Abstract
In this paper, global dynamics and bifurcations of a two-dimensional discrete-time Lotka-Volterra model have been studied in the closed first quadrant . It is proved that the discrete model has three boundary equilibria and one unique positive equilibrium under certain parametric conditions. We have investigated the local stability of boundary equilibria , , and the unique positive equilibrium , by the method of linearization. It is proved that the discrete model undergoes a period-doubling bifurcation in a small neighborhood of boundary equilibria and a Neimark-Sacker bifurcation in a small neighborhood of the unique positive equilibrium Further it is shown that every positive solution of the discrete model is bounded and the set is an invariant rectangle. It is proved that if and , then equilibrium of the discrete model is a global attractor. Finally it is proved that the unique positive equilibrium is a global attractor. Some numerical simulations are presented to illustrate theoretical results.
1. Introduction
In this paper, we study the global dynamics and bifurcations analysis of a two-dimensional discrete-time Lotka-Volterra model in the closed first quadrant , which was proposed by Waltman [1]. In this model, if two populations are growing logistically without affecting each other, then their growth can be represented by the following system of two logistic equations:where and the initial conditions are positive real numbers. Now, assume that the carrying capacity is a shared resource-each population competes for the resource and thereby interferes with the other. Then the presence of each reduces the intrinsic rate of growth of the other. We refer the reader to [1–5] for detailed discussion on the above assumption. This phenomena can be represented as follows:where are positive constants. It is convenient to change the nondimensional variables by measuring in units of , in units of , and time in units of . Then system (2) takes the following form:where for and . It is clear that for all parametric values, system (3) has three boundary equilibria and a unique positive equilibrium point if . According to continuous dynamical systems theory, it is easy to show that equilibrium is a source but never sink and saddle; equilibrium is a sink if and saddle if , but it is never source; is a sink if and saddle if , but it is never source, and the unique positive equilibrium point is locally asymptotically stable.
A discrete dynamical system is defined as a system whose state evolves over state space in discrete-time steps according to a fixed rule. These systems are represented by a system of difference equations. This is a well-known fact that difference equations existed before differential equations and have played a fundamental role in the development of the latter. During the last fifty years the theory of difference equations received attention of both mathematicians and users of mathematics and developed greatly, because of its internal mathematical beauty and applicability in almost all branches of modern science such as ecology, population dynamics, queuing problems, statistical problems, stochastic time series, number theory, geometry, neural networks, quanta in radiation, genetics in biology, economics, psychology, sociology, physics, engineering, economics, combinatorial analysis, probability theory, electrical networks, and resource management [6, 7]. Dynamics of a discrete dynamical system is studied by analyzing the behavior of the solution of the system of difference equations representing the system under study. Analyzing the behavior of solutions of a higher-order nonlinear difference equation is very interesting and attracted many researchers in recent times. Behavior of solutions means studying the equilibrium point, boundedness and persistence, existence and uniqueness of positive equilibrium point, local and global stability, periodicity nature of such difference equations or systems of difference equations (see [8–16] and references cited therein).
Discrete dynamical systems described by difference equations are more appropriate for population dynamics as compared to continuous ones. Biologists believe that the equilibrium point and its stability analysis is important to understand the population dynamics [17, 18]. Therefore, in this paper, we study the behavior of the following discrete-time Lotka-Volterra model, which is obtained by discretization of continuous-time model (3) followed by forward Euler’s method. Using forward Euler’s method, continuous-time model (3) takes the following form: After some simplification, the above system becomeswhere . It is also noted that the parameters and the initial conditions are positive real numbers.
The rest of the paper is organized as follows: In Section 2, we study the existence of equilibria of the discrete model. Section 3 deals with the study of local stability of three boundary equilibria and the unique positive equilibrium. Section 4 deals with the study of bifurcations analysis of boundary equilibria and the unique positive equilibrium. Section 5 discusses the boundedness character and the construction of invariant rectangle of the discrete model. Section 6 discusses the global behavior of and the unique positive equilibrium, whereas Section 7 is about numerical simulation to verify the obtained theoretical results. In the last section a brief conclusion is given.
2. Existence of Equilibria
In this section, we will study the existence of equilibria of the discrete model (5) in the closed first quadrant . The results about the existence of equilibria are summarized as follows.
