A predator-prey model with Beddington-DeAngelis functional response: a non-standard finite-difference method

In this paper, we transform a continuous-time predator-prey model with Beddington–DeAngelis functional response into a discrete-time model by nonstandard finite difference scheme (NSFD). The NSFD model shows complete dynamic consistency with its continuous counterpart for any step size. However, the discrete model of same continuous system obtained by Euler forward method shows dynamic inconsistency for larger step size. Extensive numerical simulations have been done to compare the dynamics of NSFD system and Euler system. Our analysis reveals that dynamics of NSFD model is independent of the step-size, whereas the dynamics of the standard discrete model completely depends on the step-size and produces spurious dynamics like chaos.     

Analysis of a fractional order eco‐epidemiological model with prey infection and type 2 functional response

In this paper, we introduce fractional order into an ecoepidemiological model, where predator consumes disproportionately large number of infected preys following type 2 response function. We prove different mathematical results like existence, uniqueness, nonnegativity, and boundedness of the solutions of fractional order system. We also prove the local and global stability of different equilibrium points of the system. The results are illustrated with several examples.     

Dynamics of a ratio-dependent eco-epidemiological system with prey harvesting

In this article, we study a ratio-dependent eco-epidemiological system where prey population is subjected to harvesting. Mathematical results like positive invariance, boundedness, stability of equilibria, and permanence of the system have been established. The dynamics of zero equilibria have been thoroughly investigated to find out conditions on the system parameters such that trajectories starting from the domain of interest can reach a zero equilibrium following any fixed direction. We have also studied suitable conditions for non-existence of a periodic solution around the interior equilibrium. Computer simulations have been carried out to illustrate different analytical findings.     

On the stability and Hopf bifurcation of a delay-induced predator–prey system with habitat complexity

We study the effect of the degree of habitat complexity and gestation delay on the stability of a predator–prey model. It is observed that there is stability switches, and Hopf bifurcation occurs when the delay crosses some critical value. By applying the normal form theory and the center manifold theorem, the explicit formulae which determine the stability and direction of the bifurcating periodic solutions are determined. The qualitative dynamical behavior of the model system is verified with the published data of Paramecium aurelia (prey) and Didinium nasutum (predator) interaction. It is observed that the quantitative level of abundance of system populations depends crucially on the delay parameter if the gestation period exceeds some critical value. However, the fluctuations in the population levels can be controlled completely by increasing the degree of habitat complexity.     

Predator–prey interaction with harvesting: mathematical study with biological ramifications

In this article, we study a simple predator–prey interaction where predator population is subjected to harvesting. Our qualitative analysis shows different outcomes including switching of stability, oscillations and deterministic extinction. Theoretically observed results have been tested with the parameter values of Paramecium aurelia and its predator Didinium nasutum. Study reveals that harvesting effort and predator’s attack rate may be used as control parameters for the system. Simulation results also indicate that the system may exhibit bistability for some parametric region. Our study also gives the possible answer to the question – why do we frequently observe coexisting predator–prey system in natural system? The study may be helpful to design control strategy for harvested predator–prey system.     

Syntheses,Structures, and Electronic Properties of Mono- and Bimetallic Thiolato Complexes Containing Unusual Coordination Modes of Thiolato Ligands

Synthesis, bonding and chemistry of mono- and bimetallic complexes supported by chelating thiolato ligands have been established. Treatment of [Cp*VCl₂]₃ ( 1 ) with [LiBH₄ ⋅ THF] followed by the addition of ethane-1,2-dithiol led to the formation of an EPR active bimetallic vanadium thiolato complex [(Cp*V){μ-(SCH₂CH₂S)-κ²S,S′)₂{V(SCH₂CH₂S-SH)}] ( 2 ). In complex 2 , two ethane-1,2-dithiolato ligands are symmetrically coordinated to two vanadium atoms through μ-S atoms. Interestingly, when similar reactions were carried out with heavier group 5 metal precursors, such as [Cp*NbCl₄] ( 3 a ), it afforded monometallic thiolato complex [Cp*Nb(SCH₂CH₂S)(SCH₂CH₂S−CH₂S)] ( 4 a ). On the other hand, the Ta-analogue [Cp*TaCl₄] ( 3 b ) yielded thiolato species [Cp*Ta(SCH₂CH₂S)(SCH₂CH₂S−CH₂S)] ( 4 b ) and [Cp*Ta(SCH₂CH₂S) (SCH₂CH₂S−S)] ( 5 ). In complexes 4 a and 4 b , one ethane-1,2-dithiolato and one trithiolato ligand are coordinated to Nb and Ta centers, respectively. Whereas, in complex 5 , one ethane-1,2-dithiolato and one 2-disulfanylethanethiolato is coordinated to the Ta center. Moreover, the photolytic reaction of 5 with [Mo(CO)₅ ⋅ THF] yielded heterobimetallic thiolato complex [(Cp*Ta){μ-(SCH₂CH₂S)-κ²S,S′}{μ-(SCH₂CH₂S−CH₂(CH₃)S)κ²S′′ : κ¹S-′′′′ : κ¹S′′′′′}{Mo(CO)₃}] ( 6 ). All the complexes have been characterized by multinuclear NMR spectroscopy and single crystal X-ray diffraction studies. Further, computational analyses were performed to provide an insight into the bonding of these complexes.