowledge, the question of the existence of such periodic solution remained open, as mentioned in [9]. Moreover, the proof of this result, which uses the stable manifold theorem, seems to us non-trivial and worthwhile. We also note that condition 1 (iii) is sufficient, but perhaps not necessary for the existence of periodic solution: in particular, this question for $d\in \{3,4\}$ remains open. Lastly, we do not know if periodic solutions do exist under the hypotheses of 2 (i) and in the second point of 2 (ii), but our study does not rule out this possibility.

We have assumed that the degradation rates of all the molecules are identical and equal to 1. As we shall see, this assumption is used in order to explicitly compute the eigenvalues of the Jacobian matrix at each of the equilibrium points of the system. One can check that all the results can easily be adapted to the case where the degradation rates are the same for each of the molecules, i.e. for system of the form $\dot{x}_{i}=\alpha _{i}f_{i}(x_{i-1})-k x_{i}, \quad i\in \{1,\ldots, d \}$, with $k>0$. More generally, the reasoning developed can be applied provided that we are able to determine the sign of the real part of the eigenvalues of the Jacobian matrix at the equilibrium points (see the matrix $M_{x}^{\alpha}$ introduced in Sect. 3.

This article, entirely dedicated to the proof of Theorem 1, is organised as follows: after characterising the equilibrium points of system (1) in Sect. 2, we prove the theorem when $n$ is even (Sect. 3), before dealing with the case where $n$ is odd, which is simpler, in the last section.

2 Characterisation of Fixed Points

Throughout this note, we assume that $f_{1}, \ldots, f_{d} \in \mathcal{C}^{3}(\mathbb{R}_{+}, \mathbb{R}_{+}^{*})$ are non-negative, monotone and that at least one of these functions is bounded and at least one of them is positive. Note that, due to the regularity of these functions, the Cauchy-Lipschitz theorem ensures the local existence and the uniqueness of the solution of this equation for any initial condition $x_{0}\in \mathbb{R}^{d}_{+}$. Furthermore, the positivity of the functions guarantees that the solutions remain in $\mathbb{R}^{d}_{+}$. Lastly, using the fact that one of them is bounded, we easily prove the global existence of solutions and the existence of a compact attractor set i.e. the existence of a compact set $K\subset \mathbb{R}^{d}_{+}$ such that for any initial condition $x_{0}\in \mathbb{R}_{+}^{d}$, there exists $T\geq 0$ such that $x(t)\in K$ for all $t\geq T$.

We now make the additional assumption that $f_{1},\ldots, f_{d}$ are $\gamma ^{1/2}$-convex, and that at least one of them is strictly $\gamma ^{1/2}$-convex, i.e. they satisfy the following definition:

Definition 1

($\mathbf{\gamma ^{1/2}-}$convexity)

Let $f \in \mathcal{C}^{3}\left (\mathbb{R}_{+}, \mathbb{R}_{+}\right )$ a non-negative and monotone function. We say that $f$ is (strictly) $\gamma ^{1/2}-$ convex if $\lvert f'\rvert >0$ and $\frac{1}{\sqrt{\lvert f' \rvert }}$ is (strictly) convex.

Note that this definition can be related to the definition of the Schwartzian derivative of $f$ defined by $S(f)=\frac{f'''}{f'}-\frac{3}{2}\left (\frac{f''}{f'}\right )^{2}$ by noting that $(\frac{1}{\sqrt{\lvert f' \rvert }})''=-\frac{1}{2} \frac{1}{\sqrt{\lvert f' \rvert }} S(f)$.

We relate here the equilibrium points of system (1) to the fixed points of an auxiliary function $\tilde{f}$. The $\gamma ^{1/2}$-convexity of the functions $f_{i}$ ensures that the number of equilibrium points cannot exceed three, and provides a criterion which characterises this exact number of equilibria. A similar approach was used (with the Schwartzian derivative) in [19] and [6].

We start by recalling some key properties of the $\gamma ^{1/2}$-convexity, which have been established in [12].