Lemma 1. Under certain parametric conditions, system (5) has at least three boundary equilibria and one unique positive equilibrium in the closed first quadrant . More precisely,(i)system (5) has a unique boundary equilibrium if and ;(ii)system (5) has two boundary equilibrium if and ;(iii)system (5) has three boundary equilibrium if and ;(iv)system (5) has three boundary equilibrium and one interior equilibrium if and . Additionally if and , then is the unique positive equilibrium of system (5).
Proof. In order to find equilibria of system (5) in the interior of , we need to solve the following algebraic equations:If , then (6) are identically satisfied, and hence for all parametric values is the unique equilibrium of this system. If then the second equation of (6) is identically satisfied, and from the first equation we get . Thus if , then is one boundary equilibrium point of system (5). If , then first equation of (6) is identically satisfied, and from the second equation we get . Thus if , then is again a boundary equilibrium of system (5).
On the other hand, we consider the existence of unique positive equilibrium of system (5) in the interior of . For this assume that if and , then system (6) takes the following form:Solving system (7) for and , one gets . Hence if and , then is the unique positive equilibrium of system (5).
Remark 2. The discrete model (5) has three boundary equilibria and one interior equilibrium if and , where . Now, using the values , the equilibria of continuous-time model (3), , and can be recovered with the same conditions on the parameters .
3. Local Stability
The Jacobian matrix of linearized system of (5) about equilibrium is
3.1. Local Stability of Boundary Equilibria
Hereafter we will study the topological classification of the boundary equilibria. The results regarding the local stability of the boundary equilibria are summarized as follows.
Theorem 3. For equilibrium point , the following statements hold:(i)The equilibrium point of system (5) is a sink if and ;(ii)The equilibrium point of system (5) is a source if and ;(iii)The equilibrium point of system (5) is a saddle if and ;(iv)The equilibrium point of system (5) is nonhyperbolic if or .
Theorem 4. For equilibrium point , the following statements hold:(i)The equilibrium of system (5) is a sink if and ;(ii)The equilibrium of system (5) is never source;(iii)The equilibrium of system (5) is a saddle if and ;(iv)The equilibrium of system (5) is nonhyperbolic if .
Theorem 5. For equilibrium point , the following statements hold:(i)The equilibrium of system (5) is a sink if and ;(ii)The equilibrium of system (5) is never source;(iii)The equilibrium of system (5) is a saddle if and ;(iv)The equilibrium of system (5) is nonhyperbolic if .
Now in the following we will study the local stability of the unique positive equilibrium by using Remark .3.1 of [7].
3.2. Local Stability of the Unique Positive Equilibrium
Theorem 6. For the unique positive equilibrium of system (5), the following statements hold:(i)The unique positive equilibrium point of system (5) is locally asymptotically stable if(ii) The unique positive equilibrium point of system (5) is unstable if where
Proof. (i) The Jacobian matrix of linearized system of (5) about is The characteristic polynomial of about is where Assume that , and using Remark .3.1 of [7] one gets Therefore of system (5) is locally asymptotically stable.
(ii) Similarly it is easy to show that of system (5) is unstable if .
Hereafter we will compute the necessary and sufficient condition(s) for the unique positive equilibrium of system (5) to be locally asymptotically stable, repeller, saddle, and nonhyperbolic, respectively.
Theorem 7. For equilibrium of system (5), the following statements hold:(i)Equilibrium of system (5) is locally asymptotically stable if and only if(ii)Equilibrium of system (5) is a repeller if and only if (iii)Equilibrium of system (5) is a saddle if and only if(iv)Equilibrium of system (5) is nonhyperbolic if and only if
Proof. (i) More precisely the characteristic equation of about is given by where Then, it follows from Theorem .1.1 of [12] that the unique positive equilibriumof system (1) is locally asymptotically stable if and only if Similarly, one can prove (ii), (iii), and (iv).
