Series Preface * Preface to the Third Edition * 1 Linear Systems * 2 Nonlinear Systems: Local Theory * 3 Nonlinear Systems: Global Theory * 4 Nonlinear Systems: Bifurcation Theory * References * Index

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Differential Equations and Dynamical Systems

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Λιόλιος Ξερολιβαδιώτης : ένας αγνοημένος Μακεδόνας αγωνιστής του '21 και λίγα για την ιστορία του Ξερολιβάδου/ Γιώργου Χ. Χιονίδη

Effect of hunting cooperation and fear in a predator-prey model

In this article, a predator-prey model has been evolved in the form of a system of fractional order differential equations incorporating two important factors, namely, fear factor and prey refuge factor. Here, the fractional calculus has been taken into consideration to investigate the dynamical behaviour of the solutions of the proposed model system as the changes in life cycle of prey species are of memory bound. Biological validation and well-posedness such as positivity and boundedness of solutions of the model system have been proved analytically. Stability analysis of all the feasible equilibrium points of the model system has been performed in a systematic way. Some important dynamical features of the model system (such as transition of stability of the system) have been demonstrated through rigorous numerical simulation. It is observed that our proposed model system experiences Hopf-bifurcation around the interior equilibrium point with respect to both the parameters f and m 1 , which are linked with amount of predator induced fear and rate of prey refuge, respectively. The system dynamics is more likely to be stable in the framework of fractional order derivative in comparison to integer-order derivative. The high amount of predator induced fear f and prey refuge rate m 1 are independently capable to make the system dynamics to be stable in integer order model system. On the other hand, the dynamics of the model system shifts towards the stability from its unstable behaviour when we continuously reduce the order of the model system; especially under the scenario of low level of predator induced fear and prey refuge rate. Thus, our comprehensive mathematical findings reveal the fact that fading memory can play a contributory role towards stable coexistence of the predator-prey system whereas strong memory of the species deteriorates the stable coexistence of the model system.

Impact of predator incited fear and prey refuge in a fractional order prey predator model

In this work, we have formulated a fractional-order predator-prey system with fear effect, where the death rate of the prey population is predator density-dependent. Generally, most of the ecological study considers the direct killing of prey in a predator’s presence, but they ignore the effect of predators’ presence on prey. Some experimental studies confirmed that fear affects the reproduction rate of the prey population, but a few studies are there, which conclude that fear also affects the death rate of the prey population. Thus our main aim in this work is to investigate the influence of the fear effect produced by a predator on the reproduction rate and death rate of the prey population. We first prove the existence, uniqueness, non-negativity, and boundedness of the considered model solutions. After that, we show a detailed analysis of different equilibria and their stability criteria based on some conditions where we specifically investigated the global stability of the interior equilibrium point. Besides that, the existence of Hopf bifurcation and the persistence of the system is derived. We also observed how fractional-order derivative also has an impact on our proposed system. Finally, some numerical simulation is performed to validate our findings.

Effects of fear in a fractional-order predator-prey system with predator density-dependent prey mortality

Abstract In the present article, we investigate the impact of fear effect in a predator–prey model, where predator–prey interaction follows Beddington–DeAngelis functional response. We consider that due to fear of predator the birth rate of prey population reduces. Mathematical properties, such as persistence, equilibria analysis, local and global stability analysis, and bifurcation analysis, have been investigated. We observe that an increase in the cost of fear destabilizes the system and produces periodic solutions via supercritical Hopf bifurcation. However, with further increase in the strength of fear, system undergoes another Hopf bifurcation and becomes stable. The stability of the Hopf-bifurcating periodic solutions is obtained by computing the first Lyapunov coefficient. Our results suggest that fear of predation risk can have both stabilizing and destabilizing effects.

