Dynamics of a two-dimensional system of rational difference equations of Leslie--Gower type

Abstract

We investigate global dynamics of the following systems of difference equations

$$\left\{\begin{array}{ccc}\hfill x_{n+1}\hfill & \hfill =\hfill & \hfill \frac{{\alpha }_1+{\beta }_1{x}_n}{A_1+y_n}\hfill \\ \hfill y_{n+1}\hfill & \hfill =\hfill & \hfill \frac{{\gamma }_2{y}_n}{A_2+B_2{x}_n+y_n}\hfill \end{array}\right.,n=0,1,2,\dots$$

where the parameters α ₁, β ₁, A ₁, γ ₂, A ₂, B ₂ are positive numbers, and the initial conditions x ₀ and y ₀ are arbitrary nonnegative numbers. We show that this system has rich dynamics which depends on the region of parametric space. We show that the basins of attractions of different locally asymptotically stable equilibrium points or non-hyperbolic equilibrium points are separated by the global stable manifolds of either saddle points or non-hyperbolic equilibrium points. We give examples of a globally attractive non-hyperbolic equilibrium point and a semi-stable non-hyperbolic equilibrium point. We also give an example of two local attractors with precisely determined basins of attraction. Finally, in some regions of parameters, we give an explicit formula for the global stable manifold.

Mathematics Subject Classification (2000)

Primary: 39A10, 39A11 Secondary: 37E99, 37D10

1 Introduction

In this paper, we study the global dynamics of the following rational system of difference equations

$$\left\{\begin{array}{ccc}\hfill x_{n+1}\hfill & \hfill =\hfill & \hfill \frac{{\alpha }_1+{\beta }_1{x}_n}{A_1+y_n}\hfill \\ \hfill y_{n+1}\hfill & \hfill =\hfill & \hfill \frac{{\gamma }_2{y}_n}{A_2+B_2{x}_n+y_n}\hfill \end{array}\right.,n=0,1,2,\dots$$

(1)

where the parameters α ₁, β ₁, A ₁, γ ₂, A ₂, B ₂ are positive numbers and initial conditions x ₀ and y ₀ are arbitrary nonnegative numbers.

System (1) was mentioned in [1] as one of three systems of Open Problem 3, which asked for a description of the global dynamics of some rational systems of difference equations. In notation used to label systems of linear fractional difference equations used in [1], System (1) is referred to as (29, 38). This system is dual to the system where the roles of x _n and y _n are interchanged, which is labeled as (29, 38) in [1], and so all results proven here extend to the latter system. In this paper, we provide a precise description of the global dynamics of the System (1). We show that System (1) may have between zero and three equilibrium points, which may have different local character. If System (1) has one equilibrium point, then this point is either locally asymptotically stable or saddle point or non-hyperbolic equilibrium point. If System (1) has two equilibrium points, then they are either locally asymptotically stable and non-hyperbolic, or locally asymptotically stable and saddle point. If System (1) has three equilibrium points, then two of equilibrium points are locally asymptotically stable and the third point, which is between these two points in southeast ordering defined below, is a saddle point. The major problem for global dynamics of the System (1) is determining the basins of attraction of different equilibrium points. The difficulty in analyzing the behavior of all solutions of the System (1) lies in the fact that there are many regions of parameters where this system possesses different equilibrium points with different local character and that in several cases, the equilibrium point is non-hyperbolic. However, all these cases can be handled by using recent results from [2].

System (1) is a competitive system, and our results are based on recent results about competitive systems in the plane, see [2, 3]. System (1) can be used as a mathematical model for competition in population dynamics. In fact, second equation in (1) is of Leslie-Gower type, and first equation can be considered to be of Leslie-Gower type with stocking which is represented with the term α ₁, see [4–6].

In the next section, we present some general results about competitive systems in the plane. Section 3 contains some basic facts such as the non-existence of period-two solution of System (1). Section 4 analyzes local stability which is fairly complicated for this system. Finally, Section 5 gives global dynamics for all values of parameters.

2 Preliminaries

A first-order system of difference equations

$$\left\{\begin{array}{ccc}\hfill x_{n+1}\hfill & \hfill =\hfill & \hfill f\left(x_n,y_n\right)\hfill \\ \hfill y_{n+1}\hfill & \hfill =\hfill & \hfill g\left(x_n,y_n\right)\hfill \end{array}\right.,n=0,1,2,\dots$$

(2)

where $\mathcal{S}$ ⊂ ℝ², (f, g): $\mathcal{S}$ → $\mathcal{S}$, f, g are continuous functions is competitive if f(x, y) is non-decreasing in x and non-increasing in y, and g(x, y) is non-increasing in x and non-decreasing in y. If both f and g are non-decreasing in x and y, the System (2) is cooperative. Competitive and cooperative maps are defined similarly. Strongly competitive systems of difference equations or strongly competitive maps are those for which the functions f and g are coordinate-wise strictly monotone.

Competitive and cooperative systems have been investigated by many authors, see [2, 3, 5–19]. Special attention to discrete competitive and cooperative systems in the plane was given in [2, 3, 5–7, 10, 12, 17, 20]. One of the reasons for paying special attention to two-dimensional discrete competitive and cooperative systems is their applicability and the fact that many examples of mathematical models in biology and economy which involve competition or cooperation are models which involve two species. Another reason is that the theory of two-dimensional discrete competitive and cooperative systems is very well developed, unlike such theory for three and higher dimensional systems. Part of the reason for this situation is de Mottoni and Schiaffino theorem given below, which provides relatively simple scenarios for possible behavior of many two-dimensional discrete competitive and cooperative systems. However, this does not mean that one cannot encounter chaos in such systems as has been shown by Smith, see [17].

If v = (u, v) ∈ ℝ², we denote with $\mathcal{Q}$ _l (v), ℓ ∈ {1, 2, 3, 4}, the four quadrants in ℝ² relative to v, i.e., $\mathcal{Q}$ ₁ (v) = {(x, y) ℝ²: x ≥ u, y ≥ v}, $\mathcal{Q}$ ₂ (v) = {(x, y) ∈ ℝ²: x ≤ u, y ≥ v}, and so on. Define the South-East partial order ≼ _se on ℝ² by (x, y) ≼ _se (s, t) if and only if x ≤ s and y ≥ t. Similarly, we define the North-East partial order ≼ _ne on ℝ² by (x, y) ≼ _ne (s, t) if and only if x ≤ s and y ≤ t. For $\mathcal{A}$ ⊂ ℝ² and x ∈ ℝ², define the distance from x to $\mathcal{A}$ as dist(x, $\mathcal{A}$) = inf{||x-y||: y ∈ $\mathcal{A}$}. By int $\mathcal{A}$, we denote the interior of a set $\mathcal{A}$.

It is easy to show that a map F is competitive if it is non-decreasing with respect to the South-East partial order, that is, if the following holds:

$$\left(\begin{array}{c}\hfill x^1\hfill \\ \hfill y^1\hfill \end{array}\right){\preccurlyeq }_{se}\left(\begin{array}{c}\hfill x^2\hfill \\ \hfill y^2\hfill \end{array}\right)\Rightarrow F\left(\begin{array}{c}\hfill x^1\hfill \\ \hfill y^1\hfill \end{array}\right){\preccurlyeq }_{se}F\left(\begin{array}{c}\hfill x^2\hfill \\ \hfill y^2\hfill \end{array}\right).$$

(3)

For standard definitions of attracting fixed point, saddle point, stable manifold, and related notions see [11].

We now state three results for competitive maps in the plane. The following definition is from [17].

Definition 1 Let $\mathcal{S}$ be a nonempty subset of ℝ². A competitive map T : $\mathcal{S}$ → $\mathcal{S}$ is said to satisfy condition (O+) if for every x, y in $\mathcal{S}$ , T(x) ≼ _ne T (y) implies x ≼ _ne y, and T is said to satisfy condition (O-) if for every x, y in $\mathcal{S}$ , T(x) ≼ _ne T (y) implies y ≼ _ne x.

The following theorem was proved by de Mottoni and Schiaffino [20] for the Poincaré map of a periodic competitive Lotka-Volterra system of differential equations. Smith [14, 15] generalized the proof to competitive and cooperative maps.

Theorem 1 Let $\mathcal{S}$ be a nonempty subset of ℝ² . If T is a competitive map for which (O+) holds then for all x ∈ $\mathcal{S}$, {T ⁿ (x)} is eventually componentwise monotone. If the orbit of x has compact closure, then it converges to a fixed point of T. If instead (O-) holds, then for all x ∈ $\mathcal{S}$, {T ²ⁿ(x)} is eventually componentwise monotone. If the orbit of x has compact closure in $\mathcal{S}$ , then its omega limit set is either a period-two orbit or a fixed point.

The following result is from [17], with the domain of the map specialized to be the cartesian product of intervals of real numbers. It gives a sufficient condition for conditions (O+) and (O-).

