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.
Highlights
The investigation of the interactions between populations of predators and preys in nature is a highly transited topic of research in applied mathematics currently
We can mention studies that report on the modeling and analysis of predator–prey models with disease in the prey [1], the analysis of stochastic systems with modified
Michaelis–Menten-type predator harvesting [4], synthetic Escherichia coli predator–prey ecosystems [5], the analytical investigation of stage-structured predator–prey models depending on maturation delay and death rate [6], and non-autonomous ratio dependent models with Holling-type functional response with temporal delay [7], among other interesting topics [8]
Summary
The investigation of the interactions between populations of predators and preys in nature is a highly transited topic of research in applied mathematics currently. Mathematics 2019, 7, 1172 on dynamically consistent nonstandard finite-difference schemes to solve predator–prey models [10], while some of those methods have been implemented in MATLAB with the aim of being available to the scientific community [11]. Recent reports have studied fractional models of predators and preys that incorporate feedback control and a constant prey refuge [27], bifurcations of delayed fractional systems with incommensurate orders [28], periodic solutions and control optimization of models with two types of harvesting [29], and fractional predator–prey systems with delay and Holling type-II functional response [30], among other recent works available in the literature. The search for better algorithms to simulate fractional systems (which provide fast results with minimal computer resources) is still an open problem of research Motivated by this background, we will consider a predator–prey system with fractional diffusion that considers reaction functionals.
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