4. Bifurcations Analysis
In this section, we will study the bifurcation analysis of discrete model (5) about the equilibria and . From theoretical results obtained in Section 3, we conclude the following:(i)If condition (iv), that is, , of Theorem 3 holds, then one of the eigenvalues of about is and so fold bifurcation may occur when parameters vary in a small neighborhood of . The condition (iv) of Theorem 3 can be rewritten as the following set: (ii)From Theorem 4, we can see that one of the eigenvalues of about is and other is neither nor when the parameters of the discrete model (5) are located in the following set: Therefore, can undergo flip or period-doubling bifurcation when all parameters of the discrete model (5) vary in a small neighborhood of . When the parameters are in , a center manifold of the discrete model (5) is , and thus (5) restricted to this central manifold is This shows that the predator becomes extension and prey undergoes period-doubling bifurcation to chaos on choosing bifurcation parameter .(iii)From Theorem 5, it is easy to verify that one of the eigenvalues of about is and the other is neither nor when the parameters of the discrete model (5) are located in the following set: Therefore, undergoes period-doubling bifurcation when all parameters of the discrete model (5) vary in a small neighborhood of . When the parameters are in , a center manifold of the discrete model (5) is , and thus (5) restricted to this central manifold is This shows that the prey becomes extension and predator undergoes period-doubling bifurcation to chaos on choosing bifurcation parameter .
Now before studying the bifurcation analysis of the discrete model (5) about the unique positive equilibrium point , first we will prove that there exist different topological types of this equilibrium point. Recall that eigenvalues of about arewhere Hereafter we state the topological classification about of system (5) as follows.
Theorem 8. For of system (5), the following topological classification holds:(i) is a locally asymptotically stable focus if and (ii) is an unstable focus if and(iii) is nonhyperbolic if and
We can see that the eigenvalues of are a pair of complex conjugates with modulus 1 when parameters of model (5) are located in the following set: Therefore, undergoes Neimark-Sacker bifurcation when all parameters of model (5) vary in a small neighborhood of .
Hereafter we will study the Neimark-Sacker bifurcation of model (5) about
For given parameters , letIt is easy to verify that , then by implicit function theorem we obtain that such that , and therefore we can choose as a bifurcation parameter.
Now consider parameter in a small neighborhood of ; that is, , where ; then system (5) becomesThe characteristic equation of about of system (41) is where The roots of characteristic equation of about are where Moreover and . Additionally, we required that when , , , which corresponds to . Since and , then and hence . Thus, we only require that , which holds true by computation.
Let ; then of system (5) transforms into . By calculation, we obtainwhere and . Hereafter when , normal form of system (48) is studied. Expanding (48) up to second-order about by Taylor series, we getwhere
Now, let and invertible matrix is defined by Using the following translation (49) giveswhereIn addition,
To guarantee the supercritical Neimark-Sacker bifurcation for (54), we require the following discriminatory quantity, that is, (see [19–23]):where
A calculation revealsBased on this analysis and Neimark-Sacker bifurcation Theorem discussed in [19, 20], we reach the following Theorem.
Theorem 9. If , then discrete model (5) undergoes a Neimark-Sacker bifurcation about as the parameters go through . Additionally, attracting (resp., repelling) invariant closed curve bifurcates from the unique positive equilibrium if (resp., ).
5. Boundedness and Construction of Invariant Rectangle
In this section we will study boundedness character and construction of invariant rectangle of positive solution of the discrete model (5).
Theorem 10. For every positive solution of the discrete model (5), the following holds:(i)Every positive solution of the discrete model (5) is bounded.(ii)The set is an invariant rectangle.
Proof. (i) Let be any positive solution of the discrete model (5). From (5), we have Hence, for every solution of the discrete model (5), one has (ii) For any positive solution of the discrete model (5) with initial conditions and , we have from (5)Hence, and . Similarly, one can show that if and , then and .
6. Global Stability
Now in the following we will investigate global dynamics of the discrete model (5) about and the unique positive equilibrium .
Theorem 11. If and , then equilibrium of the discrete model (5) is globally asymptotically stable.
Proof. According to the conclusion (i) in Lemma 1, discrete model (5) has a unique equilibrium in the first quadrant and is a sink by Theorem 3. Moreover every positive solution of the discrete model (5) in the first quadrant satisfies which leads to and . Hence the boundary equilibrium of the discrete model (5) is globally asymptotically stable in the first quadrant .
Hereafter we use Theorem .16 of [14] to determine the global dynamics of the discrete model (5) about .
Theorem 12. The unique positive equilibrium point of the discrete model (5) is a global attractor.
