^{1,2}

^{1}

^{2}

The objective of this paper is to systematically study the stability and oscillation of the discrete delay annual plants model. In particular, we establish some sufficient conditions for global stability of the unique positive fixed point and establish an explicit sufficient condition for oscillation of the positive solutions about the fixed point. Some illustrative examples and numerical simulations are included to demonstrate the validity and applicability of the results.

Most populations live in seasonal environments and, because of this, have annual rhythms of reproduction and death. In addition, measurements are often made annually because interest is centered on population changes from year to year rather than on the obvious and predictable changes that occur seasonally. Continuous differential equations are not well suited to these kinds of processes and data. Thus, practical ecologists have long employed discrete-time difference equations for studying the dynamics of resource and pest populations. In particular one can consider the difference equation

population have the potential to increase exponentially;

there is a density-dependent feedback that progressively reduces the actual rate of increase.

In fact, in population dynamics, there is a tendency for that variable

Population model shape.

In recent decades the dynamics of discrete models in different areas have been extensively investigated by many authors. For contributions, we refer the reader to [

For population models of plants, Watkinson [

Watkinson in [

In [

For the delay equations, for completeness, we present some global stability conditions of the zero solution of the delay difference equation

In this section, we establish some sufficient conditions for local and global stability of the positive fixed-point

Assume that (

To prove the main global stability results for (

Assume that (

First we will show the upper bound in (

One of the techniques used in the proof of the global stability of the zero solution of the nonlinear equation

Assume that (

First, we prove that every positive solution

From Theorem

Assume that (

Also by employing the result due to Kovácsvölgy [

Assume that (

The result by Yu and Cheng [

Assume that (

We illustrate the main results with the following examples.

Consider the model

The iterations in the phase plane

Time series

Consider the model

The iterations in the phase plane of (

The time series of (

Note that the condition of local stability of (

Consider the model

The iterations in the phase plane of (

The time series of (

We note that the results in [

In this section, we establish an explicit sufficient condition for oscillation of (

We, first prove the following theorem which proves that oscillation of (

Assume that (

Without loss of generality we assume that (

To establish the condition for oscillation of all positive solution of (

If

Theorem

Assume that (

To illustrate the main result of Theorem

Consider the model

Iterations of (

Time series of (

(1) We note that the condition (

The author thanks the Research Centre in College of Science in King Saud University for encouragements and supporting this project which has taken the number Math. 2009/33.