Analysis of Structure-Preserving Discrete Models for Predator-Prey Systems with Anomalous Diffusion

Abstract

In this work, we investigate numerically a system of partial differential equations that describes the interactions between populations of predators and preys. The system considers the effects of anomalous diffusion and generalized Michaelis–Menten-type reactions. For the sake of generality, we consider an extended form of that system in various spatial dimensions and propose two finite-difference methods to approximate its solutions. Both methodologies are presented in alternative forms to facilitate their analyses and computer implementations. We show that both schemes are structure-preserving techniques, in the sense that they can keep the positive and bounded character of the computational approximations. This is in agreement with the relevant solutions of the original population model. Moreover, we prove rigorously that the schemes are consistent discretizations of the generalized continuous model and that they are stable and convergent. The methodologies were implemented efficiently using MATLAB. Some computer simulations are provided for illustration purposes. In particular, we use our schemes in the investigation of complex patterns in some two- and three-dimensional predator–prey systems with anomalous diffusion.

A Host-parasitoid Dynamics with Allee and Refuge effects

“…Discrete dynamic systems (DDS) in general have been studied extensively and particularly in the recent period in view of the Coronavirus disease pandemic [16], [21], [1], [17], [22]. In addition, it has been shown that population dynamics implementing DDS can produce efficient computational results for host-parasitoid interactions with numerical simulations [18], [9], [20], see also [11].…”

A Density-Dependent Host-Parasitoid Model with Stability, Bifurcation and Chaos Control

“…In [20] and [21], the influence of a refuge effect was explored for certain classes of host-parasitoid models. In [22], the authors numerically investigated a system of partial differential equations that describe the interactions between populations of predators and prey. Suryanto et al [23] considered a model of predator-prey interaction at fractional-order with a ratio-dependent functional response.…”

Computational Mathematics and Neural Systems

“…In [3], a system of partial differential equations describing the interactions between populations of predators and preys was investigated. The system took into account the effects of anomalous diffusion and the generalized Michaelis-Menten-type reaction, and it was extended in various spatial dimensions.…”