Proof. Let and . Then, it is easy to see that is nondecreasing in and nonincreasing in for every . Moreover, is nonincreasing in and nondecreasing in for each . Let be a positive solution of system Then, one hasFrom (66), one hasFrom (67), one hasOn subtracting (68), one getsSimilarly, subtracting (69) one getsFrom (70) and (71) one gets . Hence, , and similarly one has . Hence, from Theorem .16 of [14] the equilibrium of the discrete model (5) is a global attractor.
Corollary 13. Under the conditions (10) and (18), the unique positive equilibrium point of the discrete model (5) is globally asymptotically stable.
7. Numerical Simulations
In the following we present numerical simulations that represent different types of qualitative behavior of the discrete model (5).
Example 1. If , then discrete model (5) with initial values can be written asIn this case , , . This shows the correctness of the conditions for the unique positive equilibrium. A straightforward computation shows that condition (10) of Theorem 6, that is, = 0.07435317508000001 < = 0.0957613829600004, under which the unique positive equilibrium point is locally asymptotically stable, holds. Moreover for these arbitrary chosen values of parameters, the necessary and sufficient condition, under which the unique positive equilibrium point is locally asymptotically stable, is also satisfied; that is, //. This verifies the condition for which the unique positive equilibrium is locally asymptotically stable. Also , and hence the parametric conditions under which every positive solution is bounded hold true. Moreover, in Figure 1 the plot of is shown in Figure 1(a), the plot of is shown in Figure 1(b), and attractor of system (73) is shown in Figure 1(c).
Example 2. If , then discrete model (5) with initial values can be written asIn this case , . This shows the correctness of the conditions of the unique positive equilibrium point. A straightforward computation shows that condition (10) of Theorem 6, that is, = 4035.5030151773317 < = 4035.5030151773317, holds. Moreover the necessary and sufficient condition under which the unique positive equilibrium point of the system is locally asymptotically stable is also satisfied; that is, / = 0.36407043421653185 < / = 0.9701211132646083 < 2. This verifies the condition for which the unique positive equilibrium is locally asymptotically stable. Also , and hence the parametric conditions under which every positive solution is bounded hold true. Moreover, in Figure 2 the plot of is shown in Figure 2(a), the plot of is shown in Figure 2(b), and attractor of system (74) is shown in Figure 2(c).
Example 3. If , then discrete model (5) with initial values can be written asIn this case , . This shows the correctness of the conditions of the unique positive equilibrium point. A computation shows that condition (10) of Theorem 6, that is, = 9280.610848888002 < , holds. Moreover for these arbitrary chosen values of parameters the necessary and sufficient condition, under which the unique positive equilibrium point is locally asymptotically stable, is also satisfied; that is, / = 0.30727881013351915 < / = 0.9862925899775977 < 2. This verifies the condition for which the unique positive equilibrium is locally asymptotically stable. Also , and hence the parametric conditions under which every positive solution is bounded hold true. Moreover, in Figure 3 the plot of is shown in Figure 3(a), the plot of is shown in Figure 3(b), and attractor of system (75) is shown in Figure 3(c).
8. Conclusion
This work is related to the global dynamics and bifurcations analysis of a two-dimensional discrete-time Lotka-Volterra model in the closed first quadrant . We proved that the discrete model (5) has three boundary equilibria and the unique positive equilibrium under certain parametric conditions. The method of linearization is used to prove the local asymptotic stability of these equilibria, and conclusions are presented in Table 1. We proved that boundary equilibrium undergoes fold bifurcation when parameters vary in a small neighborhood of and both and undergo period-doubling bifurcation when parameters of the discrete model (5) are, respectively, located in the following sets:We have also shown that undergoes Neimark-Sacker bifurcation when parameters of the discrete model (5) are located in the following set: It is proved that every positive solution of the discrete model (5) is bounded and the set is an invariant rectangle. The most interesting aspect in the theory of dynamical systems is to predict the global dynamics about equilibria. In this paper, we proved that if and , then equilibrium of the discrete model (5) is globally asymptotically stable. Furthermore, we have investigated the global stability of the unique positive equilibrium point of the discrete model (5). Some numerical examples are provided to support our theoretical results. These examples provide experimental verifications of the theoretical discussions.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the Higher Education Commission (HEC) of Pakistan.