Proposition 1

Let $f, g$ be two $\gamma ^{1/2}$-convex functions, $c>0$.

• (i) $f\circ g$ and $cf$ are $\gamma ^{1/2}$-convex. Moreover, if $f$ or $g$ is strictly $\gamma ^{1/2}$-convex, then $f\circ g$ is strictly $\gamma ^{1/2}$-convex.

Let us now assume that $f$ is strictly $\gamma ^{1/2}$-convex. Then:

• (ii) $cf$ is strictly $\gamma ^{1/2}$-convex.

• (iii) $f$ has at most three fixed points.

• (iv) If all the fixed points $x$ of $f$ satisfy $f'(x)<1$, then $f$ has a unique fixed point.

• (v) If there exists a fixed point of $f$ (denoted $x$) such that $f'(x)>1$, then $f$ has exactly three fixed points and the other two fixed points (denoted $y$, $z$) satisfy $f'( y)<1$ and $f'(z)<1$.

Example 1

For any $r\geq 1$, $a,b,c,d\geq 0$ such that $ad-bc\neq 0$, the function $x\mapsto \frac{ax^{r}+b}{cx^{r}+d}$ is $\gamma ^{1/2}-$convex. Moreover, if $r>1$, then it is strictly $\gamma ^{1/2}-$convex. In particular, affine functions and Hill functions are $\gamma ^{1/2}-$convex.²

By definition, the point $\bar{x}= (\bar{x}_{1}, \ldots, \bar{x}_{d})\in \mathbb{R}_{+}^{d}$ is an equilibrium point of (1) if and only if

$$\begin{aligned} \textstyle\begin{cases} \bar{x}_{1}=\alpha _{1}f_{1}(\bar{x}_{d}), \\ \bar{x}_{2}=\alpha _{2}f_{2}(\bar{x}_{1}), \\ \vdots \\ \bar{x}_{d} = \alpha _{d} f_{d}(\bar{x}_{d-1}). \end{cases}\displaystyle \Longleftrightarrow \quad \textstyle\begin{cases} \bar{x}_{1}=\alpha _{1}f_{1}(\bar{x}_{d}), \\ \bar{x}_{2}=\alpha _{2} f_{2}(\bar{x}_{1}), \\ \vdots \\ \bar{x}_{d-1}=\alpha _{d-1} f_{d-1}(\bar{x}_{d-2}), \\ \bar{x}_{d}=\alpha _{d} f_{d} \circ \alpha _{d-1}f_{d-1}\circ \cdots \circ \alpha _{1} f_{1}(\bar{x}_{d}). \end{cases}\displaystyle . \end{aligned}$$

Thus, the number of equilibrium points of (1) is equal to the number of fixed points of $\tilde{f}:=\alpha _{d} f_{d} \circ \cdots \circ \alpha _{1} f_{1}$. Since $\tilde{f}(0)>0$ and $\tilde{f}$ is bounded, it proves in particular that (1) has at least one equilibrium point. Moreover, the hypothese on the functions $f_{1},\ldots f_{d}$ ensure that all the equilibrium points are in $\left (\mathbb{R}_{+}^{*}\right )^{d}$. Note that, according to the first two properties of Proposition 1, $\tilde{f}$ is strictly $\gamma ^{1/2}-$ convex, and a direct computation shows that

$$ \tilde{f}'(\bar{x}_{d})=\prod _{i=1}^{d}{\alpha _{i} f_{i}'( \bar{x}_{i-1})}. $$

We can thus apply the fourth and the fifth properties of Proposition 1 to $\tilde{f}$ to derive the following lemma:

Lemma 1

Let us denote, for all $\alpha , x \in \mathbb{R}_{+}^{d}$, $p_{x}^{\alpha}=\prod _{i=1}^{d}{\alpha _{i} f_{i}'( x_{i-1})}$.

• If all the equilibrium points $\bar{x}$ of system (1) satisfy $p_{\bar{x}}^{\alpha}<1$, then this system has a unique equilibrium point.

