Abstract
Motivated by the fact that the restrictive conditions for a Turing instability are relaxed in subdiffusive regime, we investigate the effects of subdiffusion in the predator - prey model with toxins under the homogeneous Neumann boundary condition. First, the stability analysis of the corresponding ordinary differential equation is carried out. From this analysis, it follows that stability is closely related to the coefficient of toxicity. In addition, the temporal fractional derivative does not systematically widen the range of parameters to maintain a point in the stability domain. Furthermore, we derive the condition which links the Turing instability to the coefficient of toxicity in the subdiffusive regime. System parameters are varied in order to test our mathematical predictions while comparing them to ecological literature. It turns out that the memory effects, linked to the transport process can, depending on the parameters, either stabilize an ecosystem or make a completely different configuration.
Highlights
During evolution, venomous animals have produces a panoply of peptide toxins, formidable weapons for defense against predators or for the capture of prey
Subdiffusion has acquired relevance in the past decades since it has been experimentally detected in several systems such as porous media [4], glasses [5], transport through cell membranes [1, 2], and other biological systems [3]
This lead to the conclusion that the toxic coefficient is important for the occurence of the complex spatial dynamics
Summary
Venomous animals have produces a panoply of peptide toxins, formidable weapons for defense against predators or for the capture of prey. Interest was given to the link between toxic substances and Turing bifurcation, Hopf bifurcation and Steady − State bifurcation [10]. This lead to the conclusion that the toxic coefficient is important for the occurence of the complex spatial dynamics. When it comes to subdiffusive entities undergoing reactions, reaction and diffusion processes are no longer separable because of the strong memory effects in the transport mechanism [11], which means that the non-Markovian nature of subdiffusion results in a nontrivial combination of reactions and spatial dispersal In this situation, can toxicity still be considered as an important parameter responsible for complex spatial dynamics? If we consider instead the inequality (6), and we choose β = 0.09 as in [10], we will have three positive equilibrium points E1∗, E2∗ and E3∗. It would be interesting to note that this stability, depending on the system parameters, can either be modified or maintained
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