# Blow-Up Dynamics and Synchronization in Tri-Trophic Food Chain Models
## 6 Possible Causes of a Lack of Synchronization  
We investigate the possible causes of no synchronization, as seen in Figure 6. Although the HP model has bounded solutions for any initial condition [20], this is not true in the case of the UR model. The UR model possesses the dynamics of a finite-time blow-up [24]. We present a few details for completeness. We first prove a variation of a classical result for this system:  
Figure 5: GS with small initial data. (**a**) \(y_{1}-g_{1}\) plot showing a 1:1 correlation. (**b**) \(y_{2}-g_{2}\) plot showing a 1:1 correlation. (**c**) \(y_{3}-g_{3}\) plot showing a 1:1 correlation. (**d**) Simulation showing time evolution for \(x_{1}\) and \(y_{1}\) after transients die out. (**e**) Simulation showing time evolution for \(g_{1}\) and \(y_{1}\) after transients die out. (**f**) Time series plot showing no blow-up in \(x_{3}\) after transients die out. Figure 6: No GS with large initial data. (**a**) \(y_{1}-g_{1}\) plot showing no 1:1 correlation. (**b**) \(y_{2}-g_{2}\) plot showing no 1:1 correlation. (**c**) \(y_{3}-g_{3}\) plot showing no 1:1 correlation. (**d**) Simulation showing time evolution for \(x_{1}\) and \(y_{1}\). (**e**) Simulation showing time evolution for \(g_{1}\) and \(y_{1}\). (**f**) Time series plot showing blow-up occurring in \(x_{3}\) and estimated at time \(T^{*}\approx 0.00048\). **Theorem 1**. : _The solution \(x_{3}(t)\) to the top predator equation in Equation (5) blows up in a finite time such that_  
\[\limsup_{t\to T^{*}<\infty}|x_{3}(t)|=\infty, \tag{14}\]  
_for the initial data \((x_{1}(0),x_{2}(0),x_{3}(0))\), which is sufficiently large._  
**Remark 2**. : _The proof follows [24]._  
Proof. : Consider Equation (5) with positive initial conditions. By integrating the top predator equation for \(x_{3}\), we obtain  
\[-\frac{1}{x_{3}}+\frac{1}{x_{3}(0)}=-ct+w_{3}\int_{0}^{t}\frac{ds}{x_{1}(s)+x_ {2}(s)+D_{3}}\]  
which gives  
\[x_{3}=\frac{1}{\frac{1}{x_{3}(0)}-ct+w_{3}\int_{0}^{t}\frac{ds}{x_{1}(s)+x_{2}( s)+D_{3}}}.\]  
We can then show the following function:  
\[\psi(t)=\frac{1}{\frac{1}{x_{3}(0)}-ct+w_{3}\int_{0}^{t}\frac{ds}{x_{1}(s)+x_{ 2}(s)+D_{3}}}\to 0\;\text{as}\;t\to T^{*}<\infty. \tag{15}\]  
This will then show that the solution \(x_{3}\) will blow up at the finite time \(t=T^{*}\). Essentially, for a chosen \((x_{1}(0),x_{2}(0))\) that is sufficiently large, there exists a \(\delta>0\) such that  
\[\frac{1}{\frac{1}{x_{3}(0)}-ct+w_{3}\int_{0}^{t}\frac{ds}{x_{1}(s )+x_{2}(s)+D_{3}}}\] \[<\frac{1}{x_{3}(0)}-\frac{c}{2}t,\quad\text{for all}\;t\in(0, \delta). \tag{16}\]  
This is because under the continuity of the state variables \(x_{1},x_{2}\), we have the following for when \(\delta\) is sufficiently small and \(t\in(0,\delta)\):  
\[w_{3}\int_{0}^{t}\frac{ds}{x_{1}(s)+x_{2}(s)+D_{3}}<\frac{ct}{2} \tag{17}\]  
and thus  
Figure 5:0  
If \(x_{3}(0)\) is then chosen to be sufficiently large, then we can find \(T^{**}\in(0,\delta)\) such that  
Figure 5:1  
Now, through application of the classical intermediate value theorem on the continuous function \(\psi\), we obtain the existence of some \(T^{*}\in(0,\delta)\), \(T^{*}<T^{**}\), s.t \(\psi(T^{*})=0\). This implies that \(x_{3}(t)\), the solution to the top predator equation (Equation (5)), blows up in a finite time at \(t=T^{*}\), and the theorem is proven. **Remark 3**. : _Note that an explicit representation of the blow-up time \(T^{*}\) is derived in terms of the initial conditions. Essentially, \(T^{*}=\frac{2}{cx_{3}(0)}\), and thus the larger the initial condition for the top predator \(x_{3}\) in Equation (5), the quicker the blow-up occurs. Additionally, through the \"large\" initial data in Theorem 1, we mean we can find a constant large \(C\) such that \(||(x_{1}(0),x_{2}(0),x_{3}(0))||_{\infty}>C\)._  
**Theorem 2**. : _The models in Equations (5) and (6) cannot synchronize for arbitrarily large initial conditions, where Equation (5) is considered the driver system._  
Proof. : We proceed by contradiction.