On the dynamics of a nonlinear reaction–diffusion duopoly model

Mathematical Problems in Engineering

2019

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A binary mixture saturating a horizontal porous layer, with large pores and uniformly heated from below, is considered. The instability of a vertical fluid motion (throughflow) when the layer is salted by one salt (either from above or from below) is analyzed. Ultimately boundedness of solutions is proved, via the existence of positively invariant and attractive sets (i.e. absorbing sets). The critical Rayleigh numbers at which steady or oscillatory instability occurs are recovered. Sufficient conditions guaranteeing that a secondary steady motion or a secondary oscillatory motion can be observed after the loss of stability are found. When the layer is salted from above, a condition guaranteeing the occurrence of “cold” instability is determined. Finally, the influence of the velocity module on the increasing/decreasing of the instability thresholds is investigated.

Section: Introductionmentioning

confidence: 99%

Instability of Vertical Constant Through Flows in Binary Mixtures in Porous Media with Large Pores

Capone

Luca

Mathematical Problems in Engineering

2019

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“…Such a modeling provides challenges and ideas in many other fields of applied mathematics in which nonlinear mathematical models having a similar structure are considered [18][19][20][21][22]. After reviewing in Section 2 the main prerequisites on fractional calculus, the model is formulated in Section 3 and existence and boundedness of solutions is proved.…”

Section: Introductionmentioning

confidence: 99%

A Fractional-in-Time Prey–Predator Model with Hunting Cooperation: Qualitative Analysis, Stability and Numerical Approximations

Carfora

2021

Axioms

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A prey–predator system with logistic growth of prey and hunting cooperation of predators is studied. The introduction of fractional time derivatives and the related persistent memory strongly characterize the model behavior, as many dynamical systems in the applied sciences are well described by such fractional-order models. Mathematical analysis and numerical simulations are performed to highlight the characteristics of the proposed model. The existence, uniqueness and boundedness of solutions is proved; the stability of the coexistence equilibrium and the occurrence of Hopf bifurcation is investigated. Some numerical approximations of the solution are finally considered; the obtained trajectories confirm the theoretical findings. It is observed that the fractional-order derivative has a stabilizing effect and can be useful to control the coexistence between species.

“…In the present paper we have extended [18], introducing self-and cross-diffusion terms. The spatial diffusion plays an important role in the process of population evolution, not only in ecology but also in many other fields of applied mathematics such as biochemistry or economics and the effect of self-and cross-diffusion on the population dynamics has been widely investigated theoretically by many mathematicians ( [19][20][21][22][23][24] and references therein). Self-diffusion terms model the random movement of individuals in both prey and predator populations.…”

Section: Introductionmentioning

confidence: 99%

Cross-Diffusion-Driven Instability in a Predator-Prey System with Fear and Group Defense

Carfora

2020

Mathematics

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In this paper, a reaction-diffusion prey-predator system including the fear effect of predator on prey population and group defense has been considered. The conditions for the onset of cross-diffusion-driven instability are obtained by linear stability analysis. The technique of multiple time scales is employed to deduce the amplitude equation near Turing bifurcation threshold by choosing the cross-diffusion coefficient as a bifurcation parameter. The stability analysis of these amplitude equations leads to the identification of various Turing patterns driven by the cross-diffusion, which are also investigated through numerical simulations.