Theorem 2 Let ℛ ⊂ ℝ² be the cartesian product of two intervals in ℝ. Let T: ℛ → ℛ be a C ¹ competitive map. If T is injective and det J _T (x) > 0 for all x ∈ ℛ then T satisfies (O+). If T is injective and det J _T (x) < 0 for all x ∈ ℛ then T satisfies (O-).

The following result is a direct consequence of the Trichotomy Theorem of Dancer and Hess, see [3] and [21] and is helpful for determining the basins of attraction of the equilibrium points.

Corollary 1 If the nonnegative cone of ≼ is a generalized quadrant in ℝ ⁿ , and if T has no fixed points in 〚u ₁ , u ₂〛 other than u ₁ and u ₂ , then the interior of 〚u ₁ , u ₂〛 is either a subset of the basin of attraction of u ₁ or a subset of the basin of attraction of u ₂.

Next result is well-known global attractivity result that holds in partially ordered Banach spaces as well, see [21].

Theorem 3 Let T be a monotone map on a closed and bounded rectangular region ℛ ⊂ ℝ² . Suppose that T has a unique fixed point $\bar{e}$ in ℛ. Then $\bar{e}$ is a global attractor of T on ℛ.

The following theorems were proved by Kulenović and Merino [2] for competitive systems in the plane, when one of the eigenvalues of the linearized system at an equilibrium (hyperbolic or non-hyperbolic) is by absolute value smaller than 1 while the other has an arbitrary value. These results are useful for determining basins of attraction of fixed points of competitive maps.

Theorem 4 Let T be a competitive map on a rectangular region ℛ ⊂ ℝ² . Let $\overline{\mathsf{\text{x}}}\in ℛ$ be a fixed point of T such that Δ: = ℛ ∩ int $\left({\mathcal{Q}}_{\mathsf{\text{1}}}\left(\overline{\mathsf{\text{x}}}\right)\cup {\mathcal{Q}}_{\mathsf{\text{3}}}\left(\overline{\mathsf{\text{x}}}\right)\right)$ is nonempty (i.e., $\overline{\mathsf{\text{x}}}$ is not the NW or SE vertex of ℛ), and T is strongly competitive on Δ. Suppose that the following statements are true.

• a. The map T has a C ¹ extension to a neighborhood of $\overline{\mathsf{\text{x}}}$.

• b. The Jacobian $J_T\left(\overline{\mathsf{\text{x}}}\right)$ of T at $\overline{\mathsf{\text{x}}}$ has real eigenvalues λ , μ such that 0 < |λ| < μ, where |λ| < 1, and the eigenspace E ^λ associated with λ is not a coordinate axis.

Then there exists a curve $\mathcal{C}$⊂ ℛ through $\overline{\mathsf{\text{x}}}$ that is invariant and a subset of the basin of attraction of $\overline{\mathsf{\text{x}}}$ , such that $\mathcal{C}$ is tangential to the eigenspace E ^λ at $\overline{\mathsf{\text{x}}}$ , and $\mathcal{C}$ is the graph of a strictly increasing continuous function of the first coordinate on an interval. Any endpoints of $\mathcal{C}$ in the interior of ℛ are either fixed points or minimal period-two points. In the latter case, the set of endpoints of $\mathcal{C}$ is a minimal period-two orbit of T.

The situation where the endpoints of $\mathcal{C}$ are boundary points of ℛ is of interest. The following result gives a sufficient condition for this case.

Theorem 5 For the curve $\mathcal{C}$ of Theorem 4 to have endpoints in ∂ℛ, it is sufficient that at least one of the following conditions is satisfied.

• i. The map T has no fixed points nor periodic points of minimal period-two in Δ.

• ii. The map T has no fixed points in Δ, det $J_T\left(\overline{\mathsf{\text{x}}}\right)>0$, and $T\left(\overline{\mathsf{\text{x}}}\right)=\mathsf{\text{x}}$ has no solutions x ∈ Δ.

• iii. The map T has no points of minimal period-two in Δ, det $J_T\left(\overline{\mathsf{\text{x}}}\right)<0$, and $T\left(\mathsf{\text{x}}\right)=\overline{\mathsf{\text{x}}}$ has no solutions x ∈ Δ.

The next result is useful for determining basins of attraction of fixed points of competitive maps.

Theorem 6 (A) Assume the hypotheses of Theorem 4, and let $\mathcal{C}$ be the curve whose existence is guaranteed by Theorem 4. If the endpoints of $\mathcal{C}$ belong to ∂ℛ, then $\mathcal{C}$ separates ℛ into two connected components, namely

$${\mathcal{W}}_{-}:=\left\{\mathsf{\text{x}}\in ℛ\backslash \mathcal{C}:\exists \mathsf{\text{y}}\in \mathcal{C}with\mathsf{\text{x}}{\preccurlyeq }_{se}\mathsf{\text{y}}\right\}\mathsf{\text{and}}{\mathcal{W}}_{+}:=\left\{\mathsf{\text{x}}\in ℛ\backslash \mathcal{C}:\exists \mathsf{\text{y}}\in \mathcal{C}with\mathsf{\text{y}}{\preccurlyeq }_{se}\mathsf{\text{x}}\right\},$$

(4)

such that the following statements are true.

• (i) $\mathcal{W}$ _- is invariant, and $\mathsf{\text{dist}}\left(T^n\left(\mathsf{\text{x}}\right),{\mathcal{Q}}_{\mathsf{\text{2}}}\left(\overline{\mathsf{\text{x}}}\right)\right)\to 0asn\to \infty forevery\mathsf{\text{x}}\in \mathcal{W}\text{\_}$.

• (ii) $\mathcal{W}$ ₊ is invariant, and $\mathsf{\text{dist}}\left(T^n\left(\mathsf{\text{x}}\right),{\mathcal{Q}}_{\mathsf{\text{4}}}\left(\overline{\mathsf{\text{x}}}\right)\right)\to 0asn\to \infty forevery\mathsf{\text{x}}\in {\mathcal{W}}_{+}$.

• (B) If, in addition to the hypotheses of part (A), $\overline{\mathsf{\text{x}}}$ is an interior point of ℛ and T is C ² and strongly competitive in a neighborhood of $\overline{\mathsf{\text{x}}}$, then T has no periodic points in the boundary of $\left({\mathcal{Q}}_{\mathsf{\text{1}}}\left(\overline{\mathsf{\text{x}}}\right)\cup {\mathcal{Q}}_{\mathsf{\text{3}}}\left(\overline{\mathsf{\text{x}}}\right)\right)$ except for $\overline{\mathsf{\text{x}}}$ , and the following statements are true.

• (iii) For every x ∈ $\mathcal{W}$ _- there exists n ₀ ∈ ℕ such that T ⁿ (x) ∈ int ${\mathcal{Q}}_{\mathsf{\text{2}}}\left(\overline{\mathsf{\text{x}}}\right)$ for n ≥ n ₀.

• (iv) For every x ∈ $\mathcal{W}$ ₊ there exists n ₀ ∈ ℕ such that T ⁿ (x) ∈ int ${\mathcal{Q}}_{\mathsf{\text{4}}}\left(\overline{\mathsf{\text{x}}}\right)$ for n ≥ n ₀.

If T is a map on a set ℛ and if $\overline{\mathsf{\text{x}}}$ is a fixed point of T, the stable set ${\mathcal{W}}^s\left(\overline{\mathsf{\text{x}}}\right)$ of $\overline{\mathsf{\text{x}}}$ is the set $\left\{x\in ℛ:T^n\left(\mathsf{\text{x}}\right)\to \overline{\mathsf{\text{x}}}\right\}$ and unstable set ${\mathcal{W}}^u\left(\overline{\mathsf{\text{x}}}\right)$ of $\overline{\mathsf{\text{x}}}$ is the set

$$\left\{\mathsf{\text{x}}\in ℛ:\mathsf{\text{there}}\mathsf{\text{exists}}{\left\{{\mathsf{\text{x}}}_n\right\}}_{n=-\infty }^0\subset ℛ\mathsf{\text{s}}\mathsf{\text{.t}}.T\left({\mathsf{\text{x}}}_n\right)={\mathsf{\text{x}}}_{n+1},{\mathsf{\text{x}}}_0=\mathsf{\text{x}},\mathsf{\text{and}}\underset{n\to -\infty }{lim}{\mathsf{\text{x}}}_n=\overline{\mathsf{\text{x}}}\right\}$$

When T is non-invertible, the set ${\mathcal{W}}^s\left(\overline{\mathsf{\text{x}}}\right)$ may not be connected and made up of infinitely many curves, or ${\mathcal{W}}^u\left(\overline{\mathsf{\text{x}}}\right)$ may not be a manifold. The following result gives a description of the stable and unstable sets of a saddle point of a competitive map. If the map is a diffeomorphism on ℛ, the sets ${\mathcal{W}}^s\left(\overline{\mathsf{\text{x}}}\right)$ and ${\mathcal{W}}^u\left(\overline{\mathsf{\text{x}}}\right)$ are the stable and unstable manifolds of $\overline{\mathsf{\text{x}}}$.

