Dynamic properties of a discrete population model with diffusion

We study the dynamical properties of a discrete population model with diffusion. We survey the transcritical, pitchfork, and flip bifurcations of nonhyperbolic fixed points by using the center manifold theorem. For the degenerate fixed point with eigenvalues ±1 of the model, we obtain the normal form of the mapping by using the coordinate transformation. Then we give an approximating system of the normal form via an approximation by a flow. We give the local behavior near a degenerate equilibrium of the vector field by the blowup technique. By the conjugacy between the reflection of time-one mapping of a vector field and the model we obtain the stability and qualitative structures near the degenerate fixed point of the model. Finally, we carry out a numerical simulation to illustrate the analytical results of the model.

Existence and Uniqueness of Positive and Bounded Solutions of a Discrete Population Model with Fractional Dynamics

We depart from the well-known one-dimensional Fisher’s equation from population dynamics and consider an extension of this model using Riesz fractional derivatives in space. Positive and bounded initial-boundary data are imposed on a closed and bounded domain, and a fully discrete form of this fractional initial-boundary-value problem is provided next using fractional centered differences. The fully discrete population model is implicit and linear, so a convenient vector representation is readily derived. Under suitable conditions, the matrix representing the implicit problem is an inverse-positive matrix. Using this fact, we establish that the discrete population model is capable of preserving the positivity and the boundedness of the discrete initial-boundary conditions. Moreover, the computational solubility of the discrete model is tackled in the closing remarks.

Dynamics of a nonlinear discrete population model with jumps

Our aim is to investigate the global asymptotic behavior, the existence of invariant intervals, oscillatory behavior, structure of semicycles, and periodicity of a nonlinear discrete population model of the form xn+1= F(xn); for n = 0,1,...,where x0> 0; and the function F is a positive piecewise continuous function with two jump discontinuities satisfying some additional conditions. The motivation for study of this general model was inspired by the classical Williamson's discontinuous population model, some recent results about the dynamics of the discontinuous Beverton-Holt model, and applications of discontinuous maps to the West Nile epidemic model. In the first section we introduce the population model which is a focal point of this paper. We provide background information including a summary of related results, a comparison between characteristics of continuous and discontinuous population models (with and without the Allee-type effect), and a justification of hypotheses introduced in the model. In addition we review some basic concepts and formulate known results which will be used later in the paper. The second and third sections are dedicated to the study of the dynamics and the qualitative analysis of solutions of the model in two distinct cases. An example, illustrating the obtained results, together with some computer experiments that provide deeper insight into the dynamics of the model are presented in the fourth section. Finally, in the last section we formulate three open problems and provide some concluding remarks.

GLOBAL STABILITY OF A NONLINEAR DISCRETE POPULATION MODEL WITH TIME DELAYS

The global stability of a nonlinear discrete population model of Volterra type is studied. The model incorporates time delays. By linearization of the model at positive solutions and construction of Liapunov functionals, sufficient conditions are obtained to ensure that a positive solution of the model is stable and attracts all positive solutions. An example shows the feasibility of our main results.