# Blow-Up Dynamics and Synchronization in Tri-Trophic Food Chain Models
## 7 Discussion and Conclusions  
Real-world species are composed of multiple populations distributed over different geographical locations coupled by migration. Chaotic dynamics reduces the degree of synchrony among populations and thus reduces the chance of all of them being wiped out in the event of an exogenous stochastic perturbation [31]. In the current manuscript, we explored synchronization phenomena in two different but related model population systems. The existence of chaos in the linear version of the UR model suggests that ecological chaos has a larger role to play than previously thought. The effect of animal movements among different patches is replaced by prey-predator interactions. No synchronization of chaotic dynamics in the linear version of the UR model with the HP model for large initial populations affirms that generalist predators are major contributors toward the balance of natural food webs. Synchronization of chaotic dynamics in UR and HP model systems is dependent on the initial data (i.e., initial population numbers). For small initial data, the UR and HP models synchronized as well as the modified UR and HP models after coupling. This is seen in  
Figure 8: GS for modified UR model and HP model with initial data \([x_{1}(0),x_{2}(0),x_{3}(0)]=[0.001,0.001,0.001]\) and \([y_{1}(0),y_{2}(0),y_{3}(0)]=[0.001,0.001,0.001]\). (**a**) Phase plot for \(x_{1}\) and \(x_{2}\). (**b**) Phase plot for \(y_{1}\) and \(y_{2}\). (**c**) Time series plot for \(x_{3}\) and \(y_{3}\). (**d**) \(y_{1}-s_{1}\) plot showing 1:1 correlation. (**e**) \(y_{2}-s_{2}\) plot showing 1:1 correlation. (**f**) \(y_{3}-s_{3}\) plot showing 1:1 correlation. Figure 9: No synchronization for modified UR model and HP model with initial data \([x_{1}(0),x_{2}(0),x_{3}(0)]=[1000,1000,1000]\) and \([y_{1}(0),y_{2}(0),y_{3}(0)]=[1000,1000,1000]\). (**a**) \(y_{1}-s_{1}\) plot showing 1:1 correlation. (**b**) \(y_{2}-s_{2}\) plot showing no 1:1 correlation. (**c**) \(y_{3}-s_{3}\) plot showing no 1:1 correlation. (**d**) Simulation showing time evolution for \(x_{1}-y_{1}\). (**e**) Time series plot for \(y_{1}-s_{1}\). (**f**) Time series plot showing blow-up occurring in \(y_{3}\) and estimated at time \(T^{*}\approx 0.1008\). Figures 5 and 8. We saw no synchronization when the UR and HP models were coupled for the large initial data. Similarly, the modified UR and HP models did not synchronize for the large initial data after coupling. This reaffirms that the synchronization of three-species food chains with differently behaving top predators is caused solely by the initial numbers of the species, particularly the top predator. The lack of synchronization results can be seen in Figures 6 and 9. Note that the modified UR model had bounded solutions for all times, and yet we could see a lack of synchrony in Figure 9. This reaffirms that a lack of synchronization is not solely a feature of the blow-up dynamic but of the top predator being a generalist in UR models versus a specialist in HP models. It would be interesting to test this synchronization or lack thereof in other variations of the UR and HP models. Several improvements have been made to the UR class of models. The spatially explicit and ODE UR model has been shown to have bounded solutions in certain regimes of parameters or initial data via non-monotonic functional responses [32], as well as via refuge effects [33]. Cannibalism effects have also been considered herein [34], and testing the synchronization of such improved models with HP models will make for interesting future work. Another possible direction is to think of realization of the dynamics displayed by the current models via an actual hardware circuit. This has been achieved in the neuronal modeling literature [12; 13]. Although obtaining ecological data is much more of a challenge than neuronal data obtained from, for example, EEG devices, it does beg the question of how such devices could be thought about in the ecological setting.