In this article, we discuss the qualitative behavior of a two-dimensional discrete-time prey–predator model. This system is the result of the application of a nonstandard difference scheme to a system of differential equations for a prey–predator model including intraspecific competition of prey population. In particular, we evaluate the fixed points of the system and study their local asymptotic stability. We also prove the existence of a Neimark–Sacker bifurcation.

1 INTRODUCTION

Lotka–Volterra models describe the interaction between two or more species sharing habitat. The interaction between these species can take different forms: predation, mutualism, competition, and parasitism. The Lotka–Volterra prey–predator model was first introduced by Lotka¹ to study some type of chemical reactions. In 1925, Lotka² extended the model to analyze a prey-predator interaction. In 1927, Volterra³ analyzed a system composed of two associated species, of which one, finding enough food in the environment, would multiply without any limitation when left to itself, while the other would perish if left alone, as this second species feeds upon the first. The so-called Lotka–Volterra prey-predator equations are given by

\frac{d N (τ)}{d τ} = N (τ) (α - ω P (τ)), \frac{d P (τ)}{d τ} = P (τ) (ω δ N (τ) - β),

where

N (τ)

and

P (τ)

are the prey and predator population densities, respectively, at the instant of time

τ

α > 0

denotes the natural growth rate of the prey in the absence of predators,

β > 0

is the natural death rate of the predator in the absence of prey,

ω > 0

represents the effect of predation on the prey, and

δ > 0

is the conversion rate of prey into predator. This simple model was modified in order to introduce intraspecific competition of prey population, included in the equation

\frac{d N (τ)}{d τ} = N (τ) (α - ν N (τ) - ω P (τ)), \frac{d P (τ)}{d τ} = P (τ) (ω δ N (τ) - β),

()

where

ν > 0

. The dimensionless form of Equation (1) can be obtained by defining

x (t) = \frac{α δ}{β} N (τ), y (t) = \frac{ω}{β} P (τ), t = β τ, a = \frac{α}{β}, b = \frac{ν}{α δ}, c = \frac{ω}{α} .

Then, Equation (1) is written as

\frac{d x (t)}{d t} = x (t) (a - b x (t) - y (t)), \frac{d x (t)}{d t} = y (t) (c x (t) - 1),

()

where

a, b, c > 0

. The interaction between species is properly described by differential equations when different generations can overlap. If there is no overlapping between different generations, difference equations are more appropriate. May^{4, 5} has documented that discrete–time models present a very rich dynamic behavior.

Over the last decades, population models have become a research area which has attracted the interest of many scientific groups.^4-23 Din and Khan¹² have studied some of the properties of a difference equation modeling the evolution of a mosquito population, which is described by

x_{n + 1} = f (x_{n}) g (x_{n}) x_{n},

where

x_{n}

is the number of wild mosquitoes at time step n,

f (x_{n})

is the birth function, and

g (x_{n})

denotes the fraction of mosquitos that survive. In their work, Din and Khan take

f (x_{n}) = (a x_{n} + b x_{n - 1} e^{- x_{n - 1}})

and

g (x_{n}) = e^{- x_{n}}

, where

a \in (0, 1)

and

b \in [0, + \infty)

In a recent article, Elsadany and Zeng¹⁶ perform a qualitative study of a two-dimensional difference system describing the interaction between two species, prey and predator. They focus their analysis on the system given by Equation (2), which is converted into a discrete-time system by following the piecewise constant arguments method, obtaining the system

x_{n + 1} = x_{n} e^{a - b x_{n} - y_{n}}, y_{n + 1} = y_{n} e^{c x_{n} - 1} .

For these equations, the authors investigate the existence of equilibria, their local stability, and the existence of bifurcations in the system.

