Butterfly support for off diagonal coefficients and boundedness of solutions to quasilinear elliptic systems

We consider quasilinear elliptic systems in divergence form. In general, we cannot expect that weak solutions are locally bounded because of De Giorgi’s counterexample. Here we assume that off-diagonal coefficients have a “butterfly support”: this allows us to prove local boundedness of weak solutions.

Dynamics of Delayed Phytoplankton–Zooplankton System with Disease Spread Among Zooplankton

In this paper, we study the dynamics of phytoplankton–zooplankton system with delay, where delay means that releasing toxin for phytoplankton is not instantaneous. First, we prove the positivity and boundedness of solutions, discuss the Hopf bifurcation caused by delay. Furthermore, we study the property of Hopf bifurcation by center manifold and normal form. Then, we study the global existence of bifurcated periodic solution. Finally by simulation, we show the influence of delay, disease spread and recovery from infected to susceptible on the dynamics of phytoplankton–zooplankton system.

STABILITY AND BOUNDEDNESS BEHAVIOUR OF SOLUTIONS OF A CERTAIN SECOND ORDER NON-AUTONOMOUS DIFFERENTIAL EQUATIONS

In this paper, we give sufficient conditions for the stability and ultimate boundedness of solutions to a certain second order non-autonomous differential equations with damped and forced functions. Our results improve and extend some of the stability and boundedness results in the literature which themselves are extensions of some results cited therein. We give example to illustrate the result obtained.

Turing-Hopf Bifurcation in the Predator-prey Model with Cross-diffusion Considering Two Different Prey Behaviours Transition

In this paper, we study the Turing-Hopf bifurcation in the predator-prey model with cross-diffusion considering the individual behaviour and herd behaviour transition of prey population subject to homogeneous Neumann boundary condition. Firstly, we study the non-negativity and boundedness of solutions corresponding to the temporal model, spatiotemporal model and the existence and priori boundedness of solutions corresponding to the spatiotemporal model without cross-diffusion. Then by analyzing the eigenvalues of characteristic equation associated with the linearized system at the positive constant equilibrium point, we investigate the stability and instability of the corresponding spatiotemporal model. Moreover, by computing and analyzing the normal form on the center manifold associated with the Turing-Hopf bifurcation, we investigate the dynamical classification near the Turing-Hopf bifurcation point in detail. At last, some numerical simulations results are given to support our analytic results.

A New Fractional Model for Cancer Therapy with M1 Oncolytic Virus

The aim of this work is to propose and analyze a new mathematical model formulated by fractional differential equations (FDEs) that describes the dynamics of oncolytic M1 virotherapy. The well-posedness of the proposed model is proved through existence, uniqueness, nonnegativity, and boundedness of solutions. Furthermore, we study all equilibrium points and conditions needed for their existence. We also analyze the global stability of these equilibrium points and investigate their instability conditions. Finally, we state some numerical simulations in order to exemplify our theoretical results.

On the Solutions of Three-Dimensional Rational Difference Equation Systems

In this paper, we are interested in a technique for solving some nonlinear rational systems of difference equations of third order, in three-dimensional case as a special case of the following system: x n + 1 = y n z n − 1 / y n ± x n − 2 , y n + 1 = z n x n − 1 / z n ± y n − 2 , and z n + 1 = x n y n − 1 / x n ± z n − 2 with initial conditions x − 2 , x − 1 , x 0 , y − 2 , y − 1 , y 0 , z − 2 , z − 1 , and z 0 are nonzero real numbers. Moreover, we study some behavior of the systems such as the boundedness of solutions for such systems. Finally, we present some numerical examples by giving some numerical values for the initial values of each case. Some figures have been given to explain the behavior of the obtained solutions in the case of numerical examples by using the mathematical program MATLAB to confirm the obtained results.