Abstract
A new difference system is induced from a differential competition system by different discrete methods. We give theoretical analysis for local bifurcation of the fixed points and derive the conditions under which the local bifurcations such as flip occur at the fixed points. Furthermore, one- and two-dimensional diffusion systems are given when diffusion terms are added. We provide the Turing instability conditions by linearization method and inner product technique for the diffusion system with periodic boundary conditions. A series of numerical simulations are performed that not only verify the theoretical analysis, but also display some interesting dynamics.
1. Introduction
Interactions of different species may take many forms such as competition, predation, parasitism, and mutualism. One of the most important interactions is the competition relationship. The dynamic relationship between the two competition species is one of the dominant subjects in mathematical ecology due to its universal existence and importance. Lotka-Volterra competition systems are ecological models that describe the interaction among various competing species and have been extensively investigated in recent years (see [1–3] and the references therein). In the earlier literature, the two-competing species competition models are often formulated in the form of ordinary differential systems as follows: for , , where and are the quantities of the two species at time , and are growth rates of the respective species, and represent the strength of the intraspecific competition, and and represent the strength of the interspecific competition.
The discrete time models governed by difference equation are more realistic than the continuous ones when the populations have nonoverlapping generations or the population statistics are compiled from given time intervals and not continuously. Moreover, since the discrete time models can also provide efficient computational models of continuous models for numerical simulations, it is reasonable to study discrete time models governed by difference equations.
Applying forward Euler scheme to the first equation of system (1) and obtaining a discrete analog of the second equation by considering a variation with piecewise constant arguments for certain terms on the right side (exponential discrete form) [4], we obtain the following equation:
For the sake of simplicity, let
By setting and and , , and , we have the following form:
Although numerical variations of system (1) have been extensively studied (see, e.g., the work in [5–8]), some discrete analogs may be found in [9–15], regarding attractivity, persistence, global stabilities of equilibrium, and other dynamics. Up to now, to the best of our knowledge, the discrete system (4) has not been investigated.
Since the pioneering theoretical works of Skellam [15] and Turing [16], many works have focused on the effect of spatial factors which play a crucial role in the stability of populations [17–19]. Many important epidemiological and ecological phenomena are strongly influenced by spatial heterogeneities because of the localized nature of transmission or other forms of interaction. Thus, spatial models are more suitable for describing the process of population development. Impact of spatial component on system has been widely investigated (e.g., see [20–22]). It may be a case in reality that the motion of individuals is random and isotropic; that is, without any preferred direction, the individuals are also absolute ones in microscopic sense, and each isolated individual exchanges materials by diffusion with its neighbors [19, 23]. Thus, it is reasonable to consider a 1D or 2D spatially discrete reaction diffusion system to explain the population system. Corresponding to the above analysis, we can obtain the following one-dimensional diffusion systems: for , and , , and two-dimensional diffusion systems: for , and , .
In this paper, we will study the dynamical behaviors of models (4), (5), and (6). By using the theory of difference equation, the theory of bifurcation, and the center manifold theorem we will establish the series of criteria on the existence and local stability of equilibria, flip bifurcation for the system (4). For the one- or two-dimensional diffusion systems, with periodic boundary conditions, the Turing instability (or Turing bifurcation) theory analysis will be given. Turing instability conditions can then be deduced combining linearization method and inner product technique. Furthermore, by means of the numerical simulations method, we will indicate the correctness and rationality of our results.
The paper is organized as follows. In Section 2, we study the existence and stability of equilibria points and the conditions of existence for flip bifurcation are verified for system (4). Turing instability conditions will be illustrated by linearization method and inner product technique for the system (5) and (6) with periodic boundary conditions in Section 3. A series of numerical simulations are performed that not only verify the theoretical analysis, but also display some interesting dynamics. For the system (4), the bifurcation diagrams are given. The impact of the system parameters and diffusion coefficients on patterns can also be observed visually for the given diffusion systems. Finally, some conclusions are given.
2. Analysis of Equilibria and Flip Bifurcation
Clearly, the system (4) has four possible steady states; that is, , exclusion points , , and nontrivial coexistence point , where
The linearized form of (4) is then which has the Jacobian matrix
The characteristic equation of the Jacobian matrix can be written as where and .
In order to discuss the stability of the fixed points of (4), we also need the following definitions [20]:(1)if and , then steady state is called a sink and is locally asymptotically stable;(2)if and , then is called a source and is unstable;(3)if and (or and ), then is called a saddle;(4)if either and or and , then is called nonhyperbolic.
Case 1 (the fixed point ). The linearization of (4) about has the Jacobian matrix which has two eigenvalues The fact means that the system is resonance at the fixed point .
Case 2 (the fixed point ). At the fixed point, the Jacobian matrix has the form and the corresponding eigenvalues of (13) are is a bifurcation parameter. And implies , and the fixed point is nonhyperbolic.
Case 3 (the third fixed point ). The linearization of (4) about has the Jacobian matrix and the eigenvalues of (15) are is a bifurcation parameter. And implies , and the fixed point is nonhyperbolic.
Case 4 (the fixed point ). The linearization of (4) about has the Jacobian matrix
and the eigenvalues of (17) are
then, we have the following results:(1), if and only if , ;(2), if and only if , ;(3), if and only if , ;(4), if and only if , ;(5), if and only if , ;(6), if and only if ;(7), if and only if , .