• If there exists an equilibrium point of system (1) (denoted $\bar{x}$) such that $p_{\bar{x}}^{\alpha}>1$, then this system has exactly three equilibrium points, and the other two points (denoted $\bar{y},\bar{z}$) satisfy $p_{\bar{y}}^{\alpha}<1$, $p_{\bar{z}}^{\alpha}<1$.

Hence, the value of $p^{\alpha}_{\bar{x}}$ characterises the number of fixed points the system has. In the following section, we show that it also determines the dimension of the basin of attraction of $\bar{x}$.

3 Even Number of Decreasing Functions

In the case where $n$ is even, one easily checks that system (1) is an irreducible type K monotone system in the sense defined in [9]. As seen in the previous section, this system has a finite number of equilibrium points (at most three), and a compact attractor set: thus, we can apply Theorem 2.5 and Theorem 2.6 of [9] which prove that the union of the basins of attraction of the equilibrium points is dense, and that the complement of this set has Lebesgue measure zero.

In this section, we complete this result in two ways:

• We determine, for given functions $f_{1},\ldots, f_{d}$, a set of parameters $A_{\textrm{bis}}$ such that system (1) is bistable (i.e. has exactly two asymptotically stable equilibrium points) if $\alpha \in A_{\textrm{bis}}$, and monostable (i.e. has exactly one asymptotically stable equilibrium point) if $\alpha \in \overline{A_{\textrm{bis}}}^{C}$.

• We determine a set $A_{\textrm{per}}\subset A_{\textrm{bis}}$ such that system (1) has some periodic solutions if $\alpha \in A_{\textrm{per}}$.

Note that this last point does not mean that periodic solutions do not exist when $\alpha \notin A_{per}$, and that, in all cases, the set of initial conditions for which the solution converges to a periodic solution has Lebesgue measure zero (as a corollary of [9]).

In order to prove these two points, we determine the dimension of the basin of attraction of an equilibrium point $\bar{x}$, as a function of $p_{\bar{x} }^{\alpha}$. For any equilibrium point $\bar{x}$, we denote its basin of attraction $B_{\bar{x}}$. Our reasoning is based on the stable manifold theorem (the proof of which can be found for instance in [13]), that we recall:

Theorem 1

(Stable manifold)

Let $F\in \mathcal{C}^{1}(\mathbb{R}^{d}, \mathbb{R}^{d})$ a vector field, and let $\bar{x} \in \mathbb{R}^{d}$ such that $F(\bar{x})=0_{\mathbb{R}^{d}}$. If $\bar{x}$ is a hyperbolic equilibrium point, i.e. if all the eigenvalues of $\textrm{Jac}\, F(\bar{x})$ have a non-zero real part, then the basin of attraction of $\bar{x}$ is a manifold of dimension $m$, where $m$ is the number of eigenvalues of $\textrm{Jac}\, F(\bar{x})$ with a negative real part.

In order to apply this theorem, we need to compute the eigenvalues of the Jacobian matrix associated to system (1) which writes, at a given point $x\in \mathbb{R}^{d}$

$$\begin{aligned} M_{x}^{\alpha}= \begin{pmatrix} -1 & 0 & \cdots & 0 & \alpha _{1} f_{1}'( x_{d}) \\ \alpha _{2} f_{2}'(x_{1}) & -1 & 0 & \cdots &0 \\ 0 & \ddots & \ddots & \ddots & \vdots \\ \vdots & \ddots &\ddots &\ddots & 0 \\ 0 & \cdots & 0 &\alpha _{d} f_{d}'(x_{d-1}) & -1 \end{pmatrix}. \end{aligned}$$

Thus, the characteristic polynomial of $M_{x}^{\alpha}$ is easily computed to be

$$ (-1)^{d} \left ((\lambda +1)^{d}-\prod _{i=1}^{d}{\alpha _{i} f_{i}'(x_{i-1})} \right )=(-1)^{d} \left ((\lambda +1)^{d}-p_{x}^{\alpha}\right ). $$

Since $n$ is even, $p_{x}^{\alpha}>0$ and hence the spectrum of $M_{x}^{\alpha}$ is given by

$$ \mathrm{Sp}\left (M_{x}^{\alpha}\right )=\left \{(p_{x}^{\alpha})^{1/d}e^{2k \pi i/d}-1, k \in \{0,\ldots, d-1\}\right \}. $$

We deduce that

• (i) If $p_{x}^{\alpha}<1$, then all the eigenvalues of $M_{x}^{\alpha}$ have a negative real part.

