## 3 Results
### The case \(\tau=0\)  
Consider the ODE system obtained from (8) by setting \(\tau=0\):  
\[ \begin{split}\dot{x}(t)&=\alpha-\gamma x(t)-\delta x (t)y(t)+\beta\frac{x(t)^{n}}{x_{th}^{n}+x(t)^{n}}\\ \dot{y}(t)&=\rho\frac{x(t)^{m}}{y_{th}^{m}+x(t)^{m} }-\omega y(t),\end{split}  \tag{12}\]  
It is important to know the dynamics of (12), because for small delays (\(\tau>0\)), the DDE system (8) will very likely behave as the ODE system (12), _cf._[24]. Observe that the ODE system (12) and the DDE system (8) have exactly the same equilibrium points. In general, a DDE system and the associated ODE system have the same number of fixed points, but if the delay appears in the coefficients, the fixed points of the DDE system could be shifted with respect to those of the ODE system. The presence of a negative feedback in (12) leads to the hypothesis that oscillatory solutions might show up. We investigate local properties of the steady states, looking for Hopf-bifurcations. For linear (local) stability analysis, we compute the Jacobian matrix of system (12),  
\[J= \begin{pmatrix}-\gamma-\delta\tilde{y}+\beta\frac{nx_{th}^{n}x^{n-1}}{(x_{ th}^{n}+x^{n})^{2}}&-\delta\tilde{x}\\ \rho\frac{ny_{th}^{m}x^{n-1}}{(x_{th}^{m}+x^{m})^{2}}&-\omega\end{pmatrix} .\]  
In the special case \(n=m=2\) and \(x_{th}=y_{th}\), we have  
\[J= \begin{pmatrix}-\gamma-\delta\tilde{y}+\beta\frac{2x_{th}^{2}x^{n}}{(x_{th}^ {2}+x^{2})^{2}}&-\delta\tilde{x}\\ \rho\frac{2x_{th}^{2}x^{n}}{(x_{th}^{2}+x^{2})^{2}}&-\omega\end{pmatrix}  \tag{13}\]The trace and the determinant of (13) at a stationary point at \((\bar{x},\bar{y})\), with \(\bar{y}\) in (10), are given by  
\[\mathrm{Tr}(J) = \frac{2\beta x_{th}^{2}\bar{x}}{(x_{th}^{2}+\bar{x}^{2})^{2}}-\gamma -\delta\frac{\rho}{\omega}\frac{\bar{x}^{2}}{x_{th}^{2}+\bar{x}^{2}}-\omega\] \[\det(J) = -\omega\left(\frac{2\beta x_{th}^{2}\bar{x}}{(x_{th}^{2}+\bar{x}^ {2})^{2}}-\gamma-\delta\frac{\rho}{\omega}\frac{\bar{x}^{2}}{x_{th}^{2}+\bar{ x}^{2}}\right)+\delta\bar{x}\frac{2\rho x_{th}^{2}\bar{x}}{(x_{th}^{2}+\bar{x}^{2})^{ 2}}.\]  
If there was only one stationary point and this one is a repellor, one can use the fact that all solutions are bounded and stay positive, then the Poincare-Bendixson theorem yields the existence of periodic solutions. For a Hopf-bifurcation, necessary conditions are \(\mathrm{Tr}(J)=0\) and \(\Delta(J)=\mathrm{Tr}(J)^{2}-4\det(J)<0\). We choose \(\delta\), the Lactonse activity, as bifurcation parameter. From the trace condition, we get  
\[\delta\left[\frac{\rho}{\omega}\bar{x}^{2}(x_{th}^{2}+\bar{x}^{2})\right]-2 \beta x_{th}^{2}\bar{x}+\gamma(x_{th}^{2}+\bar{x}^{2})^{2}+\omega(x_{th}^{2}+ \bar{x}^{2})^{2}=0.\]  
We solve for \(\delta\) and obtain  
\[\delta=\delta(\bar{x}) = \frac{2\beta x_{th}^{2}\bar{x}-(\gamma+\omega)(x_{th}^{2}+\bar{x} ^{2})^{2}}{\frac{\rho}{\omega}\bar{x}^{2}(x_{th}^{2}+\bar{x}^{2})} \tag{14}\] \[= \frac{-\omega(\gamma+\omega)(x_{th}^{2}+\bar{x}^{2})}{\rho\bar{x} ^{2}}+\frac{2\beta x_{th}^{2}\omega}{\rho\bar{x}(x_{th}^{2}+\bar{x}^{2})}.\]  
Note that, in turn, \(\bar{x}\) also depends on \(\delta\), _cf._ Equation (11). Neglecting this for a minute, we observe that \(\lim\limits_{\bar{x}\to\infty}\delta(\bar{x})=-(\gamma+\omega)\frac{\omega}{ \rho}<0\), whereas \(\lim\limits_{\bar{x}\to 0}\delta(\bar{x})\to+\infty\). Due to the intermediate value theorem, there exists a \(\bar{x}>0\), such that \(\delta(\bar{x})>0\) for \(\bar{x}>\bar{x}\). It is possible to choose \(\bar{x}\) as the smallest positive solution of \((\gamma+\omega)(x_{th}^{2}+\bar{x}^{2})^{2}>2\beta x_{th}^{2}\bar{x}\). If a \(\bar{x}=\bar{x}(\delta)>0\) satisfies this condition, then \(\mathrm{Tr}(J)=0\) at \((\bar{x},\bar{y})\). In the next step, we check the discriminant condition (\(\Delta(J)<0\)), or equivalently, \(\det(J)>0\), as for the Hopf-bifurcation we need simultaneously \(\mathrm{Tr}(J)=0\).