Role of Fear in a Predator-Prey Model with Beddington-DeAngelis Functional Response

Numerical Study on Diffusiophoresis of a Hydrophobic Nanoparticle in a Monovalent or Multivalent Electrolyte

This study aims to quantify the impact of the dielectric permittivity of a droplet on its diffusiophoresis in different types of electrolytes. The dielectric droplet polarizes by the diffusion field along with the local electric field created by the interactions of the doublelayer with the imposed ionic concentration gradient, which generates an induced surface charge density anti-symmetrically distributed on the droplet surface. This induced surface charge influences both electrophoresis and chemiphoresis parts. Based on a low imposed concentration gradient, a simplified model is derived through a first-order perturbation technique. Dielectric polarization of the droplet attenuates the spinning force at the interface. This creates the mobility of a droplet of higher dielectric permittivity in presence of a stronger diffusion field significantly higher than that of a perfectly dielectric droplet and its value depends on the polarity of the droplet surface charge. In absence of the diffusion field, the mobility of a conducting droplet remains positive immaterial of the polarity of its surface charge density. We find that the impact of the dielectric polarization becomes significant as the surface charge density increases and attenuates with the increase of droplet viscosity. For a dielectric droplet at a thinner Debye length, a step-jump in mobility occurs at a higher value of the surface charge density. Such type of step-jump in mobility does not appear for the conducting droplet due to the absence of the Maxwell stress at the interface.

https://aip.scitation.org/doi/pdf/10.1063/5.0142875

A simplified model for the impact of dielectric polarization of a charged droplet on its diusiophoresis

The diffusiophoresis of a soft particle governed by an imposed ionic concentration gradient is studied based on a modified ion transport equation. The surface of the rigid core of the soft particle is considered to be hydrophobic, which is coated with a charged polyelectrolyte layer (PEL) whose dielectric constant is different from the dispersion medium. This step-change in the dielectric constant of the PEL and the dispersion medium creates an ion partitioning effect. The governing equations based on the coupled Navier–Stokes–Nernst–Planck–Poisson equations are modified to incorporate the ion partitioning effect. A simplified model based on the linear perturbation technique under a small applied concentration gradient in the scale of the particle radius is developed. The simplified model agrees well with the exact numerical solutions for a Debye length smaller than the particle size. We have analyzed the impact of the core slip length and PEL permittivity on the mobility of the soft particle. The interfacial slip enhances the impact of the core charge density on the diffusiophoresis. In this case, a reversal of mobility occurs at a thinner Debye length compared to a hydrophilic core. Ion partitioning attenuates the concentration of mobile counterions in PEL, leading to a reduction in screening of PEL fixed charge. We find that the diffusiophoresis of the soft particle in a non-z:z asymmetric electrolyte strongly depends on the valence of the counterions. Diffusiophoresis of the soft particle in a KCl-NaCl mixture is considered to achieve positive mobility.

https://pubs.aip.org/aip/pof/article-pdf/doi/10.1063/5.0156467/18022783/062017_1_5.0156467.pdf

Diffusiophoresis of a soft particle incorporating ion partitioning and hydrophobic core

A numerical study on the diffusiophoresis of a droplet in an electrolyte medium is carried out by solving the full set of coupled governing equations, which are based on the conservation principle. Diffusiophoresis is considered for monovalent as well as non-z:z electrolytes and mixed electrolytes. The numerical model is supplemented with a semianalytic simplified model based on first-order perturbation analysis, which agrees with the numerical model for a low to moderate range of surface potential. The mobility for a low-viscosity fluid at a thinner Debye length is dominated by the chemiphoresis part, which creates the mobility to become an even function of the surface charge density for a monovalent electrolyte. Such a pattern in mobility does not appear in a non-z:z asymmetric electrolyte. At a thinner Debye length, diffusiophoresis becomes independent of the diffusion field, hence the mobility is independent of the composition of electrolytes in a mixed monovalent electrolyte solution. Our results show that the size-based sorting of droplets is efficient when a mixed electrolyte is considered. We have also addressed the finite ion size effects by considering a modified ion transport equation. One of the key features of the present study is the simplified semianalytical model for the diffusiophoresis of a droplet in a z:z electrolyte as well as in non-z:z and mixed electrolytes, which is shown to be valid up to a moderate range of surface potential for a finite Debye length.

Subrata Majhi

Papers

Role of Fear in a Predator-Prey Model with Beddington-DeAngelis Functional Response

Numerical Study on Diffusiophoresis of a Hydrophobic Nanoparticle in a Monovalent or Multivalent Electrolyte

A simplified model for the impact of dielectric polarization of a charged droplet on its diusiophoresis

Diffusiophoresis of a soft particle incorporating ion partitioning and hydrophobic core

Diffusiophoresis of a Charged Droplet in Asymmetric as Well as Mixed Electrolytes through Numerical and Semianalytic Models.