Theorem 7 In addition to the hypotheses of part (B) of Theorem 6, suppose that μ > 1 and that the eigenspace E ^μ associated with μ is not a coordinate axis. If the curve $\mathcal{C}$ of Theorem 4 has endpoints in ∂ℛ, then $\mathcal{C}$ is the stable set ${\mathcal{W}}^s\left(\overline{\mathsf{\text{x}}}\right)$ of $\overline{\mathsf{\text{x}}}$ , and the unstable set ${\mathcal{W}}^u\left(\overline{\mathsf{\text{x}}}\right)$ of $\bar{x}$ is a curve in ℛ that is tangential to E ^μ at $\overline{\mathsf{\text{x}}}$ and such that it is the graph of a strictly decreasing function of the first coordinate on an interval. Any endpoints of ${\mathcal{W}}^u\left(\overline{\mathsf{\text{x}}}\right)$ in ℛ are fixed points of T.

The following result gives information on local dynamics near a fixed point of a map when there exists a characteristic vector whose coordinates have negative product and such that the associated eigenvalue is hyperbolic. This is a well-known result, valid in much more general setting that we include it here for completeness. A point (x, y) is a subsolution if T(x, y) ≼ _se (x, y), and (x, y) is a supersolution if (x, y) ≼ _se T(x, y). An order interval 〚(a, b), (c, d)〛 is the cartesian product of the two compact intervals [a, c] and [b, d].

Theorem 8 Let T be a competitive map on a rectangular set ℛ ⊂ ℝ² with an isolated fixed point $\overline{\mathsf{\text{x}}}\in ℛ$ such that $ℛ\cap \mathsf{\text{int }}\left({\mathcal{Q}}_{\mathsf{\text{2}}}\left(\overline{\mathsf{\text{x}}}\right)\cup {\mathcal{Q}}_{\mathsf{\text{4}}}\left(\overline{\mathsf{\text{x}}}\right)\right)\ne \varnothing$ . Suppose T has a C ¹ extension to a neighborhood of $\overline{\mathsf{\text{x}}}$ . Let v = (v⁽¹⁾, v⁽²⁾) ∈ ℝ² be an eigenvector of the Jacobian of T at $\overline{\mathsf{\text{x}}}$ , with associated eigenvalue μ∈ ℝ. If v⁽¹⁾v⁽²⁾ < 0, then there exists an order interval ℐ which is also a relative neighborhood of $\overline{\mathsf{\text{x}}}$ such that for every relative neighborhood $\mathcal{U}$ ⊂ ℐ of $\overline{\mathsf{\text{x}}}$ the following statements are true.

• i. If μ > 1, then $\mathcal{U}\cap \mathsf{\text{int }}{\mathcal{Q}}_2\left(\overline{\mathsf{\text{x}}}\right)$ contains a subsolution and $\mathcal{U}\cap \mathsf{\text{int }}{\mathcal{Q}}_4\left(\overline{\mathsf{\text{x}}}\right)$ contains a supersolution. In this case for every $\mathsf{\text{x }}\in ℐ\cap \mathsf{\text{int }}\left({\mathcal{Q}}_{\mathsf{\text{2}}}\left(\overline{\mathsf{\text{x}}}\right)\cup {\mathcal{Q}}_{\mathsf{\text{4}}}\left(\overline{\mathsf{\text{x}}}\right)\right)$ there exists N such that T ⁿ (x) ∉ ℐ for n ≥ N.

• ii. If μ < 1, then $\mathcal{U}\cap \mathsf{\text{int }}{\mathcal{Q}}^{\mathsf{\text{2}}}\left(\overline{\mathsf{\text{x}}}\right)$ contains a supersolution and $\mathcal{U}\cap \mathsf{\text{int }}{\mathcal{Q}}_4\left(\overline{\mathsf{\text{x}}}\right)$ contains a subsolution. In this case $T^n\left(\mathsf{\text{x}}\right)\to \overline{\mathsf{\text{x}}}$ for every x ∈ ℐ.

3 Some basic facts

In this section, we give some basic facts about the nonexistence of period-two solutions, local injectivity of the map T at the equilibrium point, and boundedness of solutions. See [22] for similar analysis.

3.1 Equilibrium points

The equilibrium points ($\overline{\mathsf{\text{x}}}$, $ȳ$) of System (1) satisfy

$$\bar{x}=\frac{{\alpha }_1+{\beta }_1\bar{x}}{A_1+ȳ},ȳ=\frac{{\gamma }_2ȳ}{A_2+B_2\bar{x}+ȳ}.$$

(5)

Solutions of System (5) are:

• (i) $ȳ=0$, $\bar{x}=\frac{{\alpha }_1}{A_1-{\beta }_1}$, A ₁ > β ₁, i.e. $E_1=\left(\frac{{\alpha }_1}{A_1-{\beta }_1},0\right)$. Thus, the equilibrium point $E_1=\left(\frac{{\alpha }_1}{A_1-{\beta }_1},0\right)$ exists if A ₁ > β ₁.

• (ii) If $ȳ\ne 0$, then using System (5), we obtain

  $$ȳ={\gamma }_2-A_2-B_2\bar{x},{\bar{x}}^2{B}_2-\bar{x}\left({\gamma }_2+A_1-A_2-{\beta }_1\right)+{\alpha }_1=0.$$

  (6)

Solutions of System (6) are:

$${\bar{x}}_{3,2}=\frac{{\gamma }_2+A_1-A_2-{\beta }_1\pm \sqrt{D_0}}{2B_2},{ȳ}_{2,3}=\frac{{\gamma }_2-A_2-A_1+{\beta }_1\pm \sqrt{D_0}}{2},$$

(7)

where D ₀ = (γ ₂ - A ₂ + A ₁ - β ₁)² - 4B ₂ α ₁ which gives a pair of the equilibrium points $E_2=\left({\bar{x}}_2,{ȳ}_2\right)$ and $E_3=\left({\bar{x}}_3,{ȳ}_3\right)$.

The criteria for the existence of the three equilibrium points are summarized in Table 1.

Table 1 The equilibrium points of System (1).

3.2 Injectivity

Lemma 1 Assume that $\left(\bar{x},ȳ\right)$ is an equilibrium of the map T. Then the following holds:

• 1) If

  $$B_2>\frac{A_2{\beta }_1}{{\alpha }_1},A_1\left(B_2{\alpha }_1-A_2{\beta }_1\right){\gamma }_2-\left(B_2{\alpha }_1+\left(A_1-A_2\right){\beta }_1\right)\left(A_1{A}_2-{\beta }_1{A}_2+B_2{\alpha }_1\right)=0,$$

  (8)

then $T\left(x,\frac{A_1{A}_2{\beta }_1+x{A}_1{B}_2{\beta }_1}{B_2{\alpha }_1-A_2{\beta }_1}\right)=\left(\bar{x},ȳ\right)$ for all x ≥ 0, where

$$\left(\bar{x},ȳ\right)=\left(\bar{x},\frac{A_1{A}_2{\beta }_1+\bar{x}{A}_1{B}_2{\beta }_1}{B_2{\alpha }_1-A_2{\beta }_1}\right)=\left(\frac{B_2{\alpha }_1+A_2{\beta }_1}{A_1{B}_2},\frac{-A_2{\beta }_1^2+A_1{A}_2{\beta }_1+B_2{\alpha }_1{\beta }_1}{B_2{\alpha }_1-A_2{\beta }_1}\right).$$

That is the line

$$ℐ=\left\{\left(x,\frac{A_1{A}_2{\beta }_1+x{A}_1{B}_2{\beta }_1}{B_2{\alpha }_1-A_2{\beta }_1}\right):x\ge 0\right\}$$

is invariant, equilibrium $\left(\bar{x},ȳ\right)\in ℐ$ and for (x, y) ∈ ℐ the following holds $T\left(x,y\right)=\left(\bar{x},ȳ\right)$, that is every point of this line is mapped to the equilibrium point $\left(\bar{x},ȳ\right)$.

1.i) If ${\left(B_2{\alpha }_1-A_2{\beta }_1\right)}^2-A_1^2{B}_2{\alpha }_1>0$ then $\left(\bar{x},ȳ\right)=E_3$.

1.ii) If ${\left(B_2{\alpha }_1-A_2{\beta }_1\right)}^2-A_1^2{B}_2{\alpha }_1<0$ then $\left(\bar{x},ȳ\right)=E_2$.

