Vol. 9, No. 3 (2020): Regular Issue

In this paper we derive an explicit second-order approximation of the homoclinic solutions in the Bazykin’s predator-prey model. The analytic solutions are compared with those obtained by numerical continuation.

In this proposal, a formulation for the approximate-analytical solution of a fractional-delayed damping Duffing oscillator is developed. The fractional derivative is established using the Riemann-Liouville definition. In this scheme, the solution used a homotopy perturbation. In this proposal, a transcendental frequency equation is established. Finally, an analytic solution to the complicated algebraic frequency equation is obtained. Stability conditions are formulated to maintain the structure of the oscillatory solution. The case of un-delayed damping Duffing equation is investigated through the modified homotopy technique which is assumed to be the successor to obtain the solution.

We consider an SIS spatial epidemic model with nonlinear incidence rate and diffusion, that consists of susceptible S and infective I individual populations which interact randomly in their physical environment. We use pattern formation to explain the spread and control of the epidemic over a course of time. Turing instability conditions are established and analysed for the model to exhibit spatial patterns. We find the exact Turing space in the parameter regime for these conditions to hold. An implicit pseudo-spectral method is used to numerically approximate the system and the patterns form reveal that the susceptible and infected populations behave of the same way. In some examples these populations are in isolation from each other. This happens because the susceptible individuals diffuse or move away from the infected individuals to avoid contact and the possibility of getting infected with the disease. Rigorous numerical experimentation reveals that the model has rich dynamics. We find that whenever the transmission rate β is less than the treatment r, there is no outbreak but for β ≥ r there is a possibility of having an outbreak. The results obtained extend well the findings of pattern formation in epidemic models and may have direct implications for the study of disease spread and control and perhaps the mechanistic impact of public health interventions on epidemics.

The present study deals with a prey-predator system with prey refuge depending on both species with the Beddington-DeAngelis response function. We propose a mathematical model for predator-prey interactions, allowing prey refuge in the absence of intra-specific competition and in the presence of intra-specific competition among the predators. We have analyzed the models in terms of boundedness, persistence, existence of equilibria and their stability and Hopf bifurcation. Existence of paradox of enrichments are examined well in both the cases. The analytical findings of this study are substantially validated by sufficient numerical simulations. The ecological implications of the obtained results are discussed as well.

In this paper, we investigate the existence and uniqueness of solutions for the Hadamard fractional boundary value problems with nonlocal multipoint boundary conditions. By using Leray-Schauder nonlinear alternative, Leray Schauder degree theory, Krasnoselskii fixed point theorem, Schaefer fixed point theorem, Banach fixed point theorem, Nonlinear Contractions, the existence and uniqueness of solutions are obtained. As an application, two examples are given to demonstrate our results.

In this work, evolutionary dynamical behaviour of a single-species population model in a polluted environment has been analyzed. This model system describes the effect of toxicant on a single-species population. Two discrete time delays have been incorporated for proper description. Important mathematical characteristics of the proposed model such as positivity, boundedness, stability and Hopf-bifurcation for all possible combinations of both the delays at the interior equilibriumpoint of the model system have been discussed. It is observed that increase amount of delay may lead to the change of stable behaviour of stationary points through the creation of limit cycles and higher periodic oscillations. Furthermore, it is reported that Hopf-bifurcations may also occur around stationary points for corresponding non-delayed system. Various numerical simulations are performed to validate analytical findings.

This paper explores on the qualitative analysis of a modified Leslie-Gower prey-predator model where the consumption rate of prey is according to Beddington-DeAngelis functional response and Allee effect on prey population. Moreover time-lag (τ) is established to exploit gestation period of predations. The permanence analysis of proposed system is investigated. Then we study the local stability of non-delayedmodel at all possible equilibriumpoints and it is demonstrated that the given model experiences Hopf bifurcation about interior equilibrium point with respect to delay τ . Thereafter the stability and direction of Hopf bifurcation are formulated through normal and centermanifold theorems. The derived criteria are justified with the help of numerical simulations.

We use Generalized Beta Prime distribution, also known as GB2, for fitting response time distributions. This distribution, characterized by one scale and three shape parameters, is incredibly flexible in that it canmimic behavior of many other distributions. GB2 exhibits power-law behavior at both front and tail ends and is a steady-state distribution of a simple stochastic differential equation. We apply GB2 in contrast studies between two distinct groups – in this case children with dyslexia and a control group – and show that it provides superior fitting. We compare aggregate response time distributions of the two groups for scale and shape differences (including several scale-independent measures of variability, such as Hoover index), which may in turn reflect on cognitive dynamics differences. In this approach, response time distribution of an individual can be considered as a random variate of that individual’s group distribution.

In this work, we consider a class of neutral stochastic integro-differential equations driven by Wiener process and fractional Brownian motion with impulses effects. This paper deals with the global attractiveness and quasiinvariant sets for neutral stochastic integro-differential equations driven by Wiener process and fractional Brownian motion with impulses effects in Hilbert spaces. We use new integral inequalities combined with theoriesof resolvent operators to establish a set of sufficient conditions for the exponential p-stability of the mild solution of the considered equations. An example is presented to demonstrate the obtained theory.