Liu and Xiao²¹ investigate a prey–predator model described by the equations

\frac{d x}{d t} = r_{0} x (1 - \frac{x}{k}) - b_{0} x y, \frac{d y}{d t} = (- d_{0} k + c x) y,

()

where x and y denote the prey and predator population densities, respectively, and

r_{0}, k, b_{0}, d_{0}, c \in ℝ_{+}

, showing the existence of a Hopf bifurcation in

ℝ_{+}^{2}

. First, the authors introduce the transformation defined by

X = \frac{x}{k}, Y = \frac{b_{0} y}{c k}, s = \frac{t}{k},

to reduce Equation (3) to

\frac{d X}{d s} = r_{0} k X (1 - X) - k^{2} c X Y, \frac{d Y}{d s} = (- d_{0} k + k^{2} c X) Y .

()

Their analysis is performed then via the discretization of Equation (4) by using the Euler scheme, obtaining the system

x_{n + 1} = x_{n} + h (r x_{n} (1 - x_{n}) - b x_{n} y_{x}), y_{n + 1} = y_{n} + h (- d + b x_{n}) y_{n},

being h the step size.

In this article, we carry out a discretization of Equation (2) to obtain a discrete-time system version of the continuous counterpart, by applying the nonstandard finite difference scheme proposed by Mickens,²⁴ given by

\frac{x_{n + 1} - x_{n}}{h} = a x_{n} - b x_{n} x_{n + 1} - y_{n} x_{n + 1}, \frac{y_{n + 1} - y_{n}}{h} = a x_{n} y_{n} - y_{n + 1},

()

where h the step size. By means of discretization (5), we get the discrete-time

x_{n + 1} = \frac{(1 + h a) x_{n}}{1 + h b x_{n} + h y_{n}}, y_{n + 1} = \frac{y_{n} (1 + h c x_{n})}{1 + h},

()

being

h, a, b, c > 0

. The aim of this article is to carry out a qualitative study of this prey-predator discrete-time model, obtaining the parametric conditions for local asymptotic stability of the positive equilibrium, and the existence of a Neimark–Sacker bifurcation. Numerical simulations are provided in order to illustrate the theoretical result we have obtained.

2 LOCAL STABILITY OF THE EQUILIBRIA OF THE SYSTEM

The equilibria of system (6) can be obtained by solving the two-dimensional algebraic system given by

x = \frac{(1 + h a) x}{1 + h b x + h y}, y = \frac{y (1 + h c x)}{1 + h} .

()

The solutions to system (7) are $O = (0, 0)$ , $E = (\frac{a}{b}, 0)$ and, if $a c > b$ , $P = (\frac{1}{c}, \frac{a c - b}{c})$ .

Lemma 1.The equilibrium $O = (0, 0)$ is unstable. The equilibrium $E = (\frac{a}{b}, 0)$ is locally asymptotically stable (stable sink) if $a c < b$ , unstable if $a c > b$ , and a non-hyperbolic point if $a c = b$ .

Proof.The Jacobian matrix of this system (6) reads

J (x, y) = (\begin{array}{cc} \frac{1 + h y + h^{2} a y + h a}{{(1 + h b x + h y)}^{2}} & - \frac{h (1 + h a) x}{{(1 + h b x + h y)}^{2}} \\ \frac{h c y}{1 + h} & \frac{1 + h c x}{1 + h} \end{array}) .

()

The evaluation of matrix (8) at the equilibrium point $O = (0, 0)$ gives

J (O) = (\begin{array}{cc} 1 + h a & 0 \\ 0 & \frac{1}{1 + h} \end{array}),

which reveals that

O = (0, 0)

is unstable, as the eigenvalues of

J (O)

are given by

λ_{1} = 1 + h a > 1, λ_{2} = \frac{1}{1 + h} < 1,

being

h > 0

. Then, the origin is a saddle point.

The evaluation of $J (x, y)$ at E results

J (E) = (\begin{array}{cc} \frac{1}{1 + h a} & \frac{1}{b (1 + h a)} \\ 0 & \frac{1 + \frac{h c a}{b}}{1 + h} \end{array}) .