1.iii) If ${\left(B_2{\alpha }_1-A_2{\beta }_1\right)}^2-A_1^2{B}_2{\alpha }_1=0$ then $\left(\bar{x},ȳ\right)=E_3=E_2$.

• 2) If

  $$B_2\le \frac{A_2{\beta }_1}{{\alpha }_1}or{A}_1\left(B_2{\alpha }_1-A_2{\beta }_1\right){\gamma }_2-\left(B_2{\alpha }_1+\left(A_1-A_2\right){\beta }_1\right)\left(A_1{A}_2-{\beta }_1{A}_2+B_2{\alpha }_1\right)\ne 0,$$

then the following holds.

$$T\left(x,y\right)=\left(\bar{x},ȳ\right)\Rightarrow \left(x,y\right)=\left(\bar{x},ȳ\right).$$

Proof $T\left(x,y\right)=\left(\bar{x},ȳ\right)$ is equivalent to

$$\left(\frac{{\alpha }_1+{\beta }_{1}x}{A_1+y},\frac{{\gamma }_{2}y}{A_2+B2x+y}\right)=\left(\bar{x},ȳ\right).$$

(9)

Since $\left(\bar{x},ȳ\right)$ is the equilibrium point of the map T then System (9) is equivalent to

$$\left(\frac{{\alpha }_1+{\beta }_{1}x}{A_1+y},\frac{{\gamma }_{2}y}{A_2+B2x+y}\right)=\left(\frac{{\alpha }_1+{\beta }_1\bar{x}}{A_1+ȳ},\frac{{\gamma }_2ȳ}{A_2+B2\bar{x}+ȳ}\right).$$

(10)

System (10) is equivalent to

$$-y{\alpha }_1+ȳ{\alpha }_1-y\bar{x}{\beta }_1+xȳ{\beta }_1+x{A}_1{\beta }_1-\bar{x}{A}_1{\beta }_1=0$$

(11)

$$y{A}_2{\gamma }_2-ȳA_2{\gamma }_2+y\bar{x}{B}_2{\gamma }_2-xȳB_2{\gamma }_2=0.$$

(12)

Equation 11 implies

$$y=\frac{ȳ{\alpha }_1+xȳ{\beta }_1+x{A}_1{\beta }_1-\bar{x}{A}_1{\beta }_1}{{\alpha }_1+\bar{x}{\beta }_1}.$$

and Equation 12 is equivalent to

$$\left(x-\bar{x}\right)\left(-ȳB_2{\alpha }_1+ȳA_2{\beta }_1+A_1{A}_2{\beta }_1+\bar{x}{A}_1{B}_2{\beta }_1\right){\gamma }_2=0.$$

(13)

We conclude the following: If $\left(-ȳB_2{\alpha }_1+ȳA_2{\beta }_1+A_1{A}_2{\beta }_1+\bar{x}{A}_1{B}_2{\beta }_1\right)\ne 0$, then $x=\bar{x}$ and $y=ȳ$.

On the other hand, if $\left(-ȳB_2{\alpha }_1+ȳA_2{\beta }_1+A_1{A}_2{\beta }_1+\bar{x}{A}_1{B}_2{\beta }_1\right)=0,$ since $\left(\bar{x},ȳ\right)$ is the equilibrium of the map T, then

$$B_2>\frac{A_2{\beta }_1}{{\alpha }_1},ȳ=-\frac{A_1\left(A_2+\bar{x}{B}_2\right){\beta }_1}{A_2{\beta }_1-B_2{\alpha }_1}$$

and

$$\left(\bar{x},ȳ\right)=\left(\frac{{\alpha }_1+{\beta }_1\bar{x}}{A_1+ȳ},\frac{{\gamma }_2ȳ}{A_2+B2\bar{x}+ȳ}\right).$$

Using these equations, we have

$$\bar{x}=\frac{B_2{\alpha }_1-A_2{\beta }_1}{A_1{B}_2},ȳ=\frac{{\beta }_1\left(-A_1{A}_2+{\beta }_1{A}_2-B_2{\alpha }_1\right)}{A_2{\beta }_1-B_2{\alpha }_1}$$

and

$$A_1\left(B_2{\alpha }_1-A_2{\beta }_1\right){\gamma }_2-\left(B_2{\alpha }_1+\left(A_1-A_2\right){\beta }_1\right)\left(A_1{A}_2-{\beta }_1{A}_2+B_2{\alpha }_1\right)=0,$$

(14)

which completes the proof of lemma.

3.3 Period-two solutions

In this section, we prove that System (1) has no minimal period-two solutions which will be essential for application of Theorem 4 and Corollary 6.

Lemma 2 System (1) has no minimal period-two solution.

Proof Period-two solution satisfies T ²(x, y) = (x, y), that is

$$T^2\left(x,y\right)=\left(\frac{{\alpha }_1+\frac{{\beta }_1\left({\alpha }_1+x{\beta }_1\right)}{y+A_1}}{A_1+\frac{y{\gamma }_2}{y+A_2+x{B}_2}},\frac{y{\gamma }_2^2}{\frac{\left(y+A_2+x{B}_2\right)\left(\left(y+A_1\right)A_2+B_2\left({\alpha }_1+x{\beta }_1\right)\right)}{y+A_1}+y{\gamma }_2}\right)=\left(x,y\right).$$

This is equivalent to

$$-\frac{\left(y+A_2+x{B}_2\right)\left(-x{\beta }_1^2-{\alpha }_1{\beta }_1+\left(y+A_1\right)\left(x{A}_1-{\alpha }_1\right)\right)+xy\left(y+A_1\right){\gamma }_2}{\left(y+A_1\right)\left(A_1\left(y+A_2+x{B}_2\right)+y{\gamma }_2\right)}=0$$

and

$$\frac{y\left(\left(y+A_1\right){\gamma }_2^2-y\left(y+A_1\right){\gamma }_2+\left(-y-A_2-x{B}_2\right)\left(\left(y+A_1\right)A_2+B_2\left({\alpha }_1+x{\beta }_1\right)\right)\right)}{\left(y+A_2+x{B}_2\right)\left(\left(y+A_1\right)A_2+B_2\left({\alpha }_1+x{\beta }_1\right)\right)+y\left(y+A_1\right){\gamma }_2}=0,$$

which is equivalent to

$$\left(y+A_2+x{B}_2\right)\left(-x{\beta }_1^2-{\alpha }_1{\beta }_1+\left(y+A_1\right)\left(x{A}_1-{\alpha }_1\right)\right)+xy\left(y+A_1\right){\gamma }_2=0$$

(15)

$$y\left(\left(y+A_1\right){\gamma }_2^2-y\left(y+A_1\right){\gamma }_2+\left(-y-A_2-x{B}_2\right)\left(\left(y+A_1\right)A_2+B_2\left({\alpha }_1+x{\beta }_1\right)\right)\right)=0$$

(16)

If y = 0, we substitute in (15) to obtain the first fixed point, that is $x=\frac{{\alpha }_1}{A_1-{\beta }_1}$ i $x=-\frac{A_2}{B_2}$. Assume

$$\left(y+A_1\right){\gamma }_2^2-y\left(y+A_1\right){\gamma }_2+\left(-y-A_2-x{B}_2\right)\left(\left(y+A_1\right)A_2+B_2\left({\alpha }_1+x{\beta }_1\right)\right)=0.$$

(17)

From (17) we calculate x ². We have

$$\begin{array}{c}{x}^2=-\frac{\left(y+A_1\right)A_2^2+\left(y^2+A_1\left(y+x{B}_2\right)+B_2\left({\alpha }_1+x\left(y+{\beta }_1\right)\right)\right)A_2}{B_2^2{\beta }_1}\\ -\frac{x{B}_2^2{\alpha }_1+y{B}_2\left({\alpha }_1+x{\beta }_1\right)+\left(y+A_1\right)\left(y-{\gamma }_2\right){\gamma }_2}{B_2^2{\beta }_1}.\end{array}$$

(18)

Put (18) into (15), we have that (15) is equivalent to

$$y+A_1=0$$

(19)

or

$$\begin{array}{c}\left(A_1\left(y+A_1\right)-{\beta }_1^2\right){\gamma }_2^2+y\left({\beta }_1^2+x{B}_2{\beta }_1-A_1\left(y+A_1\right)\right){\gamma }_2\\ +\left(-y-A_2-x{B}_2\right)\left(-A_2{\beta }_1^2+B_2{\alpha }_1{\beta }_1+A_1\left(\left(y+A_1\right)A_2+B_2\alpha {.}_1\right)\right)=0\end{array}$$

(20)

