A fractional model for predator-prey with omnivore

Abstract

We consider the model of interaction of predator and prey with omnivore using three different waiting time distributions. The first waiting time is induced by the power law distribution which is the generator of Pareto statistics. The second waiting time is induced by exponential decay law with a particular property of Delta Dirac distribution when the fractional order tends to 1, this distribution is link to the Poison distribution. While the last waiting distribution, induced by the Mittag-Leffler distribution, presents a crossover from exponential to power law. For each model, we presented the conditions under which the existence of unique set of exact solutions is reached using the fixed-point Picard’s method. Making use of a recent suggested numerical scheme, we solved each model numerically and some numerical simulations were generated for different values of fractional orders. We noticed a new attractor which can be considered as a combination of the Brownian motion and power law distribution in the model with the Atangana-Baleanu fractional derivative. With the aim to capture more attractors, we modified the model and presented also some numerical simulations. Our new model provides more attractors than the existing one even for fractional differential cases. We presented finally the Maximal Lyapunov exponent and the bifurcation diagrams. The comparative study shows that modeling with non-local and non-singular kernel fractional derivative leads to more attractors as this kernel is able to capture more physical problems.

Integral inequalities for a fractional operator of a function with respect to another function with nonsingular kernel

“…Also, they have their own properties consistent with those of the ML functions [5,15,18,27]. On the other hand, if we look carefully from an applied science point of view, we can deduce that these models of fractional calculus can be matched with real data where power law behavior disappears [14,20,58].…”

Simulating the Influence of Time Delay on Fractional Differential Equations based on Predictor-Corrector Scheme

“…Recently researchers have been dedicating their efforts to studying the dynamic behavior of fractional-order ecosystems. These ecosystems, characterized by their fractional-order models, exhibit a unique characteristic where the next state is not only determined by the current state 1 but also influenced by all previous historical states [7]. Predators often require a certain time delay to complete digestion.…”

Mathematical Analysis of Discrete Fractional Prey-Predator Model with Fear Effect and Square Root Functional Response

“…Based on observed ecological interactions among individuals of the species at various trophic levels, mathematical modelling is a helpful tool for understanding and predicting the long-term survival of various species. There are different types of preypredator models, such as the continuous model [1][2], discrete model [3][4][5], fractional model [6][7][8][9], etc. Nowadays, the fractional-order system can explain more natural phenomena that were previously ignored by the classical theory of the integer-order dynamical system.…”