The roots of the characteristic polynomial of

J (E)

are

λ_{1} = \frac{1}{1 + h a} < 1, λ_{2} = \frac{1}{1 + h} (1 + \frac{h a c}{b}) .

Thus,

E = (\frac{a}{b}, 0)

is locally asymptotically stable if

a c < b

, unstable if

a c > b

, and a nonhyperbolic point if

a c = b

Now, we will analyze the stability of the equilibrium P, with the help of the following Lemma:²¹

Lemma 2.Let $ρ (ℓ) = ℓ^{2} + B ℓ + C$ , where $ρ (1) > 0$ , and $ℓ_{1}$ and $ℓ_{2}$ are the two roots of $ρ (ℓ) = 0$ . Then,

$|ℓ_{1}| < 1$ and $|ℓ_{2}| < 1$ if and only if $ρ (- 1) > 0$ and $ρ (0) < 1$ .
$|ℓ_{1}| < 1$ and $|ℓ_{2}| > 1$ (or $|ℓ_{1}| > 1$ and $|ℓ_{2}| < 1$ ) if and only if $ρ (- 1) < 0$ .
$|ℓ_{1}| > 1$ and $| ℓ_{2} | > 1$ if and only if $ρ (- 1) > 0$ and $ρ (0) > 1$ .
$ℓ_{1} = - 1$ and $| ℓ_{2} | \neq 1$ if and only if $ρ (- 1) = 0$ and $ρ (0) \neq \pm 1$ .
$ℓ_{1}$ and $ℓ_{2}$ are complex numbers and $| ℓ_{1} | = | ℓ_{2} | = 1$ if and only if $B^{2} - 4 C < 0$ and $ρ (0) = 1$ .

The Jacobian matrix (8) evaluated at the equilibrium

P = (\frac{1}{c}, \frac{a c - b}{c})

is given by

J (P) = (\begin{array}{cc} 1 - \frac{h b}{c (1 + h a)} & - \frac{h}{c (1 + h a)} \\ \frac{h (a c - b)}{1 + h} & 1 \end{array}),

()

and its characteristic polynomial is

ρ (λ) = λ^{2} - (2 - \frac{h b}{c (1 + h a)}) λ + 1 + \frac{h^{2} a c - 2 h^{2} b - h b}{c (1 + h a) (1 + h)} .

()

From Equation (10), it follows that

ρ (1) = \frac{h^{2} (a c - b)}{c (1 + h a) (1 + h)}, ρ (- 1) = 4 + \frac{h^{2} a c - 3 h^{2} b - 2 h b}{c (1 + h a) (1 + h)},

and

ρ (0) = 1 + \frac{h^{2} a c - 2 h^{2} b - h b}{c (1 + h a) (1 + h)} .

Then, taking into account that

a c > b

, it follows that

ρ (1) > 0

and, thus, we can apply Lemma 2 to state the following result:

Lemma 3.Let us assume that $a c > b$ , so that $P = (\frac{1}{c}, \frac{a c - b}{c})$ is an equilibrium of Equation (6). Then,

1.
P is locally asymptotically stable if
$4 c + 4 h c - 2 h b + 4 h a c + 5 h^{2} a c - 3 h^{2} b > 0$
and
$c + h c + h a c + 2 h^{2} b + h b > 0 .$
2.
P is a saddle point if
$4 c + 4 h c - 2 h b + 4 h a c + 5 h^{2} a c - 3 h^{2} b < 0 .$
3.
P is unstable if
$4 c + 4 h c - 2 h b + 4 h a c + 5 h^{2} a c - 3 h^{2} b > 0$
and
$c + h c + h a c + 2 h^{2} b + h b < 0 .$
4.
The roots of equation $ρ (λ) = 0$ are complex numbers with modulus one if
$h^{2} b^{2} + h^{3} b^{2} - c (4 h^{2} a c - 6 h^{2} b - 2 h b) (1 + h a) < 0$
and
$h = \frac{b}{a c - 2 b} .$

