Differential Equations and Dynamical Systems

A one-dimensional optimal control problem with a state-dependent cost and a unimodular integrand is considered. It is shown that, under some standard assumptions, this problem can be solved without using the Pontryagin maximum principle, by simple methods of the classical analysis, basing on the Tchyaplygin comparison theorem. However, in some modifications of the problem, the usage of Pontryagin’s maximum principle is preferable. The optimal synthesis for the problem and for its modifications is obtained.

In this paper, uniformly convergent finite difference schemes with piecewise linear interpolation on Shishkin meshes are suggested to solve singularly perturbed boundary value problems for second order ordinary delay differential equations of convection-diffusion type. Error estimates are derived and are found to be of almost first order. Numerical results are provided to illustrate the theoretical results.

In this paper we have developed a five compartmental HIV/AIDS epidemic model with two infectious stages before full-blown AIDS defined, i.e., asymptomatic phase and symptomatic phase. Boundedness and non-negativity analysis of the solutions, existence and stability analysis of the model at various equilibrium points are discussed thoroughly. Basic reproduction number ( $$R_0$$ ) is calculated using next generation matrix method. It is found that the system is locally as well as globally asymptotically stable at disease free equilibrium $$E_0$$ when $$R_0<1$$ . But when $$R_0>1$$ , endemic equilibrium $$E^*$$ exists and the system becomes locally asymptotically stable at $$E^*$$ if Routh–Hurwitz criterion is satisfied. We also obtain necessary conditions for the global asymptotic stability of the system at $$E^*$$ by geometrical approach. The effect of single discrete time delay on the model is also discussed. We have considered the time delay $$\tau $$ as a time lag due to the development of the infection until signs or symptoms first appear. The length of delay preserving the stability is estimated using Nyquist criteria and existence conditions of the Hopf-bifurcation for the time delay are investigated by choosing the time delay $$\tau $$ as a bifurcation parameter. Important analytic results are numerically verified using MATLAB, which shows the reliability of our model from the practical point of view.

Ecological and biological conservation of living systems has been an active area of research over the years by agriculturalists, biologists and mathematicians. One of the studies involves additional food supplement feeding (also called as diversionary feeding) for the purpose of biological (wildlife in some cases) conservation. The idea in this approach is to distract (thereby supplement) the wildlife from predating upon the other species with the end goal of wildlife conservation. On the other hand in agricultural entomology, insect control and optimization, additional food is supplemented as a tool for effective pest control thereby achieving the biological control. The study of these ecosystems is usually done using the predator–prey systems. In nature, we find situations wherein the group defense (toxicity) of the prey reduces the predator’s predation rate. This type of behaviour of the prey is also known as inhibitory effect of the prey. Biological conservation of such predator prey systems in the presence of additional food supplements is quite challenging and interesting. In this paper, we consider an additional food provided predator–prey system which is a variation of the standard predator–prey model in the presence of the inhibitory effect of the prey. The predators functional response is assumed to be of Holling type IV (considering the inhibitory effect). This model is analyzed to understand the inherent dynamics of the system. The findings suggest that the quality and quantity of additional food provided to the predators, play a very significant role in determining the eventual state of the ecosystem. The outcomes of the analysis suggests eco friendly strategies to eco-managers for biological conservation of living systems.

It is the purpose of this paper to give oscillation criteria for the third order nonlinear functional dynamic equation $$ \left( {a\left( t \right)\left[ {\left( {r\left( t \right)x^\Delta \left( t \right)} \right)^\Delta } \right]^\gamma } \right)^\Delta + f\left( {t,x\left( {g\left( t \right)} \right)} \right) = 0 $$ on a time scale $$ \mathbb{T} $$ , where γ is the quotient of odd positive integers, a and r are positive rd-continuous functions on $$ \mathbb{T} $$ , and the function g: $$ \mathbb{T} \to \mathbb{T} $$ satisfies limt→∞ g(t) = ∞ and f ∈ C $$ \left( {\mathbb{T} \times \mathbb{R}, \mathbb{R}} \right) $$ . Our results are new for third order delay dynamic equations and extend many known results for oscillation of third order dynamic equations. Some examples are given to illustrate the main results.

For permanent and partially permanent, uniformly bounded Lotka–Volterra systems, we apply the Split Lyapunov function technique developed for competitive Lotka–Volterra systems to find new conditions that an interior or boundary fixed point of a Lotka–Volterra system with general species–species interactions is globally asymptotically stable. Unlike previous applications of the Split Lyapunov technique to competitive Lotka–Volterra systems, our method does not require the existence of a carrying simplex.

In this work, we have discussed the oscillation properties of first order neutral impulsive difference equations with constant coefficients by using pulsatile constant. Also, we have made an effort to apply our constant coefficient results to nonlinear impulsive difference equations with variable coefficients.

The Kummer–Schwarz Equation, $$2 y'y''' - 3 y''{}^2 = 0$$, (the prime denotes differentiation with respect to the independent variable x) is well known from its connection to the Schwartzian Derivative and in its own right for its interesting properties in terms of symmetry and singularity. We examine a class of equations which are a natural generalisation of the Kummer–Schwarz Equation and find that the algebraic and singularity properties of this class of equations display an attractive set of patterns. We demonstrate that all members of this class are readily integrable.