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In this paper, a predator–prey model with Allee effect and seasonally forcing in the prey’s growth rate is introduced and analysed. Nonlinear analysis, using equilibrium points computed via Symbolic Math software tools, bifurcation diagrams, phase diagrams, Lyapunov exponents and Kaplan–Yorke dimension, proves that such effects lead to undesirable biological dynamics including chaos behaviours. In order to establish a certain balance in the ecosystem and avoid chaotic dynamics, the biological system is controlled in order to follow an unforced reference model via a synchronization approach. Simulation results prove the efficiency of the proposed approach to allow the predator–prey system regaining natural dynamics.

In recent years, dynamical relationship between species in ecology has been intensively investigated and will continue to be one of the most significant themes. The dynamics of predator–prey’s systems are at the heart of these studies. Such models are generally depicted by nonlinear polynomials and exhibit many complex nonlinear phenomena. In this paper, not only a prey–predator model displaying richer dynamical behaviors is analyzed but also its electronic circuit is also designed via the MultiSIM software proving the very good agreement between biological theory considerations and electronic experiments.

In this chapter, the dynamics of a modified predator-prey BB model with Allee effects and seasonal perturbation are investigated in some detail. In addition to the Hopf bifurcation, new bifurcation points are detected such as limit point cycle and period-doubling bifurcations and those by modeling the prey's growth rate and the predator efficiency by modified algebraic structures. The latter is based on noninteger elements giving more extended set of information for the predator-prey system compared to the structure-based-integer-order elements. Furthermore, numerical simulations show that the root mean square operator is a generator of the period doubling (PD) bifurcation. Moreover, our finding reveals that seasonal effects have a key significance on the dynamic properties of the BB prey-predator system. Indeed, in presence of such forcing effect, the system loses its original stability proprieties and vividly demonstrates complex chaos phenomenon.

In this work, a bifurcation analysis of the predator-prey BB-model is presented by considering a weak Allee Effect on the prey. To bring out the difference between the proposed dynamical behavior and that of the BB-model with a strong Allee Effect, a local and global bifurcation analysis is presented. Finally, phase portraits and bifurcation diagrams are shown for the two cases studies to highlight the impact of the Allee Effects at low population density.