3 NEIMARK–SACKER BIFURCATION

In this section, we discuss the existence of a Neimark–Sacker bifurcation²⁵ in Equation (6). Let us consider this equation around the equilibrium

P = (\frac{1}{c}, \frac{a c - b}{c})

. According to Lemma 2, the characteristic equation of the Jacobian matrix of the system at P has two conjugate complex roots with modulus one if the condition 4 of Lemma 3 is satisfied. Hence, P undergoes a Neimark–Sacker bifurcation if parameters

(a, b, c, h)

vary in a neighborhood of the set

𝒩_{s} = \{(a, b, c, h) \in ℝ_{+}^{4} : h^{2} b^{2} + h^{3} b^{2} - c (4 h^{2} a c - 6 h^{2} b - 2 h b) (1 + h a) < 0, h > 0, h = \frac{b}{a c - 2 b}\} .

Let us suppose that

(a, b, c, h) \in 𝒩_{s}

. Thus, the system given in Equation (6) can be written as

(\begin{array}{cc} H \\ P \end{array}) \to (\begin{array}{cc} \frac{(1 + h a) H}{1 + h b H + h P} \\ \frac{P (1 + h c H)}{1 + h} \end{array}),

()

where

h = \frac{b}{a c - 2 b}

. Let

\overline{h}

be a perturbation parameter, where

|\overline{h}| ≪ 1

. The map (11) can be expressed as

(\begin{array}{cc} H \\ P \end{array}) \to (\begin{array}{cc} \frac{(1 + (h + \overline{h}) a) H}{1 + (h + \overline{h}) b H + (h + \overline{h}) P} \\ \frac{P (1 + (h + \overline{h}) c H)}{1 + (h + \overline{h})} \end{array}) .

()

Performing the transformation

(X, Y) = (H - \frac{1}{c}, P - \frac{a c - b}{c}),

the map given by Equation (12) can be described by the system

(\begin{array}{cc} X \\ Y \end{array}) \to (\begin{array}{cc} 1 - \frac{(h + \overline{h})}{c (1 + (h + \overline{h}) a)} & - \frac{h}{c (1 + (h + \overline{h}) a)} \\ \frac{(h + \overline{h}) (a c - b)}{1 + (h + \overline{h})} & 1 \end{array}) (\begin{array}{cc} X \\ Y \end{array}) + (\begin{array}{cc} f_{1} (X, Y) \\ f_{2} (X, Y) \end{array}),

()

where

\begin{align} f_{1} (X, Y) & = \frac{1}{2} a_{13} X^{2} + 2 a_{14} X Y + \frac{1}{2} a_{15} Y^{2} + \frac{1}{6} b_{1} X^{3} + \frac{1}{2} b_{2} X^{2} Y + \frac{1}{2} b_{3} X Y^{2} + \frac{1}{6} b_{4} Y^{3} + R_{f_{1}, 4} (X, Y), \\ f_{2} (X, Y) & = \frac{1}{2} a_{23} X^{2} + 2 a_{24} X Y + \frac{1}{2} a_{25} Y^{2} + \frac{1}{6} d_{1} X^{3} + \frac{1}{2} d_{2} X^{2} Y + \frac{1}{2} d_{3} X Y^{2} + \frac{1}{6} d_{4} Y^{3} + R_{f_{2}, 4} (X, Y), \end{align}

being

R_{f_{1}, 4} (X, Y)

and

R_{f_{2}, 4} (X, Y)