The following theorem is the case that the fixed point is a flip bifurcation point.
Theorem 1. The positive fixed point undergoes a flip bifurcation at the threshold .
Proof. Let , , and , and parameter is a new and dependent variable; the system (4) becomes
Let
then
By the following transformation:
the system (19) can be changed into
where
Then, we can consider
which must satisfy
By calculating, we can get that
and the system (19) is restricted to the center manifold, which is given by
Since
system (4) undergoes a flip bifurcation at . The proof is completed.
3. Turing Bifurcation
In this section, we discuss the Turing bifurcation. Turing's theory shows that diffusion could destabilize an otherwise stable equilibrium of the reaction-diffusion system and lead to nonuniform spatial patterns. This kind of instability is usually called Turing instability or diffusion-driven instability [16].
3.1. One-Dimensional Case
We consider the following diffusion system: with the periodic boundary conditions for and , where is a positive integer,
In order to study Turing instability of (30) and (31), we firstly consider eigenvalues of the following equation: with the periodic boundary conditions
By calculating, the eigenvalue problem (33)-(34) has the eigenvalues
We linearise at the steady state, to get with the periodic boundary conditions where
Then, respectively, taking the inner product of (36) with the corresponding eigenfunction of the eigenvalue , we see that Let and and use the periodic boundary conditions (34) and (37); then we have or
Thus, the following fact can be obtained.
Proposition 2. If is a solution of the problem of (30) and (31), then is a solution of (41), where is some eigenvalue of (33)-(34) and is the corresponding eigenfunction. For some eigenvalue of (33)-(34), if is a solution of the system (41), then is a solution of (30) with the periodic boundary conditions (31).
Proposition 3. If there exist positive numbers , and the eigenvalue of the problem (33)-(34) such that one of the conditions or holds, then the problem (30) and (31) at the fixed point is unstable, where
For the system (5), we have the following results about instability of the positive equilibrium of system.
Theorem 4. , and Proposition 3 mean or show that the problem (30) and (31) is diffusion-driven unstable or Turing unstable.
3.2. Two-Dimensional Case
In this subsection, we will pay our attention to the Turing instability analysis for the following two-dimensional system: with the periodic boundary conditions for and , where is a positive integer,
The following theorem will show that the system (47) also undergoes Turing instability. Since the analysis is very similar to the one-dimensional case, the proof is omitted.
Theorem 5. If there exist positive numbers , and the eigenvalue of the corresponding characteristic equation such that one of the conditions or holds and , then the problem (47)-(48) at the fixed point is diffusion-driven unstable or Turing unstable, where
4. Numerical Simulation
As is known to all, the bifurcation diagram provides a general view of the evolution process of the dynamical behaviors by plotting a state variable with the abscissa being one parameter. As a parameter varies, the dynamics of the system we concerned change through a local or global bifurcation which leads to the change of stability at the same time.
Now, is considered as a parameter with the range 0.5–3.5 for the system (4). Since the bifurcation diagrams of are similar to the bifurcation diagrams of , we will only show the former which can be seen from Figure 1.
Next, we performed a series of simulations for the reaction-diffusion systems, and, in each, the initial condition was always a small amplitude random perturbation 1% around the steady state. As a numerical example, we consider the bifurcation of the two-dimensional system (47)-(48). It is well known that Turing instability (bifurcation) is diffusion-driven instability; thus the diffusion rate is vital to the pattern formation. To investigate the effect of diffusion coefficients on patterns, by keeping all the other parameters of the system fixed (, , and ), we change a diffusion coefficient in the Turing instability region (parameter space which satisfies Turing instability). Figures 2(a)–2(f) exhibit in detail the different distribution of patterns with varying values of . If we let , a stable pattern of square shapes, namely, stationary wave, is observed in Figure 2(a). With the increase of , some tips appear, which attempt to form self-centered spiral waves (Figure 2(b)). Then some tips vanish and some tend to evolve to regular spiral waves, shown in Figure 2(c). Then, with the parameter evolution proceeding, the size of spiral waves rises, but the density of them decreases (see Figure 2(d)). If is further increased, we observe that, in parts of patterns, spiral waves begin to break up. Hardly can spiral waves be seen, disorder and chaotic structure are depicted in Figures 2(e) and 2(f).
(a)
(b)
(c)
(d)
(e)
(f)
5. Discussion and Conclusion
In this paper, we have applied different discrete schemes to convert the continuous Lotka-Volterra competition model to a new discrete model and studied the dynamical characteristic of the discrete model. Our theoretical analysis and numerical simulations have demonstrated that the discrete competition model undergoes flip bifurcation. Furthermore, when the effects of spatial factors are considered, we discuss the Turing instability conditions combining linearization method and inner product technique. The impact of the diffusion coefficients on patterns can also be observed visually, and some interesting situations can be observed. Indeed, the new discrete model can result in a rich set of patterns and we expect that it is more effective in practice.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors thank Dr. Y. D. Jiao for the valuable suggestions and thank the referees for their helpful comments. This work was financially supported by Tianjin University of Commerce with the Grant no. X0803, the National Natural Science Foundation of China under Grant nos. 11001072, 11371277, and Cultivation Program for Excellent Youth Teacher in University, Tianjin (507-125RCPY0314).