In this section, we will present the global stability analysis of an equilibrium point of (1) with and without Allee effect. We shall require the following global stability theorem and definition (see, for instance, [15]) for (1).

**Definition 7** {N}^{\ast} is said to be globally asymptotically stable if it is globally attractive and locally stable.

**Theorem 8** *Let the function* *F* *at* (1) *be continuous such that* F:[0,p)\to [0,p), 0<p\le \mathrm{\infty}, *if* 0<F(N)<N *for all* N\in (0,p), *then the origin is globally asymptotically stable*.

We then obtain the following theorem.

**Theorem 9**
*The equilibrium point*
{N}^{\ast}
*is globally asymptotically stable if*

{f}^{\prime}\left({N}^{\ast}\right)>-r.

(6)

*Proof* The function of *F* at (1) is continuous such that F:[0,p)\to [0,p). The linearized equation about {N}^{\ast} is

{u}_{T+1}=[r+{f}^{\prime}\left({N}^{\ast}\right)]{u}_{T}

such that {u}_{T}={N}_{t}-{N}^{\ast}. Let us take {F}_{l}=r+{f}^{\prime}({N}^{\ast}). Since {f}^{\prime}({N}^{\ast})<0 from condition (i) with 0<r<1, we can write

\begin{array}{c}r+{f}^{\prime}\left({N}^{\ast}\right)-r<0\hfill \\ \phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}(r+{f}^{\prime}\left({N}^{\ast}\right)-r)u<0\hfill \\ \phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}(r+{f}^{\prime}\left({N}^{\ast}\right))u<ru\hfill \\ \phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}(r+{f}^{\prime}\left({N}^{\ast}\right))u<u\hfill \\ \phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}{F}_{l}(u)<u.\hfill \end{array}

Now, we must satisfy 0<{F}_{l}(u) for the global stability of the zero equilibrium point (origin) as follows:

\begin{array}{c}(r+{f}^{\prime}\left({N}^{\ast}\right))u>0\hfill \\ \phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}r+{f}^{\prime}\left({N}^{\ast}\right)>0\hfill \\ \phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}{f}^{\prime}\left({N}^{\ast}\right)>-r.\hfill \end{array}

It follows that 0<{F}_{l}(u)<u. By Theorem 8, the origin is globally asymptotically stable. Therefore, the equilibrium point {N}^{\ast} of (1) is globally asymptotically stable. □

### 3.1 The Allee effect at time *t*

We will now consider the following nonlinear discrete-time dynamical system with Allee effect at time *t*, given by (2):

{N}_{t+1}={r}^{\ast}{N}_{t}\alpha ({N}_{t})+f({N}_{t})={F}_{\alpha}({r}^{\ast},\alpha ,{N}_{t}).

(7)

According to the information, {F}_{\alpha} has a unique positive equilibrium point {N}^{\ast}\in (0,p).

We then obtain the following theorem.

**Theorem 10** *The equilibrium point* {N}^{\ast} *of* (7) *is globally asymptotically stable if*

-r[1+{N}^{\ast}\frac{{\alpha}^{\prime}({N}^{\ast})}{\alpha ({N}^{\ast})}]<{f}^{\prime}\left({N}^{\ast}\right)\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}r[1+{N}^{\ast}\frac{{\alpha}^{\prime}({N}^{\ast})}{\alpha ({N}^{\ast})}]<1.

(8)

*Proof* The function {F}_{\alpha} at (7) is continuous function such that {F}_{\alpha}:[0,p)\to [0,p). The linearized equation about {N}^{\ast} is

{u}_{T+1}=[{f}^{\prime}\left({N}^{\ast}\right)+r{N}^{\ast}\frac{{\alpha}^{\prime}({N}^{\ast})}{\alpha ({N}^{\ast})}+r]{u}_{T}

such that {u}_{T}={N}_{t}-{N}^{\ast}. Since {f}^{\prime}({N}^{\ast})<0 from condition (i), we can write

\begin{array}{c}{f}^{\prime}\left({N}^{\ast}\right)+r{N}^{\ast}\frac{{\alpha}^{\prime}({N}^{\ast})}{\alpha ({N}^{\ast})}+r<r{N}^{\ast}\frac{{\alpha}^{\prime}({N}^{\ast})}{\alpha ({N}^{\ast})}+r\hfill \\ \phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}[{f}^{\prime}\left({N}^{\ast}\right)+r{N}^{\ast}\frac{{\alpha}^{\prime}({N}^{\ast})}{\alpha ({N}^{\ast})}+r]u<r[{N}^{\ast}\frac{{\alpha}^{\prime}({N}^{\ast})}{\alpha ({N}^{\ast})}+1]u\hfill \end{array}

such that r{N}^{\ast}\frac{{\alpha}^{\prime}({N}^{\ast})}{\alpha ({N}^{\ast})}+r>0. If r[1+{N}^{\ast}\frac{{\alpha}^{\prime}({N}^{\ast})}{\alpha ({N}^{\ast})}]<1, then

[{f}^{\prime}\left({N}^{\ast}\right)+r{N}^{\ast}\frac{{\alpha}^{\prime}({N}^{\ast})}{\alpha ({N}^{\ast})}+r]u<u.

If -r<[{f}^{\prime}({N}^{\ast})+r{N}^{\ast}\frac{{\alpha}^{\prime}({N}^{\ast})}{a({N}^{\ast})}], we have

\begin{array}{c}-r<[{f}^{\prime}\left({N}^{\ast}\right)+r{N}^{\ast}\frac{{\alpha}^{\prime}({N}^{\ast})}{\alpha ({N}^{\ast})}]\hfill \\ \phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}0<[r+{f}^{\prime}\left({N}^{\ast}\right)+r{N}^{\ast}\frac{{\alpha}^{\prime}({N}^{\ast})}{\alpha ({N}^{\ast})}]\hfill \\ \phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}0<[r+{f}^{\prime}\left({N}^{\ast}\right)+r{N}^{\ast}\frac{{\alpha}^{\prime}({N}^{\ast})}{\alpha ({N}^{\ast})}]u.\hfill \end{array}

Let us take {F}_{l,a}=r+{f}^{\prime}({N}^{\ast})+r{N}^{\ast}\frac{{\alpha}^{\prime}({N}^{\ast})}{\alpha ({N}^{\ast})}, provided that 0<{F}_{l,a}(u)<u. It is clear from Theorem 8 that the origin is globally asymptotically stable. Then {N}^{\ast} is globally asymptotically stable. Note that {F}_{l,a}>0 is always true from Theorem 5. Namely, -r[1+{N}^{\ast}\frac{{\alpha}^{\prime}({N}^{\ast})}{\alpha ({N}^{\ast})}]<{f}^{\prime}({N}^{\ast}) always. Since this inequality is related with [1+{N}^{\ast}\frac{{\alpha}^{\prime}({N}^{\ast})}{\alpha ({N}^{\ast})}]<1, we must take this inequality as stability conditions. □

**Corollary 11** *The Allee effect at time* *t* *decreases the global stability of an equilibrium point*. (*If* (6) *and* (8) *are considered*, *it can easily be seen*.)

**Corollary 12** *It is clear from* (1) *that* {N}^{\ast} *is locally stable but not globally stable if the inequality*

-1-r<f\left({N}^{\ast}\right)<-r

*holds*. (*The proof is clear from Theorems* 4 *and* 9.)