the terms of order larger than 3 in the Taylor expansion of

f_{1}

and

f_{2}

, respectively,

\begin{align} a_{13} & = \frac{- 2 (h + \overline{h}) b (c + 2 (h + \overline{h}) a c + {(h + \overline{h})}^{2} a^{2} c - (h + \overline{h}) b - {(h + \overline{h})}^{2} a b)}{c {(1 + (h + \overline{h}) a)}^{3}}, \\ a_{14} & = \frac{2 {(h + \overline{h})}^{3} a b + 2 {(h + \overline{h})}^{2} b - (h + \overline{h}) c - 2 {(h + \overline{h})}^{2} a c - {(h + \overline{h})}^{3} a^{2} c}{c {(1 + (h + \overline{h}) a)}^{3}}, \\ a_{15} & = \frac{2 {(h + \overline{h})}^{2}}{c {(1 + (h + \overline{h}) a)}^{2}}, \\ b_{1} & = \frac{6 {(h + \overline{h})}^{2} b^{2} (c + 2 (h + \overline{h}) a c + {(h + \overline{h})}^{2} a^{2} c - b (h + \overline{h}) - {(h + \overline{h})}^{2} b a)}{c {(1 + (h + \overline{h}) a)}^{4}}, \\ b_{2} & = \frac{2 {(h + \overline{h})}^{4} (2 a b c - 2 b^{2} + 2 a^{2} b c - 3 b^{2} a) + 2 {(h + \overline{h})}^{3} (2 a b c - b^{2}) + 4 {(h + \overline{h})}^{2} b c}{c {(1 + (h + \overline{h}) a)}^{4}}, \\ b_{3} & = \frac{2 {(h + \overline{h})}^{4} (a^{2} c - 3 b c) + 2 {(h + \overline{h})}^{3} (2 a c - 3 b) + 2 {(h + \overline{h})}^{2} c}{c {(1 + (h + \overline{h}) a)}^{4}}, \\ b_{4} & = \frac{- 6 (h + \overline{h})}{c {(1 + (h + \overline{h}) a)}^{3}}, \\ a_{24} & = \frac{c (h + \overline{h})}{(1 + (h + \overline{h}))}, \end{align}

and

a_{23} = a_{25} = d_{1} = d_{2} = d_{3} = d_{4} = 0 .

If we evaluate the Jacobian matrix of system (13) at the equilibrium

(0, 0)

, its characteristic equation is given by

ν^{2} - T (\overline{h}) ν + D (\overline{h}) = 0,

()

where

T (\overline{h}) = 2 - \frac{(h + \overline{h}) b}{c (1 + (h + \overline{h}) a)}

and

D (\overline{h}) = 1 + \frac{{(h + \overline{h})}^{2} a c - 2 {(h + \overline{h})}^{2} b - (h + \overline{h}) b}{c (1 + (h + \overline{h}) a) (1 + (h + \overline{h}))} .

Since

(a, b, c, h) \in 𝒩_{s}

and Equation (14) has pair of complex conjugate roots with unit modulus, given by

ν_{1} = \frac{\sqrt{1 + h + \overline{h}} (2 c (1 + (h + \overline{h}) a) - b (h + \overline{h})) + i (4 c (1 + (h + \overline{h}) a) ({(h + \overline{h})}^{2} a c - {(h + \overline{h})}^{2} b) - b^{2} (h + \overline{h}) (1 + h + \overline{h}))}{2 c (1 + (h + \overline{h}) a) \sqrt{1 + h + \overline{h}}},

and

ν_{2} = \frac{\sqrt{1 + h + \overline{h}} (2 c (1 + (h + \overline{h}) a) - b (h + \overline{h})) - i (4 c (1 + (h + \overline{h}) a) ({(h + \overline{h})}^{2} a c - {(h + \overline{h})}^{2} b) - b^{2} (h + \overline{h}) (1 + h + \overline{h}))}{2 c (1 + (h + \overline{h}) a) \sqrt{1 + h + \overline{h}}},

it follows that

| ν_{1} | = | ν_{2} | = \sqrt{D (\tilde{h})} = \sqrt{1 + \frac{{(h + \overline{h})}^{2} a c - 2 {(h + \overline{h})}^{2} b - (h + \overline{h}) b}{c (1 + (h + \overline{h}) a) (1 + (h + \overline{h}))}},

and

{(\frac{d | ν_{2} |}{d \overline{h}})}_{\overline{h} = 0} = {(\frac{d | ν_{1} |}{d \overline{h}})}_{\overline{h} = 0} = \frac{1}{2 \sqrt{D (0)}} \times (\frac{2 h a c + h^{2} a^{2} c + h^{2} a c - 4 h b - 2 h^{2} - b - 2 h^{3} a^{2} c + 4 h^{3} a b}{c (1 + h a) (1 + h) 2}) \neq 0 .