• (ii) If $d\in \{3,4\}$ and $p_{x}^{\alpha}>1$, or if $d\geq 5$ and $p_{x}^{\alpha}\in \left (1, \frac{1}{\cos \left ( \frac{2\pi}{d}\right )^{d}}\right )$, then $M_{x}^{\alpha}$ has exactly $d-1$ eigenvalues with a negative real part, and one with a positive real part.

• (iii) If $d\geq 5$, and if $p_{x}^{\alpha}>\frac{1}{\cos \left ( \frac{2\pi}{d}\right )^{d}}$, then $M_{x}^{\alpha}$ has at most $d-3$ eigenvalues with a negative real part. Moreover, if $d\leq 8$, or $d\in [\!\![4j+1,4j+4 ]\!\!]$ ($j \in \mathbb{N}\backslash \{0,1\}$), and for any $k\in \{2,\ldots, j\}$, $p_{x}^{\alpha}\neq \frac{1}{\cos \left ( \frac{2\pi k}{d}\right )^{d}}$, then all eigenvalues of $M_{x}^{\alpha}$ have a non-zero real part (where for all $a,b\in \mathbb{R}$, $a< b$, $[\!\![a,b ]\!\!]=[a,b]\cap \mathbb{N}$.)

Thus, the stable manifold theorem yields

Lemma 2

Let us assume that $n$ is even, and let $\bar{x}\in \mathbb{R}^{d}$ be an equilibrium point of (1).

• (i) If $p_{\bar{x}}^{\alpha}<1$, then $\mathrm{dim}\left (B_{\bar{x}}\right )=d$, i.e. $B_{\bar{x}}$ is an open set.

• (ii) If $d\in \{3,4\}$ and $p_{\bar{x}}^{\alpha}>1$, or if $d\geq 5$ and $p_{\bar{x}}^{\alpha}\in \left (1, \frac{1}{\cos \left ( \frac{2\pi}{d}\right )^{d}}\right )$, then $\mathrm{dim}\left (B_{\bar{x}}\right )=d-1$.

• (iii) If $d\geq 5$, $p_{\bar{x}}^{\alpha}> \frac{1}{\cos \left ( \frac{2\pi}{d}\right )^{d}}$, and $p_{\bar{x}}^{\alpha }\notin S_{d}$, with

  $$\begin{aligned} S_{d}:= \textstyle\begin{cases} \quad \emptyset \quad &\textrm{if} \quad d\in \{5,6,7,8\} \\ \left \{ \frac{1}{\cos \left ( \frac{2\pi k}{d}\right )^{d}}, k\in \{2,\ldots j\} \right \} \quad &\textrm{if}\quad d\in [\!\![4j+1,4j+4 ]\!\!], \quad j\geq 2 \end{cases}\displaystyle , \end{aligned}$$

  then $\mathrm{dim}\left (B_{\bar{x}}\right )\leq d-3$.

Moreover, we easily check that, in all cases, $M_{\bar{x}}^{\alpha}$ has an odd number of eigenvalues with a positive real part. Thus, since $n$ is even, we get from the main theorem of [14] that any solution converges to an equilibrium point or to a periodic orbit.

We will now use Lemmas 2 and 3 to prove Theorem 1 in a more precise form which specifies the sets $A_{\mathrm{bis}}$ and $A_{\mathrm{per}}$, in the case where $n$ is even. Before stating it, we give a last lemma linking the dimension of the basin of attraction of the unstable equilibrium point to the existence of divergent solutions (which thus converge to periodic solutions), in the bistable case.

