On detecting chaos in a prey-predator model with prey's counter-attack on juvenile predators

Chaotic nonlinear systems are systems whose main feature is extreme sensitivity to noise in the problem or the corresponding initial conditions. In this paper, we examine the chaotic nature of a biological system involving a differential operator based on exponential kernel law, namely the Caputo-Fabrizio derivative. To approximate the solutions of the system, an implicit algorithm using the product-integration rule and two-point Lagrange interpolation is adopted. Some basic properties of the model, including the equilibrium points and their stability analysis, are discussed. Also, we used two well-known tools to detect chaos, including smaller alignment index (SALI), and the 0-1 test. Through considering different choices of parameters in the model, several meaningful numerical simulations and chaos analysis are presented. By observing the results obtained in both tests for detecting chaos, it is clear that the predicted behaviors are completely consistent with the approximate results obtained for the system solutions.It is found that hiring a new derivative operator dramatically increases the reliability of the biological system in predicting different scenarios in real situations.