()

The condition

T (0) \neq 0, 1

leads to

h b \neq 2 c (1 + h a), h b \neq c (1 + h a) .

()

Then, $ν_{1}^{m}$ , $ν_{2}^{m} \neq 1$ , for all $m = 1, 2, 3, 4$ .

In order to obtain the normal form of (13) at

\overline{h}

, we take

α = \frac{T (0)}{2}

β = \frac{1}{2} \sqrt{4 D (0) - T^{2} (0)}

, and consider the following transformation:

(\begin{array}{cc} X \\ Y \end{array}) = (\begin{array}{cc} - \frac{h + \overline{h}}{c (1 + (h + \overline{h}) a)} & 0 \\ α - 1 + \frac{(h + \overline{h}) b}{c (1 + (h + \overline{h}) a)} & - β \end{array}) (\begin{array}{cc} u \\ v \end{array}) .

()

Hence, by using transformation (17) on Equation (13), we get

(\begin{array}{cc} u \\ v \end{array}) \to (\begin{array}{cc} α & - β \\ β & α \end{array}) (\begin{array}{cc} u \\ v \end{array}) + (\begin{array}{cc} \overline{p} (u, v) \\ \overline{q} (u, v) \end{array}),

()

where

\overline{p} (u, v) = \frac{a_{13}}{2 a_{12}} X^{2} + \frac{2 a_{14}}{a_{12}} X Y + \frac{a_{15}}{2 a_{12}} Y^{2} + \frac{b_{1}}{6 a_{12}} X^{3} + \frac{b_{2}}{2 a_{12}} X^{2} Y + \frac{b_{3}}{2 a_{12}} X Y^{2} + \frac{b_{4}}{6 a_{12}} Y^{3} + R_{\overline{p}, 4} (X, Y)

and

\begin{align} \overline{q} (u, v) & = \frac{a_{13} (α - a_{11})}{2 a_{12}} X^{2} + (\frac{2 a_{14} (α - a_{11})}{a_{12}} - \frac{2 a_{24}}{β}) X Y + \frac{a_{15} (α - a_{11})}{2 a_{12}} Y^{2} + \frac{b_{1} (α - a_{11})}{6 a_{12}} X^{3} \\ + \frac{b_{2} (α - a_{11})}{2 a_{12}} X^{2} Y + \frac{b_{3} (α - a_{11})}{2 a_{12}} X Y^{2} + \frac{b_{4} (α - a_{11})}{6 a_{12}} Y^{3} + R_{\overline{q}, 4} (X, Y), \end{align}

being

R_{\overline{p}, 4} (X, Y)

and

R_{\overline{q}, 4} (X, Y)

the terms of order larger than 3 in the Taylor expansion of

\overline{p}

and

\overline{q}

, respectively,

X = a_{12} u, Y = (α - a_{11}) u - β v,

and

a_{11} = 1 - \frac{(h + \overline{h})}{c (1 + (h + \overline{h}) a)}, a_{12} = - \frac{h}{c (1 + (h + \overline{h}) a)} .