If (19) holds, then we obtain a negative solution. Now, assume that (20) holds. We have

$$\begin{array}{c}x=\frac{\left(A_1\left(y+A_1\right)-{\beta }_1^2\right){\gamma }_2^2-y\left(A_1\left(y+A_1\right)-{\beta }_1^2\right){\gamma }_2}{B_2\left(A_2{A}_1^2+\left(y{A}_2+B_2{\alpha }_1\right)A_1-{\beta }_1\left(-B_2{\alpha }_1+A_2{\beta }_1+y{\gamma }_2\right)\right)}\\ +\frac{\left(-y-A_2\right)\left(-A_2{\beta }_1^2+B_2{\alpha }_1{\beta }_1+A_1\left(\left(y+A_1\right)A_2+B_2{\alpha }_1\right)\right)}{B_2\left(A_2{A}_1^2+\left(y{A}_2+B_2{\alpha }_1\right)A_1-{\beta }_1\left(-B_2{\alpha }_1+A_2{\beta }_1+y{\gamma }_2\right)\right)}.\end{array}$$

(21)

Put (21) into (18), we obtain that (18) is equivalent to

$$-y^2+\left(-A_1-A_2+{\beta }_1+{\gamma }_2\right)y-B_2{\alpha }_1+{\beta }_1\left(A_2-{\gamma }_2\right)+A_1\left({\gamma }_2-A_2\right)=0$$

(22)

or

$$\begin{array}{c}-\left(A_2+{\gamma }_2\right)\left(A_1^2+\left({\beta }_1-A_2\right)A_1+{\beta }_1{\gamma }_2\right)y^2-\left(A_1+{\beta }_1\right)\left(A_1-A_2+{\beta }_1-{\gamma }_2\right)\\ \times \left(B_2{\alpha }_1+A_1\left(A_2+{\gamma }_2\right)-{\beta }_1\left(A_2+{\gamma }_2\right)\right)y+{\left(A_1+{\beta }_1\right)}^2{\gamma }_2\left(B_2{\alpha }_1+A_1\left(A_2+{\gamma }_2\right)-{\beta }_1\left(A_2+{\gamma }_2\right)\right)=0.\end{array}$$

(23)

If (22) holds, we obtain the fixed points. So, we assume that (23) holds. Set

$$\begin{array}{cc}\hfill \Delta :& ={\left(A_1+{\beta }_1\right)}^2\left(B_2{\alpha }_1+\left(A_1-{\beta }_1\right)\left(A_2+{\gamma }_2\right)\right)\hfill \\ \times \left(\left(B_2{\alpha }_1+\left(A_1-{\beta }_1\right)\left(A_2+{\gamma }_2\right)\right){\left(A_1-A_2+{\beta }_1-{\gamma }_2\right)}^2+4{\gamma }_2\left(A_2+{\gamma }_2\right)\left(A_1\left(A_1-A_2+{\beta }_1\right)+{\beta }_1{\gamma }_2\right)\right).\hfill \end{array}$$

(24)

If Δ ≥ 0 and A ₁(A ₁ - A ₂ + β ₁) + β ₁ γ ₂ ≠ 0 hold, we obtain the real solution of the form

$$\begin{array}{c}{y}_1=-\frac{\left({\Delta }_1-\sqrt{\Delta }\right)}{2\left(A_2+{\gamma }_2\right)\left(A_1\left(A_1-A_2+{\beta }_1\right)+{\beta }_1{\gamma }_2\right)}\\ y_2=-\frac{\left({\Delta }_1+\sqrt{\Delta }\right)}{2\left(A_2+{\gamma }_2\right)\left(A_1\left(A_1-A_2+{\beta }_1\right)+{\beta }_1{\gamma }_2\right)}\end{array}$$

where

$${\Delta }_1:=\left(A_1+{\beta }_1\right)\left(A_1-A_2+{\beta }_1-{\gamma }_2\right)\left(B_2{\alpha }_1+\left(A_1-{\beta }_1\right)\left(A_2+{\gamma }_2\right)\right).$$

Substituting this into (21), we have that the corresponding solutions are

$$\begin{array}{c}{x}_1=\frac{\left({\Delta }_2-\sqrt{\Delta }\right)}{2B_2\left(A_1+{\beta }_1\right)\left(A_1\left(A_1-A_2+{\beta }_1\right)+{\beta }_1{\gamma }_2\right)}\\ x_2=\frac{\left({\Delta }_2+\sqrt{\Delta }\right)}{2B_2\left(A_1+{\beta }_1\right)\left(A_1\left(A_1-A_2+{\beta }_1\right)+{\beta }_1{\gamma }_2\right)}\end{array}$$

where

$${\Delta }_2:=\left(A_1+{\beta }_1\right)\left(-\left(A_1+{\beta }_1\right){\gamma }_2^2-\left({\left(A_1+{\beta }_1\right)}^2+B_2{\alpha }_1\right){\gamma }_2+\left(-A_1+A_2-{\beta }_1\right)\left(A_2\left(A_1+{\beta }_1\right)-B_2{\alpha }_1\right)\right).$$

(25)

   □

Claim 1 Assume Δ ≥ 0. Then we have:

• a) If x ₁ ≥ 0 then y ₁ < 0.

• b) If x ₂ ≥ 0 then y ₂ < 0.

Proof. Since T : [0, ∞)² → [0, ∞)², T(x ₁, y ₁) = (x ₂, y ₂) and T(x ₂, y ₂) = (x ₁, y ₁), it is obvious that if (x _i , y _i ) ∈ [0, ∞)² holds then T(x _i , y _i ) ∈ [0, ∞)² for i = 1, 2. It is enough to show that the assumptions (x ₁, y ₁), (x ₂, y ₂) ∈ [0, ∞)² and T(x ₁, y ₁) = (x ₂, y ₂) ≠ (x ₁, y ₁) lead to a contradiction.

Indeed, if A ₁(A ₁ - A ₂ + β ₁) + β ₁ γ ₂ > 0 then (x ₁, y ₁) ≺ _se (x ₂, y ₂). Since T is strongly competitive map then (x ₂, y ₂) = T(x ₁, y ₁) << _se T(x ₂, y ₂) = (x ₁, y ₁) which is impossible since (x ₁, y ₁) ≺ _se (x ₂, y ₂).

If A ₁(A ₁ - A ₂ + β ₁) + β ₁ γ ₂ < 0 then (x ₂, y ₂) ≺ _se (x ₁, y ₁) Similarly, we have the same conclusion if A ₁(A ₁ - A ₂ + β ₁) + β ₁ γ ₂ = 0.   □

3.4 Boundedness of solutions

Lemma 3 Assume that y ₀ = 0, x ₀ ∈ ℝ⁺ . Then the following statements are true.

• (i) If A ₁ > β ₁ then $y_n=0x_n\to \frac{{\alpha }_1}{A_1-{\beta }_1}$, n → ∞.

• (ii) If A ₁ < β ₁ then y _n = 0, x _n → ∞, n → ∞.

• (iii) If A ₁ = β ₁, then $x_n=x_0+\frac{{\alpha }_1}{A_1}n$ and y _n = 0, x _n → ∞.

Assume that y ₀ ≠ 0 and $\left(x_0,y_0\right)\in {ℝ}_2^{+}$. Then the following statements are true.

• (iv) $x_{n+1}\le \frac{{\alpha }_1}{A_1}+\frac{{\beta }_1}{A_1}{x}_n$ for all n = 0, 1, 2,...

• (v) y _n ≤ γ ₂, n ≥ N, $y_{n+1}\le C{\left(\frac{{\gamma }_2}{A_2}\right)}^n$ and

• (a) $x_n\to \frac{{\alpha }_1}{A_1-{\beta }_1}$, A ₁ > β ₁.

• (b) $x_n\le \frac{{\alpha }_1}{A_1-{\beta }_1}+\epsilon$, ε > 0, A ₁ > β ₁.

