Abstract
In this paper, we study the equilibrium points, local asymptotic stability of equilibrium points, and global behavior of equilibrium points of a discrete Lotka-Volterra model given by
where parameters , and initial conditions , are positive real numbers. Moreover, the rate of convergence of a solution that converges to the unique positive equilibrium point is discussed. Some numerical examples are given to verify our theoretical results.
MSC:39A10, 40A05.
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1 Introduction and preliminaries
Many authors investigated the ecological competition systems governed by differential equations of Lotka-Volterra type. Many interesting results related with the global character and local asymptotic stability have been obtained. We refer to [1, 2] and the references therein. Already, many authors [3, 4] have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations are of non-overlapping generations. Particularly, the persistence, boundedness, local asymptotic stability, global character, and the existence of positive periodic solutions.
The discrete Lotka-Volterra models have many applications in applied sciences. Such models were first established in mathematical biology, and then their applications were spread to other fields [5–8]. Several variations of the Lotka-Volterra predator-prey model have been proposed that offer more realistic descriptions of the interactions of the populations. If the population of rabbits is always much larger than the number of foxes, then the considerations that entered into the development of the logistic equation may come into play. If the number of rabbits becomes sufficiently great, then the rabbits may be interfering with each other in their quest for food and space. One way to describe this effect mathematically is to replace the original model by the more complicated system. Most predators feed on more than one type of food. If the foxes can survive on an alternative resource, although the presence of their natural prey (rabbits) favors growth, a possible alternative model is the discrete dynamical system
where parameters , and initial conditions , are positive real numbers.
It is a well-known fact that the discrete-time type models described by difference equations are more suitable than the continuous-time models. Nonlinear difference equations of order greater than one are of paramount importance in applications. Such equations also appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations which model various diverse phenomena in biology, ecology, physiology, physics, engineering, and economics. Rational difference equations are a special form of nonlinear difference equations. We refer to [9–14] for basic theory of difference equations and rational difference equations. Recently, many authors have discussed the dynamics of rational difference equations [15–27].
2 Linearized stability
Let us consider a two-dimensional discrete dynamical system of the form
where and are continuously differentiable functions and I, J are some intervals of real numbers. Furthermore, a solution of the system (2.1) is uniquely determined by initial conditions . An equilibrium point of (2.1) is a point that satisfies
Definition 2.1 Let be an equilibrium point of the system (2.1).
-
(i)
An equilibrium point is said to be stable if for every there exists such that for every initial condition if implies for all , where is the usual Euclidean norm in .
-
(ii)
An equilibrium point is said to be unstable if it is not stable.
-
(iii)
An equilibrium point is said to be asymptotically stable if there exists such that and as .
-
(iv)
An equilibrium point is called a global attractor if as .
-
(v)
An equilibrium point is called an asymptotic global attractor if it is a global attractor and stable.
Definition 2.2 Let be an equilibrium point of a map , where f and g are continuously differentiable functions at . The linearized system of (2.1) about the equilibrium point is given by
where and is a Jacobian matrix of the system (2.1) about the equilibrium point .
Let be an equilibrium point of the system (1.1), then
Hence, , , , and are equilibrium points of the system (1.1). Then, clearly, is the unique positive equilibrium point of the system (1.1), if , , or , , .
The Jacobian matrix of the linearized system of (1.1) about the fixed point is given by
Theorem 2.3 For the system , , of difference equations such that is a fixed point of F. If all eigenvalues of the Jacobian matrix about lie inside the open unit disk , then is locally asymptotically stable. If one of them has a modulus greater than one, then is unstable.
3 Main results
Theorem 3.1 Assume that and , then the following statements are true.
-
(i)
The equilibrium point is locally asymptotically stable.
-
(ii)
The equilibrium point is unstable.
-
(iii)
The equilibrium point is unstable.
Proof (i) The Jacobian matrix of the linearized system of (1.1) about the fixed point is given by
Moreover, the eigenvalues of the Jacobian matrix about are and . Hence, the equilibrium point is locally asymptotically stable.
-
(ii)
The Jacobian matrix of the linearized system of (1.1) about the fixed point is given by
The eigenvalues of the Jacobian matrix about are and .
-
(iii)
The Jacobian matrix of the linearized system of (1.1) about the fixed point is given by
The eigenvalues of the Jacobian matrix about are and . □
Theorem 3.2 The following statements are true.
-
(i)
If , , and , then the equilibrium point is locally asymptotically stable.
-
(ii)
If and , then the equilibrium point is locally asymptotically stable.
Theorem 3.3 Assume that , , and , then the unique equilibrium point is locally asymptotically stable if
where
Proof Assume that , , and . Let . Then a characteristic polynomial of the Jacobian matrix about the unique equilibrium point is given by
where
and
Let
Assume that . Then one has
Then, by Rouche’s theorem, and have the same number of zeroes in an open unit disk . Hence, the unique positive equilibrium point P is locally asymptotically stable. □
3.1 Global character
Theorem 3.4 Let and be real intervals, and let and be continuous functions. Consider the system (2.1) with initial conditions . Suppose that the following statements are true.