In order to determine the conditions for the existence of a Neimark–Sacker bifurcation, we consider the following nonzero real number,²⁵

L = {(Re (ν_{2} ξ_{21}) - Re (\frac{(1 - 2 ν_{1}) ν_{2}^{2}}{1 - ν_{1}} ξ_{20} ξ_{11}) - \frac{1}{2} | ξ_{11} |^{2} - | ξ_{02} |^{2})}_{\tilde{h} = 0},

where

\begin{align} ξ_{20} & = \frac{1}{8} ({\overline{p}}_{u u} - {\overline{p}}_{v v} + 2 {\overline{q}}_{u v} + i ({\overline{q}}_{u u} - {\overline{q}}_{v v} - 2 {\overline{p}}_{u v})), \\ ξ_{11} & = \frac{1}{4} ({\overline{p}}_{u u} + {\overline{p}}_{v v} + i ({\overline{q}}_{u u} + {\overline{q}}_{v v})), \\ ξ_{02} & = \frac{1}{8} ({\overline{p}}_{u u} - {\overline{p}}_{v v} - 2 {\overline{q}}_{u v} + i ({\overline{q}}_{u u} - {\overline{q}}_{v v} + 2 {\overline{p}}_{u v})), \\ ξ_{21} & = \frac{1}{16} ({\overline{p}}_{u u u} + {\overline{p}}_{u v v} + {\overline{q}}_{u u v} + {\overline{q}}_{v v v} + i ({\overline{q}}_{u u u} + {\overline{q}}_{u v v} - {\overline{p}}_{u u v} - {\overline{p}}_{v v v})), \end{align}

being

{\overline{p}}_{u u} = \frac{\partial^{2} \overline{p}}{\partial u^{2}}, {\overline{p}}_{v v} = \frac{\partial^{2} \overline{p}}{\partial v^{2}}, {\overline{p}}_{u v} = \frac{\partial^{2} \overline{p}}{\partial u \partial v}, \dots .

Hence, the conclusions about the existence of a Neimark–Sacker bifurcation can be summarize in the following theorem, according to the calculations described above.²⁵

Theorem 1.Let us assume conditions (16) are satisfied, and $L \neq 0$ . Then, system (6) undergoes a Neimark–Sacker bifurcation at the equilibrium point $P = (\frac{1}{c}, a - \frac{b}{c})$ if h varies in a neighborhood of

h_{0} = \frac{b}{a c - 2 b} .

L < 0

, the equilibrium point bifurcates in an attracting invariant closed curve, for

h > h_{0}

. If

L > 0

, a repelling invariant closed curve bifurcates from the equilibrium point, for

h < h_{0}

4 NUMERICAL COMPUTATIONS AND DISCUSSION

In this section, we show the existence of a Neimark–Sacker bifurcation in system (6), by taking the following values of the parameters: $a = 5.4$ , $b = 4.1$ , $c = 2.5$ , and taking $h \in (0.5, 2.4)$ as bifurcation parameter. For these values, $P = (0.4, 3.76)$ . The bifurcation diagrams for $x_{n}$ and $y_{n}$ are depicted in Figures 1 and 2, respectively. These figures show that system (6) undergoes a Neimark–Sacker bifurcation as h varies in the vicinity of $h ≃ 0.7736$ . The bifurcation diagrams depicted in Figures 1 and 2 show that the stability of P holds for $h < 0.7736$ , it loses its stability at $h = 7736$ , and an attracting invariant curve appears if $h > 7736$ . In Figure 3, we show the evolution of $x_{n}$ (left panel) and $y_{n}$ (right panel), for $h = 0.8$ .

Details are in the caption following the image — **FIGURE 1**
Open in figure viewer PowerPoint

Bifurcation diagram for $x_{n}$

Figures 4 and 5 show some phase portraits, for the values of the bifurcation parameter given by $h = 0.7$ , 0.76, 0.7736, 0.779, 0.8, 0.9, 1 and 1.4. These figures depict how a invariant closed curve emerges from the stable equilibrium $P = (0.4, 3.76)$ . For values of h larger than 0.7736, a closed curve enclosing the fixed point P emerges. The radius of this curve grows with h. We also observe that when the bifurcation parameter passes a critical bifurcation value, the stability of P changes from stable to unstable and a Neimark–Sacker bifurcation takes place for this critical value.