• (c) If γ ₂ < A ₂ then y _n → 0, n → ∞

Proof. Take y ₀ = 0 and x ₀ ∈ ℝ⁺. Then, we have y _n = 0, for all n ∈ ℕ, and

$$x_{n+1}=\frac{{\alpha }_1}{A_1}+\frac{{\beta }_1}{A_1}{x}_n.$$

(26)

Solution of Equation 26 is

$$x_n=c{\left(\frac{{\beta }_1}{A_1}\right)}^n+\frac{{\alpha }_1}{A_1-{\beta }_1}$$

(27)

From $y_{n+1}=\frac{{\gamma }_2{y}_n}{A_2+B_2{x}_n+y_n}$ it follows that $y_{n+1}\le \frac{{\gamma }_2}{A_2}{y}_n$, y _n+1≤ γ ₂, n ≥ 0. The proof of Lemma 3 follows from (27).   □

4 Linearized stability analysis

The map T associated to System (1) is given by

$$T\left(x,y\right)=\left(\frac{{\alpha }_1+{\beta }_{1}x}{A_1+y},\frac{{\gamma }_{2}y}{A_2+B_{2}x+y}\right).$$

The Jacobian matrix of the map T has the form:

$$J_T=\left(\begin{array}{cc}\hfill \frac{{\beta }_1}{A_1+y}\hfill & \hfill -\frac{{\alpha }_1+{\beta }_{1}x}{{\left(A_1+y\right)}^2}\hfill \\ \hfill -\frac{B_2{\gamma }_{2}y}{{\left(A_2+B_{2}x+y\right)}^2}\hfill & \hfill \frac{{\gamma }_2{A}_2+{\gamma }_2{B}_{2}x}{{\left(A_2+B_{2}x+y\right)}^2}\hfill \end{array}\right).$$

(28)

The value of the Jacobian matrix of T at the equilibrium point $E=\left(\bar{x},ȳ\right)$ is

$$J_T\left(\bar{x},ȳ\right)=\left(\begin{array}{cc}\hfill \frac{{\beta }_1}{A_1+ȳ}\hfill & \hfill -\frac{\bar{x}}{A_1+ȳ}\hfill \\ \hfill -\frac{B_2ȳ}{A_2+B_2\bar{x}+ȳ}\hfill & \hfill \frac{{\gamma }_2{A}_2+{\gamma }_2{B}_2\bar{x}}{{\left(A_2+B_2\bar{x}+ȳ\right)}^2}\hfill \end{array}\right).$$

(29)

The determinant of (29) is given by

$$det{J}_T\left(\bar{x},ȳ\right)=\frac{{\beta }_1}{A_1+ȳ}\frac{{\gamma }_2{A}_2+{\gamma }_2{B}_2\bar{x}}{{\left(A_2+B_2\bar{x}+ȳ\right)}^2}-\frac{\bar{x}}{A_1+ȳ}\frac{B_2ȳ}{A_2+B_2\bar{x}+ȳ}.$$

The trace of (29) is

$$\mathsf{\text{Tr}}{J}_T\left(\bar{x},ȳ\right)=\frac{{\beta }_1}{A_1+ȳ}+\frac{{\gamma }_2{A}_2+{\gamma }_2{B}_2\bar{x}}{{\left(A_2+B_2\bar{x}+ȳ\right)}^2}.$$

The characteristic equation has the form

$${\lambda }^2-\lambda \left(\frac{{\beta }_1}{A_1+\overline{y}}+\frac{{\gamma }_2{A}_2+{\gamma }_2{B}_2\overline{x}}{(A_2+B_2\overline{x}+\overline{y})}\right)+\frac{{\beta }_1({\gamma }_2{A}_2+{\gamma }_2{B}_2\overline{x})}{(A_1+\overline{y}){(A_2+B_1\overline{x}+\overline{y})}^2}-\frac{B_1\overline{x}\overline{y}}{(A_1+\overline{y})(A_2+B_2\overline{x}+\overline{y})}=0.$$

Theorem 9 Assume that A ₁ > β ₁. Then there exists the equilibrium point E ₁ and:

• (i) E ₁ is locally asymptotically stable if ${\gamma }_2-A_2<\frac{B_2{\alpha }_1}{A_1-{\beta }_1}$.

• (ii) E ₁ is a saddle point if ${\gamma }_2-A_2>\frac{B_2{\alpha }_1}{A_1-{\beta }_1}$. The eigenvalues are

  $${\lambda }_1=\frac{{\beta }_1}{A_1},{\lambda }_2=\frac{\left(A_1-{\beta }_1\right){\gamma }_2}{B_2{\alpha }_1+A_2\left(A_2-{\beta }_1\right)}.$$

The corresponding eigenvectors, respectively, are

$$v_1=\left(1,0\right),v_2=\left(\frac{{\alpha }_1}{A_1\left(A_1-{\beta }_1\right)\left(\frac{{\beta }_1}{{\alpha }_1}-\frac{\left(A_1-{\beta }_1\right)}{A_1{A}_2+B_2{\alpha }_1-A_2{\beta }_1}\right)},1\right).$$

• (iii) E ₁ is non-hyperbolic if ${\gamma }_2-A_2=\frac{B_2{\alpha }_1}{A_1-{\beta }_1}.$ The eigenvalues are ${\lambda }_1=\frac{{\beta }_1}{A_1}$, λ ₂ = 1. The corresponding eigenvectors are $\left(-\frac{{\alpha }_1}{{\left(A_1-{\beta }_1\right)}^2},1\right)$ and (1, 0).

Proof. Evaluating Jacobian (29) at the equilibrium point E ₁(α ₁/(A ₁ - β ₁), 0),

$$J_T\left(E_1\right)=\left(\begin{array}{cc}\hfill \frac{{\beta }_1}{A_1}\hfill & \hfill -\frac{{\alpha }_1}{A_1\left(A_1-{\beta }_1\right)}\hfill \\ \hfill 0\hfill & \hfill \frac{\left(A_1-{\beta }_1\right){\gamma }_2}{A_2\left(A_1-{\beta }_1\right)+B_2{\alpha }_1}\hfill \end{array}\right).$$

(30)

The determinant of (30) is given by

$$det{J}_T\left(\bar{x},ȳ\right)=\frac{{\beta }_1{\gamma }_2\left(A_1-{\beta }_1\right)}{A_1\left[A_2\left(A_1-{\beta }_1\right)+B_2{\alpha }_1\right]}.$$

The trace of (30) is

$$\mathsf{\text{Tr}}{J}_T\left(\bar{x},ȳ\right)=\frac{{\beta }_1}{A_1}+\frac{\left(A_1-{\beta }_1\right){\gamma }_2}{A_2\left(A_1-{\beta }_1\right)+B_2{\alpha }_1}.$$

The characteristic equation associated to System (1) at E ₁ has the form

$$\left(\frac{{\beta }_1}{A_1}-\lambda \right)\left(\frac{\left(A_1-{\beta }_1\right){\gamma }_2}{A_2\left(A_1-{\beta }_1\right)+B_2{\alpha }_1}-\lambda \right)=0.$$

(31)

From Equation 31 we have

$${\lambda }_1=\frac{{\beta }_1}{A_1},{\lambda }_2=\frac{\left(A_1-{\beta }_1\right){\gamma }_2}{A_2\left(A_1-{\beta }_1\right)+B_2{\alpha }_1}.$$

• (i) If A ₁ > β ₁ and ${\gamma }_2-A_2<\frac{B_2{\alpha }_1}{A_1-{\beta }_1}$ then λ ₁ < 1 and λ ₂ < 1. Hence, E ₁ is a sink.

• (ii) If A ₁ > β ₁ and ${\gamma }_2-A_2>\frac{B_2{\alpha }_1}{A_1-{\beta }_1}$. Then λ ₁ < 1, and λ ₂ < 1. Hence, E ₁ is a saddle.

• (iii) If A ₁ > β ₁ and ${\gamma }_2-A_2=\frac{B_2{\alpha }_1}{A_1-{\beta }_1}$. Then, using Equation 31, we have that λ ₁ < 1 and λ ₂ < 1.

From (30) we obtain the eigenvectors that correspond to these eigenvalues.   □

We now perform a similar analysis for the other cases in table.

Theorem 10 Assume

$$A_1>{\beta }_1,A_1+A_2<{\beta }_1+{\gamma }_2,\frac{\left(A_1-{\beta }_1\right)\left({\gamma }_2-A_2\right)}{B_2}<{\alpha }_1<\frac{{\left(A_1-A_2-{\beta }_1+{\gamma }_2\right)}^2}{4B_2}.$$

Then E ₁, E ₂, E ₃ exist and:

• (i) Equilibrium E ₁ is locally asymptotically stable.

• (ii) Equilibrium E ₃ is a saddle point. The eigenvalues are

  $${\lambda }_1=\frac{-{ȳ}_3\left(A_1+{ȳ}_3\right)+{\gamma }_2\left(A_1+{\beta }_1+{ȳ}_3\right)-\sqrt{\mathcal{D}}}{2{\gamma }_2\left(A_1+{ȳ}_3\right)}$$

and

$${\lambda }_2=\frac{-{ȳ}_3\left(A_1+{ȳ}_3\right)+{\gamma }_2\left(A_1+{\beta }_1+{ȳ}_3\right)+\sqrt{\mathcal{D}}}{2{\gamma }_2\left(A_1+{ȳ}_3\right)},$$

and |λ ₁| < 1, |λ ₂| > 1, where

$$\mathcal{D}={ȳ}_3^2{\left(A_1+{ȳ}_3\right)}^2-2{\gamma }_2{ȳ}_3\left(A_1-{\beta }_1-2B_2{\bar{x}}_3+{ȳ}_3\right)\left(A_1+{ȳ}_3\right)+{\gamma }_2^2{\left(A_1-{\beta }_1+{ȳ}_3\right)}^2.$$

The corresponding eigenvectors, respectively, are

$$\begin{array}{c}{v}_1=\left(-{ȳ}_3\left(A_1+{ȳ}_3\right)+{\gamma }_2\left(A_1-{\beta }_1+{ȳ}_3\right)+\sqrt{D},2B_2{ȳ}_3\left(A_1+{ȳ}_3\right)\right)\\ v_2=\left(-{ȳ}_3\left(A_1+{ȳ}_3\right)+{\gamma }_2\left(A_1-{\beta }_1+{ȳ}_3\right)-\sqrt{\mathcal{D}},2B_2{ȳ}_3\left(A_1+{ȳ}_3\right)\right).\end{array}$$

• (iii) Equilibrium E ₂ is locally asymptotically stable.

