# Blow-Up Dynamics and Synchronization in Tri-Trophic Food Chain Models
## 1 Introduction  
Synchronization is a multi-disciplinary phenomenon with a wide range of applications in chemical, physical, and biological systems [1, 2]. Synchronization is pivotal to our understanding of some natural systems, such as natural rhythms of clapping [3], cardiac pacemakers [4], neuronal processing [5], and cell cycles [6]. Meanwhile, the excess of synchronization in some systems may result in adverse effects such as epilepsy [7], Parkinson's disease [8], and bridge oscillations [9]. Synchronization is the agreement of different processes in time. It is a technique used in achieving related dynamics between chaotic systems by coupling. The related dynamics are asymptotic with time. The trajectories of the response system with that of the driver system are tracked. In 1990, [PERSON] and [PERSON] showed that two chaotic systems were able to synchronize with each other [10]. Since then, a wide variety of methods have been devised for the synchronization of chaotic systems. These include generalized synchronization (GS), complete synchronization (CS), and phase synchronization (PS), among others [11]. Recently, numerical and experimental studies on multistability and synchronization in heterogeneous interconnected networks were carried out in [12]. The synchronous behaviors of coupled biological neurons and four-dimensional electronic neurons, known to produce chaotic spiking-bursting behavior, were been observed in [13]. In this paper, we focus our attention on GS. When two coupled systems are under investigation, GS arises when the state or behavior of one of the coupled systems is governed or influenced by the behavior of the other system. The two systems may not necessarily be identical. GS is often seen in systems with unidirectional coupling [14]. [PERSON] et al. [15] used a reversed engineering technique with the objective of obtaining aGS state as a functional relation between a driver and a response oscillator. They were able to design an appropriate coupling between the driver and the response using an open-plus-closed-loop (OPCL) coupling and guaranteed the stability of the GS state. Their results showed that the OPCL method was robust to mismatching in oscillators and also viable for nonidentical oscillators. The OPCL method is applicable to any model-based system where \(\dot{x}=F(\mathbf{x},t),\mathbf{x}\in\mathbb{R}^{n}\), which can be driven [16] and particularly chaos control related [17]. Other techniques of exploring GS in coupled systems such as the mutual false nearest neighbor were thoroughly discussed in [18]. [PERSON] et al. [19] also studied GS in drive-response systems using the auxiliary system approach. The method uses a second identical response in monitoring synchronized motions. It can also detect and characterize forced GS. Chaotic dynamics have been observed in some food chain dynamical systems. [PERSON] and [PERSON] derived a three-species food chain system that exhibited chaotic dynamics [20]. Shortly thereafter, [PERSON] and [PERSON] derived yet another three-species food chain model exhibiting chaos. These models have been extremely thoroughly investigated, as they exhibit a variety of rich dynamics ranging from extinction to periodic dynamics to chaos and, most recently, finite-time blow-ups [21; 22; 23; 24; 25; 26]. Species extinction and loss of biodiversity are two coupled issues which have become increasingly relevant in ecology. Thus, in an attempt to gain insight into an ecosystem, where there may be interactions among different food chains, [PERSON] and [PERSON] [11] explored synchronization of the population fluctuations of their model with those of [PERSON] and [PERSON] [20]. One of the key differences between the two models is that the former's top predator is a generalist, and the latter's predator at the same tri-trophic level is a specialist. A generalist predator has several food choices. Of course, it may have a food item (prey) of its liking. On the other hand, a specialist predator dies out exponentially due to a lack of additional food choices.