Lemma 3

Let ‘$\dot{x} =F(x)$’ be an ODE which has exactly three equilibrium points, (denoted $\bar{x}$, $\bar{y}$, $\bar{z}$) assumed hyperbolic, and let us assume that for all initial condition $x_{0}\in \mathbb{R}^{d}$, the solution of this ODE is defined on $\mathbb{R}_{+}$ and converges. If $\bar{y}$, $\bar{z}$ are asymptotically stable, then $\mathrm{dim}\left (B_{\bar{x}}\right )=d-1$.

Proof

First, let us note that, according to the stable manifold theorem, $B_{\bar{y}}$ and $B_{\bar{z}}$ are two open sets, and that $B_{\bar{x}}$ is a manifold. Since, by hypothesis, all the solutions converge, $\mathbb{R}_{+}^{d}=B_{\bar{x}}\cup B_{\bar{y}} \cup B_{\bar{z}}$.

Since $\mathbb{R}_{+}^{d}$ is a connected space, this is possible only if $\mathrm{dim}\left (B_{\bar{x}}\right )=d-1$, as proved in [20] (Corollary 1 of Theorem IV 4). □

Before stating our theorem, let us introduce the functions $\Gamma$ and $G$, defined for any $x\in \mathbb{R}_{+}^{d}$ by $\Gamma (x)=\left ( \frac{x_{1}}{f_{1}(x_{d})}, \frac{x_{2}}{f_{2}(x_{1})}, \ldots, \frac{x_{d}}{f_{d}(x_{d-1})} \right )$ and $G(x)=\prod _{i=1}^{d}{\frac{x_{i}f'(x_{i})}{f(x_{i})}}$, and the sets $E_{\mathrm{bis}}=\left \{ x\in \mathbb{R}_{+}^{d}: G(x) > 1 \right \}$ and if $d\geq 5$, $E_{\textrm{per}}=\left \{ x\in \mathbb{R}_{+}^{d}: G(x)> \frac{1}{\cos \left (\frac{2 \pi }{d}\right )^{d}}, G(x)\notin S_{d} \right \}$. Note that $E_{\textrm{per}}\subset E_{\textrm{bis}}$.

Theorem 2

Let us assume that $n$ is even.

• (i) If $\alpha \in \overline{\Gamma (E_{\mathrm{bis}})}^{C}$, then (1) has a unique equilibrium point which is globally asymptotically stable.

• (ii) If $\alpha \in \Gamma (E_{\mathrm{bis}})$, then (1) has exactly three equilibrium points, among which two are asymptotically stable and one is unstable. Moreover, the union of the basins of attraction of the two stable equilibria is a dense open subset of $\mathbb{R}_{+}^{d}$.

• (iii) If $d\geq 5$, and if $\alpha \in \Gamma (E_{\mathrm{per}})$, then there exist periodic solutions of (1).

Proof

First, let us note that, according to the definitions of $\Gamma$ and $G$, $\bar{x}\in \mathbb{R}_{+}^{d}$ is a fixed point of system (1) if and only if $\alpha =\Gamma (\bar{x})$, and that for any equilibrium point $\bar{x}$ of (1),

$$ p_{\bar{x}}^{\alpha}=p_{\bar{x}}^{\Gamma (\bar{x})}=G(\bar{x}). $$

• (i) Let us assume that $\alpha \in \overline{\Gamma (E_{\mathrm{bis}})}^{C}$, and let $\bar{x}$ be a fixed point of (1). Since $\Gamma (\overline{E_{\mathrm{bis}}}) \subset \overline{\Gamma (E_{\mathrm{bis}})}$, and $\alpha =\Gamma (\bar{x})$, $\bar{x} \in \overline{\Gamma (E_{\mathrm{bis}})}^{C}$, which means, by definition of $E_{\mathrm{bis}}$, that $p^{\alpha}_{\bar{x}}=G(\bar{x})<1$. Since this equality holds for any equilibrium point, we conclude by Lemma 1, that system (1) has a unique equilibrium point, which is asymptotically stable, by Lemma 2. Since (1) is an irreducible type K monotone system, this unique equilibrium is in fact globally asymptotically stable [9].