Proof. By Theorem 9 (i) holds.

Equilibrium E ₃ is a saddle if and only if the following conditions are satisfied

$$\left|\mathsf{\text{Tr}}{J}_T\left(\bar{x},ȳ\right)\right|>|1+det{J}_T\left(\bar{x},ȳ\right)|\mathsf{\text{and}}{\mathsf{\text{Tr}}}^2{J}_T\left(\bar{x},ȳ\right)-4det{J}_T\left(\bar{x},ȳ\right)>0.$$

The first condition is equivalent to

$$\frac{{\beta }_1}{A_1+ȳ}+\frac{{\gamma }_2{A}_2+{\gamma }_2{B}_2\bar{x}}{{\left(A_2+B_2\bar{x}+ȳ\right)}^2}>\left|1+\frac{{\beta }_1}{\left(A_1+ȳ\right)}\frac{{\gamma }_2{A}_2+{\gamma }_2{B}_2\bar{x}}{{\left(A_2+B_2\bar{x}+ȳ\right)}^2}-\frac{B_2\bar{x}ȳ}{\left(A_1+ȳ\right)\left(A_2+B_2\bar{x}+ȳ\right)}\right|$$

which is equivalent to

$$\begin{array}{l}{\beta }_1{(A_2+B_2\overline{x}+\overline{y})}^2+(A_1+\overline{y})({\gamma }_2{A}_2+{\gamma }_2{B}_2\overline{x})\\ >(A_1+\overline{y}){(A_2+B_2\overline{x}+\overline{y})}^2+{\beta }_1{\gamma }_2(A_2+B_2\overline{x})-B_2\overline{x}\overline{y}(A_2+B_2\overline{x}+\overline{y}).\end{array}$$

This is equivalent to

$$\begin{array}{ll}\hfill {\left(A_2+B_2\bar{x}+ȳ\right)}^2\left({\beta }_1-A_1-ȳ\right)+{\gamma }_2\left(A_2+B_2\bar{x}\right)\left(A_1+ȳ-{\beta }_1\right)& >-B_2\bar{x}ȳ\left(A_2+B_2\bar{x}+ȳ\right)\\ \hfill {\gamma }_2^2\left({\beta }_1-A_1-ȳ\right)+{\gamma }_2\left(A_2+B_2\bar{x}\right)\left(A_1+ȳ-{\beta }_1\right)& >-B_2{\gamma }_2\bar{x}ȳ\\ \hfill \left(A_1-{\beta }_1+ȳ\right)\left(A_2+B_2\bar{x}-{\gamma }_2\right)& >-B_2\bar{x}ȳ\\ \hfill \left({\beta }_1-A_1-ȳ\right)\left(A_2+B_2\bar{x}-{\gamma }_2\right)& <B_2\bar{x}ȳ.\end{array}$$

We have to prove that $\left({\beta }_1-A_1-{ȳ}_3\right)\left(A_2+B_2{\bar{x}}_3-{\gamma }_2\right)<B_2{\bar{x}}_3{ȳ}_3$. Notice that

$${\beta }_1-A_1-{ȳ}_3=-B_2{\bar{x}}_2\mathsf{\text{and}}{A}_2+B_2{\bar{x}}_3-{\gamma }_2=-{ȳ}_3.$$

Now,

$$\left({\beta }_1-A_1-{ȳ}_3\right)\left(A_2+B_2{\bar{x}}_3-{\gamma }_2\right)<B_2{\bar{x}}_3{ȳ}_3$$

is equivalent to $B_2{\bar{x}}_2{ȳ}_3<B_2{\bar{x}}_3{ȳ}_3$. This implies ${\bar{x}}_2<{\bar{x}}_3$ which is true. Condition

$${\mathsf{\text{Tr}}}^2{J}_T\left(\bar{x},ȳ\right)-4det{J}_T\left(\bar{x},ȳ\right)>0$$

is equivalent to

$${\left(\frac{{\beta }_1}{A_1+ȳ}-\frac{{\gamma }_2{A}_2+{\gamma }_2{B}_2\bar{x}}{{\left(A_2+B_2\bar{x}+ȳ\right)}^2}\right)}^2+\frac{4B_2\bar{x}ȳ}{\left(A_1+ȳ\right)\left(A_2+B_2\bar{x}+ȳ\right)}>0$$

which is clearly satisfied. Hence, E ₃ is a saddle.

Now, we prove that E ₂ is locally asymptotically stable. Notice that

$$\left|\mathsf{\text{Tr}}{J}_T\left(\bar{x},ȳ\right)\right|<1+det{J}_T\left(\bar{x},ȳ\right)<2$$

implies ${\bar{x}}_3>{\bar{x}}_2$ which is true.

The second condition is equivalent to

$$\frac{{\beta }_1}{\left(A_1+ȳ\right)}\frac{{\gamma }_2{A}_2+{\gamma }_2{B}_2\bar{x}}{{\left(A_2+B_2\bar{x}+ȳ\right)}^2}-\frac{B_2\bar{x}ȳ}{\left(A_1+ȳ\right)\left(A_2+B_2\bar{x}+ȳ\right)}<1.$$

This implies the following

$${\beta }_1{\gamma }_2(A_2+B_2\overline{x})-B_2\overline{x}\overline{y}(A_2+B_2\overline{x}+\overline{y})<(A_1+\overline{y}){(A_2+B_2\overline{x}+\overline{y})}^2.$$

Now, using Equation 5, we obtain

$$\begin{array}{ll}\hfill {\beta }_1{\gamma }_2\left({\gamma }_2-ȳ\right)-B_2\bar{x}ȳ{\gamma }_2& <\left(A_1+ȳ\right){\gamma }_2^2\\ \hfill -\left({\beta }_1ȳ+B_2\bar{x}ȳ\right)& <\left(A_1-{\beta }_1+ȳ\right){\gamma }_2\end{array}$$

which is true, since the left side is always negative, while the right side is always positive.

Theorem 11 Assume

$$A_1>{\beta }_1,A_1+A_2<{\beta }_1+{\gamma }_2,{\alpha }_1=\frac{{\left(A_1-A_2-{\beta }_1+{\gamma }_2\right)}^2}{4B_2}.$$

Then E ₁(α ₁/(A ₁ - β ₁), 0) and $E_2=E_3=\left(\frac{{\gamma }_2-A_2+A_1-{\beta }_1}{2B_2},\frac{{\gamma }_2-A_2-A_1+{\beta }_1}{2}\right)$ exist and

• (i) Equilibrium E ₁ is locally asymptotically stable.

• (ii) Equilibrium E ₂ is non-hyperbolic. The eigenvalues are

  $${\lambda }_1=1,{\lambda }_2=\frac{A_1^2-A_2^2+2A_2{\beta }_1-{\beta }_1^2+2A_2{\gamma }_2+2{\beta }_1{\gamma }_2-{\gamma }_2^2}{2{\gamma }_2\left(A_1-A_2+{\beta }_1+{\gamma }_2\right)}.$$

The corresponding eigenvectors are

$$\left(-1∕B_2,1\right),\left(\frac{2{\gamma }_2\left(A_1-A_2-{\beta }_1+{\gamma }_2\right)}{B_2\left(-A_1-A_2+{\beta }_1+{\gamma }_2\right)\left(A_1-A_2+{\beta }_1+{\gamma }_2\right)},1\right).$$

Proof. By Theorem 9, E ₁ is locally asymptotically stable.

Now, we prove that E ₂ is non-hyperbolic.

