ON THE STABILITY OF DELAY POPULATION DYNAMICS RELATED WITH ALLEE EFFECTS

In recent years, many scientists have focus on the studies of the Allee effect in population dynamics. This paper presents the stability analysis of equilibrium points of population dynamics with Allee effect which occurs at low population density.

Allee effect was first defined by Allee as negative density dependence when the growth rate of the population decreases in low population density.This effect can consist of social dysfunction at small population size, inbreeding depression, food exploitation, predator avoidance of defence and difficulties finding in mates.Authors have studied the stability of different population models within the framework of these effects and developed similar models.Besides, stability analysis is an important research topic in such studies.
In this present study, our purpose is to investigate and compare the stability of equilibrium point with and without Allee effect by considering a more general state of the model studied in [3].Let's look at the nonlinear general delay difference equation ) , , , ( where  is per capita growth rate which is always positive, t N represents the population density at time t and T is the time for sexual maturity.Also, F has the following form: 0 ), , ( ) , , , ( is the function describing interactions (competitions) among mature individuals.
This paper is organized as follows: In section 2, first of all, we give a characterization of the stability of the equilibrium points of Eq.(1).In section 3, we work on the stability analysis of the equilibrium points in Eq.( 1) with the Allee effect.In section 4, we present numerical simulations that support the analytical result.Finally the last section of the paper includes conclusions.

STABILITY ANALYSIS OF Eq.(1)
Before we give the main results of this paper, we shall remind the following Schur-Cohn criterion (see references [2,4,15]).

Theorem 1. (Schur-Cohn
Criteria) The roots of the characteristic polynomial, The characteristic polynomial which is getting from linearization of Eq.( 1) around * N will be 32 ( ) .
Assume that Eq.( 1) has an equilibrium points as * .N Then we get the following theorem.

Theorem 2. *
N is locally stable if and only if the inequalities 2 ) , ( Proof.From the equilibrium point definition of Eq.(1), we have Let's take ) , , ( ), , , ( by Theorem 1.If we write the values of , p q and r in the first inequality of Eq.( 6), we obtain It is easy to see that in this case are negative values for [0, ), N  the last inequality is always provided.Therefore, (2) is confirmed.Now, if the values of , p q and r are written in the second inequality in (6), we get If the last expressions is written in the form of two inequalities, we can write as confirmed.

ALLEE EFFECTS ON THE DISCRETE DELAY MODEL (1)
In this section, we study the local stability analysis of the equilibrium points of Eq.( 1) with the addition of Allee effect at time 2, t  t and ( , 2).tt 3.1.Allee effect at time t-2 We consider the following non-linear delay difference equation by the addition of Allee effect to discrete delay model Eq.( 1) where the function f satisfies the properties ( 1) and ( 2).The conclusion of the biological facts requires the following assumption on α.
For the second inequality in (6), we arrive at 1 ) , (

Allee effect at time t-1
Let us consider the following non-linear delay difference equation by the addition of Allee effect to discrete delay model (1) ).
* N equilibrium point of Eq.( 11) is positive equilibrium point of Eq.( 7).Then we have the following theorem.
Proof.According to Eq.( 11), the values of , p q and r are as follows


Firstly, from the first inequality in (6), we obtain

Allee effect at time t
We now incorporate an Allee effect into the discrete delay model as follows: * N equilibrium point of Eq.( 15) is positive equilibrium point of Eq.( 7).Then we can state the following theorem.
Proof.If the values of , p q and r are written in Eq.( 15), we have ) , Let's consider stability conditions in (6).Thus, we get  16), ( 17) and (18) Eq.( 1) is stable if and only if If the values of , x y and z are written in the stability conditions of Eq.( 7), Eq.( 11) and Eq.( 15), we obtain respectively.It is clear that for each value of x and , y which provides inequality (19), at least one of the conditions (20), ( 21) and ( 22) is not satisfied (for each z>0).In other words, stable equilibrium point of Eq.( 2) is not stable for equations (7), (11), (15).

NUMERICAL SIMILATIONS
In this section, we numerically present our the analytical result obtained in the former sections by using MATLAB programming.We graph the 2D trajectories of the population dynamics model (1) with and without Allee effect at time 1 , 2   t t and t in Fig. 1, Fig. 2 and Fig. 3, respectively.In this figures we take the function (see, for instance [7]) and the Allee function ), /( ) ( and , t where  is a positive constant.It is obvious from the graph that the comparisons of the population density diagrams also verify the stabilizing impact of the Allee effects.In these computations, the initial conditions are taken as 21 0.  1.9, ( ) / ( ), 0.03, ( ).
Fig. 2. Density-time graphs of the models)1 ( 2 1 1       t t t t N N N N  is, there is no reproduction without partners. (1)    that is, Allee effect vanishes at high densities.Eq.(7) has the same positive equilibrium points with(1), since *  is normalized growth rate such that * /.     Then we get the following theorem.Theorem 3. * N is locally stable if and only if the inequalities , are confirmed.