The characteristic equation of (6) at the positive equilibrium is given by

ν^{2} - 1.755005 ν + 1.00062 = 0,

with roots

ν_{1} = 0.8775675 + 0.4802178 i, ν_{2} = 0.8775675 - 0.4802178 i,

both with modulus one, so that

(a, b, c) = (5.4, 4.1, 2.5) \in 𝒩_{s}

. Next, we observe that

T (0) = 1.7550 \neq 0, 1

and, thus, condition (16) is satisfied. Moreover, the value of

ρ (1)

ρ (1) = \frac{h^{2} (a c - b)}{c (1 + h a) (1 + h)} = 0.2450 > 0,

with

f_{1} (x, y) = - 0.074 x^{2} - 0.0244 x y + 0.00355 y^{2} + 0.01813 x^{3} + 0.0071 x^{2} y + 0.00085 x y^{2} + R_{f_{1}, 4} (x, y)

and

f_{2} (x, y) = 2.1808 x y + R_{f_{2}, 4} (x, y) .

Then,

\overline{p} (u, v) = 0.477 u^{2} - 0.2121 u v - 0.013 v^{2} - 0.3349 u^{3} + 0.063 v u^{2} + R_{\overline{p}, 4} (u, v),

and

\overline{q} (u, v) = 1.128 u^{2} - 1.339 u v + 0.2024 v^{2} + 1.707 u^{3} + 0.6452 u^{2} v + 0.675 u v^{2} - 0.017 v^{3} + R_{\overline{q}, 4} (u, v) .

5 CONCLUSIONS

In this article, we investigate the stability and the existence of a bifurcation in a prey–predator model described by a difference equation, showing that the only equilibrium solution representing the coexistence of both species presents a Neimark–Sacker bifurcation when the bifurcation parameter varies in the vicinity of a critical value. This result has been verified by means of a numerical analysis of the system.

CONFLICT OF INTEREST

The authors declare no potential conflict of interests.

Biographies

Messaoud Berkal is currently working on his doctoral dissertation in the field of Difference Equations and Discrete Dynamical Systems. He has published several papers devoted to analytical solutions to some type of rational high order difference systems.
Juan F. Navarro completed his Ph.D. degree from the University of Alicante, Alicante, Spain, in 2002. He is a recipient of the Extraordinary Award of the University of Alicante for his master thesis on the rotation of the rigid Earth. This work took part in the project Pinpoint positioning in a wobbly world awarded with the Descartes Prize in 2003, an annual award in science given by the European Union to outstanding scientific achievements resulting from European collaborative research. He is currently Professor at the Department of Applied Mathematics, University of Alicante, member of the Scientific Group on Space Geodesy and Space Dynamics of the University of Alicante, and member of the International Astronomical Union. His scientific interests include different problems in Celestial Mechanics, such as the study of the rotational motion of the Earth, the escape of particles from galactic potentials, and the numerical exploration of the N-body ring problem.

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Qualitative behavior of a two-dimensional discrete-time prey–predator model

Abstract

1 INTRODUCTION

2 LOCAL STABILITY OF THE EQUILIBRIA OF THE SYSTEM

3 NEIMARK–SACKER BIFURCATION

4 NUMERICAL COMPUTATIONS AND DISCUSSION

5 CONCLUSIONS

CONFLICT OF INTEREST

Biographies

REFERENCES

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Qualitative behavior of a two-dimensional discrete-time prey–predator model

Abstract

1 INTRODUCTION

2 LOCAL STABILITY OF THE EQUILIBRIA OF THE SYSTEM

3 NEIMARK–SACKER BIFURCATION

4 NUMERICAL COMPUTATIONS AND DISCUSSION

5 CONCLUSIONS

CONFLICT OF INTEREST

Biographies

REFERENCES

Citing Literature

Figures

References

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