Abstract
The Kolmogorov model has been applied to many biological and environmental problems. We are particularly interested in one of its variants, that is, a Gauss-type predator–prey model that includes the Allee effect and Holling type-III functional response. Instead of using classic first order differential equations to formulate the model, fractional order differential equations are utilized. The existence and uniqueness of a nonnegative solution as well as the conditions for the existence of equilibrium points are provided. We then investigate the local stability of the three types of equilibrium points by using the linearization method. The conditions for the existence of a Hopf bifurcation at the positive equilibrium are also presented. To further affirm the theoretical results, numerical simulations for the coexistence equilibrium point are carried out.
Highlights
Fractional calculus is an extension of classical calculus that generalizes the order of derivatives and integrals to a non-integer order
The above results indicate that equilibrium point E3 = (30, 42) loses its stability when the order α is increased to pass through the critical value α∗, which implies that a Hopf bifurcation occurs
6 Conclusions In this paper, we have studied the dynamic behavior of a fractional Gauss-type predator– prey model with Allee effect and Holling type-III functional response
Summary
Fractional calculus is an extension of classical calculus that generalizes the order of derivatives and integrals to a non-integer order. In 2010, Eduardo et al [72] studied a Gauss-type predator–prey model with Allee effect on prey and Holling type-III functional response and examined the global behavior of this model. They identified three important assumptions as regards the interactions between prey and predator:. The Caputo definition is very useful because in this case the derivative of a constant is zero and the initial conditions for the fractional order differential equations can be provided in the same manner as for the classical integer case, which has a clear physical meaning [78].
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