Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

Solutions for semilinear parabolic equations in L[p] and regularity of weak solutions of the Navier-Stokes system

Stability and bifurcation of a delayed generalized fractional-order prey–predator model with interspecific competition

Controlling bifurcation in a delayed fractional predator-prey system with incommensurate orders

The paper studies the dynamical behaviors of a discrete predator–prey system with nonmonotonic functional response. The local stability of equilibria of the model is obtained. The model undergoes flip bifurcation and Hopf bifurcation by using the center manifold theorem and the bifurcation theory. Numerical simulations not only illustrate our results, but also exhibit the complex dynamical behaviors of the model, such as the period-doubling bifurcation in periods 2, 4 and 8, and quasi-periodic orbits and chaotic sets. The most interesting aspect is choosing the same parameters and the initial value of the model; then we vary the parameter K , and obtain series bifurcations, such as flip bifurcation and Hopf bifurcation.

Stability and bifurcation analysis of a discrete predator–prey model with nonmonotonic functional response

In this paper, we study the stability of a discrete-time predator–prey system with and without Allee effect. By analyzing both systems, we first obtain local stability conditions of the equilibrium points without the Allee effect and then exhibit the impact of the Allee effect on stability when it is imposed on prey population. We also show the stabilizing effect of Allee effect by numerical simulations and verify that when the prey population is subject to an Allee effect, the trajectory of the solutions approximates to the corresponding equilibrium point much faster. Furthermore, for some fixed parameter values satisfying necessary conditions, we show that the corresponding equilibrium point moves from instability to stability under the Allee effect on prey population.

Allee effect in a discrete-time predator-prey system

In this paper, we consider a predator–prey system with time delay where the predator dynamics is logistic with the carrying capacity proportional to prey population. We study the impact of the time delay on the stability of the model and by choosing the delay time τ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay time τ passes some critical values. Using normal form theory and central manifold argument, we also establish the direction and the stability of Hopf bifurcation. Finally, we perform numerical simulations to support our theoretical results.

The stability and Hopf bifurcation for a predator–prey system with time delay

In this paper, we consider a ratio dependent predator–prey system with time delay where the dynamics is logistic with the carrying capacity proportional to prey population. By considering the time delay as bifurcation parameter, we analyze the stability and the Hopf bifurcation of the system based on the normal form approach and the center manifold theory. Finally, we illustrate our theoretical results by numerical simulations.

Hopf bifurcation of a ratio-dependent predator–prey system with time delay

In this paper, we study the stability analysis of equilibrium points of population dynamics with delay when the Allee effect occurs at low population density. Mainly, our mathematical results and numerical simulations point to the stabilizing effect of the Allee effects on population dynamics with delay.

Allee effects on population dynamics with delay

In this article, we consider the nonlinear heat equation on with some boundary conditions and the initial condition , where and p > 1. For the one dimensional case, it is well known that for p < 3 this problem has a local solution for any initial condition . But the existence and uniqueness of a local solution in L 1for the critical exponent p= 3 was wide open and this work is to answer this open question. First, we prove that for the Cauchy problem there is no local solution in L 1for some u 0∈ L 1. Then using the nonlocal existence of Cauchy problem by a cutoff function argument, we prove the nonlocal existence of a solution for the Dirichlet problem which answers this open question. Moreover, we generalize the nonlocal existence result for n-dimensional case with the critical exponent . More general nonlinearity is also considered for Dirichlet boundary value problems. Finally, we prove the same result for the mixed boundary condition with the same initial data u 0.

Canan Çelik

Papers

Allee effect in a discrete-time predator-prey system

The stability and Hopf bifurcation for a predator–prey system with time delay

Hopf bifurcation of a ratio-dependent predator–prey system with time delay

Allee effects on population dynamics with delay

No Local L 1Solution for a Nonlinear Heat Equation