Erratum to: Stable periodicity and negative circuits in differential systems

1 Erratum to: J. Math. Biol. (2011) 63:593–600 DOI 10.1007/s00285-010-0388-y

In a private communication, Frederic Beck (University of Mainz) pointed out that, in the originally published article, when proving that $\varOmega$ contains a stable periodic solution we use a wrong argument, the following: $(\partial f_i/\partial x_i)(x)<0$ for all $x\in \varOmega$, $i=1,2$. Indeed, for instance, if $x_1\in ]2-\varepsilon ,2+\varepsilon [$ and $x_2\in \times [1+\varepsilon ,3-\varepsilon ]\setminus ]2-\varepsilon ,2+\varepsilon [$ then $(\partial f_i/\partial x_i)(x)=4\varphi _2'(x_1)-1$, and this term can not be less than zero for all $x_1\in ]2-\varepsilon ,2+\varepsilon [$ because $\varphi '_2$ has to be greater than $1/2\varepsilon >1$ somewhere in this region. So there are $x\in \varOmega$ with $(\partial f_i/\partial x_i)(x)>0$.

However, using slightly more involved arguments, we proved here that the system is still a counter-example of Conjecture 2$'$. More precisely, we prove that if $\varepsilon \le 1/8$ then there is a stable periodic solution in the domain $\varOmega '=[0,4]^2\!\setminus \!\Gamma$, where $\Gamma$ is the interior of the convex hull of the set containing the points $A=(1-\varepsilon ,3-\varepsilon )$, $B=(2-\varepsilon ,3+\varepsilon )$, $C=(3-\varepsilon ,3+\varepsilon )$, $D=(3+\varepsilon ,2+\varepsilon )$, $E=(3+\varepsilon ,1+\varepsilon )$, $F=(2+\varepsilon ,1-\varepsilon )$, $G=(1+\varepsilon ,1-\varepsilon )$, and $H=(1-\varepsilon ,2-\varepsilon )$; see Fig. 1 for an illustration.

Fig. 1

The gray region is an illustration of $\varOmega '$

First, since $\varOmega '\subseteq \varOmega$, there is no equilibrium point in $\varOmega '$. Suppose now that $\varepsilon \le 1/8$, and let us prove that all the solutions starting in $\varOmega '$ remain in $\varOmega '$. As showed in the originally published article, no solution starting in $[0,4]^2$ leaves $[0,4]^2$, thus it is sufficient to prove that no solution starting in $\varOmega '$ reaches the interior of the convex hull $\Gamma$. Consider first the line segment $L$ with endpoints $A$ and $B$. For all $x\in L$ we have

$$\begin{aligned} f_1(x)&= 4\varphi _3(x_2)-x_1\le 4-x_1\le 3+\varepsilon \\ f_2(x)&= 4-x_2\ge 1-\varepsilon . \end{aligned}$$

Thus for all $x\in L$ the scalar product between $f(x)$ and the vector $v=(-2\varepsilon ,1)$ is at least $-2\varepsilon (3+\varepsilon )+(1-\varepsilon )=1-7\varepsilon -2\varepsilon ^2$, and this term is positive since $\varepsilon \le 1/8$. Since $v$ is orthogonal to $L$ and is pointing outside $\Gamma$, this means that if a solution starts in $\varOmega '$, then it cannot reach $\Gamma$ by crossing the line segment $L=AB$. Also, for all $x$ that lies in the line segment $BC$ we have $f_2(x)=1-\varepsilon >0$. Thus if a solution starts in $\varOmega '$, then it cannot reach $\Gamma$ by crossing $BC$. Reasoning similarly with the segments $CD$, $DE$, $EF$, $FG$, $GH$, and $HA$, we deduce that all the solutions starting in $\varOmega '$ remains in $\varOmega '$. Thus, following the Poincaré–Bendixon theorem, there exists a periodic solution $\psi$ of period $T>0$ starting in $\varOmega '$.

Finally, let us prove that $\psi$ is stable. For all $x\in {\mathbb {R}}^2$ we have $(\partial f_1(x)/\partial x_1)(x)=4\varphi _2'(x_1)(\varphi _1(x_2)-\varphi _3(x_2))-1$. Thus $(\partial f_1(x)/\partial x_1)(x)\ge 0$ implies $4\varphi _2'(x_1)(\varphi _1(x_2)-\varphi _3(x_2))>0$ which implies that $x$ belongs to the domain $]2-\varepsilon ,2+\varepsilon [\times ]1-\varepsilon ,3+\varepsilon [$ which is disjoint from $\varOmega '$. Thus $(\partial f_1/\partial x_1)(x)<0$ for all $x\in \varOmega '$, and we prove with similar arguments that $(\partial f_2/\partial x_2)(x)<0$ for all $x\in \varOmega '$. Thus

$$\begin{aligned} \int _0^T \frac{\partial f_1}{\partial x_1}(\psi (t))+ \frac{\partial f_2}{\partial x_2}(\psi (t))\mathrm{d}t<0 \end{aligned}$$

and we deduce that $\psi$ is stable.