• (ii) Let us assume that $\alpha \in \Gamma (E_{\textrm{bis}})$. Then, there exists $\bar{x} \in E_{\textrm{bis}}$ such that $\alpha =\Gamma (\bar{x})$. The point $\bar{x}$ is thus an equilibrium point of (1) which satisfies $p_{\bar{x}}^{\alpha}=G(\bar{x})>1$. Therefore, Lemmas 1, 2 and the result of [9] mentioned at the beginning of the section yield the result.

• (iii) Let us assume that $\alpha \in \Gamma (E_{\textrm{per}})$. Since $E_{\textrm{per}}\subset E_{\textrm{bis}}$, (1) has exactly three equilibrium points, denoted $\bar{x}$, $\bar{y}$ and $\bar{z}$, with $\bar{x} \in E_{\textrm{per}}$, and $\bar{y}$ and $\bar{z}$ which are asymptotically stable. By definition of $E_{\textrm{per}}$, $\textrm{dim}\left (B_{\bar{x}}\right )\leq d-3$. By the contrapositive of Lemma 3, system (1) has some divergent solutions, which thus converge to a periodic orbit, according to [14].

 □

Remark 1

As mentioned above, this theorem is a more precise version of Theorem 1: we find the statement of the latter by defining $A_{\textrm{bis}}=\Gamma \left (E_{\textrm{bis}} \right )$ and $A_{\textrm{per}}=\Gamma \left ( E_{\textrm{per}} \right )$, and by noting that $E_{\textrm{bis}}$ (resp. $E_{\textrm{per}}$) is empty if and only if $D\leq 1$ (resp. $D\leq \frac{1}{\cos \left ( \frac{2\pi}{d} \right )^{d}}$).

4 Odd Number of Decreasing Functions

We now deal with the case where $n$ is odd. This case is simpler, since the system has a unique equilibrium point under this hypothesis. Nevertheless, we make weaker conclusions regarding the global behaviour of the system, since it is not an irreducible type K monotone system (see [9]). Thus, we simply study the linearised system at the neighbourhood of the equilibrium point, and we conclude with [14], which guarantees that the solutions either converge to this equilibrium point, or to a periodic orbit.

Let us denote $E_{unst}:=\left \{ x\in \mathbb{R}^{d}: G(x)<- \frac{1}{\cos \left ( \frac{\pi}{d} \right )^{d}} \right \}$.

We get the following result:

Theorem 3

Let us assume that $n$ is odd, and that $d\geq 3$. Then, system (1) has a unique equilibrium point. Moreover,

• (i) If $\alpha \in \overline{\Gamma \left (E_{\textrm{unst}}\right )}^{C}$, then this equilibrium point is asymptotically stable. Moreover, all the solutions of (1) either converge to this point or to a periodic orbit.

• (ii) If $\alpha \in \Gamma (E_{\textrm{unst}})$, then this equilibrium point is unstable, Moreover, the set of initial conditions for which the solution converges to a periodic solution is a dense open subset of $(\mathbb{R}_{+}^{*})^{d}$, and its complement, which is the set of initial conditions for which the solution converges to the equilibrium point, has Lebesgue measure zero.

Proof

We use the same notations as in the previous section. Since $n$ is odd, $\tilde{f}$ is decreasing with $\tilde{f}(0)>0$, it has a unique fixed point, which implies that system (1) has a unique equilibrium point, that we denote $\bar{x}$. This point is asymptotically stable if all the eigenvalues of $M_{\bar{x}}^{\alpha}$ have a negative real part, and unstable if at least one of these eigenvalues has a positive real part. Since $n$ is odd,

$$ \mathrm{Sp}\left (M_{\bar{x}}^{\alpha}\right )=\left \{(\lvert p_{x}^{ \alpha}\rvert )^{1/d}e^{(2k+1)\pi i/d}-1, k \in \{0,\ldots, d-1\}\right \}, $$

