We consider the following system of difference equations:

The following difference equation is known as the Beverton-Holt model:

This model was introduced by Beverton and Holt in 1957. It depicts density dependent recruitment of a population with limited resources which are not shared equally. The model assumes that the

The Beverton-Holt model is well studied and understood and exhibits the following properties.

Equation (

All solutions of (

If

If

Both equilibrium points are globally asymptotically stable in the corresponding regions of parameters

All these properties can be derived from the explicit form of the solution of (

The following difference equation,

Equation (

Equation (

There exist a zero equilibrium and two positive equilibria,

All solutions of (

If

If

If

In other words, the smaller positive equilibrium serves as the boundary between two basins of attraction. The zero equilibrium has the basin of attraction

The equilibrium points

The two dimensional analogue of (

Two species can interact in several different ways through competition, cooperation, or host-parasitoid interactions. For each of these interactions, we obtain variations of system (

One such variation that exhibits competitive interaction is the following model, known as the Leslie-Gower model, which was considered in Cushing et al. [

The two dimensional analogue of system (

A variation of system (

Our proofs use some recent general results for competitive systems of difference equations of the form:

Competitive systems of the form (

Here we give some basic notions about monotonic maps in the plane.

We define a

Similarly, we define North-East ordering as

A map

For each

For

The following definition is from [

Let

The following theorem was proved by de Mottoni and Schiaffino [

Let

It is well known that a stable period-two orbit and a stable fixed point may coexist; see Hess [

The following result is from [

Let

Theorems

The following result is from [

Let

Suppose that the following statements are true.

The map

The Jacobian matrix of

Either

or

Then there exists a curve

the endpoints of

The following result is a direct consequence of the Trichotomy Theorem of Dancer and Hess (see [

If the nonnegative cone of

The next results give the existence and uniqueness of invariant curves emanating from a nonhyperbolic point of unstable type, that is, a nonhyperbolic point where second eigenvalue is outside interval

Let

Then there exist curves

for

for

Let

Then

Let a map

First we present the local stability analysis of the equilibrium points. It is interesting that the local stability analysis is the more difficult part of our analysis.

The equilibrium points of system (

All solutions of system (

The equilibrium points with strictly positive coordinates satisfy the following system of equations:

From (

The next result gives the necessary and sufficient conditions for (

Let

Assume that

If

If

If

If

If

If

If

If

The discrimination matrix [

Geometrically solutions of system (

The equilibrium curves of system (

Consequently when

When a positive equilibrium point is nonhyperbolic we will refer to it as

The map associated with system (

The Jacobian matrix of

It is worth noting that

Using the equilibrium condition (

The characteristic equation of the matrix (

whose solutions are the eigenvalues

The corresponding eigenvectors of (

We will now consider two lemmas that will be used to prove the local stability character of the positive equilibrium points of system (

The following conditions hold for the coordinates of the positive equilibrium points,

For

For

For

For

This is clear from geometry. See Figure

Local stability.

The following conditions hold for the coordinates of the positive equilibrium points,

For

For

For

(i) Let

See Figure

Therefore

The proofs for the remaining case in (i) and all cases in (ii) and (iii) are similar and will be omitted.

The following conditions hold for the equilibrium points

(i) The eigenvalues of (

(ii) The eigenvalues of (

(iii) The eigenvalues of (

Note that when

Therefore

Note that when

In both cases, the conclusion follows.

(iv) We need to show that

By (

Therefore

By (

Therefore

(v) We need to show that

(vi) We need to show that

Therefore

Therefore

(vii) By (

By (

(viii) The proof of (viii) is similar to the proof of (vii) and will be omitted.

In this section we combine the results from Sections

When

When

It follows from (

The following conditions hold for solutions

When no equilibrium points exist on the

When

if

if

When

if

if

if

(i) When

(ii) In this case

By (

(iii) In this case,

By (

(iv) In this case,

By (

The map

Indeed,

The map

Assume that

The last inequality is equivalent to

Thus we conclude that all solutions of system (

Consequently, all solutions of system (

In this section we show that there are seven dynamic scenarios for global dynamics of system (

Global stability.

Global stability.

Assume that

Local stability of all equilibrium points follows from Theorem

Assume that

Local stability of all equilibrium points follows from Theorem

Now, let

Assume that

Local stability of all equilibrium points follows from Theorem

By Theorem

Let

Finally, let

Assume that

Local stability of all equilibrium points follows from Theorem

By Theorem

Let

Next, assume that

Now, let

Next, assume that

Finally, let

Based on our numerical simulations we believe that

Assume that

Local stability of all equilibrium points follows from Theorem

The existence of the global stable manifold is guaranteed by Theorems

By Theorem

Let

Next, assume that

Now, let

Next, assume that

Assume that

Local stability of all equilibrium points follows from Theorem

The existence of the global stable manifolds are guaranteed by Theorems

The proofs of the basins of attractions

Assume that

Local stability of all equilibrium points follows from Theorem

The authors declare that there is no conflict of interests regarding the publication of this paper.