-
(i)
is non-decreasing in x and non-increasing in y.
-
(ii)
is non-decreasing in both arguments.
-
(iii)
If is a solution of the system
such that and , then there exists exactly one equilibrium point of the system (2.1) such that .
Proof According to the Brouwer fixed point theorem, the function defined by has a fixed point , which is a fixed point of the system (2.1).
Assume that , , , such that
and
Then
and
Moreover, one has
and
We similarly have
and
Now observe that for each ,
and
Hence, , and for . Let , , , and . Then and . By the continuity of f and g, one has
Hence, , . □
Theorem 3.5 Assume that , then the unique positive equilibrium point P of the system (1.1) is a global attractor.
Proof Let and . Then it is easy to see that is non-decreasing in x and non-increasing in y. Moreover, is non-decreasing in both x and y. Let be a positive solution of the system
Then one has
and
From (3.1), one has
On subtraction, (3.3) implies that
Similarly, from (3.2), one has
On subtraction, (3.5) implies that
Comparing (3.4) and (3.6), one has
Then one has and . Hence, from Theorem 3.4 the equilibrium point of the system (1.1) is a global attractor. □
Theorem 3.6 Assume that , , and . Then the unique positive equilibrium point is globally asymptotically stable.
Proof The proof follows from Theorem 3.3 and Theorem 3.5. □
3.2 Rate of convergence
In this section we determine the rate of convergence of a solution that converges to the unique positive equilibrium point of the system (1.1).
The following result gives the rate of convergence of solutions of a system of difference equations:
where is an m-dimensional vector, is a constant matrix, and is a matrix function satisfying
as , where denotes any matrix norm which is associated with the vector norm
Proposition 3.7 (Perron’s theorem [28])
Suppose that condition (3.8) holds. If is a solution of (3.7), then either for all large n or
exists and is equal to the modulus of one of the eigenvalues of matrix A.
Proposition 3.8 [28]
Suppose that condition (3.8) holds. If is a solution of (3.7), then either for all large n or
exists and is equal to the modulus of one of the eigenvalues of matrix A.
Let be any solution of the system (1.1) such that , and , where . To find the error terms, one has from the system (1.1)
and
Let and , then one has
and
where
Moreover,
Now the limiting system of error terms can be written as
which is similar to the linearized system of (1.1) about the equilibrium point .
Using Proposition 3.7, one has following result.
Theorem 3.9 Assume that is a positive solution of the system (1.1) such that and , where
Then the error vector of every solution of (1.1) satisfies both of the following asymptotic relations:
where are the characteristic roots of the Jacobian matrix .
4 Examples
In this section, we consider some numerical examples which show that under a suitable choice of parameters α, β, γ, δ, ϵ, η, the unique positive equilibrium point of the system (1.1) is globally asymptotically stable.
Example Let , , , , , . Then the system (1.1) can be written as
with initial conditions , .
In this case, the unique positive equilibrium point P of the system (4.1) is given by
Moreover, the plot is shown in Figure 1.
Example Let , , , , , . Then the system (1.1) can be written as
with initial conditions , .
In this case, the unique equilibrium point P of the system (4.2) is given by
Moreover, the plot is shown in Figure 2.
Example Let , , , , , . Then the system (1.1) can be written as
with initial conditions , .
In this case, the unique equilibrium point P of the system (4.3) is given by
Moreover, the plot is shown in Figure 3.
Example Let , , , , , . Then the system (1.1) can be written as
with initial conditions , .
In this case, the unique positive equilibrium point P of the system (4.4) is given by
Moreover, the plot of the system (4.4) is shown in Figure 4. An attractor of the system is shown in Figure 5.
5 Conclusions
This work is related to the qualitative behavior of a discrete-time Lotka-Volterra model. The continuous form of this model is given by
where a, b, c, m, n, p are positive constants. Moreover, the discrete form (1.1) of the continuous model is obtained by using some nonstandard difference scheme such that the equilibrium points in both cases are conserved. We proved that the system (1.1) has four equilibrium points, which are locally asymptotically stable under certain conditions. The main contribution in this paper is to prove that the unique positive equilibrium point
of the system (1.1) is globally asymptotically stable. Furthermore, we have investigated the rate of convergence of the solution that converges to the unique positive equilibrium point of the system (1.1). Some numerical examples are provided to support our theoretical results. These examples are experimental verifications of theoretical discussions.
Author’s contributions
The author carried out the proof of the main results and approved the final manuscript.
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The author would like to thank the main editor and anonymous referees for their valuable comments and suggestions leading to improvement of this paper. This work was supported by the Higher Education Commission of Pakistan.
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Din, Q. Dynamics of a discrete Lotka-Volterra model. Adv Differ Equ 2013, 95 (2013). https://doi.org/10.1186/1687-1847-2013-95
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DOI: https://doi.org/10.1186/1687-1847-2013-95