which implies that $\bar{x}$ is stable if $\lvert p_{\bar{x}}^{\alpha }\rvert < \frac{1}{\cos \left ( \frac{\pi}{d} \right )^{d}}$, and unstable if $\lvert p_{\bar{x}}^{\alpha }\rvert > \frac{1}{\cos \left ( \frac{\pi}{d} \right )^{d}}$. We conclude by noting that $\lvert p_{\bar{x}}^{\alpha}\rvert =-p_{\bar{x}}^{\alpha}=- p_{\bar{x}}^{ \Gamma (\bar{x})} =-G(\bar{x})$, and by applying the main theorem of [14] for the first point, and Theorem 4.3 of this same article for the second one. □

We recover the result of Theorem 1 by defining $A_{\textrm{unst}}=\Gamma (E_{\textrm{unst}})$.

5 Perspectives

This article was dedicated to the study of the global stability of cyclic feedback loops of the form

$$\begin{aligned} \forall i \in \{1, \dots , d\}, \quad \dot{x}_{i}=\alpha _{i}f_{i}(x_{i-1})-x_{i}. \end{aligned}$$

We have shown that, under a convexity hypothesis on the functions $f_{1}, \dots , f_{d}$, the global stability of this system depends on the number of decreasing functions among $f_{1}, \dots , f_{d}$ (denoted $n$), the value of $D=\prod _{k=1}^{d}{\underset{x>0}{\sup}\left \lvert \frac{xf_{k}'(x)}{f_{k}(x)}\right \rvert}$ and on the parameters $\alpha _{1}, \dots ,\alpha _{d}$. The results obtained are synthesised in Theorem 1. Nevertheless, several points remain to be clarified.

First, in the case where $n$ is even, we only provided a sufficient condition ensuring the existence of periodic solutions. In particular, the question of the existence of periodic solutions in the cases $d=3$ and $d=4$ remains open. In the case where $n$ is odd, similar questions arise: under the conditions for which the unique equilibrium point is stable, the existence of periodic orbits has not be ruled out, while numerical simulations seem to show that the equilibrium point is globally asymptotically stable in this case. As well, in the cases where this point is unstable, we do not have a criterion which grantees the existence of a unique periodic orbit.

Moreover, let us note that, in Theorem 1, all the inequalities for $D$ are strict, and that we do not consider the cases where $\alpha$ is a boundary point of $A_{bis}$, $A_{per}$ or $A_{unst}$. Indeed, in these cases, one of the equilibrium points is not hyperbolic, which prevents us from determining its stability, the possible existence of other equilibrium points and applying the stable manifold theorem. The global stability of the system is thus not clear in these cases.

Throughout this paper, we have assumed that the degradation rates are the same for all the molecules of the network. Computing the eigenvalues of the matrix $M_{x}^{\alpha}$ (see section (3)) without this hypothesis is intricate, and this is why we do not generalise our study to more general system of the form

$$ \forall i \in \{1,\ldots, d\}, \quad \dot{x}_{i}=\alpha _{i} f_{i} (x_{i-1})-k_{i} x_{i}, $$

which would be the natural next step in understanding these models.

From a modelling point of view, one of the main limitations of this model is that it does not take into account self-regulation i.e. the fact that the synthesis rate of a given molecule is regulated by its own level. Mathematically speaking, taking self-regulation into account means studying systems of the form

$$ \forall i\in \{1,\ldots, d\}, \quad \dot{x}_{i}=\alpha _{i} f_{i}(x_{i-1}) \phi _{i}(x_{i})-x_{i}, $$

where the functions $\phi _{1}, \dots \phi _{d}$ are increasing in the case of self-activation, and decreasing in the case of self-inhibition. The study of such systems seems out of reach with our method: in particular, the characterisation of the fixed points given in Sect. 2 does not longer hold. Numerical simulations have shown that such systems with self-activation can be tristable [21]: determining a criterion ensuring the tristability of such models and determining the maximal number of stable equilibria it can have is another natural perspective in the understanding of cyclic feedback loops.