Evaluating Jacobian (29) at the equilibrium point $E_2=\left(\frac{{\gamma }_2-A_2+A_1-{\beta }_1}{2B_2},\frac{{\gamma }_2-A_2-A_1+{\beta }_1}{2}\right)$,

$$J_T\left(E_2\right)=\left(\begin{array}{cc}\hfill \frac{{\beta }_1}{A_1+ȳ}\hfill & \hfill -\frac{\bar{x}}{A_1+ȳ}\hfill \\ \hfill -\frac{B_2ȳ}{{\gamma }_2}\hfill & \hfill \frac{A_2+B_2\bar{x}}{{\gamma }_2}\hfill \end{array}\right)=\left(\begin{array}{cc}\hfill \frac{2{\beta }_1}{A_1+{\gamma }_2-A_2+{\beta }_1}\hfill & \hfill \frac{-{\gamma }_2+A_2-A_1+{\beta }_1}{B_2\left(A_1+{\gamma }_2-A_2+{\beta }_1\right)}\hfill \\ \hfill \frac{-B_2\left({\gamma }_2-A_2-A_1+{\beta }_1\right)}{2{\gamma }_2}\hfill & \hfill \frac{A_2+{\gamma }_2+A-1-{\beta }_1}{2{\gamma }_2}\hfill \end{array}\right).$$

(32)

The eigenvalues of (32) are

$${\lambda }_1=1,\mathsf{\text{and}}{\lambda }_2=\frac{A_1^2-A_2^2+2A_2{\beta }_1-{\beta }_1^2+2A_2{\gamma }_2+2{\beta }_1{\gamma }_2-{\gamma }_2^2}{2{\gamma }_2\left(A_1-A_1+{\beta }_1+{\gamma }_2\right)}.$$

Notice that |λ ₂| < 1. Hence, E ₂ is non-hyperbolic.

Theorem 12 Assume

$$\begin{array}{c}{A}_1>{\beta }_1,A_2<{\gamma }_2<A_1+A_2-{\beta }_1,\frac{\left(A_1-{\beta }_1\right)\left({\gamma }_2-A_2\right)}{B_2}<{\alpha }_1\le \frac{{\left(A_1-A_2-{\beta }_1+{\gamma }_2\right)}^2}{4B_2}\\ A_1>{\beta }_1,A_2>{\gamma }_2,{\alpha }_1\le \frac{{\left(A_1-A_2-{\beta }_1+{\gamma }_2\right)}^2}{4B_2},A_1+{\gamma }_2\ne A_2+{\beta }_1\\ A_1>{\beta }_1,A_2={\gamma }_2,{\alpha }_1\le \frac{{\left(A_1-A_2-{\beta }_1+{\gamma }_2\right)}^2}{4B_2}\\ A_1>{\beta }_1,{\alpha }_1>\frac{{\left(A_1-A_2-{\beta }_1+{\gamma }_2\right)}^2}{4B_2}\end{array}$$

Then there exists a unique equilibrium E ₁ (α ₁/(A ₁ - β),0) which is locally asymptotically stable.

Proof. Observe that the assumption of Theorem 12 implies that the y coordinates of the equilibrium E ₂ and E ₃ are less then zero. By Theorem 9 E ₁ is locally asymptotically stable.

Theorem 13 Assume

$$A_1>{\beta }_1,A_2<{\gamma }_2,{\alpha }_1<\frac{\left(A_1-{\beta }_1\right)\left({\gamma }_2-A_2\right)}{B_2}.$$

Then then there exist two equilibrium points E ₁ and E ₂. E ₁ is a saddle point. The eigenvalues are

$${\lambda }_1=\frac{{\beta }_1}{A_1},{\lambda }_2=\frac{\left(A_1-{\beta }_1\right){\gamma }_2}{B_2{\alpha }_1+A_2\left(A_2-{\beta }_1\right)}.$$

The corresponding eigenvectors, respectively, are

$$v_1=\left(1,0\right),v_2=\left(\frac{{\alpha }_1}{A_1\left(A_1-{\beta }_1\right)\left(\frac{{\beta }_1}{{\alpha }_1}-\frac{\left(A_1-{\beta }_1\right)}{A_1{A}_2+B_2{\alpha }_1-A_2{\beta }_1}\right)},1\right).$$

The equilibrium E ₂ is locally asymptotically stable.

Proof. By Theorem 9 (ii), E ₁ is a saddle point.

Now, we check the sign of coordinates of the equilibrium point E ₂. We have that ${\bar{x}}_2>0$, since all parameters are positive. Consider ${ȳ}_2.$ Since

$$\frac{{\left(A_1-A_2-{\beta }_1+{\gamma }_2\right)}^2}{4B_2}-\frac{\left(A_1-{\beta }_1\right)\left({\gamma }_2-A_2\right)}{B_2}=\frac{{\left(A_1+A_2-{\beta }_1-{\gamma }_2\right)}^2}{4B_2}>0,$$

we have that (γ ₂ - A ₂ + A ₁ - β ₁)² - 4α ₁ B ₂ > 0.

$${ȳ}_1>0\iff {\gamma }_2-A_2+{\beta }_1-A_1+\sqrt{{\left({\gamma }_2-A_2+A_1-{\beta }_1\right)}^2-4{\alpha }_1{B}_2}>0.$$

This implies

$$\sqrt{{\left({\gamma }_2-A_2+A_1-{\beta }_1\right)}^2-4{\alpha }_1{B}_2}>\left(A_1-{\beta }_1\right)-\left({\gamma }_2-A_2\right).$$

(33)

From Equation 33, we see that inequality is always true if A ₁ - β ₁ < γ ₂ - A ₂. If A ₁ - β ₁ > γ ₂ - A ₂, then

$$\begin{array}{l}\begin{array}{ccc}{({\gamma }_2-A_2)}^2+2({\gamma }_2-A_2)(A_1-{\beta }_1)+{(A_1-{\beta }_1)}^2-4{\alpha }_1{B}_1& >& {(A_1-{\beta }_1)}^2-2(A_1-{\beta }_1)({\gamma }_2-A_2)\end{array}\\ \begin{array}{ccc}({\gamma }_2-A_2)(A_1-{\beta }_1)& >& {\alpha }_1{B}_2\end{array}\end{array}$$

which is true, since $A_1-{\beta }_1>\frac{B_2{\alpha }_1}{{\gamma }_2-A_2}$. So, in both cases ${\bar{x}}_2>0$ and ${ȳ}_2>0.$

Notice, that ${\bar{x}}_3>0.$ Now, we check the sign of ${ȳ}_3.$ Assume that ${ȳ}_3>0.$ Then, we have

$$\begin{array}{ll}\hfill {ȳ}_2>0& \iff \left({\gamma }_2-A_2\right)-\left(A_1-{\beta }_1\right)>\sqrt{{\left({\gamma }_2-A_2+A_1-{\beta }_1\right)}^2-4{\alpha }_1{B}_2}.\\ \iff \left({\gamma }_2-A_2\right)\left(A-1-{\beta }_1\right)<{\alpha }_1{B}_2.\end{array}$$

This is a contradiction with the assumption of theorem and so E ₃ is not in considered domain.

By Theorem 10, E ₂ is a locally asymptotically stable.

Theorem 14 Assume

$$A_1>{\beta }_1,A_1+A_2<{\beta }_1+{\gamma }_2,{\alpha }_1=\frac{\left(A_1-{\beta }_1\right)\left({\gamma }_2-A_2\right)}{B_2}.$$

Then there exist two equilibrium points $E_1\equiv E_3=\left(\frac{{\alpha }_1}{A_1-{\beta }_1},0\right)$ and $E_2=\left(\frac{{\alpha }_1}{{\gamma }_2-A_2},\frac{{\gamma }_2-A_2-A_1+{\beta }_1}{2}\right),$ and E ₁ ≡ E ₃ is non-hyperbolic. The eigenvalues are ${\lambda }_1=\frac{{\beta }_1}{A_1}$, λ ₂ = 1. The corresponding eigenvectors are $\left(-\frac{{\alpha }_1}{{\left(A_1-{\beta }_1\right)}^2},1\right)$ and (1. 0) The equilibrium point E ₂ is locally asymptotically stable.

Proof. By Theorem 10, E ₂ is locally asymptotically stable. By Theorem 9 (iii), E ₁ is non-hyperbolic.

Now, we consider the special case of System (1) when A ₁ = β ₁.

In this case, System (1) becomes

$$\left\{\begin{array}{ccc}\hfill x_{n+1}\hfill & \hfill =\hfill & \hfill \frac{{\alpha }_1+A_1{x}_n}{A_1+y_n}\hfill \\ \hfill y_{n+1}\hfill & \hfill =\hfill & \hfill \frac{{\gamma }_2{y}_n}{A_2+B_2{x}_n+y_n}\hfill \end{array}\right.,n=0,1,2,\dots$$

(34)

The map T associated to System (34) is given by

$$T\left(x,y\right)=\left(\frac{{\alpha }_1+A_{1}x}{A_1+y},\frac{{\gamma }_{2}y}{A_2+B_{2}x+y}\right).$$

The Jacobian matrix of the map T has the form:

$$J_T=\left(\begin{array}{cc}\hfill \frac{A_1}{A_1+y}\hfill & \hfill -\frac{{\alpha }_1+A_{1}x}{{\left(A_1+y\right)}^2}\hfill \\ \hfill -\frac{{\beta }_2{\gamma }_{2}y}{{\left(A_2+B_{2}x+y\right)}^2}\hfill & \hfill \frac{{\gamma }_2{A}_2+{\gamma }_2{B}_{2}x}{{(A_2}^{}}\hfill \end{array}\right)$$